Surface-wave magnitude

<h2 id="definition">Definition</h2>
<p>The formula to calculate surface wave magnitude is:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>

M
          
            s
          
        
        =
        
          log
          
            10
          
        
        ⁡
        
          
            (
            
              
                A
                T
              
            
            )
          
          
            max
          
        
        +
        σ
        (
        Δ
        )
        
        ,
      
    
    {\displaystyle M_{s}=\log _{10}\left({\frac {A}{T}}\right)_{\text{max}}+\sigma (\Delta )\,,}

<p>where A is the maximum <a href="/facts/Particle_displacement/FjGlAwP1">particle displacement</a> in surface waves (<a href="/facts/Vector_sum/bCLrdksy">vector sum</a> of the two horizontal displacements) in <a href="/facts/Micrometre/7eDFxD2L">μm</a>, T is the corresponding <a href="/facts/Period_(physics)/QUSbPaW8">period</a> in <a href="/facts/Second/jtFxnJXx">s</a> (usually 20 ±2 seconds), Δ is the <a href="/facts/Epicentral_distance/8nwmjncG">epicentral distance</a> in <a href="/facts/Degree_(angle)/f48Ns8SX">°</a>, and
</p>

σ
        (
        Δ
        )
        =
        1.66
        ⋅
        
          log
          
            10
          
        
        ⁡
        (
        Δ
        )
        +
        3.5
        
        .
      
    
    {\displaystyle \sigma (\Delta )=1.66\cdot \log _{10}(\Delta )+3.5\,.}

<p>Several versions of this equation were derived throughout the 20th century, with minor variations in the constant values.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> Since the original form of 
  
    
      
        
          M
          
            s
          
        
      
    
    {\displaystyle M_{s}}
  
 was derived for use with <a href="/facts/Teleseismic/HvDfDlYr">teleseismic</a> waves, namely shallow earthquakes at distances >100 km from the seismic receiver, corrections must be added to the computed value to compensate for epicenters deeper than 50 km or less than 20° from the receiver.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>
</p><p>For official use by the Chinese government,<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> the two horizontal displacements must be measured at the same time or within 1/8 of a period; if the two displacements have different periods, a weighted sum must be used:
</p>

T
        =
        
          
            
              
                T
                
                  N
                
              
              
                A
                
                  N
                
              
              +
              
                T
                
                  E
                
              
              
                A
                
                  E
                
              
            
            
              
                A
                
                  N
                
              
              +
              
                A
                
                  E
                
              
            
          
        
        
        ,
      
    
    {\displaystyle T={\frac {T_{N}A_{N}+T_{E}A_{E}}{A_{N}+A_{E}}}\,,}

<p>where AN is the north–south displacement in μm,　AE is the east–west displacement in μm,　TN is the period corresponding to AN in s, and TE is the period corresponding to AE in s.
</p>
<h2 id="other-studies">Other studies</h2>
<p>Vladimír Tobyáš and Reinhard Mittag proposed to relate surface wave magnitude to <a href="/facts/Local_magnitude_scale/UDhqkAX1">local magnitude scale</a> ML, using<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>
</p>

M
          
            s
          
        
        =
        −
        3.2
        +
        1.45
        
          M
          
            L
          
        
      
    
    {\displaystyle M_{s}=-3.2+1.45M_{L}}

<p>Other formulas include three revised formulae proposed by CHEN Junjie et al.:<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>
</p>

M
          
            s
          
        
        =
        
          log
          
            10
          
        
        ⁡
        
          (
          
            
              
                A
                
                  max
                
              
              T
            
          
          )
        
        +
        1.54
        ⋅
        
          log
          
            10
          
        
        ⁡
        (
        Δ
        )
        +
        3.53
      
    
    {\displaystyle M_{s}=\log _{10}\left({\frac {A_{\text{max}}}{T}}\right)+1.54\cdot \log _{10}(\Delta )+3.53}

M
          
            s
          
        
        =
        
          log
          
            10
          
        
        ⁡
        
          (
          
            
              
                A
                
                  max
                
              
              T
            
          
          )
        
        +
        1.73
        ⋅
        
          log
          
            10
          
        
        ⁡
        (
        Δ
        )
        +
        3.27
      
    
    {\displaystyle M_{s}=\log _{10}\left({\frac {A_{\text{max}}}{T}}\right)+1.73\cdot \log _{10}(\Delta )+3.27}

M
          
            s
          
        
        =
        
          log
          
            10
          
        
        ⁡
        
          (
          
            
              
                A
                
                  max
                
              
              T
            
          
          )
        
        −
        6.2
        ⋅
        
          log
          
            10
          
        
        ⁡
        (
        Δ
        )
        +
        20.6
      
    
    {\displaystyle M_{s}=\log _{10}\left({\frac {A_{\text{max}}}{T}}\right)-6.2\cdot \log _{10}(\Delta )+20.6}

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Seismic_magnitude_scales/g5uSHyQe">Seismic magnitude scales</a></li></ul>
<h2 id="notes-and-references">Notes and references</h2>

