https://pt.mxrap.workers.dev/assets/js/488a7d67.e57d6c84.js

1"use strict";(self.webpackChunkmxrap_com=self.webpackChunkmxrap_com||[]).push([["20219"],{60807(s,e,a){a.r(e),a.d(e,{metadata:()=>n,default:()=>d,frontMatter:()=>t,contentTitle:()=>m,toc:()=>h,assets:()=>r});var n=JSON.parse('{"id":"documentation/theory/hazard-assessment/mul-explained","title":"What on Earth is MUL?","description":"MUL refers to the truncating magnitude of the Gutenberg-Richter distribution","source":"@site/docs/documentation/theory/hazard-assessment/mul-explained.mdx","sourceDirName":"documentation/theory/hazard-assessment","slug":"/documentation/theory/hazard-assessment/mul-explained","permalink":"/documentation/theory/hazard-assessment/mul-explained","draft":false,"unlisted":false,"tags":[{"inline":false,"label":"Frequently Asked Questions","permalink":"/tags/faqs","description":"Common questions, troubleshooting, and quick answers."},{"inline":false,"label":"Seismicity","permalink":"/tags/seismic","description":"Seismic monitoring, microseismic analysis, and earthquake detection."},{"inline":false,"label":"Theory","permalink":"/tags/theory","description":"Theoretical concepts, scientific principles, and academic foun

1dations."},{"inline":false,"label":"Hazard Assessment","permalink":"/tags/hazard-assessment","description":"Seismic hazard evaluation, risk assessment, and safety analysis."}],"version":"current","lastUpdatedAt":1772755200000,"sidebarPosition":1,"frontMatter":{"title":"What on Earth is MUL?","description":"MUL refers to the truncating magnitude of the Gutenberg-Richter distribution","draft":false,"unlisted":false,"hide_title":true,"hide_table_of_contents":false,"sidebar_position":1,"sidebar_custom_props":{"myEmoji":"\u{1F4C4}"},"tags":["faqs","seismic","theory","hazard-assessment"],"keywords":["MUL","Mmax","upper limit magnitude","Gutenberg-Richter","truncating magnitude","maximum magnitude"],"last_update":{"date":"2026-03-06"}},"sidebar":"theorySidebar","previous":{"title":"Seismic hazard - sensitivity to b-value","permalink":"/documentation/theory/hazard-assessment/hazard-assessment-b-value"},"next":{"title":"Hazard Isosurfaces","permalink":"/documentation/theory/hazard-assessment/hazard-isos"}}'),i=a(11684),l=a(77025);let t={title:"What on Earth is MUL?",description:"MUL refers to the truncating magnitude of the Gutenberg-Richter distribution",draft:!1,unlisted:!1,hide_title:!0,hide_table_of_contents:!1,sidebar_position:1,sidebar_custom_props:{myEmoji:"\u{1F4C4}"},tags:["faqs","seismic","theory","hazard-assessment"],keywords:["MUL","Mmax","upper limit magnitude","Gutenberg-Richter","truncating magnitude","maximum magnitude"],last_update:{date:"2026-03-06"}},m="What on Earth is M_{UL}?",r={},h=[{value:"Reasons for confusion",id:"reasons-for-confusion",level:2},{value:"Why do we need an upper-limit or truncating magnitude?",id:"why-do-we-need-an-upper-limit-or-truncating-magnitude",level:2},{value:"How do we estimate M_{UL}?",id:"how-do-we-estimate-m_ul",level:2},{value:"Conclusion",id:"conclusion",level:2}];function c(s){let e={a:"a",admonition:"admonition",annotation:"annotation",em:"em",figure:"figure",h1:"h1",h2:"h2",header:"header",img:"img",li:"li",math:"math",mi:"mi",mo:"mo",mrow:"mrow",msub:"msub",ol:"ol",p:"p",section:"section",semantics:"semantics",span:"span",strong:"strong",sup:"sup",ul:"ul",...(0,l.R)(),...s.components};return(0,i.jsxs)(i.Fragment,{children:[(0,i.jsx)(e.header,{children:(0,i.jsxs)(e.h1,{id:"what-on-earth-is-m_ul",children:["What on Earth is ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{UL}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"},children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsx)(e.span,{className:"mord mtight",children:(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"UL"})})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]}),"?"]})}),"\n",(0,i.jsxs)(e.p,{children:["Yes, this is a frequently asked question.... ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{UL}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"}

