https://docs-vrgeo-version3.netlify.app/assets/js/6b3f6dbb.b842cda5.js

1"use strict";(globalThis.webpackChunkvrgs_docs_v_3_1=globalThis.webpackChunkvrgs_docs_v_3_1||[]).push([[29138],{11467(s,e,a){a.r(e),a.d(e,{assets:()=>c,contentTitle:()=>m,default:()=>d,frontMatter:()=>t,metadata:()=>n,toc:()=>r});const n=JSON.parse('{"id":"general/wells-seismic/seismic-modelling-method","title":"Synthetic Seismic Modelling from Classified Meshes: Method and Implementation","description":"A detailed, paper-style account of how VRGS forward-models a band-limited synthetic seismic section from a classified triangulated mesh: reflector extraction, the 2-D extrusion reflectivity model, the Kirchhoff diffraction stack, co-located facies/impedance attributes, and the rendering pipeline.","source":"@site/docs/general/wells-seismic/seismic-modelling-method.md","sourceDirName":"general/wells-seismic","slug":"/general/wells-seismic/seismic-modelling-method","permalink":"/docs/next/general/wells-seismic/seismic-modelling-method","draft":false,"unlisted":false,"editUrl":"https://github.com/vrgeoscience/vrgs_docs_v3/tree/master/docs/general/wells-seismic/seismic-modelling-method.md","tags":[],"version":"current","sidebarPosition":4,"frontMatter":{"id":"seismic-modelling-method","title":"Synthetic Seismic Modelling from Classified Meshes: Method and Implementation","sidebar_label":"Seismic Modelling \u2014 Method","sidebar_position":4,"description":"A detailed, paper-style account of how VRGS forward-models a band-limited synthetic seismic section from a classified triangulated mesh: reflector extraction, the 2-D extrusion reflectivity model, the Kirchhoff diffraction stack, co-located facies/impedance attributes, and the rendering pipeline.","keywords":["seismic","forward modelling","kirchhoff","diffraction stack","reflectivity","facies","impedance","attributes","outcrop","synthetic seismic"]},"sidebar":"docs","previous":{"title":"Seismic Modelling","permalink":"/docs/next/general/wells-seismic/seismic-modelling"},"next":{"title":"Voxel Volumes (CT / MRI)","permalink":"/docs/next/general/wells-seismic/voxel-volumes"}}');var l=a(74848),i=a(28453);const t={id:"seismic-modelling-method",title:"Synthetic Seismic Modelling from Classified Meshes: Method and Implementation",sidebar_label:"Seismic Modelling \u2014 Method",sidebar_position:4,description:"A detailed, paper-style account of how VRGS forward-models a band-limited synthetic seismic section from a classified triangulated mesh: reflector extraction, the 2-D extrusion reflectivity model, the Kirchhoff diffraction stack, co-located facies/impedance attributes, and the rendering pipeline.",keywords:["seismic","forward modelling","kirchhoff","diffraction stack","reflectivity","facies","impedance","attributes","outcrop","synthetic seismic"]},m="Synthetic Seismic Modelling from Classified Meshes",c={},r=[{value:"Abstract",id:"abstract",level:2},{value:"1. Introduction",id:"1-introduction",level:2},{value:"2. Method",id:"2-method",level:2},{value:"2.1 Inputs",id:"21-inputs",level:3},{value:"2.2 Reflector extraction",id:"22-reflector-extraction",level:3},{value:"2.3 Reflectivity",id:"23-reflectivity",level:3},{value:"2.4 The two-dimensional extrusion assumption",id:"24-the-two-dimensional-extrusion-assumption",level:3},{value:"2.5 The Kirchhoff diffraction stack",id:"25-the-kirchhoff-diffraction-stack",level:3},{value:"2.6 Wavelet and band-limitation",id:"26-wavelet-and-band-limitation",level:3},{value:"2.7 Acquisition geometry",id:"27-acquisition-geometry",level:3},{value:"2.8 Co-located facies and impedance attributes (flattened overlay)",id:"28-co-located-facies-and-impedance-attributes-flattened-overlay",level:3},{value:"3. Implementation",id:"3-implementation",level:2},{value:"3.1 Data model",id:"31-data-model",level:3},{value:"3.2 Rendering",id:"32-rendering",level:3},{value:"3.3 Attribute layers, histograms, and filters",id:"33-attribute-layers-histograms-and-filters",level:3},{value:"3.4 Concurrency",id:"34-concurrency",level:3},{value:"4. Parameter reference (theory view)",id:"4-parameter-reference-theory-view",level:2},{value:"5. Assumptions and limitations",id:"5-assumptions-and-limitations",level:2},{value:"6. Future work",id:"6-future-work",level:2},{value:"References",id:"references",level:2},{value:"Related",id:"related",level:2}];function h(s){const e={a:"a",annotation:"annotation",blockquote:"blockquote",code:"code",em:"em",h1:"h1",h2:"h2",h3:"h3",header:"header",li:"li",math:"math",mfrac:"mfrac",mi:"mi",mn:"mn",mo:"mo",mover:"mover",mrow:"mrow",mspace:"mspace",mstyle:"mstyle",msub:"msub",msup:"msup",mtext:"mtext",munder:"munder",ol:"ol",p:"p",path:"path",semantics:"semantics",span:"span",strong:"strong",svg:"svg",table:"table",tbody:"tbody",td:"td",th:"th",thead:"thead",tr:"tr",ul:"ul",...(0,i.R)(),...s.components};return(0,l.jsxs)(l.Fragment,{children:[(0,l.jsx)(e.header,{children:(0,l.jsx)(e.h1,{id:"synthetic-seismic-modelling-from-classified-meshes",children:"Synthetic Seismic Modelling from Classified Meshes"})}),"\n",(0,l.jsxs)(e.blockquote,{children:["\n",(0,l.jsxs)(e.p,{children:[(0,l.jsx)(e.strong,{children:"Scope."})," This page is the ",(0,l.jsx)(e.em,{children:"methodological"})," companion to the\n",(0,l.jsx)(e.a,{href:"/docs/next/general/wells-seismic/seismic-modelling",children:"Seismic Modelling"})," user guide. It documents the physics,\nthe numerical scheme, and the softw

1are architecture in enough detail to\nreproduce or extend the method. The user guide covers the workflow and the\ndialog; here we explain ",(0,l.jsx)(e.em,{children:"why"})," the result looks the way it does."]}),"\n"]}),"\n",(0,l.jsx)(e.h2,{id:"abstract",children:"Abstract"}),"\n",(0,l.jsxs)(e.p,{children:["We describe a forward operator that turns a ",(0,l.jsx)(e.strong,{children:"classified triangulated surface\nmesh"})," \u2014 an outcrop or interpreted horizon model carrying per-vertex ",(0,l.jsx)(e.em,{children:"facies"})," and\n",(0,l.jsx)(e.em,{children:"acoustic impedance"})," \u2014 into a ",(0,l.jsx)(e.strong,{children:"band-limited synthetic seismic section"}),". Contacts\nbetween facies are treated as reflecting elements; their reflectivity follows from\nthe impedance contrast across the contact, and their orientation follows from the\nin-plane impedance gradient under an explicit ",(0,l.jsx)(e.strong,{children:"two-dimensional extrusion\nassumption"}),". A ",(0,l.jsx)(e.strong,{children:"Kirchhoff diffraction stack"})," sums the contribution of every\nreflecting element into each trace using straight-ray traveltimes in a constant\nbackground velocity, geometric spreading, and an obliquity weight; each trace is\nthen convolved with a chosen wavelet. The same traversal records, co-located with\nthe modelled reflections, the ",(0,l.jsx)(e.strong,{children:"facies and impedance that produced them"}),", which\nare exposed as filterable attribute layers on the resulting section. We close with\nthe data model, the colour-mapped panel renderer, and the assumptions and\nlimitations of the approach."]}),"\n",(0,l.jsx)(e.h2,{id:"1-introduction",children:"1. Introduction"}),"\n",(0,l.jsxs)(e.p,{children:["Tying outcrop interpretation to the seismic scale is a recurring problem in\nreservoir geology: a contact that is obvious on a cliff face may image strongly,\nweakly, or only as a diffraction once it is band-limited and viewed under a\nparticular acquisition geometry. Rather than exporting to an external modelling\npackage, VRGS computes a synthetic section ",(0,l.jsx)(e.em,{children:"in situ"})," from the same classified mesh\nthe interpreter is already working with. The method is deliberately lightweight \u2014\na single-scattering (Born/Kirchhoff) approximation with straight rays \u2014 so that it\nruns interactively on a background worker and answers the qualitative question\n",(0,l.jsx)(e.em,{children:'"how would this geometry image?"'})," rather than producing a calibrated amplitude\nprediction."]}),"\n",(0,l.jsx)(e.h2,{id:"2-method",children:"2. Method"}),"\n",(0,l.jsx)(e.h3,{id:"21-inputs",children:"2.1 Inputs"}),"\n",(0,l.jsx)(e.p,{children:"The operator consumes a triangulated mesh with two per-vertex scalar layers:"}),"\n",(0,l.jsxs)(e.table,{children:[(0,l.jsx)(e.thead,{children:(0,l.jsxs)(e.tr,{children:[(0,l.jsx)(e.th,{children:"Symbol"}),(0,l.jsx)(e.th,{children:"Quantity"}),(0,l.jsx)(e.th,{children:"Role"})]})}),(0,l.jsxs)(e.tbody,{children:[(0,l.jsxs)(e.tr,{children:[(0,l.jsx)(e.td,{children:(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"c"}),(0,l.jsx)(e.mi,{children:"i"})]})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"c_i"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",children:"c"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3117em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",children:"i"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]})]})})]})}),(0,l.jsxs)(e.td,{children:["facies / class id at vertex ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"i"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"i"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6595em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"i"})]})})]})]}),(0,l.jsxs)(e.td,{children:["identifies contacts (where ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"c"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"c"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"c"})]})})]})," differs across a facet)"]})]}),(0,l.jsxs)(e.tr,{children:[(0,l.jsx)(e.td,{children:(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"Z"}),(0,l.jsx)(e.mi,{children:"i"})]})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"Z_i"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.07153em"},children:"Z"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.311