<h2 id="external-links">External links</h2>
<ul><li>Robert E. Wallace, ed. (1991). <a href="http://www.johnmartin.com/earthquakes/eqsafs/">"The San Andreas Fault System, California (Professional Paper 1515)"</a>. United States Geological Survey. Retrieved 2008-09-14.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>William L. Ellsworth (1991). "SURFACE-WAVE MAGNITUDE (MS) AND BODY-WAVE MAGNITUDE (mb)". United States Geological Survey. Retrieved 2008-09-14. <a href="http://www.johnmartin.com/earthquakes/eqsafs/safs_694.htm" target="_blank">http://www.johnmartin.com/earthquakes/eqsafs/safs_694.htm</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Kanamori, Hiroo (April 1983). "Magnitude scale and quantification of earthquakes". Tectonophysics. 93 (3–4): 185–199. Bibcode:1983Tectp..93..185K. doi:10.1016/0040-1951(83)90273-1. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>XU Shaokui, LU Yuanzhong, GUO Lucan, CHEN Shanpei, XU Zhonghuai, XIAO Chengye, FENG Yijun (许绍燮、陆远忠、郭履灿、陈培善、许忠淮、肖承邺、冯义钧) (1999-04-26). "Specifications on Seismic Magnitudes (地震震级的规定)" (in Chinese). General Administration of Quality Supervision, Inspection, and Quarantine of P.R.C. Archived from the original on 2009-04-24. Retrieved 2008-09-14.{{cite web}}:  CS1 maint: multiple names: authors list (link) <a href="https://web.archive.org/web/20090424211418/http://www.dccdnc.ac.cn/html/zcfg/gfxwj1.jsp" target="_blank">https://web.archive.org/web/20090424211418/http://www.dccdnc.ac.cn/html/zcfg/gfxwj1.jsp</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>XU Shaokui, LU Yuanzhong, GUO Lucan, CHEN Shanpei, XU Zhonghuai, XIAO Chengye, FENG Yijun (许绍燮、陆远忠、郭履灿、陈培善、许忠淮、肖承邺、冯义钧) (1999-04-26). "Specifications on Seismic Magnitudes (地震震级的规定)" (in Chinese). General Administration of Quality Supervision, Inspection, and Quarantine of P.R.C. Archived from the original on 2009-04-24. Retrieved 2008-09-14.{{cite web}}:  CS1 maint: multiple names: authors list (link) <a href="https://web.archive.org/web/20090424211418/http://www.dccdnc.ac.cn/html/zcfg/gfxwj1.jsp" target="_blank">https://web.archive.org/web/20090424211418/http://www.dccdnc.ac.cn/html/zcfg/gfxwj1.jsp</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Kanamori, Hiroo (April 1983). "Magnitude scale and quantification of earthquakes". Tectonophysics. 93 (3–4): 185–199. Bibcode:1983Tectp..93..185K. doi:10.1016/0040-1951(83)90273-1. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Bath, M (1966). "Earthquake energy and magnitude". In Ahrens, L. H.; Press, F.; Runcorn, S. (eds.). Physics and Chemistry of the Earth. Pergamon Press. pp. 115–165. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Bath, M (1966). "Earthquake energy and magnitude". In Ahrens, L. H.; Press, F.; Runcorn, S. (eds.). Physics and Chemistry of the Earth. Pergamon Press. pp. 115–165. <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>XU Shaokui, LU Yuanzhong, GUO Lucan, CHEN Shanpei, XU Zhonghuai, XIAO Chengye, FENG Yijun (许绍燮、陆远忠、郭履灿、陈培善、许忠淮、肖承邺、冯义钧) (1999-04-26). "Specifications on Seismic Magnitudes (地震震级的规定)" (in Chinese). General Administration of Quality Supervision, Inspection, and Quarantine of P.R.C. Archived from the original on 2009-04-24. Retrieved 2008-09-14.{{cite web}}:  CS1 maint: multiple names: authors list (link) <a href="https://web.archive.org/web/20090424211418/http://www.dccdnc.ac.cn/html/zcfg/gfxwj1.jsp" target="_blank">https://web.archive.org/web/20090424211418/http://www.dccdnc.ac.cn/html/zcfg/gfxwj1.jsp</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Vladimír Tobyáš and Reinhard Mittag (1991-02-06). "Local magnitude, surface wave magnitude and seismic energy". Studia Geophysica et Geodaetica. 35 (4): 354. Bibcode:1991StGG...35..354T. doi:10.1007/BF01613981. S2CID 128567958. Archived from the original on 2013-01-04. Retrieved 2008-09-14. <a href="https://archive.today/20130104222201/http://www.springerlink.com/content/lv140444x5m01362/" target="_blank">https://archive.today/20130104222201/http://www.springerlink.com/content/lv140444x5m01362/</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>CHEN Junjie, CHI Tianfeng, WANG Junliang, CHI Zhencai (陈俊杰, 迟天峰, 王军亮, 迟振才) (2002-01-01). "Study of Surface Wave Magnitude in China (中国面波震级研究)" (in Chinese). Journal of Seismological Research (《地震研究》). Archived from the original on April 26, 2009. Retrieved 2008-09-14.{{cite web}}:  CS1 maint: multiple names: authors list (link) <a href="https://archive.today/20090426022217/http://scholar.ilib.cn/Abstract.aspx?A=dzyj200201016" target="_blank">https://archive.today/20090426022217/http://scholar.ilib.cn/Abstract.aspx?A=dzyj200201016</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
</ol>

Surface-wave magnitude open-in-new

Surface-wave magnitude