<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE article PUBLIC "-//NLM//DTD Journal Publishing with OASIS Tables v3.0 20080202//EN" "journalpub-oasis3.dtd">
<article xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:oasis="http://docs.oasis-open.org/ns/oasis-exchange/table" dtd-version="3.0"><?xmltex \makeatother\@nolinetrue\makeatletter?>
  <front>
    <journal-meta>
<journal-id journal-id-type="publisher">SE</journal-id>
<journal-title-group>
<journal-title>Solid Earth</journal-title>
<abbrev-journal-title abbrev-type="publisher">SE</abbrev-journal-title>
<abbrev-journal-title abbrev-type="nlm-ta">Solid Earth</abbrev-journal-title>
</journal-title-group>
<issn pub-type="epub">1869-9529</issn>
<publisher><publisher-name>Copernicus GmbH</publisher-name>
<publisher-loc>Göttingen, Germany</publisher-loc>
</publisher>
</journal-meta>

    <article-meta>
      <article-id pub-id-type="doi">10.5194/se-5-1169-2014</article-id><title-group><article-title>Wave-equation-based travel-time seismic tomography – <?xmltex \hack{\newline}?>Part 2: Application to the 1992 Landers earthquake (<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi mathvariant="normal">w</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> 7.3) area</article-title>
      </title-group><?xmltex \runningtitle{Part 2: Application}?><?xmltex \runningauthor{P.~Tong et~al.}?>
      <contrib-group>
        <contrib contrib-type="author" corresp="yes" rid="aff1">
          <name><surname>Tong</surname><given-names>P.</given-names></name>
          <email>tongping85@gmail.com</email>
        </contrib>
        <contrib contrib-type="author" corresp="no" rid="aff2">
          <name><surname>Zhao</surname><given-names>D.</given-names></name>
          
        </contrib>
        <contrib contrib-type="author" corresp="no" rid="aff3">
          <name><surname>Yang</surname><given-names>D.</given-names></name>
          
        </contrib>
        <contrib contrib-type="author" corresp="no" rid="aff4">
          <name><surname>Yang</surname><given-names>X.</given-names></name>
          
        </contrib>
        <contrib contrib-type="author" corresp="no" rid="aff4">
          <name><surname>Chen</surname><given-names>J.</given-names></name>
          
        </contrib>
        <contrib contrib-type="author" corresp="no" rid="aff1">
          <name><surname>Liu</surname><given-names>Q.</given-names></name>
          
        </contrib>
        <aff id="aff1"><label>1</label><institution>Department of Physics, University of Toronto, Toronto, M5S 1A7, Ontario, Canada</institution>
        </aff>
        <aff id="aff2"><label>2</label><institution>Department of Geophysics, Tohoku University, Sendai, Japan</institution>
        </aff>
        <aff id="aff3"><label>3</label><institution>Department of Mathematical Sciences, Tsinghua University, Beijing, China</institution>
        </aff>
        <aff id="aff4"><label>4</label><institution>Department of Mathematics, University of California, Santa Barbara, California, USA</institution>
        </aff>
      </contrib-group>
      <author-notes><corresp id="corr1">P. Tong (tongping85@gmail.com)</corresp></author-notes><pub-date><day>26</day><month>November</month><year>2014</year></pub-date>
      
      <volume>5</volume>
      <issue>2</issue>
      <fpage>1169</fpage><lpage>1188</lpage>
      <history>
        <date date-type="received"><day>10</day><month>August</month><year>2014</year></date>
           <date date-type="rev-request"><day>25</day><month>August</month><year>2014</year></date>
           <date date-type="rev-recd"><day>19</day><month>October</month><year>2014</year></date>
           <date date-type="accepted"><day>24</day><month>October</month><year>2014</year></date>
           
      </history>
      <permissions>
<license license-type="open-access">
<license-p>This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit <ext-link ext-link-type="uri" xlink:href="http://creativecommons.org/licenses/by/3.0/">http://creativecommons.org/licenses/by/3.0/</ext-link></license-p>
</license>
</permissions>

      <self-uri xlink:href="https://se.copernicus.org/articles/.html">This article is available from https://se.copernicus.org/articles/.html</self-uri>
<self-uri xlink:href="https://se.copernicus.org/articles/.pdf">The full text article is available as a PDF file from https://se.copernicus.org/articles/.pdf</self-uri>
<abstract>
    <p>High-resolution 3-D <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave crustal velocity
and Poisson's ratio models of the 1992 Landers earthquake (<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi mathvariant="normal">w</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> 7.3)
area are determined iteratively by a wave-equation-based travel-time
seismic tomography (WETST) technique. The details of data selection, synthetic arrival-time
determination, and trade-off analysis of damping and smoothing
parameters are presented to show the performance of this new
tomographic inversion method. A total of 78 523 <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave and
46 999 <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave high-quality arrival-time data from 2041
local earthquakes recorded by 275 stations during the period of
1992–2013 are used to obtain the final tomographic models, which cost
around 10 000 CPU hours. Checkerboard resolution tests are
conducted to verify the reliability of inversion results for the
chosen seismic data and the wave-equation-based travel-time seismic
tomography method. Significant structural heterogeneities are revealed
in the crust of the 1992 Landers earthquake area which may be closely
related to the local seismic activities. Strong variations of velocity
and Poisson's ratio exist in the source regions of the Landers and
three other nearby strong earthquakes. Most seismicity occurs in
areas with high-velocity and low Poisson's ratio, which may be
associated with the seismogenic layer. Pronounced low-velocity
anomalies revealed in the lower crust along the Elsinore, the San
Jacinto, and the San Andreas faults may reflect the existence of fluids
in the lower crust. The recovery of these strong heterogeneous
structures is facilitated by the use of full wave equation solvers
and WETST and verifies their ability in generating high-resolution
tomographic models.</p>
  </abstract>
    </article-meta>
  </front>
<body>
      