17em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.0715em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",children:"i"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]})]})})]})}),(0,l.jsxs)(e.td,{children:["acoustic impedance at vertex ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"i"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"i"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6595em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"i"})]})})]})]}),(0,l.jsx)(e.td,{children:"drives reflectivity and reflector orientation"})]})]})]}),"\n",(0,l.jsxs)(e.p,{children:["Impedance is the product of bulk density and P-wave velocity, ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"Z"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mi,{children:"\u03c1"}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"V"}),(0,l.jsx)(e.mi,{children:"p"})]})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"Z=\\rho V_p"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.07153em"},children:"Z"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.9694em",verticalAlign:"-0.2861em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"\u03c1"}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.22222em"},children:"V"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.2222em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",children:"p"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.2861em"},children:(0,l.jsx)(e.span,{})})})]})})]})]})]})]}),". The\nlayers are produced upstream (classification, painting, or the ",(0,l.jsx)(e.em,{children:"Add Test Facies +\nImpedance"})," helper) and are read by name; the operator does not classify."]}),"\n",(0,l.jsx)(e.h3,{id:"22-reflector-extraction",children:"2.2 Reflector extraction"}),"\n",(0,l.jsxs)(e.p,{children:["The mesh is traversed facet by facet. For a facet with vertices\n",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mo,{stretchy:"false",children:"{"}),(0,l.jsx)(e.mi,{children:"i"}),(0,l.jsx)(e.mo,{separator:"true",children:","}),(0,l.jsx)(e.mi,{children:"j"}),(0,l.jsx)(e.mo,{separator:"true",children:","}),(0,l.jsx)(e.mi,{children:"k"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"}"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\{i,j,k\\}"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mopen",children:"{"}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"i"}),(0,l.jsx)(e.span,{className:"mpunct",children:","}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.05724em"},children:"j"}),(0,l.jsx)(e.span,{className:"mpunct",children:","}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03148em"},children:"k"}),(0,l.jsx)(e.span,{className:"mclose",children:"}"})]})})]})," we group the vertices by facies id. A facet whose three vertices share\na single facies is interior to a unit and ",(0,l.jsx)(e.strong,{children:"contributes nothing"}),". Otherwise the\nfacet ",(0,l.jsx)(e.em,{children:"straddles a contact"})," and becomes a reflecting element with:"]}),"\n",(0,l.jsxs)(e.ul,{children:["\n",(0,l.jsxs)(e.li,{children:["a ",(0,l.jsx)(e.strong,{children:"centroid"})," ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"x"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsxs)(e.mfrac,{children:[(0,l.jsx)(e.mn,{children:"1"}),(0,l.jsx)(e.mn,{children:"3"})]}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"p"}),(0,l.jsx)(e.mi,{children:"i"})]}),(0,l.jsx)(e.mo,{children:"+"}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"p"}),(0,l.jsx)(e.mi,{children:"j"})]}),(0,l.jsx)(e.mo,{children:"+"}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"p"}),(0,l.jsx)(e.mi,{children:"k"})]}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\mathbf{x}=\\tfrac{1}{3}(\\mathbf{p}_i+\\mathbf{p}_j+\\mathbf{p}_k)"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4444em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"x"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1.1901em",verticalAlign:"-0.345em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mopen nulldelimiter"}),(0,l.jsx)(e.span,{className:"mfrac",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.8451em"}

1,children:[(0,l.jsxs)(e.span,{style:{top:"-2.655em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"3"})})})]}),(0,l.jsxs)(e.span,{style:{top:"-3.23em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,l.jsxs)(e.span,{style:{top:"-3.394em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"1"})})})]})]}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.345em"},children:(0,l.jsx)(e.span,{})})})]})}),(0,l.jsx)(e.span,{className:"mclose nulldelimiter"})]}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathbf",children:"p"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3117em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",children:"i"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"+"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8694em",verticalAlign:"-0.2861em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathbf",children:"p"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3117em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.05724em"},children:"j"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.2861em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"+"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathbf",children:"p"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3361em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.03148em"},children:"k"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mclose",children:")"})]})]})]})," in true world coordinates;"]}),"\n",(0,l.jsxs)(e.li,{children:["a true 3-D ",(0,l.jsx)(e.strong,{children:"area"})," ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"a"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsxs)(e.mfrac,{children:[(0,l.jsx)(e.mn,{children:"1"}),(0,l.jsx)(e.mn,{children:"2"})]}),(0,l.jsx)(e.mo,{stretchy:"false",children:"\u2225"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"p"}),(0,l.jsx)(e.mi,{children:"j"})]}),(0,l.jsx)(e.mo,{children:"\u2212"}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"p"}),(0,l.jsx)(e.mi,{children:"i"})]}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"}),(0,l.jsx)(e.mo,{children:"\xd7"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"p"}),(0,l.jsx)(e.mi,{children:"k"})]}),(0,l.jsx)(e.mo,{children:"\u2212"}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"p"}),(0,l.jsx)(e.mi,{children:"i"})]}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"\u2225"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"a=\\tfrac12\\lVert(\\mathbf{p}_j-\\mathbf{p}_i)\\times(\\mathbf{p}_k-\\mathbf{p}_i)\\rVert"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"a"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1.1901em",verticalAlign:"-0.345em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mopen nulldelimiter"}),(0,l.jsx)(e.span,{className:"mfrac",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.8451em"}

1,children:[(0,l.jsxs)(e.span,{style:{top:"-2.655em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"2"})})})]}),(0,l.jsxs)(e.span,{style:{top:"-3.23em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,l.jsxs)(e.span,{style:{top:"-3.394em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"1"})})})]})]}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.345em"},children:(0,l.jsx)(e.span,{})})})]})}),(0,l.jsx)(e.span,{className:"mclose nulldelimiter"})]}),(0,l.jsx)(e.span,{className:"mopen",children:"\u2225("}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathbf",children:"p"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3117em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.05724em"},children:"j"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.2861em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u2212"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathbf",children:"p"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3117em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",children:"i"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mclose",children:")"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\xd7"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathbf",children:"p"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3361em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.03148em"},children:"k"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u2212"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathbf",children:"p"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.311

17em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",children:"i"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mclose",children:")\u2225"})]})]})]}),";"]}),"\n",(0,l.jsxs)(e.li,{children:["a ",(0,l.jsx)(e.strong,{children:"reflection coefficient"})," and a ",(0,l.jsx)(e.strong,{children:"normal"})," (below)."]}),"\n"]}),"\n",(0,l.jsx)(e.h3,{id:"23-reflectivity",children:"2.3 Reflectivity"}),"\n",(0,l.jsxs)(e.p,{children:["Each facies group has a mean impedance. For a facet touching two facies the contact\nis unambiguous; for a facet touching three, the pair with the ",(0,l.jsx)(e.strong,{children:"largest impedance\ncontrast"})," is selected (a deterministic, vertex-order tie-break). With\n",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"Z"}),(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"normal",children:"h"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"i"})]})]})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"Z_{\\mathrm{hi}}"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.07153em"},children:"Z"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3361em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.0715em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord mathrm mtight",children:"hi"})})})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]})]})})]})," and ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"Z"}),(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"normal",children:"l"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"o"})]})]})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"Z_{\\mathrm{lo}}"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.07153em"},children:"Z"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3361em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.0715em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord mathrm mtight",children:"lo"})})})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]})]})})]})," the mean impedances on the two sides, the\n",(0,l.jsx)(e.strong,{children:"normal-incidence reflection coefficient"})," is"]}),"\n",(0,l.jsx)(e.span,{className:"katex-display",children:(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"R"}),(0,l.jsx)(e.mtext,{children:"\u2005\u200a"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mtext,{children:"\u2005\u200a"}),(0,l.jsxs)(e.mfrac,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"Z"}),(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"normal",children:"h"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"i"})]})]}),(0,l.jsx)(e.mo,{children:"\u2212"}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"Z"}),(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"normal",children:"l"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"o"})]})]})]}),(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"Z"}),(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"normal",children:"h"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"i"})]})]}),(0,l.jsx)(e.mo,{children:"+"}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"Z"}),(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"normal",children:"l"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"o"})]})]})]})]}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"."})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"R \\;=\\; \\frac{Z_{\\mathrm{hi}} - Z_{\\mathrm{lo}}}{Z_{\\mathrm{hi}} + Z_{\\mathrm{lo}}}."})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.00773em"},children:"R"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"2.1963em",verticalAlign:"-0.836em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mopen nulldelimiter"}),(0,l.jsx)(e.span,{className:"mfrac",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsxs)(e.span,{className:"vlist",style:{height:"1.3603em"}

1,children:[(0,l.jsxs)(e.span,{style:{top:"-2.314em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.07153em"},children:"Z"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3361em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.0715em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord mathrm mtight",children:"hi"})})})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"+"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.07153em"},children:"Z"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3361em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.0715em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord mathrm mtight",children:"lo"})})})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]})]})]}),(0,l.jsxs)(e.span,{style:{top:"-3.23em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,l.jsxs)(e.span,{style:{top:"-3.677em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.07153em"},children:"Z"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3361em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.0715em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord mathrm mtight",children:"hi"})})})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u2212"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.07153em"},children:"Z"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3361em"}

1,children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.0715em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord mathrm mtight",children:"lo"})})})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]})]})]})]}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.836em"},children:(0,l.jsx)(e.span,{})})})]})}),(0,l.jsx)(e.span,{className:"mclose nulldelimiter"})]}),(0,l.jsx)(e.span,{className:"mord",children:"."})]})]})]})}),"\n",(0,l.jsx)(e.p,{children:"This is the standard acoustic, normal-incidence form; angle-dependent (AVO) effects\nare not modelled."}),"\n",(0,l.jsx)(e.h3,{id:"24-the-two-dimensional-extrusion-assumption",children:"2.4 The two-dimensional extrusion assumption"}),"\n",(0,l.jsxs)(e.p,{children:["The reflecting orientation is ",(0,l.jsx)(e.strong,{children:"not"})," taken from the cliff-face (mesh) normal.\nInstead it is the ",(0,l.jsx)(e.strong,{children:"impedance gradient across the facet, projected into the\nacquisition plane"})," and normalised:"]}),"\n",(0,l.jsx)(e.span,{className:"katex-display",children:(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.mover,{accent:"true",children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"n"}),(0,l.jsx)(e.mo,{children:"^"})]}),(0,l.jsx)(e.mtext,{children:"\u2005\u200a"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mtext,{children:"\u2005\u200a"}),(0,l.jsxs)(e.mfrac,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"P"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u03a0"})]}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u2207"}),(0,l.jsx)(e.mi,{children:"Z"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mo,{stretchy:"false",children:"\u2225"}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"P"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u03a0"})]}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u2207"}),(0,l.jsx)(e.mi,{children:"Z"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"\u2225"})]})]}),(0,l.jsx)(e.mo,{separator:"true",children:","}),(0,l.jsx)(e.mspace,{width:"2em"}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"P"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u03a0"})]}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"v"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"v"}),(0,l.jsx)(e.mo,{children:"\u2212"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"v"}),(0,l.jsx)(e.mo,{children:"\u22c5"}),(0,l.jsxs)(e.mover,{accent:"true",children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"m"}),(0,l.jsx)(e.mo,{children:"^"})]}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"}),(0,l.jsx)(e.mtext,{children:"\u2009"}),(0,l.jsxs)(e.mover,{accent:"true",children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"m"}),(0,l.jsx)(e.mo,{children:"^"})]}),(0,l.jsx)(e.mo,{separator:"true",children:","})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\hat{\\mathbf{n}} \\;=\\; \\frac{P_\\Pi(\\nabla Z)}{\\lVert P_\\Pi(\\nabla Z)\\rVert},\n\\qquad\nP_\\Pi(\\mathbf{v}) = \\mathbf{v} - (\\mathbf{v}\\cdot\\hat{\\mathbf{m}})\\,\\hat{\\mathbf{m}},"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.7079em"}}),(0,l.jsx)(e.span,{className:"mord accent",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.7079em"}