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1,children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsxs)(e.span,{className:"mord mtight",children:[(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"ma"}),(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"x"})]})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]})," in the Hazard Assessment app and on the frequency-magnitude chart but we found there was confusion caused by ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"m"}),(0,i.jsx)(e.mi,{children:"a"}),(0,i.jsx)(e.mi,{children:"x"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{max}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsxs)(e.span,{className:"mord mtight",children:[(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"ma"}),(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"x"})]})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]})," being used to describe multiple things. Hopefully if we refer to ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{UL}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"},children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsx)(e.span,{className:"mord mtight",children:(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"UL"})})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]})," or the upper-limit magnitude, this will clear up the terminology a little."]}),"\n",(0,i.jsx)(e.p,{children:"A quick review on the terminology that concerns the frequency-magnitude chart and the Gutenberg-Richter distribution:"}),"\n",(0,i.jsxs)(e.ul,{children:["\n",(0,i.jsxs)(e.li,{children:[(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"m"}),(0,i.jsx)(e.mi,{children:"i"}),(0,i.jsx)(e.mi,{children:"n"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{min}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.311

17em"},children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsx)(e.span,{className:"mord mtight",children:(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"min"})})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]})," - The magnitude of completeness, the dataset is considered complete above this magnitude (property of the data)."]}),"\n",(0,i.jsxs)(e.li,{children:[(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"b"}),(0,i.jsx)(e.mo,{children:"\u2212"}),(0,i.jsx)(e.mi,{children:"v"}),(0,i.jsx)(e.mi,{children:"a"}),(0,i.jsx)(e.mi,{children:"l"}),(0,i.jsx)(e.mi,{children:"u"}),(0,i.jsx)(e.mi,{children:"e"})]}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"b-value"})]})})}),(0,i.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.7778em",verticalAlign:"-0.0833em"}}),(0,i.jsx)(e.span,{className:"mord mathnormal",children:"b"}),(0,i.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,i.jsx)(e.span,{className:"mbin",children:"\u2212"}),(0,i.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.6944em"}}),(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"v"}),(0,i.jsx)(e.span,{className:"mord mathnormal",children:"a"}),(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.01968em"},children:"l"}),(0,i.jsx)(e.span,{className:"mord mathnormal",children:"u"}),(0,i.jsx)(e.span,{className:"mord mathnormal",children:"e"})]})]})]})," - The slope of the Gutenberg-Richter distribution, describes how the frequency of events scales with magnitude (property of the statistical model)."]}),"\n",(0,i.jsxs)(e.li,{children:[(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"X"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"m"}),(0,i.jsx)(e.mi,{children:"a"}),(0,i.jsx)(e.mi,{children:"x"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"X_{max}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.07847em"},children:"X"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.0785em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsxs)(e.span,{className:"mord mtight",children:[(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"ma"}),(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"x"})]})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]})," - The largest magnitude event in the dataset (property of the data)."]}),"\n",(0,i.jsxs)(e.li,{children:[(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"a"}),(0,i.jsx)(e.mi,{mathvariant:"normal",children:"/"}),(0,i.jsx)(e.mi,{children:"b"})]}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"a/b"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,i.jsx)(e.span,{className:"mord mathnormal",children:"a"}),(0,i.jsx)(e.span,{className:"mord",children:"/"}),(0,i.jsx)(e.span,{className:"mord mathnormal",children:"b"})]})})]})," - The magnitude at N = 1 of the Gutenberg-Richter distribution (property of the model, maximum likelihood, see previous ",(0,i.jsx)(e.a,{href:"/documentation/theory/to-a-b-or-not-to-a-b",children:"this document"}),")."]}),"\n",(0,i.jsxs)(e.li,{children:[(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"m"}),(0,i.jsx)(e.mi,{children:"a"}),(0,i.jsx)(e.mi,{children:"x"}),(0,i.jsx)(e.mo,{stretchy:"false",children:"("}),(0,i.jsx)(e.mi,{children:"m"}),(0,i.jsx)(e.mo,{separator:"true",children:","}),(0,i.jsx)(e.mi,{children:"n"}),(0,i.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"max(m,n)"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,i.jsx)(e.span,{className:"mord mathnormal",children:"ma"}),(0,i.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,i.jsx)(e.span,{className:"mopen",children:"("}),(0,i.jsx)(e.span,{className:"mord mathnormal",children:"m"}),(0,i.jsx)(e.span,{className:"mpunct",children:","}),(0,i.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,i.jsx)(e.span,{className:"mord mathnormal",children:"n"}),(0,i.jsx)(e.span,{className:"mclose",children:")"})]})})]})," - This is the probability density function, given n events, of the largest event in that n events. This is a property of the Gutenberg-Richter statistical model. In other words, given a certain Gutenberg-Richter model, if you record N events, what is the largest event? This is not a single number but a likelihood distribution. The maximum likelihood of the largest event is the ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"a"}),(0,i.jsx)(e.mi,{mathvariant:"normal",children:"/"}),(0,i.jsx)(e.mi,{children:"b"})]}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"a/b"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,i.jsx)(e.span,{className:"mord mathnormal",children:"a"}),(0,i.jsx)(e.span,{className:"mord",children:"/"}),(0,i.jsx)(e.span,{className:"mord mathnormal",children:"b"})]})})]})," value."]}),"\n",(0,i.jsxs)(e.li,{children:[(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{UL}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"}