<sec id="Ch1.S1" sec-type="intro">
  <title>Introduction</title>
      <p>In <xref ref-type="bibr" rid="bib1.bibx40" id="text.1"/> (hereinafter referred to as paper I), we introduced
a new tomographic method, the so called wave-equation-based travel-time
seismic tomography (WETST) which is a “2-D–3-D” adjoint tomography
technique based upon a high-order finite-difference solver. This
approach restricts each forward modeling in a 2-D vertical plane
containing the source and the receiver, while tomographic unknowns
such as velocity perturbations are specified on a 3-D inversion
grid. Comparing with the “3-D–3-D” wave-equation travel-time inversion <xref ref-type="bibr" rid="bib1.bibx21" id="paren.2"/> or the “3-D–3-D” adjoint
tomography based on spectral-element numerical solvers <xref ref-type="bibr" rid="bib1.bibx42 bib1.bibx6 bib1.bibx30" id="paren.3"><named-content content-type="pre">e.g.,</named-content></xref>, the theoretical disadvantage of this
“2-D–3-D” tomographic method is that it ignores the influence of
the off-plane structures on seismic arrivals. However, from the
computational aspect, WETST is generally much more efficient. This is
essential for tomographic problems involving large data sets, which is
important for increasing the illumination of subsurface
structures. Because the off-ray finite-frequency effects within the
2-D vertical plane are considered, WETST has a theoretical advantage
over simple ray-based tomographic methods. In this second paper, we
choose the 1992 Landers earthquake area as our study area and test the
performance of WETST in a realistic application.</p>
      <p>The 1992 Landers earthquake with a magnitude of 7.3 occurred on 28 June 1992
in the Mojave Desert of southern California (Fig. <xref ref-type="fig" rid="Ch1.F1"/>). The source
area is also within the southern part of the eastern California shear zone,
a <?xmltex \hack{\mbox\bgroup}?>major<?xmltex \hack{\egroup}?><?xmltex \hack{\mbox\bgroup}?>tectonic<?xmltex \hack{\egroup}?> element of the transform plate boundary zone
between the Pacific and North America Plates <xref ref-type="bibr" rid="bib1.bibx28" id="paren.4"/>. The epicenter
was located at 34.161<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula> N, 116.396<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula> W and its focal depth
was 7.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> <xref ref-type="bibr" rid="bib1.bibx52" id="paren.5"/>. This large earthquake had
a right-lateral strike slip focal mechanism, agreeing with the regional
deformation of the Mojave block <xref ref-type="bibr" rid="bib1.bibx43 bib1.bibx25" id="paren.6"/>. It caused
a surface rupture of approximate 80 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> across a series of complex
fault intersections <xref ref-type="bibr" rid="bib1.bibx43" id="paren.7"/>. More than 40 000 foreshocks,
preshocks, and aftershocks to the Landers earthquake were reported by the
Southern California Seismographic Network (SCSN) in the year 1992
<xref ref-type="bibr" rid="bib1.bibx28" id="paren.8"/>. The Landers earthquake sequence itself is the largest
sequence recorded by SCSN since the monitoring began in 1920s
<xref ref-type="bibr" rid="bib1.bibx9" id="paren.9"/>. Besides the Landers mainshock, the Joshua Tree
foreshock (<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi mathvariant="normal">w</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> 6.1) and the Big Bear aftershock
(<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi mathvariant="normal">w</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> 6.2) are two other main events of this sequence
(Fig. <xref ref-type="fig" rid="Ch1.F1"/>). The 1999 Hector Mine earthquake (<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi mathvariant="normal">w</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> 7.1)
which is considered to be triggered by the 1992 Landers earthquake is another
large earthquake in the study area from the past 20 years
<xref ref-type="bibr" rid="bib1.bibx24" id="paren.10"/>. To gain insights into the earthquake sequences and local
crustal heterogeneities, many researchers have investigated the Landers
mainshock, the corresponding sequence, and the structures of the source area
using different techniques <xref ref-type="bibr" rid="bib1.bibx9 bib1.bibx52 bib1.bibx7 bib1.bibx22 bib1.bibx2" id="paren.11"><named-content content-type="pre">e.g.,</named-content></xref>. Seismic tomography has shown to be
one of the most promising tools in revealing the heterogeneous structures of
the Earth's interior <xref ref-type="bibr" rid="bib1.bibx33 bib1.bibx51 bib1.bibx26 bib1.bibx20" id="paren.12"><named-content content-type="pre">e.g.,</named-content></xref>. With the large number of high-quality seismic data recorded by
SCSN, it is possible to explore the Landers earthquake area by tomographic
techniques. Additionally, the detailed tomographic structures may then
improve our understanding of the relationship between the occurrence of large
crustal earthquakes and local structural heterogeneities <xref ref-type="bibr" rid="bib1.bibx52 bib1.bibx18" id="paren.13"/>.</p>
      <p>The seismic velocity structures beneath southern California have been
investigated by numerous researchers <xref ref-type="bibr" rid="bib1.bibx52 bib1.bibx18 bib1.bibx35 bib1.bibx30 bib1.bibx31 bib1.bibx1" id="paren.14"><named-content content-type="pre">e.g.,</named-content></xref>. These tomographic results
generally show that strong structural heterogeneities exist in the crust and
upper mantle under southern California <xref ref-type="bibr" rid="bib1.bibx55 bib1.bibx31" id="paren.15"><named-content content-type="pre">e.g.,</named-content></xref>.
Furthermore, <xref ref-type="bibr" rid="bib1.bibx18" id="text.16"/> observed a weak correlation between earthquake
occurrence and seismic velocities, with upper-crust earthquakes mostly
occurring in high <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> velocity regions and mid-crustal earthquakes occurring
in low <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> velocity regions. For the source area of the 1992 Landers
earthquake, <xref ref-type="bibr" rid="bib1.bibx52" id="text.17"/> and <xref ref-type="bibr" rid="bib1.bibx56" id="text.18"/> successively mapped out
detailed <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave tomographic images, both of which showed strong
heterogeneous velocity structures and suggested that the earthquake
occurrence may be closely related to crustal heterogeneities.
<xref ref-type="bibr" rid="bib1.bibx14" id="text.19"/> reached the same conclusion through tomographic inversion of
<inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave arrival times from aftershocks of the 1992 Landers earthquake.
<xref ref-type="bibr" rid="bib1.bibx34" id="text.20"/> simultaneously determined <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave velocity and
Poisson's ratio models for the Landers earthquake area. They showed
a correlation between the seismic activity and crustal heterogeneities and
suggested that the existence of crustal fluids may have weakened the fault
zone and thus triggered the Landers earthquake.</p>
      <p>Taking these previous tomographic results as references, we test the
performance of WETST in imaging crustal structures of the Landers earthquake
source area. The tomographic images inverted by WETST may help shed some new
lights on local heterogeneous structures and the nucleation of large crustal
earthquakes.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F1"><caption><p>The tectonic conditions and surface topography around southern
California. The blue box indicates the present study area. The red star
represents the epicenter of the 1992 Landers earthquake
(<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi mathvariant="normal">w</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>7.3</mml:mn></mml:mrow></mml:math></inline-formula>), the two blue stars show epicenters of the 1992 Joshua
Tree earthquake (<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi mathvariant="normal">w</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>6.1</mml:mn></mml:mrow></mml:math></inline-formula>) and the 1992 Big Bear earthquake
(<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi mathvariant="normal">w</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>6.2</mml:mn></mml:mrow></mml:math></inline-formula>), and the brown star denotes the epicenter of the 1999
Hector Mine earthquake (<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi mathvariant="normal">w</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>7.2</mml:mn></mml:mrow></mml:math></inline-formula>). Active regional faults and
volcanic centers are indicated by grey curves and black triangles,
respectively.</p></caption>
        <?xmltex \igopts{width=241.848425pt}?><graphic xlink:href="https://www.solid-earth.net/5/1169/2014/se-5-1169-2014-f01.jpg"/>

      </fig>

</sec>
<sec id="Ch1.S2">
  <title>Practical implementation</title>
      <p>Taking the 1992 Landers earthquake area as our test field, in this
section we show the implement of WETST in real data applications. The
detailed theory of the WETST method is fully presented in paper I, and
only key results of paper I are summarized as follows.
<?xmltex \hack{\newpage}?>
Wave-equation-based travel-time seismic tomography is rooted in the
following tomographic equation
          <disp-formula content-type="numbered" id="Ch1.E1"><mml:math display="block"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs</mml:mtext></mml:msup><mml:mo>-</mml:mo><mml:msup><mml:mi>T</mml:mi><mml:mtext>syn</mml:mtext></mml:msup><mml:mo>=</mml:mo><mml:munder><mml:mo movablelimits="false">∫</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi></mml:munder><mml:mi>K</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>;</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mi mathvariant="normal">r</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mi mathvariant="normal">s</mml:mi></mml:msub><mml:mo>)</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi mathvariant="italic">δ</mml:mi><mml:mi>c</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mfrac></mml:mstyle><mml:mi mathvariant="normal">d</mml:mi><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>
        where <inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs</mml:mtext></mml:msup></mml:mrow></mml:math></inline-formula> is the arrival time of the interested seismic
phase picked on recorded seismogram, <inline-formula><mml:math display="inline"><mml:mrow><mml:mi>c</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> is the
<inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> or <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> velocity model based on which synthetic
arrival time <inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>syn</mml:mtext></mml:msup></mml:mrow></mml:math></inline-formula> is calculated, and
<inline-formula><mml:math display="inline"><mml:mrow><mml:mi>K</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>;</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mi mathvariant="normal">r</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mi mathvariant="normal">s</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> is the travel-time
sensitivity kernel constructed based on the interactions of forward
wavefield <inline-formula><mml:math display="inline"><mml:mrow><mml:mi>u</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and adjoint wavefield <inline-formula><mml:math display="inline"><mml:mrow><mml:mi>q</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> by