1,children:[(0,l.jsxs)(e.span,{style:{top:"-3em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"n"})]}),(0,l.jsxs)(e.span,{style:{top:"-3.0134em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"accent-body",style:{left:"-0.25em"},children:(0,l.jsx)(e.span,{className:"mord",children:"^"})})]})]})})})}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"2.363em",verticalAlign:"-0.936em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mopen nulldelimiter"}),(0,l.jsx)(e.span,{className:"mfrac",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsxs)(e.span,{className:"vlist",style:{height:"1.427em"},children:[(0,l.jsxs)(e.span,{style:{top:"-2.314em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mopen",children:"\u2225"}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"P"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.1389em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"\u03a0"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsx)(e.span,{className:"mord",children:"\u2207"}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.07153em"},children:"Z"}),(0,l.jsx)(e.span,{className:"mclose",children:")\u2225"})]})]}),(0,l.jsxs)(e.span,{style:{top:"-3.23em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,l.jsxs)(e.span,{style:{top:"-3.677em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"P"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.1389em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"\u03a0"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsx)(e.span,{className:"mord",children:"\u2207"}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.07153em"},children:"Z"}),(0,l.jsx)(e.span,{className:"mclose",children:")"})]})]})]}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.936em"},children:(0,l.jsx)(e.span,{})})})]})}),(0,l.jsx)(e.span,{className:"mclose nulldelimiter"})]}),(0,l.jsx)(e.span,{className:"mpunct",children:","}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"2em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"P"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"}

1,children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.1389em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"\u03a0"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsx)(e.span,{className:"mord mathbf",style:{marginRight:"0.01597em"},children:"v"}),(0,l.jsx)(e.span,{className:"mclose",children:")"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",style:{marginRight:"0.01597em"},children:"v"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u2212"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsx)(e.span,{className:"mord mathbf",style:{marginRight:"0.01597em"},children:"v"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u22c5"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord accent",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.7079em"},children:[(0,l.jsxs)(e.span,{style:{top:"-3em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"m"})]}),(0,l.jsxs)(e.span,{style:{top:"-3.0134em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"accent-body",style:{left:"-0.25em"},children:(0,l.jsx)(e.span,{className:"mord",children:"^"})})]})]})})})}),(0,l.jsx)(e.span,{className:"mclose",children:")"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord accent",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.7079em"},children:[(0,l.jsxs)(e.span,{style:{top:"-3em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"m"})]}),(0,l.jsxs)(e.span,{style:{top:"-3.0134em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"accent-body",style:{left:"-0.25em"},children:(0,l.jsx)(e.span,{className:"mord",children:"^"})})]})]})})})}),(0,l.jsx)(e.span,{className:"mpunct",children:","})]})]})]})}),"\n",(0,l.jsxs)(e.p,{children:["where ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u2207"}),(0,l.jsx)(e.mi,{children:"Z"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\nabla Z"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u2207"}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.07153em"},children:"Z"})]})})]})," is the constant gradient of the linearly-interpolated impedance\nover the triangle, ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsxs)(e.mover,{accent:"true",children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"m"}),(0,l.jsx)(e.mo,{children:"^"})]})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\hat{\\mathbf{m}}"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.7079em"}}),(0,l.jsx)(e.span,{className:"mord accent",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.7079em"}

1,children:[(0,l.jsxs)(e.span,{style:{top:"-3em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"m"})]}),(0,l.jsxs)(e.span,{style:{top:"-3.0134em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"accent-body",style:{left:"-0.25em"},children:(0,l.jsx)(e.span,{className:"mord",children:"^"})})]})]})})})})]})})]})," is the horizontal normal of the vertical\nacquisition plane, and ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"P"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u03a0"})]})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"P_\\Pi"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"P"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.1389em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"\u03a0"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]})]})})]})," removes the out-of-plane component. The gradient is\nundefined (returns ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mn,{mathvariant:"bold",children:"0"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\mathbf 0"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"0"})]})})]}),") for a degenerate triangle or vanishing contrast."]}),"\n",(0,l.jsxs)(e.p,{children:["Geologically this asserts that ",(0,l.jsx)(e.strong,{children:"each contact extends perpendicular to the\nsection"})," \u2014 a 2-D structure swept along strike. It is what makes a ",(0,l.jsx)(e.em,{children:"horizontal bed"}),"\nreflect strongly to a source directly above it (its normal points up, into the\nplane), while a ",(0,l.jsx)(e.em,{children:"vertical fault"})," is weak at zero offset (its normal is near-\nhorizontal) yet still ",(0,l.jsx)(e.strong,{children:"diffracts off its truncated tips"}),". Where measured\nper-vertex dip vectors are available they can be substituted for the computed\ngradient, turning the assumption into data."]}),"\n",(0,l.jsx)(e.h3,{id:"25-the-kirchhoff-diffraction-stack",children:"2.5 The Kirchhoff diffraction stack"}),"\n",(0,l.jsxs)(e.p,{children:["Every reflecting element is treated as a Huygens secondary source. For a\nsource\u2013receiver pair ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"s"}),(0,l.jsx)(e.mo,{separator:"true",children:","}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"g"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"(\\mathbf s,\\mathbf g)"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"s"}),(0,l.jsx)(e.span,{className:"mpunct",children:","}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",style:{marginRight:"0.01597em"},children:"g"}),(0,l.jsx)(e.span,{className:"mclose",children:")"})]})})]})," defining trace ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"T"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"T"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"T"})]})})]}),", and an element at\n",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{mathvariant:"bold",children:"x"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\mathbf x"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4444em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"x"})]})})]})," with reflectivity ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"R"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"R"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.00773em"},children:"R"})]})})]}),", area ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"a"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"a"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"a"})]})})]}),", and unit normal ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsxs)(e.mover,{accent:"true",children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"n"}),(0,l.jsx)(e.mo,{children:"^"})]})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\hat{\\mathbf n}"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.7079em"}}),(0,l.jsx)(e.span,{className:"mord accent",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.7079em"}

1,children:[(0,l.jsxs)(e.span,{style:{top:"-3em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"n"})]}),(0,l.jsxs)(e.span,{style:{top:"-3.0134em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"accent-body",style:{left:"-0.25em"},children:(0,l.jsx)(e.span,{className:"mord",children:"^"})})]})]})})})})]})})]}),", let"]}),"\n",(0,l.jsx)(e.span,{className:"katex-display",children:(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"r"}),(0,l.jsx)(e.mi,{children:"s"})]}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mo,{stretchy:"false",children:"\u2225"}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"x"}),(0,l.jsx)(e.mo,{children:"\u2212"}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"s"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"\u2225"}),(0,l.jsx)(e.mo,{separator:"true",children:","}),(0,l.jsx)(e.mspace,{width:"2em"}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"r"}),(0,l.jsx)(e.mi,{children:"g"})]}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mo,{stretchy:"false",children:"\u2225"}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"x"}),(0,l.jsx)(e.mo,{children:"\u2212"}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"g"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"\u2225"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"."})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"r_s=\\lVert\\mathbf x-\\mathbf s\\rVert,\\qquad r_g=\\lVert\\mathbf x-\\mathbf g\\rVert."})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.02778em"},children:"r"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.0278em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",children:"s"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mopen",children:"\u2225"}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"x"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u2212"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1.0361em",verticalAlign:"-0.2861em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"s"}),(0,l.jsx)(e.span,{className:"mclose",children:"\u2225"}),(0,l.jsx)(e.span,{className:"mpunct",children:","}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"2em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.02778em"},children:"r"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"}

1,children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.0278em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.03588em"},children:"g"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.2861em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mopen",children:"\u2225"}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"x"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u2212"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",style:{marginRight:"0.01597em"},children:"g"}),(0,l.jsx)(e.span,{className:"mclose",children:"\u2225"}),(0,l.jsx)(e.span,{className:"mord",children:"."})]})]})]})}),"\n",(0,l.jsx)(e.p,{children:"The element deposits an impulse of amplitude"}),"\n",(0,l.jsx)(e.span,{className:"katex-display",children:(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"A"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"x"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"}),(0,l.jsx)(e.mtext,{children:"\u2005\u200a"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mtext,{children:"\u2005\u200a"}),(0,l.jsxs)(e.mfrac,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"R"}),(0,l.jsx)(e.mtext,{children:"\u2009"}),(0,l.jsx)(e.mi,{children:"a"}),(0,l.jsx)(e.mtext,{children:"\u2009"}),(0,l.jsx)(e.mi,{mathvariant:"script",children:"O"})]}),(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"r"}),(0,l.jsx)(e.mi,{children:"s"})]}),(0,l.jsx)(e.mtext,{children:"\u2009"}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"r"}),(0,l.jsx)(e.mi,{children:"g"})]})]})]}),(0,l.jsx)(e.mo,{separator:"true",children:","}),(0,l.jsx)(e.mspace,{width:"2em"}),(0,l.jsx)(e.mi,{mathvariant:"script",children:"O"}),(0,l.jsx)(e.mtext,{children:"\u2005\u200a"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mtext,{children:"\u2005\u200a"}),(0,l.jsx)(e.mstyle,{displaystyle:"false",scriptlevel:"0",children:(0,l.jsxs)(e.mfrac,{children:[(0,l.jsx)(e.mn,{children:"1"}),(0,l.jsx)(e.mn,{children:"2"})]})}),(0,l.jsx)(e.mtext,{children:"\u2009\u2063"}),(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mo,{fence:"true",children:"("}),(0,l.jsx)(e.mo,{fence:"true",stretchy:"true",minsize:"1.2em",maxsize:"1.2em",children:"\u2223"}),(0,l.jsxs)(e.msub,{children:[(0,l.jsxs)(e.mover,{accent:"true",children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"d"}),(0,l.jsx)(e.mo,{children:"^"})]}),(0,l.jsx)(e.mi,{children:"s"})]}),(0,l.jsx)(e.mtext,{children:"\u2009\u2063"}),(0,l.jsx)(e.mo,{children:"\u22c5"}),(0,l.jsxs)(e.mover,{accent:"true",children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"n"}),(0,l.jsx)(e.mo,{children:"^"})]}),(0,l.jsx)(e.mo,{fence:"true",stretchy:"true",minsize:"1.2em",maxsize:"1.2em",children:"\u2223"}),(0,l.jsx)(e.mo,{children:"+"}),(0,l.jsx)(e.mo,{fence:"true",stretchy:"true",minsize:"1.2em",maxsize:"1.2em",children:"\u2223"}),(0,l.jsxs)(e.msub,{children:[(0,l.jsxs)(e.mover,{accent:"true",children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"d"}),(0,l.jsx)(e.mo,{children:"^"})]}),(0,l.jsx)(e.mi,{children:"g"})]}),(0,l.jsx)(e.mtext,{children:"\u2009\u2063"}),(0,l.jsx)(e.mo,{children:"\u22c5"}),(0,l.jsxs)(e.mover,{accent:"true",children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"n"}),(0,l.jsx)(e.mo,{children:"^"})]}),(0,l.jsx)(e.mo,{fence:"true",stretchy:"true",minsize:"1.2em",maxsize:"1.2em",children:"\u2223"}),(0,l.jsx)(e.mo,{fence:"true",children:")"})]}),(0,l.jsx)(e.mo,{separator:"true",children:","})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"A(\\mathbf x) \\;=\\; \\frac{R\\,a\\,\\mathcal{O}}{r_s\\,r_g},\n\\qquad\n\\mathcal{O} \\;=\\; \\tfrac12\\!\\left(\\bigl|\\hat{\\mathbf d}_s\\!\\cdot\\hat{\\mathbf n}\\bigr| + \\bigl|\\hat{\\mathbf d}_g\\!\\cdot\\hat{\\mathbf n}\\bigr|\\right),"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"A"}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"x"}),(0,l.jsx)(e.span,{className:"mclose",children:")"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"2.3324em",verticalAlign:"-0.9721em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mopen nulldelimiter"}),(0,l.jsx)(e.span,{className:"mfrac",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsxs)(e.span,{className:"vlist",style:{height:"1.3603em"}