1,children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsx)(e.span,{className:"mord mtight",children:(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"UL"})})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]})," - The upper-limit magnitude of the ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"m"}),(0,i.jsx)(e.mi,{children:"a"}),(0,i.jsx)(e.mi,{children:"x"}),(0,i.jsx)(e.mo,{stretchy:"false",children:"("}),(0,i.jsx)(e.mi,{children:"m"}),(0,i.jsx)(e.mo,{separator:"true",children:","}),(0,i.jsx)(e.mi,{children:"n"}),(0,i.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"max(m,n)"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,i.jsx)(e.span,{className:"mord mathnormal",children:"ma"}),(0,i.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,i.jsx)(e.span,{className:"mopen",children:"("}),(0,i.jsx)(e.span,{className:"mord mathnormal",children:"m"}),(0,i.jsx)(e.span,{className:"mpunct",children:","}),(0,i.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,i.jsx)(e.span,{className:"mord mathnormal",children:"n"}),(0,i.jsx)(e.span,{className:"mclose",children:")"})]})})]})," distribution. It is an estimate only and a property of the statistical model. The probability of an event greater than ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{UL}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"},children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsx)(e.span,{className:"mord mtight",children:(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"UL"})})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]})," is, by definition, zero."]}),"\n",(0,i.jsxs)(e.li,{children:[(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"m"}),(0,i.jsx)(e.mi,{children:"a"}),(0,i.jsx)(e.mi,{children:"x"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{max}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"}

1,children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsxs)(e.span,{className:"mord mtight",children:[(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"ma"}),(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"x"})]})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]})," is the physical maximum event that nature and mining can produce."]}),"\n"]}),"\n",(0,i.jsxs)(e.admonition,{type:"tip",children:[(0,i.jsxs)(e.p,{children:[(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{UL}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t 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",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"m"}),(0,i.jsx)(e.mi,{children:"a"}),(0,i.jsx)(e.mi,{children:"x"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{max}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t 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It is not the number to tell your manager!"]}),(0,i.jsxs)(e.p,{children:["It is best to be conservative. You can set ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{UL}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"}