              <disp-formula content-type="numbered" specific-use="align"><mml:math display="block"><mml:mtable displaystyle="true"><mml:mlabeledtr id="Ch1.E2"><mml:mtd/><mml:mtd/><mml:mtd><mml:mrow><mml:mi>K</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>;</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mi mathvariant="normal">r</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mi mathvariant="normal">s</mml:mi></mml:msub><mml:mo>)</mml:mo><mml:mo>=</mml:mo></mml:mrow></mml:mtd></mml:mlabeledtr><mml:mtr><mml:mtd/><mml:mtd><mml:mrow><mml:mo>-</mml:mo><mml:munderover><mml:mo movablelimits="false">∫</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mi>T</mml:mi></mml:munderover><mml:mfenced close="]" open="["><mml:mn mathvariant="normal">2</mml:mn><mml:msup><mml:mi>c</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo><mml:mi mathvariant="normal">∇</mml:mi><mml:mi>q</mml:mi><mml:mo>(</mml:mo><mml:mi>T</mml:mi><mml:mo>-</mml:mo><mml:mi mathvariant="italic">τ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo><mml:mo>⋅</mml:mo><mml:mi mathvariant="normal">∇</mml:mi><mml:mi>u</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="italic">τ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo></mml:mfenced><mml:mi mathvariant="normal">d</mml:mi><mml:mi mathvariant="italic">τ</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>

          The forward wavefield <inline-formula><mml:math display="inline"><mml:mrow><mml:mi>u</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and adjoint wavefield
<inline-formula><mml:math display="inline"><mml:mrow><mml:mi>q</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> satisfy the forward and adjoint acoustic wave
equations as
          <disp-formula content-type="numbered" id="Ch1.E3"><mml:math display="block"><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:msup><mml:mo>∂</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:msup><mml:mi>u</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>=</mml:mo><mml:mi mathvariant="normal">∇</mml:mi><mml:mo>⋅</mml:mo><mml:mfenced close="]" open="["><mml:msup><mml:mi>c</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo><mml:mi mathvariant="normal">∇</mml:mi><mml:mi>u</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo></mml:mfenced><mml:mo>+</mml:mo><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mi mathvariant="italic">δ</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mi mathvariant="normal">s</mml:mi></mml:msub><mml:mo>)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>
        and

              <disp-formula content-type="numbered" specific-use="align"><mml:math display="block"><mml:mtable displaystyle="true"><mml:mlabeledtr id="Ch1.E4"><mml:mtd/><mml:mtd/><mml:mtd><mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mo>∂</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:msup><mml:mi>q</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi mathvariant="normal">∇</mml:mi><mml:mo>⋅</mml:mo><mml:mfenced open="[" close="]"><mml:msup><mml:mi>c</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo><mml:mi mathvariant="normal">∇</mml:mi><mml:mi>q</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo></mml:mfenced></mml:mrow></mml:mtd></mml:mlabeledtr><mml:mtr><mml:mtd/><mml:mtd><mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>w</mml:mi><mml:mo>(</mml:mo><mml:mi>T</mml:mi><mml:mo>-</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mfenced open="[" close="]"><mml:mo>∂</mml:mo><mml:mi>u</mml:mi><mml:mo>(</mml:mo><mml:mi>T</mml:mi><mml:mo>-</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo><mml:mo>/</mml:mo><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mi>T</mml:mi></mml:msubsup><mml:mi>w</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mi>u</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mfenced open="[" close="]"><mml:msup><mml:mo>∂</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:msup><mml:mi>u</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>/</mml:mo><mml:mo>∂</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mfenced><mml:mi mathvariant="normal">d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mi mathvariant="italic">δ</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mi mathvariant="normal">r</mml:mi></mml:msub><mml:mo>)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>

          where <inline-formula><mml:math display="inline"><mml:mrow><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> is the source time function and <inline-formula><mml:math display="inline"><mml:mrow><mml:mi>w</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> is the time
window function used to isolate a particular seismic phase (such as
first <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> or <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> arrival in this study). We assume that
seismic waves propagate in the vertical plane which contains the
source <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mi mathvariant="normal">s</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and receiver <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mi mathvariant="normal">r</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and satisfy 2-D
acoustic wave Eq. (<xref ref-type="disp-formula" rid="Ch1.E3"/>). This 2-D approximation is mainly
invoked to reduce computational cost and enable the use of as many
seismic data as possible. Given a reference velocity model
<inline-formula><mml:math display="inline"><mml:mrow><mml:mi>c</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, the purpose of WETST is to find the relative velocity
perturbation <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="italic">δ</mml:mi><mml:mi>c</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo><mml:mo>/</mml:mo><mml:mi>c</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> which can be then
used to obtain the updated model <inline-formula><mml:math display="inline"><mml:mrow><mml:mi>c</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo><mml:mo>+</mml:mo><mml:mi mathvariant="italic">δ</mml:mi><mml:mi>c</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>
that best explains travel-time data <inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs</mml:mtext></mml:msup></mml:mrow></mml:math></inline-formula>. To this end, we
select seismic phases to make measurements <inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs</mml:mtext></mml:msup></mml:mrow></mml:math></inline-formula>, recast
tomographic Eq. (<xref ref-type="disp-formula" rid="Ch1.E1"/>) on a set of inversion grid nodes, and
solve an optimization problem.</p>
<sec id="Ch1.S2.SS1">
  <title>Data</title>
      <p>Our initial data consist of <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave arrival
times of local earthquakes recorded by the SCSN, compiled by the
Southern California Earthquake Data Center and obtained
through the Seismogram Transfer Program
(<uri>http://www.data.scec.org/research-tools/stp-index.html</uri>). In the study
area (blue box in Fig. <xref ref-type="fig" rid="Ch1.F1"/>), SCSN data analysts have picked
the phase data (first <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> arrivals) of nearly
30 000 earthquakes with magnitudes between 2.0 and 4.0 occurring
during a period from January 1992 to November 2013. Since it is very
computationally intensive and also unnecessary to include all these
events, we only choose a small subset of them for our tomographic
inversion.</p>
      <p>To ensure that the chosen seismic data illuminate the study region
well, events and corresponding phase records are carefully selected
based on the following six criteria: (1) to guarantee the quality of
seismic data and validity of point source assumption for forward
modeling, the magnitudes of the selected events should be within the
range <inline-formula><mml:math display="inline"><mml:mrow><mml:mo>[</mml:mo><mml:mn>2.0</mml:mn><mml:mo>,</mml:mo><mml:mn>4.0</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula>; (2) to reduce the influence of mislocation errors
on tomographic inversion, we only choose events with more than 20
<inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> and more than 10 <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> arrivals; (3) the focal depth
of each chosen event is greater than 3.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>; (4) to ensure
that picking errors of selected phase data are within an acceptable
range, the misfit between the observed arrival time and the synthetic
arrival time in the 1-D reference model (discussed later) is required
to be less than 1.0 s for <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave or 1.5 s for
<inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave; (5) to save on computation, we only use seismic records
whose epicentral distances are less than 100 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>; (6) to avoid
event clustering and keep a uniform distribution of hypocenter
locations, we divide the Landers earthquake source area (the blue box
in Fig. <xref ref-type="fig" rid="Ch1.F1"/>) into
2 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>×</mml:mo></mml:math></inline-formula> 2 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>×</mml:mo></mml:math></inline-formula> 2 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>
blocks and only choose one event in each
block that is recorded by the maximal number of stations if it
exists. As a result, our selected data set includes 78 523 first
<inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave and 46 999 first <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave arrival times
recorded by 275 SCSN stations (Fig. <xref ref-type="fig" rid="Ch1.F2"/>b) for 2041 local
earthquakes (Fig. <xref ref-type="fig" rid="Ch1.F2"/>a).</p>