1,children:[(0,l.jsxs)(e.span,{style:{top:"-2.314em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.02778em"},children:"r"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.0278em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",children:"s"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.02778em"},children:"r"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.0278em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.03588em"},children:"g"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.2861em"},children:(0,l.jsx)(e.span,{})})})]})})]})]})]}),(0,l.jsxs)(e.span,{style:{top:"-3.23em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,l.jsxs)(e.span,{style:{top:"-3.677em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.00773em"},children:"R"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"a"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord mathcal",style:{marginRight:"0.02778em"},children:"O"})]})]})]}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.9721em"},children:(0,l.jsx)(e.span,{})})})]})}),(0,l.jsx)(e.span,{className:"mclose nulldelimiter"})]}),(0,l.jsx)(e.span,{className:"mpunct",children:","}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"2em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord mathcal",style:{marginRight:"0.02778em"},children:"O"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1.8em",verticalAlign:"-0.65em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mopen nulldelimiter"}),(0,l.jsx)(e.span,{className:"mfrac",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.8451em"}

1,children:[(0,l.jsxs)(e.span,{style:{top:"-2.655em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"2"})})})]}),(0,l.jsxs)(e.span,{style:{top:"-3.23em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,l.jsxs)(e.span,{style:{top:"-3.394em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"1"})})})]})]}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.345em"},children:(0,l.jsx)(e.span,{})})})]})}),(0,l.jsx)(e.span,{className:"mclose nulldelimiter"})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"-0.1667em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsxs)(e.span,{className:"minner",children:[(0,l.jsx)(e.span,{className:"mopen delimcenter",style:{top:"0em"},children:(0,l.jsx)(e.span,{className:"delimsizing size2",children:"("})}),(0,l.jsx)(e.span,{className:"mopen",children:(0,l.jsx)(e.span,{className:"delimsizing mult",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.85em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.85em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3.2em"}}),(0,l.jsx)(e.span,{style:{width:"0.333em",height:"1.200em"},children:(0,l.jsx)(e.svg,{xmlns:"http://www.w3.org/2000/svg",width:"0.333em",height:"1.200em",viewBox:"0 0 333 1200",children:(0,l.jsx)(e.path,{d:"M145 15 v585 v0 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z"})})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.35em"},children:(0,l.jsx)(e.span,{})})})]})})}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord accent",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.9579em"},children:[(0,l.jsxs)(e.span,{style:{top:"-3em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"d"})]}),(0,l.jsxs)(e.span,{style:{top:"-3.2634em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"accent-body",style:{left:"-0.0833em"},children:(0,l.jsx)(e.span,{className:"mord",children:"^"})})]})]})})})}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",children:"s"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"-0.1667em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u22c5"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mord accent",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.7079em"}

1,children:[(0,l.jsxs)(e.span,{style:{top:"-3em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"n"})]}),(0,l.jsxs)(e.span,{style:{top:"-3.0134em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"accent-body",style:{left:"-0.25em"},children:(0,l.jsx)(e.span,{className:"mord",children:"^"})})]})]})})})}),(0,l.jsx)(e.span,{className:"mclose",children:(0,l.jsx)(e.span,{className:"delimsizing mult",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.85em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.85em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3.2em"}}),(0,l.jsx)(e.span,{style:{width:"0.333em",height:"1.200em"},children:(0,l.jsx)(e.svg,{xmlns:"http://www.w3.org/2000/svg",width:"0.333em",height:"1.200em",viewBox:"0 0 333 1200",children:(0,l.jsx)(e.path,{d:"M145 15 v585 v0 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z"})})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.35em"},children:(0,l.jsx)(e.span,{})})})]})})}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"+"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mopen",children:(0,l.jsx)(e.span,{className:"delimsizing mult",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.85em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.85em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3.2em"}}),(0,l.jsx)(e.span,{style:{width:"0.333em",height:"1.200em"},children:(0,l.jsx)(e.svg,{xmlns:"http://www.w3.org/2000/svg",width:"0.333em",height:"1.200em",viewBox:"0 0 333 1200",children:(0,l.jsx)(e.path,{d:"M145 15 v585 v0 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z"})})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.35em"},children:(0,l.jsx)(e.span,{})})})]})})}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord accent",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.9579em"},children:[(0,l.jsxs)(e.span,{style:{top:"-3em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"d"})]}),(0,l.jsxs)(e.span,{style:{top:"-3.2634em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"accent-body",style:{left:"-0.0833em"},children:(0,l.jsx)(e.span,{className:"mord",children:"^"})})]})]})})})}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.03588em"},children:"g"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.2861em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"-0.1667em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u22c5"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mord accent",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.7079em"}

1,children:[(0,l.jsxs)(e.span,{style:{top:"-3em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"n"})]}),(0,l.jsxs)(e.span,{style:{top:"-3.0134em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"accent-body",style:{left:"-0.25em"},children:(0,l.jsx)(e.span,{className:"mord",children:"^"})})]})]})})})}),(0,l.jsx)(e.span,{className:"mclose",children:(0,l.jsx)(e.span,{className:"delimsizing mult",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.85em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.85em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3.2em"}}),(0,l.jsx)(e.span,{style:{width:"0.333em",height:"1.200em"},children:(0,l.jsx)(e.svg,{xmlns:"http://www.w3.org/2000/svg",width:"0.333em",height:"1.200em",viewBox:"0 0 333 1200",children:(0,l.jsx)(e.path,{d:"M145 15 v585 v0 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z"})})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.35em"},children:(0,l.jsx)(e.span,{})})})]})})}),(0,l.jsx)(e.span,{className:"mclose delimcenter",style:{top:"0em"},children:(0,l.jsx)(e.span,{className:"delimsizing size2",children:")"})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mpunct",children:","})]})]})]})}),"\n",(0,l.jsxs)(e.p,{children:["where ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.msub,{children:[(0,l.jsxs)(e.mover,{accent:"true",children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"d"}),(0,l.jsx)(e.mo,{children:"^"})]}),(0,l.jsx)(e.mi,{children:"s"})]}),(0,l.jsx)(e.mo,{separator:"true",children:","}),(0,l.jsxs)(e.msub,{children:[(0,l.jsxs)(e.mover,{accent:"true",children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"d"}),(0,l.jsx)(e.mo,{children:"^"})]}),(0,l.jsx)(e.mi,{children:"g"})]})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\hat{\\mathbf d}_s,\\hat{\\mathbf d}_g"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1.244em",verticalAlign:"-0.2861em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord accent",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.9579em"},children:[(0,l.jsxs)(e.span,{style:{top:"-3em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"d"})]}),(0,l.jsxs)(e.span,{style:{top:"-3.2634em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"accent-body",style:{left:"-0.0833em"},children:(0,l.jsx)(e.span,{className:"mord",children:"^"})})]})]})})})}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",children:"s"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mpunct",children:","}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord accent",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.9579em"}

1,children:[(0,l.jsxs)(e.span,{style:{top:"-3em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"d"})]}),(0,l.jsxs)(e.span,{style:{top:"-3.2634em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"accent-body",style:{left:"-0.0833em"},children:(0,l.jsx)(e.span,{className:"mord",children:"^"})})]})]})})})}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.03588em"},children:"g"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.2861em"},children:(0,l.jsx)(e.span,{})})})]})})]})]})})]})," are the unit ray directions from the\nelement to source and receiver. The factor ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mn,{children:"1"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"/"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"r"}),(0,l.jsx)(e.mi,{children:"s"})]}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"r"}),(0,l.jsx)(e.mi,{children:"g"})]}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"1/(r_s r_g)"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1.0361em",verticalAlign:"-0.2861em"}}),(0,l.jsx)(e.span,{className:"mord",children:"1/"}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.02778em"},children:"r"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.0278em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",children:"s"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.02778em"},children:"r"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.0278em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.03588em"},children:"g"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.2861em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mclose",children:")"})]})})]})," is the product of two\n3-D spherical-spreading terms; the ",(0,l.jsx)(e.strong,{children:"obliquity"})," ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{mathvariant:"script",children:"O"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\mathcal O"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(e.span,{className:"mord mathcal",style:{marginRight:"0.02778em"},children:"O"})]})})]})," down-weights elements\nseen at grazing incidence and reduces to unity when the normal is undefined."]}),"\n",(0,l.jsxs)(e.p,{children:["Traveltimes use ",(0,l.jsx)(e.strong,{children:"straight rays in a constant background velocity"})," ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"V"}),(0,l.jsx)(e.mn,{children:"0"})]})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"V_0"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.22222em"},children:"V"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3011em"}

1,children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.2222em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"0"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]})]})})]}),":"]}),"\n",(0,l.jsx)(e.span,{className:"katex-display",children:(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"t"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"x"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsxs)(e.mfrac,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"r"}),(0,l.jsx)(e.mi,{children:"s"})]}),(0,l.jsx)(e.mo,{children:"+"}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"r"}),(0,l.jsx)(e.mi,{children:"g"})]})]}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"V"}),(0,l.jsx)(e.mn,{children:"0"})]})]}),(0,l.jsx)(e.mspace,{width:"1em"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mtext,{children:"time\xa0axis"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"}),(0,l.jsx)(e.mo,{separator:"true",children:","}),(0,l.jsx)(e.mspace,{width:"2em"}),(0,l.jsx)(e.mi,{children:"d"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"x"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsxs)(e.mfrac,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"r"}),(0,l.jsx)(e.mi,{children:"s"})]}),(0,l.jsx)(e.mo,{children:"+"}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"r"}),(0,l.jsx)(e.mi,{children:"g"})]})]}),(0,l.jsx)(e.mn,{children:"2"})]}),(0,l.jsx)(e.mspace,{width:"1em"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mtext,{children:"depth\xa0axis"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"."})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"t(\\mathbf x) = \\frac{r_s + r_g}{V_0}\\quad(\\text{time axis}),\n\\qquad\nd(\\mathbf x) = \\frac{r_s + r_g}{2}\\quad(\\text{depth axis})."})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"t"}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"x"}),(0,l.jsx)(e.span,{className:"mclose",children:")"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"2.0963em",verticalAlign:"-0.836em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mopen nulldelimiter"}),(0,l.jsx)(e.span,{className:"mfrac",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsxs)(e.span,{className:"vlist",style:{height:"1.2603em"},children:[(0,l.jsxs)(e.span,{style:{top:"-2.314em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord",children:(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.22222em"},children:"V"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3011em"}

1,children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.2222em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"0"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]})})]}),(0,l.jsxs)(e.span,{style:{top:"-3.23em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,l.jsxs)(e.span,{style:{top:"-3.677em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.02778em"},children:"r"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.0278em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",children:"s"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"+"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.02778em"},children:"r"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.0278em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.03588em"},children:"g"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.2861em"},children:(0,l.jsx)(e.span,{})})})]})})]})]})]})]}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.836em"},children:(0,l.jsx)(e.span,{})})})]})}),(0,l.jsx)(e.span,{className:"mclose nulldelimiter"})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"1em"}}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsx)(e.span,{className:"mord text",children:(0,l.jsx)(e.span,{className:"mord",children:"time\xa0axis"})}),(0,l.jsx)(e.span,{className:"mclose",children:")"}),(0,l.jsx)(e.span,{className:"mpunct",children:","}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"2em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"d"}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"x"}),(0,l.jsx)(e.span,{className:"mclose",children:")"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1.9463em",verticalAlign:"-0.686em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mopen nulldelimiter"}),(0,l.jsx)(e.span,{className:"mfrac",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsxs)(e.span,{className:"vlist",style:{height:"1.2603em"}