1,children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsx)(e.span,{className:"mord mtight",children:(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"UL"})})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]})," = 6, it does not mean ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"m"}),(0,i.jsx)(e.mi,{children:"a"}),(0,i.jsx)(e.mi,{children:"x"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{max}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsxs)(e.span,{className:"mord mtight",children:[(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"ma"}),(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"x"})]})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]})," = 6."]})]}),"\n",(0,i.jsx)(e.figure,{children:(0,i.jsx)(e.img,{src:a(1639).A+"",width:"2440",height:"887"})}),"\n",(0,i.jsx)(e.figure,{children:(0,i.jsx)(e.img,{src:a(75154).A+"",width:"1024",height:"597"})}),"\n",(0,i.jsx)(e.h2,{id:"reasons-for-confusion",children:"Reasons for confusion"}),"\n",(0,i.jsxs)(e.p,{children:["The truncating magnitude has slightly different meanings in mining seismology and crustal seismology. ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{UL}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"},children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsx)(e.span,{className:"mord mtight",children:(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"UL"})})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]})," is usually referred to as ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"m"}),(0,i.jsx)(e.mi,{children:"a"}),(0,i.jsx)(e.mi,{children:"x"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{max}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"}

1,children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsxs)(e.span,{className:"mord mtight",children:[(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"ma"}),(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"x"})]})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]})," in crustal seismology literature and is generally considered constant for a particular area. In mining seismology ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{UL}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"},children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsx)(e.span,{className:"mord mtight",children:(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"UL"})})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]})," generally increases over time given the gradual increase in mining dimensions and loading of the rock mass. For this reason the definition is slightly modified in mining seismology to be the upper limit of the next largest event."]}),"\n",(0,i.jsx)(e.figure,{children:(0,i.jsx)(e.img,{src:a(32784).A+"",width:"1024",height:"330"})}),"\n",(0,i.jsx)(e.h2,{id:"why-do-we-need-an-upper-limit-or-truncating-magnitude",children:"Why do we need an upper-limit or truncating magnitude?"}),"\n",(0,i.jsx)(e.p,{children:"The truncated Gutenberg-Richter distribution, rather than the open-ended distribution, is the most common frequency-magnitude relationship used in mine seismology. If there is no upper limit given to the Gutenberg-Richter distribution, then to evaluate the total energy of events in the relevant time period, the energy tends to infinity as the relationship is integrated above Mmin. This is clearly unrealistic."}),"\n",(0,i.jsx)(e.p,{children:"We know there is a physical limit to possible magnitudes since the size of large earthquakes is related to the slip area of the fault and the physical size of faults is limited. Earthquakes on Earth above magnitude 10 (Richter) are essentially impossible given the size of known faults and a magnitude above 12 represents a fault area larger than the Earth itself!"}),"\n",(0,i.jsxs)(e.p,{children:["So it is safe to say that ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{UL}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"}

1,children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsx)(e.span,{className:"mord mtight",children:(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"UL"})})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]})," for a particular mine is going to be less than Richter Magnitude 10. The question is how much less is reasonable given the significantly reduced physical dimensions in mining."]}),"\n",(0,i.jsxs)(e.h2,{id:"how-do-we-estimate-m_ul",children:["How do we estimate ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{UL}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"},children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsx)(e.span,{className:"mord mtight",children:(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"UL"})})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]}),"?"]}),"\n",(0,i.jsx)(e.figure,{children:(0,i.jsx)(e.img,{src:a(33182).A+"",width:"619",height:"635"})}),"\n",(0,i.jsxs)(e.p,{children:["An empirical method of estimating ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{UL}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"},children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsx)(e.span,{className:"mord mtight",children:(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"UL"})})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]})," can be taken using a dataset compiled by McGarr et al. (2002)",(0,i.jsx)(e.sup,{children:(0,i.jsx)(e.a,{href:"#user-content-fn-2",id:"user-content-fnref-2","data-footnote-ref":!0,"aria-describedby":"footnote-label",children:"1"})})," of large events and the largest dimension of the human activity associated with them. The figure on the right comes from Wesseloo (2018)",(0,i.jsx)(e.sup,{children:(0,i.jsx)(e.a,{href:"#user-content-fn-3",id:"user-content-fnref-3","data-footnote-ref":!0,"aria-describedby":"footnote-label",children:"2"})})," who added a few extra points to the dataset from Australian and Canadian mines. The range applicable to mining indicates rough dimensions between 500 and 5,000m."]}),"\n",(0,i.jsxs)(e.p,{children:["Aside from the empirical approach, there are also statistical approaches to estimating ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{UL}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"}