<?xmltex \floatpos{t}?><table-wrap id="Ch1.T1"><caption><p>The starting 1-D velocity model (<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula>) used in this study.</p></caption><oasis:table frame="topbot"><oasis:tgroup cols="3">
     <oasis:colspec colnum="1" colname="col1" align="left"/>
     <oasis:colspec colnum="2" colname="col2" align="left"/>
     <oasis:colspec colnum="3" colname="col3" align="left"/>
     <oasis:thead>
       <oasis:row>  
         <oasis:entry colname="col1">Depth to</oasis:entry>  
         <oasis:entry colname="col2"><inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave</oasis:entry>  
         <oasis:entry colname="col3"><inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave</oasis:entry>
       </oasis:row>
       <oasis:row rowsep="1">  
         <oasis:entry colname="col1">surface (km)</oasis:entry>  
         <oasis:entry colname="col2">velocity (<inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="normal">km</mml:mi><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>)</oasis:entry>  
         <oasis:entry colname="col3">velocity (<inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="normal">km</mml:mi><mml:mspace linebreak="nobreak" width="0.125em"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>)</oasis:entry>
       </oasis:row>
     </oasis:thead>
     <oasis:tbody>
       <oasis:row>  
         <oasis:entry colname="col1">0.0–2.0</oasis:entry>  
         <oasis:entry colname="col2">4.800</oasis:entry>  
         <oasis:entry colname="col3">2.775</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1">2.0–5.5</oasis:entry>  
         <oasis:entry colname="col2">5.800</oasis:entry>  
         <oasis:entry colname="col3">3.353</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1">5.5–16.0</oasis:entry>  
         <oasis:entry colname="col2">6.300</oasis:entry>  
         <oasis:entry colname="col3">3.642</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1">16.0–29.2</oasis:entry>  
         <oasis:entry colname="col2">6.700</oasis:entry>  
         <oasis:entry colname="col3">3.873</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1"><inline-formula><mml:math display="inline"><mml:mo>&gt;</mml:mo></mml:math></inline-formula> 29.2</oasis:entry>  
         <oasis:entry colname="col2">7.800</oasis:entry>  
         <oasis:entry colname="col3">4.509</oasis:entry>
       </oasis:row>
     </oasis:tbody>
   </oasis:tgroup>

</oasis:table></table-wrap>

</sec>
<sec id="Ch1.S2.SS2">
  <title>Model parameterization</title>
      <p>The discrete form of tomographic Eq. (<xref ref-type="disp-formula" rid="Ch1.E1"/>) requires model
parameterization. We first need to define the forward modeling grid for the
calculation of travel-time kernel
<inline-formula><mml:math display="inline"><mml:mrow><mml:mi>K</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>;</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mi mathvariant="normal">r</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mi mathvariant="normal">s</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> in Eq. (2). Since in
this study the travel-time kernel
<inline-formula><mml:math display="inline"><mml:mrow><mml:mi>K</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>;</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mi mathvariant="normal">r</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mi mathvariant="normal">s</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> is computed on the
vertical plane passing through the source <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mi mathvariant="normal">s</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and the
receiver <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mi mathvariant="normal">r</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> by numerically solving the two acoustic wave
Eqs. (<xref ref-type="disp-formula" rid="Ch1.E3"/>) and (<xref ref-type="disp-formula" rid="Ch1.E4"/>) using a finite-difference scheme
(i.e., the high-order central difference method presented in the appendix of
paper I), the forward modeling grid should be designed to suit 2-D
finite-difference calculations. Usually, for a finite-difference calculation,
the computational domain is divided into a uniform grid where the grid size
is determined by the velocity, dominant frequency of seismic wave, and
stability condition of the numerical scheme.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F2"><caption><p><bold>(a)</bold> Hypocentral distribution of the 2041 earthquakes
(purple dots) used in this study. The stars denote the relatively large
earthquakes which occurred in and around the Landers area as shown in
Fig. <xref ref-type="fig" rid="Ch1.F1"/>. <bold>(b)</bold> Distribution of the 275 seismic stations
(blue reverse triangles) used in this study. The grey crosses represent the
inversion grid nodes.</p></caption>
          <?xmltex \igopts{width=213.395669pt}?><graphic xlink:href="https://www.solid-earth.net/5/1169/2014/se-5-1169-2014-f02.jpg"/>

        </fig>

      <p>We first define <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> as a 1-D layered velocity model that contains five
layers separated by two velocity boundaries at 2.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> and
5.5 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>, the Conrad discontinuity (16 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>), and an averaged
flat Moho (<inline-formula><mml:math display="inline"><mml:mn>29.2</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>) <xref ref-type="bibr" rid="bib1.bibx9 bib1.bibx58" id="paren.21"/>. In each layer,
the velocity structure is homogeneous and the corresponding <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave
velocities are shown in Table <xref ref-type="table" rid="Ch1.T1"/>. For this 1-D layered model, the
arrival times of the direct <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> waves, head waves refracted from the
velocity boundary at the depth of 5.5 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> observed at epicentral
distances <inline-formula><mml:math display="inline"><mml:mo>&gt;</mml:mo></mml:math></inline-formula> 40–50 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>, head wave (<inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula>*, <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula>*) refracted from the
Conrad discontinuity when the epicentral distances are in the range of
90–140 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>, and head waves (<italic>Pn</italic>, <italic>Sn</italic>) from the Moho
when the epicentral distances are greater than 140–150 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> can be
easily calculated according to the geometrical ray theory <xref ref-type="bibr" rid="bib1.bibx54 bib1.bibx55 bib1.bibx36" id="paren.22"/>. Accordingly, the synthetic first <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> arrival
times can be determined for each source–receiver pair based on its
epicentral distance for the velocity model <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula>. However, the undulated
Moho of southern California region has large lateral depth variation and
strong influence on seismic wave propagation <xref ref-type="bibr" rid="bib1.bibx58" id="paren.23"/>, and has
considerable effects on the tomographic images of the lower crust and the
uppermost mantle <xref ref-type="bibr" rid="bib1.bibx56 bib1.bibx35" id="paren.24"><named-content content-type="pre">e.g.,</named-content></xref>. Therefore, for this
study, we take into account the variation of Moho topography, and introduce
a velocity model <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> that differs <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> by adding an undulated Moho
obtained from receiver functions by <xref ref-type="bibr" rid="bib1.bibx58" id="text.25"/> as the starting model.
The synthetic travel times of the first <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> waves in <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> can be
calculated by the combined ray and cross-correlation technique discussed in
Paper I. Once the synthetic arriving times for <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> and the observed
arrival times picked from data are available, velocity structures can be
updated from <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> based on the WETST technique.</p>
      <p>For the 2-D finite-difference forward modeling, we choose a Gaussian wavelet as the
source time function <inline-formula><mml:math display="inline"><mml:mrow><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> in Eq. (<xref ref-type="disp-formula" rid="Ch1.E3"/>)