1,children:[(0,l.jsxs)(e.span,{style:{top:"-2.314em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord",children:(0,l.jsx)(e.span,{className:"mord",children:"2"})})]}),(0,l.jsxs)(e.span,{style:{top:"-3.23em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}),(0,l.jsxs)(e.span,{style:{top:"-3.677em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.02778em"},children:"r"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.0278em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",children:"s"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"+"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.02778em"},children:"r"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.0278em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.03588em"},children:"g"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.2861em"},children:(0,l.jsx)(e.span,{})})})]})})]})]})]})]}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.686em"},children:(0,l.jsx)(e.span,{})})})]})}),(0,l.jsx)(e.span,{className:"mclose nulldelimiter"})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"1em"}}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsx)(e.span,{className:"mord text",children:(0,l.jsx)(e.span,{className:"mord",children:"depth\xa0axis"})}),(0,l.jsx)(e.span,{className:"mclose",children:")"}),(0,l.jsx)(e.span,{className:"mord",children:"."})]})]})]})}),"\n",(0,l.jsxs)(e.p,{children:["The impulse is laid into the trace by ",(0,l.jsx)(e.strong,{children:"linear interpolation"})," between the two\nnearest samples at ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"\u03c4"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mi,{children:"t"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"/"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u0394"}),(0,l.jsx)(e.mi,{children:"t"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\tau=t/\\Delta t"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.1132em"},children:"\u03c4"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"t"}),(0,l.jsx)(e.span,{className:"mord",children:"/\u0394"}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"t"})]})]})]})," (or ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"d"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"/"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u0394"}),(0,l.jsx)(e.mi,{children:"t"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"d/\\Delta t"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"d"}),(0,l.jsx)(e.span,{className:"mord",children:"/\u0394"}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"t"})]})})]}),"). Summing over all elements\ngives the reflectivity series"]}),"\n",(0,l.jsx)(e.span,{className:"katex-display",children:(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"u"}),(0,l.jsx)(e.mi,{children:"T"})]}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mi,{children:"\u03c4"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"}),(0,l.jsx)(e.mtext,{children:"\u2005\u200a"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mtext,{children:"\u2005\u200a"}),(0,l.jsxs)(e.munder,{children:[(0,l.jsx)(e.mo,{children:"\u2211"}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"x"})]}),(0,l.jsx)(e.mi,{children:"A"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"x"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"}),(0,l.jsx)(e.mtext,{children:"\u2009"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u039b"}),(0,l.jsx)(e.mtext,{children:"\u2009\u2063"}),(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mo,{fence:"true",children:"("}),(0,l.jsx)(e.mi,{children:"\u03c4"}),(0,l.jsx)(e.mo,{children:"\u2212"}),(0,l.jsx)(e.mstyle,{displaystyle:"false",scriptlevel:"0",children:(0,l.jsxs)(e.mfrac,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"t"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"x"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u0394"}),(0,l.jsx)(e.mi,{children:"t"})]})]})}),(0,l.jsx)(e.mo,{fence:"true",children:")"})]}),(0,l.jsx)(e.mo,{separator:"true",children:","})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"u_T(\\tau) \\;=\\; \\sum_{\\mathbf x} A(\\mathbf x)\\,\\Lambda\\!\\left(\\tau - \\tfrac{t(\\mathbf x)}{\\Delta t}\\right),"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",children:"u"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"}

1,children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.13889em"},children:"T"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.1132em"},children:"\u03c4"}),(0,l.jsx)(e.span,{className:"mclose",children:")"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"2.4em",verticalAlign:"-1.25em"}}),(0,l.jsx)(e.span,{className:"mop op-limits",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsxs)(e.span,{className:"vlist",style:{height:"1.05em"},children:[(0,l.jsxs)(e.span,{style:{top:"-1.9em",marginLeft:"0em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3.05em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord mathbf mtight",children:"x"})})})]}),(0,l.jsxs)(e.span,{style:{top:"-3.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3.05em"}}),(0,l.jsx)(e.span,{children:(0,l.jsx)(e.span,{className:"mop op-symbol large-op",children:"\u2211"})})]})]}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"1.25em"},children:(0,l.jsx)(e.span,{})})})]})}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"A"}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"x"}),(0,l.jsx)(e.span,{className:"mclose",children:")"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u039b"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"-0.1667em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsxs)(e.span,{className:"minner",children:[(0,l.jsx)(e.span,{className:"mopen delimcenter",style:{top:"0em"},children:(0,l.jsx)(e.span,{className:"delimsizing size2",children:"("})}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.1132em"},children:"\u03c4"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u2212"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mopen nulldelimiter"}),(0,l.jsx)(e.span,{className:"mfrac",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsxs)(e.span,{className:"vlist",style:{height:"1.01em"},children:[(0,l.jsxs)(e.span,{style:{top:"-2.655em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsxs)(e.span,{className:"mord mtight",children:[(0,l.jsx)(e.span,{className:"mord mtight",children:"\u0394"}),(0,l.jsx)(e.span,{className:"mord mathnormal mtight",children:"t"})]})})]}),(0,l.jsxs)(e.span,{style:{top:"-3.23em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"frac-line",style:{borderBottomWidth:"0.04em"}})]}

1),(0,l.jsxs)(e.span,{style:{top:"-3.485em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsxs)(e.span,{className:"mord mtight",children:[(0,l.jsx)(e.span,{className:"mord mathnormal mtight",children:"t"}),(0,l.jsx)(e.span,{className:"mopen mtight",children:"("}),(0,l.jsx)(e.span,{className:"mord mathbf mtight",children:"x"}),(0,l.jsx)(e.span,{className:"mclose mtight",children:")"})]})})]})]}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.345em"},children:(0,l.jsx)(e.span,{})})})]})}),(0,l.jsx)(e.span,{className:"mclose nulldelimiter"})]}),(0,l.jsx)(e.span,{className:"mclose delimcenter",style:{top:"0em"},children:(0,l.jsx)(e.span,{className:"delimsizing size2",children:")"})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mpunct",children:","})]})]})]})}),"\n",(0,l.jsxs)(e.p,{children:["with ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u039b"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\Lambda"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u039b"})]})})]})," the unit triangular (linear-interpolation) kernel. This is a\nsingle-scattering approximation: multiples, transmission losses, mode conversion,\nand intrinsic attenuation are not represented."]}),"\n",(0,l.jsx)(e.h3,{id:"26-wavelet-and-band-limitation",children:"2.6 Wavelet and band-limitation"}),"\n",(0,l.jsxs)(e.p,{children:["Each finished trace is convolved ",(0,l.jsx)(e.strong,{children:"once"})," with a source wavelet ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"w"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"w"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.02691em"},children:"w"})]})})]})," to produce the\ndisplayed, band-limited trace ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"s"}),(0,l.jsx)(e.mi,{children:"T"})]}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"u"}),(0,l.jsx)(e.mi,{children:"T"})]}),(0,l.jsx)(e.mo,{children:"\u2217"}),(0,l.jsx)(e.mi,{children:"w"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"s_T = u_T * w"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",children:"s"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.13889em"},children:"T"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6153em",verticalAlign:"-0.15em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",children:"u"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"}

1,children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.13889em"},children:"T"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u2217"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.02691em"},children:"w"})]})]})]}),". Two wavelets are available:"]}),"\n",(0,l.jsxs)(e.ul,{children:["\n",(0,l.jsxs)(e.li,{children:["a ",(0,l.jsx)(e.strong,{children:"Ricker"})," wavelet of peak frequency ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"f"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"f"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10764em"},children:"f"})]})})]}),",\n",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"w"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mi,{children:"t"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mo,{fence:"true",stretchy:"true",minsize:"1.2em",maxsize:"1.2em",children:"("}),(0,l.jsx)(e.mn,{children:"1"}),(0,l.jsx)(e.mo,{children:"\u2212"}),(0,l.jsx)(e.mn,{children:"2"}),(0,l.jsxs)(e.msup,{children:[(0,l.jsx)(e.mi,{children:"\u03c0"}),(0,l.jsx)(e.mn,{children:"2"})]}),(0,l.jsxs)(e.msup,{children:[(0,l.jsx)(e.mi,{children:"f"}),(0,l.jsx)(e.mn,{children:"2"})]}),(0,l.jsxs)(e.msup,{children:[(0,l.jsx)(e.mi,{children:"t"}),(0,l.jsx)(e.mn,{children:"2"})]}),(0,l.jsx)(e.mo,{fence:"true",stretchy:"true",minsize:"1.2em",maxsize:"1.2em",children:")"}),(0,l.jsx)(e.mtext,{children:"\u2009"}),(0,l.jsxs)(e.msup,{children:[(0,l.jsx)(e.mi,{children:"e"}),(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mo,{children:"\u2212"}),(0,l.jsxs)(e.msup,{children:[(0,l.jsx)(e.mi,{children:"\u03c0"}),(0,l.jsx)(e.mn,{children:"2"})]}),(0,l.jsxs)(e.msup,{children:[(0,l.jsx)(e.mi,{children:"f"}),(0,l.jsx)(e.mn,{children:"2"})]}),(0,l.jsxs)(e.msup,{children:[(0,l.jsx)(e.mi,{children:"t"}),(0,l.jsx)(e.mn,{children:"2"})]})]})]}),(0,l.jsx)(e.mo,{separator:"true",children:","})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"w(t) = \\bigl(1 - 2\\pi^2 f^2 t^2\\bigr)\\,e^{-\\pi^2 f^2 t^2},"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.02691em"},children:"w"}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"t"}),(0,l.jsx)(e.span,{className:"mclose",children:")"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1.2em",verticalAlign:"-0.35em"}}

1),(0,l.jsx)(e.span,{className:"mopen",children:(0,l.jsx)(e.span,{className:"delimsizing size1",children:"("})}),(0,l.jsx)(e.span,{className:"mord",children:"1"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u2212"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1.3369em",verticalAlign:"-0.35em"}}),(0,l.jsx)(e.span,{className:"mord",children:"2"}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"\u03c0"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.8141em"},children:(0,l.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10764em"},children:"f"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.8141em"},children:(0,l.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",children:"t"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.8141em"},children:(0,l.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,l.jsx)(e.span,{className:"mclose",children:(0,l.jsx)(e.span,{className:"delimsizing size1",children:")"})}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",children:"e"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.9869em"},children:(0,l.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsxs)(e.span,{className:"mord mtight",children:[(0,l.jsx)(e.span,{className:"mord mtight",children:"\u2212"}),(0,l.jsxs)(e.span,{className:"mord mtight",children:[(0,l.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.03588em"},children:"\u03c0"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.8913em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.931em",marginRight:"0.0714em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.5em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size3 size1 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,l.jsxs)(e.span,{className:"mord mtight",children:[(0,l.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.10764em"},children:"f"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.8913em"}