1,children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsx)(e.span,{className:"mord mtight",children:(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"UL"})})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]}),". These generally take the form:"]}),"\n",(0,i.jsx)(e.span,{className:"katex-display",children:(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsxs)(e.mrow,{children:[(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]}),(0,i.jsx)(e.mo,{children:"\u2248"}),(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"X"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"m"}),(0,i.jsx)(e.mi,{children:"a"}),(0,i.jsx)(e.mi,{children:"x"})]})]}),(0,i.jsx)(e.mo,{children:"+"}),(0,i.jsx)(e.mi,{mathvariant:"normal",children:"\u0394"})]}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"\nM_{UL} \u2248 X_{max} + \u0394\n\n"})]})})}),(0,i.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"},children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsx)(e.span,{className:"mord mtight",children:(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"UL"})})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]}),(0,i.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,i.jsx)(e.span,{className:"mrel",children:"\u2248"}),(0,i.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.07847em"},children:"X"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.1514em"},children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.0785em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsxs)(e.span,{className:"mord mtight",children:[(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"ma"}),(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"x"})]})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]}),(0,i.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,i.jsx)(e.span,{className:"mbin",children:"+"}),(0,i.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,i.jsx)(e.span,{className:"mord",children:"\u0394"})]})]})]})}),"\n",(0,i.jsxs)(e.p,{children:["There are a number of different methods for calculating the ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsx)(e.mi,{mathvariant:"normal",children:"\u0394"})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"\u0394"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,i.jsx)(e.span,{className:"mord",children:"\u0394"})]})})]}

1)," value. Many of these methods are described by Kijko and Singh (2011)",(0,i.jsx)(e.sup,{children:(0,i.jsx)(e.a,{href:"#user-content-fn-4",id:"user-content-fnref-4","data-footnote-ref":!0,"aria-describedby":"footnote-label",children:"3"})}),". Most of these have been implemented in the Hazard Assessment app along with the associated uncertainty of each method as described by Lasocki and Urban (2011)",(0,i.jsx)(e.sup,{children:(0,i.jsx)(e.a,{href:"#user-content-fn-5",id:"user-content-fnref-5","data-footnote-ref":!0,"aria-describedby":"footnote-label",children:"4"})}),"."]}),"\n",(0,i.jsxs)(e.p,{children:["It is better to over-estimate ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{UL}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"},children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsx)(e.span,{className:"mord mtight",children:(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"UL"})})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]})," than to under-estimate it. In terms of probabilistic seismic hazard calculations, the truncated Gutenberg-Richter model will always give a lower hazard result than the original Gutenberg-Richter, for magnitudes approaching ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{UL}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"},children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsx)(e.span,{className:"mord mtight",children:(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"UL"})})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]}),". For magnitudes well below ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{UL}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"}

1,children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsx)(e.span,{className:"mord mtight",children:(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"UL"})})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]}),", the seismic hazard calculations are the same. In the Hazard Assessment app, the default ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{UL}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"},children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsx)(e.span,{className:"mord mtight",children:(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"UL"})})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]})," used is the maximum of each ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsxs)(e.mrow,{children:[(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]}),(0,i.jsx)(e.mo,{children:"+"}),(0,i.jsx)(e.mi,{children:"\u03C3"})]}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{UL} + \u03C3"})]})})}),(0,i.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"},children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsx)(e.span,{className:"mord mtight",children:(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"UL"})})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]}),(0,i.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,i.jsx)(e.span,{className:"mbin",children:"+"}),(0,i.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"\u03C3"})]})]})]})," estimate from multiple methods."]}),"\n",(0,i.jsx)(e.figure,{children:(0,i.jsx)(e.img,{src:a(37068).A+"",width:"728",height:"720"})}),"\n",(0,i.jsxs)(e.p,{children:["These statistical approaches assume the recorded magnitudes of large events are reliable. Moment is under-recorded for large events if there are no low-frequency sensors installed. The figure to the left comes from ",(0,i.jsx)(e.em,{children:"Morkel and Wesseloo (2017)"}),(0,i.jsx)(e.sup,{children:(0,i.jsx)(e.a,{href:"#user-content-fn-1",id:"user-content-fnref-1","data-footnote-ref":!0,"aria-describedby":"footnote-label",children:"5"})})," showing the effect on the frequency-magnitude relationship, given certain sensor bandwidth limitations."]}),"\n",(0,i.jsxs)(e.p,{children:["In cases like this it is best to override the ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{UL}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"}