                <disp-formula content-type="numbered" specific-use="align"><mml:math display="block"><mml:mtable displaystyle="true"><mml:mlabeledtr id="Ch1.E5"><mml:mtd/><mml:mtd/><mml:mtd><mml:mrow><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi>A</mml:mi><mml:mfenced open="[" close="]"><mml:mn mathvariant="normal">2</mml:mn><mml:msup><mml:mi mathvariant="italic">π</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup><mml:msubsup><mml:mi>f</mml:mi><mml:mn mathvariant="normal">0</mml:mn><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup><mml:msup><mml:mfenced close=")" open="("><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1.2</mml:mn><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mfenced><mml:mn mathvariant="normal">2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mfenced></mml:mrow></mml:mtd></mml:mlabeledtr><mml:mtr><mml:mtd/><mml:mtd><mml:mrow><mml:mi>exp⁡</mml:mi><mml:mfenced open="(" close=")"><mml:mo>-</mml:mo><mml:msup><mml:mi mathvariant="italic">π</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup><mml:msubsup><mml:mi>f</mml:mi><mml:mn mathvariant="normal">0</mml:mn><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup><mml:msup><mml:mfenced open="(" close=")"><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1.2</mml:mn><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mfenced><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mfenced><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>

            where <inline-formula><mml:math display="inline"><mml:mi>A</mml:mi></mml:math></inline-formula> is the amplitude and <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> is the dominant frequency. The
frequency spectrum of the source time function Eq. (<xref ref-type="disp-formula" rid="Ch1.E5"/>) is
mainly concentrated within <inline-formula><mml:math display="inline"><mml:mrow><mml:mo>[</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo>,</mml:mo><mml:mn>2.5</mml:mn><mml:msub><mml:mi>f</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula>. For example, the spectrum
(shown in Fig. <xref ref-type="fig" rid="Ch1.F3"/>b) for a <inline-formula><mml:math display="inline"><mml:mrow><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> with unit amplitude
<inline-formula><mml:math display="inline"><mml:mrow><mml:mi>A</mml:mi><mml:mo>=</mml:mo><mml:mn>1.0</mml:mn></mml:mrow></mml:math></inline-formula> and dominant frequency <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>2.0</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">Hz</mml:mi></mml:math></inline-formula>
(Fig. <xref ref-type="fig" rid="Ch1.F3"/>a) has significant values between 0.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">Hz</mml:mi></mml:math></inline-formula>
and 5.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">Hz</mml:mi></mml:math></inline-formula>. Correspondingly for consistency, data traces
need to be filtered between the frequency range of <inline-formula><mml:math display="inline"><mml:mrow><mml:mo>[</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo>,</mml:mo><mml:mn>2.5</mml:mn><mml:msub><mml:mi>f</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula> for
the picking of observed travel times. We specifically denote the
observed travel times picked on band-pass filtered seismograms as
<inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs,f</mml:mtext></mml:msup></mml:mrow></mml:math></inline-formula>. Since seismic waves filtered at different
frequencies have different sensitivity to heterogeneous structures,
arrival-time <inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs,f</mml:mtext></mml:msup></mml:mrow></mml:math></inline-formula> is not necessarily equal to
<inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs</mml:mtext></mml:msup></mml:mrow></mml:math></inline-formula> obtained from the travel-time data catalog (such as
the SCSN). We relate the two arrival times using the formula
            <disp-formula content-type="numbered" id="Ch1.E6"><mml:math display="block"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs,f</mml:mtext></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs</mml:mtext></mml:msup><mml:mo>+</mml:mo><mml:mi mathvariant="italic">δ</mml:mi><mml:msup><mml:mi>t</mml:mi><mml:mi>f</mml:mi></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula>
          Fortunately, <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="italic">δ</mml:mi><mml:msup><mml:mi>t</mml:mi><mml:mi>f</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula>s are found to be very small and negligible
in this study. In detail, we first choose the dominant frequencies
<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>2.0</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">Hz</mml:mi></mml:math></inline-formula> for <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> waves and <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>1.2</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">Hz</mml:mi></mml:math></inline-formula> for
<inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> waves, since the dominant parts of the seismic energy are
around these frequencies for moderate crustal earthquakes
<xref ref-type="bibr" rid="bib1.bibx8 bib1.bibx36" id="paren.26"><named-content content-type="pre">e.g.,</named-content></xref>. The wavelengths of the
<inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave are approximately equal to those of the
<inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave in the same layers. To explore the
properties of the arrival-time difference <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="italic">δ</mml:mi><mml:msup><mml:mi>t</mml:mi><mml:mi>f</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula> in
Eq. (<xref ref-type="disp-formula" rid="Ch1.E6"/>), a Butterworth filter between
<inline-formula><mml:math display="inline"><mml:mn>0.001</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">Hz</mml:mi></mml:math></inline-formula> and 5.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">Hz</mml:mi></mml:math></inline-formula> is applied to more than 50
arbitrarily selected <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave seismograms recorded for 10
earthquakes with magnitudes between <inline-formula><mml:math display="inline"><mml:mn>2.08</mml:mn></mml:math></inline-formula> and
<inline-formula><mml:math display="inline"><mml:mn>3.99</mml:mn></mml:math></inline-formula>. Figure <xref ref-type="fig" rid="Ch1.F4"/> shows three such examples of
<inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs</mml:mtext></mml:msup></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs,f</mml:mtext></mml:msup></mml:mrow></mml:math></inline-formula> picked on raw and filtered
seismograms. In all our selected examples we find that the
differences <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="italic">δ</mml:mi><mml:msup><mml:mi>t</mml:mi><mml:mi>f</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula> are generally smaller than <inline-formula><mml:math display="inline"><mml:mn>0.08</mml:mn></mml:math></inline-formula> s, which
account for the combined effect of finite-frequency measurements,
noise, and picking inaccuracy. For a regional <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave
tomography as in this study, <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="italic">δ</mml:mi><mml:msup><mml:mi>t</mml:mi><mml:mi>f</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula> less than <inline-formula><mml:math display="inline"><mml:mn>0.08</mml:mn></mml:math></inline-formula> s has
very limited effect on the final images and can be safely viewed as
noise, which will also be confirmed in the checkerboard resolution
tests shown in the Supplement (Figs. S1 and S2). Similarly, <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="italic">δ</mml:mi><mml:msup><mml:mi>t</mml:mi><mml:mi>f</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula> can also be ignored for the <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave
seismograms. Therefore, we would rather use the existing SCSN catalog
of <inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs</mml:mtext></mml:msup></mml:mrow></mml:math></inline-formula> than hand-picking large number of
<inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs,f</mml:mtext></mml:msup></mml:mrow></mml:math></inline-formula> for this tomographic study. In addition, the
spacing of the uniform forward modeling grid is <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>z</mml:mi><mml:mo>=</mml:mo><mml:mn>0.2</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>. The time steps are chosen to be <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>0.0025</mml:mn></mml:mrow></mml:math></inline-formula> s for <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave simulations and <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>0.004</mml:mn></mml:mrow></mml:math></inline-formula> s for
<inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave modelings. These parameters guarantee the stability
condition of the high-order central difference method
<xref ref-type="bibr" rid="bib1.bibx40" id="paren.27"/>.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F3"><caption><p>Panel <bold>(a)</bold> shows source time function Eq. (<xref ref-type="disp-formula" rid="Ch1.E5"/>) with unit
amplitude <inline-formula><mml:math display="inline"><mml:mrow><mml:mi>A</mml:mi><mml:mo>=</mml:mo><mml:mn>1.0</mml:mn></mml:mrow></mml:math></inline-formula> and dominant frequency <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>2.0</mml:mn></mml:mrow></mml:math></inline-formula>. Panel <bold>(b)</bold> shows frequency
spectrum for the source time function in <bold>(a)</bold>. The purple line is at
5.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">Hz</mml:mi></mml:math></inline-formula>.</p></caption>
          <?xmltex \igopts{width=227.622047pt}?><graphic xlink:href="https://www.solid-earth.net/5/1169/2014/se-5-1169-2014-f03.png"/>