1,children:(0,l.jsxs)(e.span,{style:{top:"-2.931em",marginRight:"0.0714em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.5em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size3 size1 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"2"})})]})})})})})]}),(0,l.jsxs)(e.span,{className:"mord mtight",children:[(0,l.jsx)(e.span,{className:"mord mathnormal mtight",children:"t"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.8913em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.931em",marginRight:"0.0714em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.5em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size3 size1 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"2"})})]})})})})})]})]})})]})})})})})]}),(0,l.jsx)(e.span,{className:"mpunct",children:","})]})]})]}),"\nwindowed to ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mo,{children:"\xb1"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\pm"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\xb1"})]})})]})," one period and peak-normalised;"]}),"\n",(0,l.jsxs)(e.li,{children:["a ",(0,l.jsx)(e.strong,{children:"Klauder"})," wavelet (the autocorrelation of a linear sweep ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"f"}),(0,l.jsx)(e.mn,{children:"1"})]}),(0,l.jsx)(e.mtext,{children:"\u2009\u2063"}),(0,l.jsx)(e.mo,{children:"\u2192"}),(0,l.jsx)(e.mtext,{children:"\u2009\u2063"}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"f"}),(0,l.jsx)(e.mn,{children:"2"})]})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"f_1\\!\\to\\!f_2"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10764em"},children:"f"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3011em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.1076em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"1"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"-0.1667em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"\u2192"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"-0.1667em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10764em"},children:"f"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3011em"}

1,children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.1076em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"2"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]})]})]})]})," of\nlength ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"L"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"L"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"L"})]})})]}),"), appropriate for modelling Vibroseis data."]}),"\n"]}),"\n",(0,l.jsxs)(e.p,{children:["Both kernels are symmetric and odd-length, centred on their midpoint. The wavelet\nbandwidth controls reflection thickness; the vertical resolution is governed by the\nclassic tuning limit ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"\u03bb"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"/"}),(0,l.jsx)(e.mn,{children:"4"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"V"}),(0,l.jsx)(e.mn,{children:"0"})]}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"/"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mn,{children:"4"}),(0,l.jsx)(e.mi,{children:"f"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\lambda/4 = V_0/(4f)"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"\u03bb"}),(0,l.jsx)(e.span,{className:"mord",children:"/4"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.22222em"},children:"V"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3011em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.2222em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"0"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mord",children:"/"}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsx)(e.span,{className:"mord",children:"4"}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10764em"},children:"f"}),(0,l.jsx)(e.span,{className:"mclose",children:")"})]})]})]}),"."]}),"\n",(0,l.jsx)(e.h3,{id:"27-acquisition-geometry",children:"2.7 Acquisition geometry"}),"\n",(0,l.jsxs)(e.p,{children:["The default geometry is a ",(0,l.jsx)(e.strong,{children:"zero-offset line of co-located stations"})," hung above the\nmodel, auto-centred on the mesh footprint and scaled to it: the line runs along the\nlonger horizontal axis, the record length ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"N"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u0394"}),(0,l.jsx)(e.mi,{children:"t"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"N\\Delta t"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"N"}),(0,l.jsx)(e.span,{className:"mord",children:"\u0394"}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"t"})]})})]})," is sized so the panel spans\nfrom the line down past the base of the model, and the wavelet frequency is set so\nseveral wavelengths fit the section (a fixed survey-scale default would place every\nreflection in a thin band and leave the panel flat). The geometry is fully general,\nhowever \u2014 fixed offsets or arbitrary source/receiver trajectories are supported, and\na vertical fault that is invisible at zero offset can be imaged by shooting offsets\nthat capture its tip diffractions."]}),"\n",(0,l.jsxs)(e.p,{children:["The line can also be ",(0,l.jsx)(e.strong,{children:"aligned to a measured structure"}),": choosing a strike shoots the\nline ",(0,l.jsx)(e.strong,{children:"perpendicular to it (along the dip direction)"}),", and the regional dip is then\n",(0,l.jsx)(e.strong,{children:"flattened out"})," (\xa72.8) so dipping beds image as horizontal bands. Both can be entered\ndirectly or taken from a nearby orientation measurement (whose dip-azimuth fixes the\nstrike as ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mtext,{children:"azimuth"}),(0,l.jsx)(e.mo,{children:"\u2212"}),(0,l.jsx)(e.mn,{children:"9"}),(0,l.jsxs)(e.msup,{children:[(0,l.jsx)(e.mn,{children:"0"}),(0,l.jsx)(e.mo,{children:"\u2218"})]})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\text{azimuth}-90^\\circ"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.7

1778em",verticalAlign:"-0.0833em"}}),(0,l.jsx)(e.span,{className:"mord text",children:(0,l.jsx)(e.span,{className:"mord",children:"azimuth"})}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u2212"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6741em"}}),(0,l.jsx)(e.span,{className:"mord",children:"9"}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord",children:"0"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.6741em"},children:(0,l.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mbin mtight",children:"\u2218"})})]})})})})})]})]})]})]}),", and hence the line as the dip azimuth)."]}),"\n",(0,l.jsx)(e.h3,{id:"28-co-located-facies-and-impedance-attributes-flattened-overlay",children:"2.8 Co-located facies and impedance attributes (flattened overlay)"}),"\n",(0,l.jsxs)(e.p,{children:["Alongside the amplitude, the forward model records the ",(0,l.jsx)(e.strong,{children:"facies id and impedance of\nthe high-impedance side"})," of each contact and exposes them as two auxiliary panels,\nco-registered with the reflections, that answer ",(0,l.jsx)(e.em,{children:'"which contact, of what impedance,\nproduced this event?"'})," directly on the synthetic."]}),"\n",(0,l.jsxs)(e.p,{children:["Crucially, these attributes are ",(0,l.jsx)(e.strong,{children:"not"})," placed at the Kirchhoff traveltime used for\nthe amplitude. Doing so would smear each contact's facies along its diffraction\nhyperbola; a winner-take-all colour fill then tracks the ",(0,l.jsx)(e.em,{children:"apex locus"})," of that smile,\nso a flat contact would image as an upward ",(0,l.jsx)(e.strong,{children:"arch"})," instead of a horizontal band.\nInstead the attributes are projected geometrically onto the section, at the contact's\ntrue position."]}),"\n",(0,l.jsxs)(e.p,{children:[(0,l.jsx)(e.strong,{children:"The section frame."})," The acquisition line ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsxs)(e.mover,{accent:"true",children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"t"}),(0,l.jsx)(e.mo,{children:"^"})]})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\hat{\\mathbf t}"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8984em"}}),(0,l.jsx)(e.span,{className:"mord accent",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.8984em"},children:[(0,l.jsxs)(e.span,{style:{top:"-3em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"t"})]}),(0,l.jsxs)(e.span,{style:{top:"-3.2039em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"accent-body",style:{left:"-0.1667em"},children:(0,l.jsx)(e.span,{className:"mord",children:"^"})})]})]})})})})]})})]})," is shot ~along the dip\ndirection; the depth axis ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsxs)(e.mover,{accent:"true",children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"q"}),(0,l.jsx)(e.mo,{children:"^"})]})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\hat{\\mathbf q}"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.9023em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mord accent",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.7079em"}

1,children:[(0,l.jsxs)(e.span,{style:{top:"-3em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"q"})]}),(0,l.jsxs)(e.span,{style:{top:"-3.0134em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"accent-body",style:{left:"-0.1667em"},children:(0,l.jsx)(e.span,{className:"mord",children:"^"})})]})]}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.1944em"},children:(0,l.jsx)(e.span,{})})})]})})]})})]})," is vertical ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mn,{children:"0"}),(0,l.jsx)(e.mo,{separator:"true",children:","}),(0,l.jsx)(e.mn,{children:"0"}),(0,l.jsx)(e.mo,{separator:"true",children:","}),(0,l.jsx)(e.mo,{children:"\u2212"}),(0,l.jsx)(e.mn,{children:"1"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"(0,0,-1)"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsx)(e.span,{className:"mord",children:"0"}),(0,l.jsx)(e.span,{className:"mpunct",children:","}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord",children:"0"}),(0,l.jsx)(e.span,{className:"mpunct",children:","}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u2212"}),(0,l.jsx)(e.span,{className:"mord",children:"1"}),(0,l.jsx)(e.span,{className:"mclose",children:")"})]})})]}),"; the off-plane\ndirection is ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.mover,{accent:"true",children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"m"}),(0,l.jsx)(e.mo,{children:"^"})]}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsxs)(e.mover,{accent:"true",children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"t"}),(0,l.jsx)(e.mo,{children:"^"})]}),(0,l.jsx)(e.mo,{children:"\xd7"}),(0,l.jsxs)(e.mover,{accent:"true",children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"q"}),(0,l.jsx)(e.mo,{children:"^"})]})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\hat{\\mathbf m}=\\hat{\\mathbf t}\\times\\hat{\\mathbf q}"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.7079em"}}),(0,l.jsx)(e.span,{className:"mord accent",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.7079em"},children:[(0,l.jsxs)(e.span,{style:{top:"-3em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"m"})]}),(0,l.jsxs)(e.span,{style:{top:"-3.0134em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"accent-body",style:{left:"-0.25em"},children:(0,l.jsx)(e.span,{className:"mord",children:"^"})})]})]})})})}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.9817em",verticalAlign:"-0.0833em"}}),(0,l.jsx)(e.span,{className:"mord accent",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.8984em"}

1,children:[(0,l.jsxs)(e.span,{style:{top:"-3em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"t"})]}),(0,l.jsxs)(e.span,{style:{top:"-3.2039em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"accent-body",style:{left:"-0.1667em"},children:(0,l.jsx)(e.span,{className:"mord",children:"^"})})]})]})})})}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\xd7"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.9023em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mord accent",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.7079em"},children:[(0,l.jsxs)(e.span,{style:{top:"-3em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"q"})]}),(0,l.jsxs)(e.span,{style:{top:"-3.0134em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"accent-body",style:{left:"-0.1667em"},children:(0,l.jsx)(e.span,{className:"mord",children:"^"})})]})]}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.1944em"},children:(0,l.jsx)(e.span,{})})})]})})]})]})]})," (the strike). For a\ncontact element at ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{mathvariant:"bold",children:"x"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\mathbf x"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4444em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"x"})]})})]})," and a trace with source ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{mathvariant:"bold",children:"s"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\mathbf s"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4444em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"s"})]})})]}),", write\n",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"d"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"x"}),(0,l.jsx)(e.mo,{children:"\u2212"}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"s"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\mathbf d=\\mathbf x-\\mathbf s"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6944em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"d"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"x"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u2212"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4444em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"s"})]})]})]})," and resolve"]}),"\n",(0,l.jsx)(e.span,{className:"katex-display",children:(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"z"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"d"}),(0,l.jsx)(e.mo,{children:"\u22c5"}),(0,l.jsxs)(e.mover,{accent:"true",children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"q"}),(0,l.jsx)(e.mo,{children:"^"})]}),(0,l.jsx)(e.mtext,{children:"\xa0(vertical\xa0depth)"}),(0,l.jsx)(e.mo,{separator:"true",children:","}),(0,l.jsx)(e.mspace,{width:"2em"}),(0,l.jsx)(e.mi,{children:"u"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"d"}),(0,l.jsx)(e.mo,{children:"\u22c5"}),(0,l.jsxs)(e.mover,{accent:"true",children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"t"}),(0,l.jsx)(e.mo,{children:"^"})]}),(0,l.jsx)(e.mtext,{children:"\xa0(along-line)"}),(0,l.jsx)(e.mo,{separator:"true",children:","}),(0,l.jsx)(e.mspace,{width:"2em"}),(0,l.jsx)(e.mi,{children:"p"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"d"}),(0,l.jsx)(e.mo,{children:"\u22c5"}),(0,l.jsxs)(e.mover,{accent:"true",children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"m"}),(0,l.jsx)(e.mo,{children:"^"})]}),(0,l.jsx)(e.mtext,{children:"\xa0(off-plane)"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"."})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"z = \\mathbf d\\cdot\\hat{\\mathbf q}\\ \\text{(vertical depth)},\\qquad\nu = \\mathbf d\\cdot\\hat{\\mathbf t}\\ \\text{(along-line)},\\qquad\np = \\mathbf d\\cdot\\hat{\\mathbf m}\\ \\text{(off-plane)}."})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.04398em"},children:"z"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6944em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"d"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u22c5"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord accent",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.7079em"}