1,children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsx)(e.span,{className:"mord mtight",children:(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"UL"})})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]})," as it is likely to be under-estimated with statistical methods."]}),"\n",(0,i.jsx)(e.h2,{id:"conclusion",children:"Conclusion"}),"\n",(0,i.jsxs)(e.p,{children:["While it is important to understand what ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{UL}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"},children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsx)(e.span,{className:"mord mtight",children:(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"UL"})})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]})," is and how it effects seismic hazard calculations, it is not something to use for design purposes or to communicate seismic hazard. It is just ",(0,i.jsx)(e.em,{children:"one part of how seismic hazard is defined"}),". By definition, the probability of an event exceeding ",(0,i.jsxs)(e.span,{className:"katex",children:[(0,i.jsx)(e.span,{className:"katex-mathml",children:(0,i.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,i.jsxs)(e.semantics,{children:[(0,i.jsx)(e.mrow,{children:(0,i.jsxs)(e.msub,{children:[(0,i.jsx)(e.mi,{children:"M"}),(0,i.jsxs)(e.mrow,{children:[(0,i.jsx)(e.mi,{children:"U"}),(0,i.jsx)(e.mi,{children:"L"})]})]})}),(0,i.jsx)(e.annotation,{encoding:"application/x-tex",children:"M_{UL}"})]})})}),(0,i.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,i.jsxs)(e.span,{className:"base",children:[(0,i.jsx)(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,i.jsxs)(e.span,{className:"mord",children:[(0,i.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,i.jsx)(e.span,{className:"msupsub",children:(0,i.jsxs)(e.span,{className:"vlist-t vlist-t2",children:[(0,i.jsxs)(e.span,{className:"vlist-r",children:[(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.3283em"},children:(0,i.jsxs)(e.span,{style:{top:"-2.55em",marginLeft:"-0.109em",marginRight:"0.05em"},children:[(0,i.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,i.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,i.jsx)(e.span,{className:"mord mtight",children:(0,i.jsx)(e.span,{className:"mord mathnormal mtight",children:"UL"})})})]})}),(0,i.jsx)(e.span,{className:"vlist-s",children:"\u200B"})]}),(0,i.jsx)(e.span,{className:"vlist-r",children:(0,i.jsx)(e.span,{className:"vlist",style:{height:"0.15em"},children:(0,i.jsx)(e.span,{})})})]})})]})]})})]})," is zero, so it isn't a great measure of seismic hazard."]}),"\n","\n","\n","\n",(0,i.jsxs)(e.section,{"data-footnotes":!0,className:"footnotes",children:[(0,i.jsx)(e.h2,{className:"sr-only",id:"footnote-label",children:"Footnotes"}),"\n",(0,i.jsxs)(e.ol,{children:["\n",(0,i.jsxs)(e.li,{id:"user-content-fn-2",children:["\n",(0,i.jsxs)(e.p,{children:["McGarr, A., Simpson, D., and Seeber, L. (2002). ",(0,i.jsx)(e.a,{href:"https://earthquake.usgs.gov/static/lfs/research/induced/McGarr-Simpson-Seeber-2002-International-Geophysics.pdf",children:"Case histories of induced and triggered seismicity"}),". International Geophysics, 81, 647-661. ",(0,i.jsx)(e.a,{href:"#user-content-fnref-2","data-footnote-backref":"","aria-label":"Back to reference 1",className:"data-footnote-backref",children:"\u21A9"})]}),"\n"]}),"\n",(0,i.jsxs)(e.li,{id:"user-content-fn-3",children:["\n",(0,i.jsxs)(e.p,{children:["Wesseloo, J. (2018). ",(0,i.jsx)(e.a,{href:"https://research-repository.uwa.edu.au/e