        </fig>

      <?xmltex \floatpos{t}?><fig id="Ch1.F4" specific-use="star"><caption><p><bold>(a–c)</bold> Three examples of observed <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave arrival time
picked on raw data obtained from the SCSN catalog (first row) and filtered
seismograms filtered between <inline-formula><mml:math display="inline"><mml:mn>0.001</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">Hz</mml:mi></mml:math></inline-formula> and 5.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">Hz</mml:mi></mml:math></inline-formula> (second
row). The brown lines denote the observed arrival-times <inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs</mml:mtext></mml:msup></mml:mrow></mml:math></inline-formula>
determined by data analysts, and the dashed purple lines are the possible
arrival-times <inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs,f</mml:mtext></mml:msup></mml:mrow></mml:math></inline-formula> manually picked on filtered seismograms.
Earthquake IDs (such as 11335706), <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi>L</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> magnitudes, and station names (such
as CI.CJM) are specified for each record. The observed arrival times on raw
data and on filtered seismograms are <bold>(a)</bold>
<inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs</mml:mtext></mml:msup><mml:mo>=</mml:mo><mml:mn>3.618</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">s</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs,f</mml:mtext></mml:msup><mml:mo>≈</mml:mo><mml:mn>3.558</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">s</mml:mi></mml:math></inline-formula>,
<bold>(b)</bold> <inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs</mml:mtext></mml:msup><mml:mo>=</mml:mo><mml:mn>6.558</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">s</mml:mi></mml:math></inline-formula>,
<inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs,f</mml:mtext></mml:msup><mml:mo>≈</mml:mo><mml:mn>6.508</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">s</mml:mi></mml:math></inline-formula>, and <bold>(c)</bold>
<inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs</mml:mtext></mml:msup><mml:mo>=</mml:mo><mml:mn>5.311</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">s</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs,f</mml:mtext></mml:msup><mml:mo>≈</mml:mo><mml:mn>5.261</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">s</mml:mi></mml:math></inline-formula>. The differences <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="italic">δ</mml:mi><mml:msup><mml:mi>t</mml:mi><mml:mi>f</mml:mi></mml:msup><mml:mo>=</mml:mo></mml:mrow></mml:math></inline-formula> are less than <inline-formula><mml:math display="inline"><mml:mn>0.06</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">s</mml:mi></mml:math></inline-formula>.</p></caption>
          <?xmltex \igopts{width=398.338583pt}?><graphic xlink:href="https://www.solid-earth.net/5/1169/2014/se-5-1169-2014-f04.png"/>

        </fig>

      <?xmltex \floatpos{t}?><fig id="Ch1.F5"><caption><p>Panels <bold>a</bold> and <bold>b</bold> show examples of travel-time sensitivity
kernels for the starting model <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> of <bold>(a)</bold> the direct <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave
(<italic>Pg</italic>) and <bold>(b)</bold> the direct <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave (<italic>Sg</italic>), which are
fully separated from other later phases. The star and the inverse triangle
indicate the earthquake at the depth of <inline-formula><mml:math display="inline"><mml:mn>3.14</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> and a recording
seismic station on the surface, respectively. The epicentral distance is
<inline-formula><mml:math display="inline"><mml:mn>3.75</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>. The dashed grey lines denote the velocity discontinuities
at the depth of 2.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> and 5.5 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>. Panels <bold>c</bold> and
<bold>d</bold> show the corresponding synthetic <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave seismograms
(black curves). The arrival times of the direct waves (<italic>Pg</italic> and
<italic>Sg</italic>) and reflected phases from the discontinuity at 5.5 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>
(<italic>Pr</italic> and <italic>Sr</italic>) are indicated by the blue and purple lines. The
red waveforms are the windowed and tapered seismograms used to compute the
travel-time sensitivity kernels of the direct arrivals shown <?xmltex \hack{\mbox\bgroup}?>in (<bold>a</bold> and <bold>b</bold>)<?xmltex \hack{\egroup}?>.</p></caption>
          <?xmltex \igopts{width=241.848425pt}?><graphic xlink:href="https://www.solid-earth.net/5/1169/2014/se-5-1169-2014-f05.png"/>

        </fig>

      <p>One of the main purposes of forward modeling is to compute the sensitivity
kernel as in Eq. (2). Prior to that, we need to determine the time window for
the first arriving <italic> P</italic>-phase or <italic> S</italic>-phase. Since the onset
time <inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>syn</mml:mtext></mml:msup></mml:mrow></mml:math></inline-formula> of a specific phase in an iterative model can be
calculated by using the combined ray and cross-correlation method and the
signal length (dependant on the source time Eq. 5) is about twice the
dominant period, the time window for the synthetic seismic phase is chosen to
be <inline-formula><mml:math display="inline"><mml:mrow><mml:mo>[</mml:mo><mml:msup><mml:mi>T</mml:mi><mml:mtext>syn</mml:mtext></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>T</mml:mi><mml:mtext>syn</mml:mtext></mml:msup><mml:mo>+</mml:mo><mml:mn>2.0</mml:mn><mml:mo>/</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula>. For the 1-D layered model with an
undulated Moho (<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula>) and at epicentral distances less than
100 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>, the first <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> or <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave arrival of a crustal earthquake
(depth greater than 3.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>) should be either the direct phase
<italic>Pg</italic> (<italic>Sg</italic>), head waves refracted from the velocity boundary at
the depth of 5.5 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>, or head waves <italic>P*</italic> (<italic>S*</italic>)
refracted from the Conrad discontinuity depending on the epicentral distance.
Accordingly, the travel-time sensitivity kernels for the first arrivals also
have different spatial variations. Figures <xref ref-type="fig" rid="Ch1.F5"/>a, b
and <xref ref-type="fig" rid="Ch1.F6"/>a, b show two typical sensitivity kernels for the
velocity model <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula>. For example, the kernels of <italic>Pg</italic> and
<italic>Sg</italic> waves at a distance of <inline-formula><mml:math display="inline"><mml:mn>3.75</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> for an earthquake at the
depth <inline-formula><mml:math display="inline"><mml:mn>3.14</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> (Fig. <xref ref-type="fig" rid="Ch1.F5"/>a and b) clearly display 2-D
cigar shapes <xref ref-type="bibr" rid="bib1.bibx42 bib1.bibx29" id="paren.28"><named-content content-type="pre">e.g.,</named-content></xref>. On the synthetic
seismograms (Fig. <xref ref-type="fig" rid="Ch1.F5"/>c and d), the direct <italic>Pg</italic>
(<italic>Sg</italic>) and the reflected phase <italic>Pr</italic> (<italic>Sr</italic>) from the
velocity boundary at 5.5 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> are distinguishable and almost totally
separated, which enables us to separate the direct arrivals and calculate
their kernels (Fig. <xref ref-type="fig" rid="Ch1.F5"/>a and b). The negative kernel values in
the first Fresnel zone indicate that a velocity decrease is required to delay
the synthetic arrival time <inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>syn</mml:mtext></mml:msup></mml:mrow></mml:math></inline-formula>. However, for the records of 2041
selected crustal events, it is only possible to separate the first arrival
from its coda waves on a very small fraction of synthetic seismograms. For
many synthetic seismograms, the first arrivals are closely followed or even
overlapped by other phases. For example, on the synthetic seismograms
generated by the same crustal earthquake but recorded at a distance of
<inline-formula><mml:math display="inline"><mml:mn>87.24</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> (Fig. <xref ref-type="fig" rid="Ch1.F6"/>c and d), the first arrival
<italic>Ph</italic> (<italic>Sh</italic>) refracted from the velocity boundary at
5.5 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> depth is sequentially overlapped by the direct arrival
<italic>Pg</italic> (<italic>Sg</italic>), the Conrad refracted phase <italic>P*</italic>
(<italic>S*</italic>), and the reflected wave <italic>Pr</italic> (<italic>Sr</italic>) from the
velocity boundary at 5.5 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> depth. It is difficult to separate these
phases because the later phases are in the time window of the first <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave
(<inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave) arrival. Therefore, the computed sensitivity kernels
(Fig. <xref ref-type="fig" rid="Ch1.F6"/>a and b) have significant values around the traveling
paths of all the phases that arrive within the first-arrival windows. This
feature is helpful for resolving multipathing problems which are common for
complex velocity structures <xref ref-type="bibr" rid="bib1.bibx27" id="paren.29"/>. Every <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave/<inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave
travel-time sensitivity kernel is also smoothed out by a Gaussian function
with the scaling length chosen to be the minimum <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave/<inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave wavelength
in the starting model <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> <xref ref-type="bibr" rid="bib1.bibx29 bib1.bibx39 bib1.bibx40" id="paren.30"/>.
Additionally, the sensitivity kernel
<inline-formula><mml:math display="inline"><mml:mrow><mml:mi>K</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>;</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mi mathvariant="normal">r</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mi mathvariant="normal">s</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and the relative
velocity perturbation <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="italic">δ</mml:mi><mml:mi>c</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo><mml:mo>/</mml:mo><mml:mi>c</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> are bilinearly
interpolated on the forward modeling grid <xref ref-type="bibr" rid="bib1.bibx40" id="paren.31"/> in this study.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F6"><caption><p>Panels <bold>a</bold> and <bold>b</bold> show examples of travel-time sensitivity
kernels for the starting model <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> of <bold>(a)</bold> the first arrival of
<inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> waves and <bold>(b)</bold> the first arrival of <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> waves. The
earthquake is the same one as in Fig. <xref ref-type="fig" rid="Ch1.F5"/> but the seismic
station is at an epicentral distance of <inline-formula><mml:math display="inline"><mml:mn>87.24</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>. In this case, the
first arrivals are seismic waves refracted from the <inline-formula><mml:math display="inline"><mml:mn>5.5</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>
discontinuity but overlapped by other later phases. The dashed grey lines
denote the velocity discontinuities at the depth 2.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> and
5.5 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>, and the Conrad discontinuity (<inline-formula><mml:math display="inline"><mml:mn>16.0</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>). Panels <bold>c</bold> and
<bold>d</bold> show the corresponding synthetic <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave seismograms
(black curves). The purple lines indicate the arrival times of the head waves
refracted from the discontinuity at the depth of 5.5 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>, the blue
lines denote the onset times of the direct waves (<italic>Pg</italic> and
<italic>Sg</italic>), the pink lines show the arrival times of the head waves
refracted by the Conrad discontinuity, and the brown lines denote the arrival
times of the reflected phases from the 5.5 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> discontinuity
(<italic>Pr</italic> and <italic>Sr</italic>). The red waveforms are the windowed and tapered
seismograms used to compute the travel-time sensitivity kernels of the first
arrivals shown in (<bold>a</bold> and <bold>b</bold>).</p></caption>
          <?xmltex \igopts{width=241.848425pt}?><graphic xlink:href="https://www.solid-earth.net/5/1169/2014/se-5-1169-2014-f06.png"/>