1,children:[(0,l.jsxs)(e.span,{style:{top:"-3em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"q"})]}),(0,l.jsxs)(e.span,{style:{top:"-3.0134em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"accent-body",style:{left:"-0.1667em"},children:(0,l.jsx)(e.span,{className:"mord",children:"^"})})]})]}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.1944em"},children:(0,l.jsx)(e.span,{})})})]})}),(0,l.jsx)(e.span,{className:"mspace",children:"\xa0"}),(0,l.jsx)(e.span,{className:"mord text",children:(0,l.jsx)(e.span,{className:"mord",children:"(vertical\xa0depth)"})}),(0,l.jsx)(e.span,{className:"mpunct",children:","}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"2em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"u"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6944em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"d"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u22c5"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1.1484em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord accent",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.8984em"},children:[(0,l.jsxs)(e.span,{style:{top:"-3em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"t"})]}),(0,l.jsxs)(e.span,{style:{top:"-3.2039em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"accent-body",style:{left:"-0.1667em"},children:(0,l.jsx)(e.span,{className:"mord",children:"^"})})]})]})})})}),(0,l.jsx)(e.span,{className:"mspace",children:"\xa0"}),(0,l.jsx)(e.span,{className:"mord text",children:(0,l.jsx)(e.span,{className:"mord",children:"(along-line)"})}),(0,l.jsx)(e.span,{className:"mpunct",children:","}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"2em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"p"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6944em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"d"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u22c5"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord accent",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.7079em"},children:[(0,l.jsxs)(e.span,{style:{top:"-3em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"m"})]}),(0,l.jsxs)(e.span,{style:{top:"-3.0134em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"accent-body",style:{left:"-0.25em"},children:(0,l.jsx)(e.span,{className:"mord",children:"^"})})]})]})})})}),(0,l.jsx)(e.span,{className:"mspace",children:"\xa0"}),(0,l.jsx)(e.span,{className:"mord text",children:(0,l.jsx)(e.span,{className:"mord",children:"(off-plane)"})}),(0,l.jsx)(e.span,{className:"mord",children:"."})]})]})]})}),"\n",(0,l.jsxs)(e.p,{children:[(0,l.jsx)(e.strong,{children:"Flattening."})," Each element's depth is reduced by its ",(0,l.jsx)(e.strong,{children:"own"})," along-line dip ramp,"]}),"\n",(0,l.jsx)(e.span,{className:"katex-display",children:(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"z"}),(0,l.jsx)(e.mtext,{children:"flat"})]}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mi,{children:"z"}),(0,l.jsx)(e.mo,{children:"\u2212"}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"u"}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"x"})]}),(0,l.jsx)(e.mtext,{children:"\u2009"}),(0,l.jsx)(e.mi,{children:"tan"}),(0,l.jsx)(e.mo,{children:"\u2061"}),(0,l.jsx)(e.mi,{children:"\u03b4"}),(0,l.jsx)(e.mo,{separator:"true",children:","}),(0,l.jsx)(e.mspace,{width:"2em"}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"u"}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"x"})]}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"x"}),(0,l.jsx)(e.mo,{children:"\u2212"}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"c"}),(0,l.jsx)(e.mtext,{children:"line"})]}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"}),(0,l.jsx)(e.mo,{children:"\u22c5"}),(0,l.jsxs)(e.mover,{accent:"true",children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"t"}),(0,l.jsx)(e.mo,{children:"^"})]}),(0,l.jsx)(e.mo,{separator:"true",children:","})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"z_{\\text{flat}} = z - u_{\\mathbf x}\\,\\tan\\delta,\\qquad\nu_{\\mathbf x} = (\\mathbf x - \\mathbf c_{\\text{line}})\\cdot\\hat{\\mathbf t},"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.04398em"},children:"z"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3361em"}

1,children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.044em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord text mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"flat"})})})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.04398em"},children:"z"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u2212"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",children:"u"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.1611em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord mathbf mtight",children:"x"})})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mop",children:"tan"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03785em"},children:"\u03b4"}),(0,l.jsx)(e.span,{className:"mpunct",children:","}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"2em"}}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",children:"u"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.1611em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord mathbf mtight",children:"x"})})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"x"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u2212"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathbf",children:"c"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3361em"}

1,children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord text mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"line"})})})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mclose",children:")"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u22c5"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1.0928em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mord accent",children:(0,l.jsx)(e.span,{className:"vlist-t",children:(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsxs)(e.span,{className:"vlist",style:{height:"0.8984em"},children:[(0,l.jsxs)(e.span,{style:{top:"-3em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"t"})]}),(0,l.jsxs)(e.span,{style:{top:"-3.2039em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,l.jsx)(e.span,{className:"accent-body",style:{left:"-0.1667em"},children:(0,l.jsx)(e.span,{className:"mord",children:"^"})})]})]})})})}),(0,l.jsx)(e.span,{className:"mpunct",children:","})]})]})]})}),"\n",(0,l.jsxs)(e.p,{children:["where ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"\u03b4"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\delta"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6944em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03785em"},children:"\u03b4"})]})})]})," is the regional dip and ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{mathvariant:"bold",children:"c"}),(0,l.jsx)(e.mtext,{children:"line"})]})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\mathbf c_{\\text{line}}"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.5944em",verticalAlign:"-0.15em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathbf",children:"c"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3361em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord text mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"line"})})})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]})]})})]})," the line centre.\nBecause a bed dipping ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"\u03b4"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\delta"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6944em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03785em"},children:"\u03b4"})]})})]})," along the line deepens by ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"u"}),(0,l.jsx)(e.mi,{children:"tan"}),(0,l.jsx)(e.mo,{children:"\u2061"}),(0,l.jsx)(e.mi,{children:"\u03b4"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"u\\tan\\delta"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6944em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"u"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mop",children:"tan"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03785em"},children:"\u03b4"})]})})]}),", ",(0,l.jsx)(e.strong,{children:"every"})," point\nof that bed maps to the ",(0,l.jsx)(e.em,{children:"same"})," flattened depth \u2014 a thin horizontal band. The element is\nwritten at sample ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"\u03c4"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mn,{children:"2"}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"z"}),(0,l.jsx)(e.mtext,{children:"flat"})]}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"/"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"V"}),(0,l.jsx)(e.mn,{children:"0"})]}),(0,l.jsx)(e.mtext,{children:"\u2009"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u0394"}),(0,l.jsx)(e.mi,{children:"t"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\tau = 2z_{\\text{flat}}/(V_0\\,\\Delta t)"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.1132em"},children:"\u03c4"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord",children:"2"}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.04398em"},children:"z"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3361em"}

1,children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.044em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord text mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"flat"})})})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mord",children:"/"}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.22222em"},children:"V"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3011em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.2222em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"0"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u0394"}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"t"}),(0,l.jsx)(e.span,{className:"mclose",children:")"})]})]})]})," (time) or\n",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"z"}),(0,l.jsx)(e.mtext,{children:"flat"})]}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"/"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u0394"}),(0,l.jsx)(e.mi,{children:"t"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"z_{\\text{flat}}/\\Delta t"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.04398em"},children:"z"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3361em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.044em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord text mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"flat"})})})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mord",children:"/\u0394"}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"t"})]})})]})," (depth), onto the trace nearest along the line (every trace\nwithin a short window, ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mo,{children:"\u223c"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\sim"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.3669em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"\u223c"})]})})]}),"two stations, so a dense contact fills without gaps). When\nseveral elements compete for one cell, the one with the ",(0,l.jsxs)(e.strong,{children:["smallest off-plane distance\n",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u2223"}),(0,l.jsx)(e.mi,{children:"p"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u2223"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"|p|"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u2223"}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"p"}),(0,l.jsx)(e.span,{className:"mord",children:"\u2223"})]})})]})," wins"]}),". The deposits are then ",(0,l.jsx)(e.strong,{children:"dilated to the wavelet's main-lobe half-width"}),";\nempty cells are left undefined. Doing the flatten ",(0,l.jsx)(e.em,{children:"per element"})," (not as a per-trace\npost-shift) keeps the band thin \u2014 the along-line window does not smear it."]}),"\n",(0,l.jsxs)(e.p,{children:["For ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"\u03b4"}),(0,l.jsx)(e.mo,{children:"="}),(0,l.jsx)(e.mn,{children:"0"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\delta = 0"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6944em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03785em"},children:"\u03b4"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(e.span,{className:"mrel",children:"="}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,l.jsx)(e.span,{className:"mord",children:"0"})]})]})]})," this reduces to placing each contact at its true depth on the trace\nabove it \u2014 horizontal bands, no arch. With a regional dip, dipping bed-parallel contacts\ncollapse to a constant flattened depth and image as ",(0,l.jsx)(e.strong,{children:"horizontal bands"}),". The amplitude\nis flattened to match by the corresponding per-trace shift (each trace moved up by\n",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"u"}),(0,l.jsx)(e.mtext,{children:"tr"})]}),(0,l.jsx)(e.mi,{children:"tan"}),(0,l.jsx)(e.mo,{children:"\u2061"}),(0,l.jsx)(e.mi,{children:"\u03b4"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"u_{\\text{tr}}\\tan\\delta"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8444em",verticalAlign:"-0.15em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",children:"u"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.2806em"}