1n/publications/the-spatial-assessment-of-the-current-seismic-hazard-state-for-ha",children:"The spatial assessment of the current seismic hazard state for hard rock underground mines"}),". University of Western Australia Research Repository. ",(0,i.jsx)(e.a,{href:"#user-content-fnref-3","data-footnote-backref":"","aria-label":"Back to reference 2",className:"data-footnote-backref",children:"\u21A9"})]}),"\n"]}),"\n",(0,i.jsxs)(e.li,{id:"user-content-fn-4",children:["\n",(0,i.jsxs)(e.p,{children:["Kijko, A. and Singh, M. (2011). ",(0,i.jsx)(e.a,{href:"http://dx.doi.org/10.2478/s11600-011-0012-6",children:"Statistical tools for maximum possible earthquake magnitude estimation"}),". Acta Geophysica, 59(4), 674-700. ",(0,i.jsx)(e.a,{href:"#user-content-fnref-4","data-footnote-backref":"","aria-label":"Back to reference 3",className:"data-footnote-backref",children:"\u21A9"})]}),"\n"]}),"\n",(0,i.jsxs)(e.li,{id:"user-content-fn-5",children:["\n",(0,i.jsxs)(e.p,{children:["Lasocki, S. and Urban, P. (2011). ",(0,i.jsx)(e.a,{href:"http://dx.doi.org/10.2478/s11600-010-0049-y",children:"Bias, variance and computational properties of Kijko's estimators of the upper limit of magnitude distribution, Mmax"}),". Acta Geophysica, 59(4), 659-673. ",(0,i.jsx)(e.a,{href:"#user-content-fnref-5","data-footnote-backref":"","aria-label":"Back to reference 4",className:"data-footnote-backref",children:"\u21A9"})]}),"\n"]}),"\n",(0,i.jsxs)(e.li,{id:"user-content-fn-1",children:["\n",(0,i.jsxs)(e.p,{children:["Morkel, I.G. and Wesseloo, J. (2017). ",(0,i.jsx)(e.a,{href:"https://research-repository.uwa.edu.au/en/publications/the-effect-of-sensor-bandwidth-limitations-on-the-calculation-of-",children:"The effect of sensor bandwidth limitations on the calculation of seismic hazard"}),". University of Western Australia Research Repository. ",(0,i.jsx)(e.a,{href:"#user-content-fnref-1","data-footnote-backref":"","aria-label":"Back to reference 5",className:"data-footnote-backref",children:"\u21A9"})]}),"\n"]}),"\n"]}),"\n"]})]})}function d(s={}){let{wrapper:e}={...(0,l.R)(),...s.components};return e?(0,i.jsx)(e,{...s,children:(0,i.jsx)(c,{...s})}):c(s)}},37068(s,e,a){a.d(e,{A:()=>n});let n=a.p+"assets/images/freq-e95b4e6ef9b59126354ec4be59ad6461.png"},33182(s,e,a){a.d(e,{A:()=>n});let n=a.p+"assets/images/mcgarr-d9783c45692b1e437b4e1ad5fcd2af5f.png"},32784(s,e,a){a.d(e,{A:()=>n});let n=a.p+"assets/images/mines-vs-crust-a2ec1c1c842120230cfca31a20f6e5dc.png"},1639(s,e,a){a.d(e,{A:()=>n});let n=a.p+"assets/images/mmax-227e49ea968dbba0a01965772f6880d0.png"},75154(s,e,a){a.d(e,{A:()=>n});let n=a.p+"assets/images/mul-718f9dcad99736a32022341df99af6ea.png"},77025(s,e,a){a.d(e,{R:()=>t,x:()=>m});var n=a(12888);let i={},l=n.createContext(i);function t(s){let e=n.useContext(l);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(i):s.components||i:t(s.components),n.createElement(l.Provider,{value:e},s.children)}}}]);