        </fig>

      <p>Once all the travel-time sensitivity kernels are calculated, smoothed, and
interpolated, we can invert for the relative velocity perturbation field
<inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="italic">δ</mml:mi><mml:mi>c</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo><mml:mo>/</mml:mo><mml:mi>c</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>. As discussed in paper I, the relative
velocity perturbation field <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="italic">δ</mml:mi><mml:mi>c</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo><mml:mo>/</mml:mo><mml:mi>c</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> at each forward
modeling grid node is linearly interpolated by its values at the eight
neighboring inversion grid nodes. Based on the data distribution as shown in
Fig. <xref ref-type="fig" rid="Ch1.F2"/>a, we setup the inversion grid in the study area
(Fig. <xref ref-type="fig" rid="Ch1.F2"/>b) with a horizontal grid spacing of 0.12<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula> at
the central potion and 0.15<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula> near the edges (Fig. <xref ref-type="fig" rid="Ch1.F2"/>b),
and seven vertical layers located at the depths of 1, 5, 10, 15, 21, 28, and
40 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>. The spacing of the chosen inversion grid is much larger than
that of the forward modeling grid. Additionally, the minimum wavelengths of both <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula>
and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> waves are approximately half of the minimum inversion grid size.
Generally speaking, at least four grid nodes per wavelength are needed to
fully capture the seismic wavefield with a finite-difference forward
modeling method <xref ref-type="bibr" rid="bib1.bibx49 bib1.bibx38" id="paren.32"><named-content content-type="pre">e.g.,</named-content></xref>. If an inverse algorithm
has a resolving ability at the scale of the wavelength <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">λ</mml:mi></mml:math></inline-formula>, such as
that of full waveform inversion methods <xref ref-type="bibr" rid="bib1.bibx45" id="paren.33"/>, the grid spacing
of the inversion grid should be at least 4 times that of the forward modeling
grid spacing. Since the theoretical resolving ability of WETST is at the
scale of <inline-formula><mml:math display="inline"><mml:msqrt><mml:mrow><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>L</mml:mi></mml:mrow></mml:msqrt></mml:math></inline-formula> (<inline-formula><mml:math display="inline"><mml:mi>L</mml:mi></mml:math></inline-formula> is the traveling distance)
<xref ref-type="bibr" rid="bib1.bibx45" id="paren.34"/>, the grid spacing of the inverse grid should be even
larger. Meanwhile, the resolution of the inversion results also relies on the
coverage of seismic data. Checkerboard resolution tests are good measures
on the resolving ability of an inverse algorithm with chosen seismic data and
model parameterization. Therefore, checkerboard resolution tests are
conducted to verify the chosen data, inversion grid, and the WETST algorithm
in Sect. 3. However, due to the demanding computational cost, we do not test the
limit of the inversion grid size in this study.</p>
</sec>
<sec id="Ch1.S2.SS3">
  <title>Inversion algorithm</title>
      <p>After the calculation of sensitivity kernels and the interpolation of
relative velocity perturbation <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="italic">δ</mml:mi><mml:mi>c</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo><mml:mo>/</mml:mo><mml:mi>c</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> on
inversion grid, tomographic Eq. (<xref ref-type="disp-formula" rid="Ch1.E1"/>) could be discretely
expressed as a linear system <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="bold-italic">b</mml:mi><mml:mo>=</mml:mo><mml:mtext mathvariant="bold">A</mml:mtext><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow></mml:math></inline-formula>, where
<inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="bold-italic">b</mml:mi><mml:mo>=</mml:mo><mml:mo>[</mml:mo><mml:msub><mml:mi>b</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:msub><mml:mo>]</mml:mo><mml:mrow><mml:mi>M</mml:mi><mml:mo>×</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> is the travel-time residual vector
(<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>b</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mi>T</mml:mi><mml:mi>m</mml:mi><mml:mtext>obs</mml:mtext></mml:msubsup><mml:mo>-</mml:mo><mml:msubsup><mml:mi>T</mml:mi><mml:mi>m</mml:mi><mml:mtext>syn</mml:mtext></mml:msubsup></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mi>m</mml:mi></mml:math></inline-formula> is the index for
a particular travel-time record), <inline-formula><mml:math display="inline"><mml:mrow><mml:mtext mathvariant="bold">A</mml:mtext><mml:mo>=</mml:mo><mml:mo>[</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mo>]</mml:mo><mml:mrow><mml:mi>M</mml:mi><mml:mo>×</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> is
the Fréchet matrix, and <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="bold-italic">X</mml:mi><mml:mo>=</mml:mo><mml:mo>[</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:msub><mml:mo>]</mml:mo><mml:mrow><mml:mi>N</mml:mi><mml:mo>×</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> is the
unknown velocity perturbation vector. Usually, the limited data
coverage deems this inversion an ill-posed problem, and
<inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="bold-italic">b</mml:mi><mml:mo>=</mml:mo><mml:mtext mathvariant="bold">A</mml:mtext><mml:mi mathvariant="bold-italic">X</mml:mi></mml:mrow></mml:math></inline-formula> is solved instead by minimizing the
following regularized objective function