1,children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:(0,l.jsx)(e.span,{className:"mord text mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"tr"})})})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mop",children:"tan"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03785em"},children:"\u03b4"})]})})]}),"), so the overlay stays co-registered with the wiggle\n(approximately for\noff-plane or strongly-dipping geometry)."]}),"\n",(0,l.jsx)(e.h2,{id:"3-implementation",children:"3. Implementation"}),"\n",(0,l.jsx)(e.h3,{id:"31-data-model",children:"3.1 Data model"}),"\n",(0,l.jsxs)(e.p,{children:["The engine (",(0,l.jsx)(e.code,{children:"SeismicModeller"}),") is free of UI, rendering, and mesh types: the caller\nadapts the mesh into a plain ",(0,l.jsx)(e.code,{children:"MeshView"})," (vertex positions plus the two attribute\nspans) and receives a ",(0,l.jsx)(e.code,{children:"TraceGather"})," of amplitude, facies, and impedance grids. The\ndisplayable result, ",(0,l.jsx)(e.code,{children:"CSeismicSection"}),", stores these as flat arrays laid out\n",(0,l.jsx)(e.code,{children:"[sample \xb7 n_traces + trace]"})," so they upload directly as textures, and carries the\nacquisition-line geometry as a relative-to-eye anchor and three corners for\nfloating-point-safe rendering at large coordinates."]}),"\n",(0,l.jsx)(e.h3,{id:"32-rendering",children:"3.2 Rendering"}),"\n",(0,l.jsxs)(e.p,{children:["The section is drawn as a single ",(0,l.jsx)(e.strong,{children:"textured vertical panel"})," hung from the acquisition\nline, with sample 0 at the top and time/depth increasing downward. With a regional dip\nset the gather has already been flattened (\xa72.8), so the panel itself stays vertical \u2014\nthe flattening lives in the data, not the panel geometry. The\nactive value grid is uploaded as a single-channel floating-point texture; a\nfragment shader samples it, applies clip/gain/polarity, maps the result through a\n256-entry colour lookup, and writes logarithmic depth so the panel composes\ncorrectly with the rest of the scene. For a time axis the vertical extent maps\ntwo-way time to depth using ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"V"}),(0,l.jsx)(e.mn,{children:"0"})]}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"/"}),(0,l.jsx)(e.mn,{children:"2"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"V_0/2"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.22222em"},children:"V"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3011em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.2222em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"0"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mord",children:"/2"})]})})]}),"."]}),"\n",(0,l.jsx)(e.h3,{id:"33-attribute-layers-histograms-and-filters",children:"3.3 Attribute layers, histograms, and filters"}),"\n",(0,l.jsxs)(e.p,{children:["Amplitude, facies, and impedance are exposed as sibling ",(0,l.jsx)(e.strong,{children:"attribute layers"})," reusing\nthe application's standard attribute system. Selecting a layer recolours the panel\nthrough that layer's own colour map;

1 each layer carries a histogram and a value\n",(0,l.jsx)(e.strong,{children:"filter"}),". Filtering masks the panel \u2014 samples outside the filter range (and cells\nwith no recorded attribute) are made transparent \u2014 so the interpreter can, for\nexample, isolate the reflections produced by a single facies or by impedance\ncontrasts above a threshold. Undefined cells are encoded as not-a-number and\ndiscarded in the shader, which is also how the filter mask is realised."]}),"\n",(0,l.jsx)(e.h3,{id:"34-concurrency",children:"3.4 Concurrency"}),"\n",(0,l.jsx)(e.p,{children:"The forward model runs on a background worker; the result object is published to the\nscene immediately and its render path is gated until the worker has finished\npopulating the buffers, so the UI stays responsive while a dense mesh is modelled."}),"\n",(0,l.jsx)(e.h2,{id:"4-parameter-reference-theory-view",children:"4. Parameter reference (theory view)"}),"\n",(0,l.jsxs)(e.table,{children:[(0,l.jsx)(e.thead,{children:(0,l.jsxs)(e.tr,{children:[(0,l.jsx)(e.th,{children:"Parameter"}),(0,l.jsx)(e.th,{children:"Symbol"}),(0,l.jsx)(e.th,{children:"Effect"})]})}),(0,l.jsxs)(e.tbody,{children:[(0,l.jsxs)(e.tr,{children:[(0,l.jsx)(e.td,{children:"Backgroun

1d velocity"}),(0,l.jsx)(e.td,{children:(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"V"}),(0,l.jsx)(e.mn,{children:"0"})]})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"V_0"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.22222em"},children:"V"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3011em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.2222em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"0"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]})]})})]})}),(0,l.jsx)(e.td,{children:"depth\u2194time mapping and geometric spreading"})]}),(0,l.jsxs)(e.tr,{children:[(0,l.jsx)(e.td,{children:"Sample interval"}),(0,l.jsx)(e.td,{children:(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{mathvariant:"normal",children:"\u0394"}),(0,l.jsx)(e.mi,{children:"t"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\Delta t"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(e.span,{className:"mord",children:"\u0394"}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"t"})]})})]})}),(0,l.jsxs)(e.td,{children:["vertical sampling; with ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"N"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"N"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"N"})]})})]})," sets the record length"]})]}),(0,l.jsxs)(e.tr,{children:[(0,l.jsx)(e.td,{children:"Number of samples"}),(0,l.jsx)(e.td,{children:(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"N"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"N"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"N"})]})})]})}),(0,l.jsx)(e.td,{children:"trace length / imaged depth"})]}),(0,l.jsxs)(e.tr,{children:[(0,l.jsx)(e.td,{children:"Wavelet frequency"}),(0,l.jsx)(e.td,{children:(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"f"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"f"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8889em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10764em"},children:"f"})]})})]})}),(0,l.jsxs)(e.td,{children:["bandwidth \u2192 reflection thickness, ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"\u03bb"}),(0,l.jsx)(e.mi,{mathvariant:"normal",children:"/"}),(0,l.jsx)(e.mn,{children:"4"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\lambda/4"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"\u03bb"}),(0,l.jsx)(e.span,{className:"mord",children:"/4"})]})})]})," resolution"]})]}),(0,l.jsxs)(e.tr,{children:[(0,l.jsx)(e.td,{children:"Azimuth, spacing, count, height"}),(0,l.jsx)(e.td,{children:"\u2014"}),(0,l.jsx)(e.td,{children:"acquisition line position and sampling"})]}),(0,l.jsxs)(e.tr,{children:[(0,l.jsx)(e.td,{children:"Offset mode / distance"}),(0,l.jsx)(e.td,{children:"\u2014"}),(0,l.jsx)(e.td,{children:"zero-offset vs. offset acquisition (fault-tip imaging)"})]})]})]}),"\n",(0,l.jsx)(e.h2,{id:"5-assumptions-and-limitations",children:"5. Assumptions and limitations"}),"\n",(0,l.jsxs)(e.ol,{children:["\n",(0,l.jsxs)(e.li,{children:[(0,l.jsx)(e.strong,{children:"Single scattering."})," No multiples, transmission loss, mode conversion, or\nattenuation; amplitudes are qualitative, not calibrated."]}),"\n",(0,l.jsxs)(e.li,{children:[(0,l.jsxs)(e.strong,{children:["Straight rays, constant ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"V"}),(0,l.jsx)(e.mn,{children:"0"})]})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"V_0"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.22222em"},children:"V"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3011em"}

1,children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.2222em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"0"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]})]})})]}),"."]})," No ray bending; traveltimes are exact only for\na homogeneous background."]}),"\n",(0,l.jsxs)(e.li,{children:[(0,l.jsx)(e.strong,{children:"Normal-incidence reflectivity."})," No AVO; ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"R"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"R"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.00773em"},children:"R"})]})})]})," ignores incidence angle."]}),"\n",(0,l.jsxs)(e.li,{children:[(0,l.jsx)(e.strong,{children:"2-D extrusion."})," Contacts are assumed to extend perpendicular to the section;\ngenuinely 3-D structure off-plane is not represented unless measured dip vectors\nare supplied."]}),"\n",(0,l.jsxs)(e.li,{children:[(0,l.jsx)(e.strong,{children:"Surface, not volume."})," Reflectivity comes from facies contacts on the mesh;\nthe medium between contacts is otherwise non-reflective."]}),"\n",(0,l.jsxs)(e.li,{children:[(0,l.jsx)(e.strong,{children:"Sparse attributes."})," Facies/impedance exist only where reflections were\ndeposited (co-located with events), by design."]}),"\n"]}),"\n",(0,l.jsx)(e.h2,{id:"6-future-work",children:"6. Future work"}),"\n",(0,l.jsxs)(e.p,{children:["Angle-dependent reflectivity (AVO), curved-ray or eikonal traveltimes for a\nheterogeneous ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"V"}),(0,l.jsx)(e.mn,{children:"0"})]}),(0,l.jsx)(e.mo,{stretchy:"false",children:"("}),(0,l.jsx)(e.mi,{mathvariant:"bold",children:"x"}),(0,l.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"V_0(\\mathbf x)"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.22222em"},children:"V"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,l.jsxs)(e.span,{className:"vlist-r",children:[(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.3011em"},children:(0,l.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.2222em",marginRight:"0.05em"},children:[(0,l.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(e.span,{className:"mord mtight",children:"0"})})]})}),(0,l.jsx)(e.span,{className:"vlist-s",children:"\u200b"})]}),(0,l.jsx)(e.span,{className:"vlist-r",children:(0,l.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,l.jsx)(e.span,{})})})]})})]}),(0,l.jsx)(e.span,{className:"mopen",children:"("}),(0,l.jsx)(e.span,{className:"mord mathbf",children:"x"}),(0,l.jsx)(e.span,{className:"mclose",children:")"})]})})]}),", multiple-arrival handling, and substitution of\nmeasured dip vectors for the computed impedance gradient are natural extensions, as\nis migrating the inline trace storage to a sidecar for very large gathers."]}),"\n",(0,l.jsx)(e.h2,{id:"references",children:"References"}),"\n",(0,l.jsxs)(e.ol,{children:["\n",(0,l.jsxs)(e.li,{children:["Ricker, N. (1953). ",(0,l.jsx)(e.em,{children:"The form and laws of propagation of seismic wavelets."}),"\nGeophysics, 18(1), 10\u201340."]}),"\n",(0,l.jsxs)(e.li,{children:["Klauder, J. R., Price, A. C., Darlington, S., & Albersheim, W. J. (1960).\n",(0,l.jsx)(e.em,{children:"The theory and design of chirp radars."})," Bell System Technical Journal, 39(4)."]}),"\n",(0,l.jsxs)(e.li,{children:["Schneider, W. A. (1978). ",(0,l.jsx)(e.em,{children:"Integral formulation for migration in two and three\ndimensions."})," Geophysics, 43(1), 49\u201376."]}),"\n",(0,l.jsxs)(e.li,{children:["Bleistein, N. (1987). ",(0,l.jsx)(e.em,{children:"On the imaging of reflectors in the earth."})," Geophysics,\n52(7), 931\u2013942."]}),"\n",(0,l.jsxs)(e.li,{children:["Carcione, J. M., Herman, G. C., & ten Kroode, A. P. E. (2002). ",(0,l.jsx)(e.em,{children:"Seismic\nmodeling."})," Geophysics, 67(4), 1304\u20131325."]}),"\n",(0,l.jsxs)(e.li,{children:["Sheriff, R. E., & Geldart, L. P. (1995). ",(0,l.jsx)(e.em,{children:"Exploration Seismology"})," (2nd ed.).\nCambridge University Press."]}),"\n"]}),"\n",(0,l.jsx)(e.h2,{id:"related",children:"Related"}),"\n",(0,l.jsxs)(e.ul,{children:["\n",(0,l.jsxs)(e.li,{children:[(0,l.jsx)(e.a,{href:"/docs/next/general/wells-seismic/seismic-modelling",children:"Seismic Modelling"})," \u2014 the user workflow and dialog reference."]}),"\n"]})]})}function d(s={}){const{wrapper:e}={...(0,i.R)(),...s.components};return e?(0,l.jsx)(e,{...s,children:(0,l.jsx)(h,{...s})}):h(s)}},28453(s,e,a){a.d(e,{R:()=>t,x:()=>m});var n=a(96540);const l={},i=n.createContext(l);function t(s){const e=n.useContext(i);return n.useMemo(function(){return"function"==typeof s?s(e):{...e,...s}},[e,s])}function m(s){let e;return e=s.disableParentContext?"function"==typeof s.components?s.components(l):s.components||l:t(s.components),n.createElement(i.Provider,{value:e},s.children)}}}]);