                <disp-formula content-type="numbered" specific-use="align"><mml:math display="block"><mml:mtable displaystyle="true"><mml:mlabeledtr id="Ch1.E7"><mml:mtd/><mml:mtd/><mml:mtd><mml:mrow><mml:mi mathvariant="italic">χ</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold-italic">X</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mfrac><mml:mn mathvariant="normal">1</mml:mn><mml:mn mathvariant="normal">2</mml:mn></mml:mfrac><mml:mo>(</mml:mo><mml:mi mathvariant="bold">A</mml:mi><mml:mi mathvariant="bold-italic">X</mml:mi><mml:mo>-</mml:mo><mml:mi mathvariant="bold">b</mml:mi><mml:msup><mml:mo>)</mml:mo><mml:mi mathvariant="bold">T</mml:mi></mml:msup><mml:msubsup><mml:mi mathvariant="bold">C</mml:mi><mml:mi mathvariant="bold">d</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="bold">1</mml:mn></mml:mrow></mml:msubsup><mml:mo>(</mml:mo><mml:mi mathvariant="bold">A</mml:mi><mml:mi mathvariant="bold">X</mml:mi><mml:mo>-</mml:mo><mml:mi mathvariant="bold">b</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mlabeledtr><mml:mtr><mml:mtd/><mml:mtd><mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mi mathvariant="italic">ϵ</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mfrac><mml:msup><mml:mi mathvariant="bold-italic">X</mml:mi><mml:mi>T</mml:mi></mml:msup><mml:msubsup><mml:mi mathvariant="bold">C</mml:mi><mml:mi>m</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msubsup><mml:mi mathvariant="bold-italic">X</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mi mathvariant="italic">η</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mfrac><mml:msup><mml:mi mathvariant="bold-italic">X</mml:mi><mml:mi>T</mml:mi></mml:msup><mml:msup><mml:mi mathvariant="bold">D</mml:mi><mml:mi>T</mml:mi></mml:msup><mml:mi mathvariant="bold">D</mml:mi><mml:mi mathvariant="bold-italic">X</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>

            where <bold>D</bold> is a first derivative smoothing operator, and <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">ϵ</mml:mi></mml:math></inline-formula> and
<inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">η</mml:mi></mml:math></inline-formula> are the damping parameter and the smoothing parameter, respectively
<xref ref-type="bibr" rid="bib1.bibx32 bib1.bibx15 bib1.bibx26" id="paren.35"><named-content content-type="pre">e.g.,</named-content></xref>. <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="bold">C</mml:mi><mml:mi>d</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and
<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="bold">C</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> are the a prior data and model covariance matrices and
reflect the uncertainties in the data and the initial model. Here we assume
that both <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="bold">C</mml:mi><mml:mi>d</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="bold">C</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> are identity matrices.</p>
      <p>Either the LSQR solver or non-linear conjugate-gradient method can be used to
solve the optimization problem (Eq. 7) as discussed in paper I
<xref ref-type="bibr" rid="bib1.bibx23 bib1.bibx42 bib1.bibx40" id="paren.36"/>. We choose to use the LSQR solver in
this study. The solution of the minimization problem (<xref ref-type="disp-formula" rid="Ch1.E7"/>) can be
obtained by solving the equivalent linear system using the LSQR solver
<xref ref-type="bibr" rid="bib1.bibx26" id="paren.37"/>
            <disp-formula content-type="numbered" id="Ch1.E8"><mml:math display="block"><mml:mrow><mml:mfenced close=")" open="("><mml:mtable class="matrix" columnalign="center" framespacing="0em"><mml:mtr><mml:mtd><mml:mtext>A</mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mi mathvariant="italic">ϵ</mml:mi><mml:mtext mathvariant="bold">I</mml:mtext></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mi mathvariant="italic">η</mml:mi><mml:mtext mathvariant="bold">D</mml:mtext></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mfenced><mml:mi mathvariant="bold-italic">X</mml:mi><mml:mo>=</mml:mo><mml:mfenced close=")" open="("><mml:mtable class="matrix" columnalign="center" framespacing="0em"><mml:mtr><mml:mtd><mml:mi mathvariant="bold-italic">b</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mtext>0</mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mtext>0</mml:mtext></mml:mtd></mml:mtr></mml:mtable></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula>
          The choice of the damping and smoothing parameters involves some
degree of subjectivity. Analysis of the trade-off between the data
variance reduction and the model smoothness may help the selection of
optimal damping and smoothing parameters
<xref ref-type="bibr" rid="bib1.bibx12 bib1.bibx37" id="paren.38"/>. After the <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> models are updated,
the Poisson's ratio (<inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">σ</mml:mi></mml:math></inline-formula>) image can be determined based on the
relation
            <disp-formula content-type="numbered" id="Ch1.E9"><mml:math display="block"><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi mathvariant="italic">δ</mml:mi><mml:mi mathvariant="italic">σ</mml:mi></mml:mrow><mml:mi mathvariant="italic">σ</mml:mi></mml:mfrac></mml:mstyle><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mn mathvariant="normal">2</mml:mn><mml:msubsup><mml:mi>V</mml:mi><mml:mi>p</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup><mml:msubsup><mml:mi>V</mml:mi><mml:mi mathvariant="normal">s</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mrow><mml:mrow><mml:mfenced close=")" open="("><mml:msubsup><mml:mi>V</mml:mi><mml:mi>p</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup><mml:mo>-</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:msubsup><mml:mi>V</mml:mi><mml:mi mathvariant="normal">s</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mfenced><mml:mfenced close=")" open="("><mml:msubsup><mml:mi>V</mml:mi><mml:mi>p</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup><mml:mo>-</mml:mo><mml:msubsup><mml:mi>V</mml:mi><mml:mi mathvariant="normal">s</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mfenced></mml:mrow></mml:mfrac></mml:mstyle><mml:mfenced open="(" close=")"><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi mathvariant="italic">δ</mml:mi><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>-</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi mathvariant="italic">δ</mml:mi><mml:msub><mml:mi>V</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mstyle></mml:mfenced><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>
          which is derived from the relation between Poisson's ratio and
<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mi>V</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> ratio <xref ref-type="bibr" rid="bib1.bibx55" id="paren.39"/>
            <disp-formula content-type="numbered" id="Ch1.E10"><mml:math display="block"><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>=</mml:mo><mml:msqrt><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mn mathvariant="normal">2</mml:mn><mml:mo>(</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mo>-</mml:mo><mml:mi mathvariant="italic">σ</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mn mathvariant="normal">1</mml:mn><mml:mo>-</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mi mathvariant="italic">σ</mml:mi></mml:mrow></mml:mfrac></mml:mstyle></mml:msqrt><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula>
          Clearly, the reliability of the Poisson's ratio result depends on the
accuracy of both recovered <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> structures.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F7"><caption><p>Trade-off analysis of data variance <inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula> and model
variance <inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula> for damping parameters <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">ϵ</mml:mi></mml:math></inline-formula> ranging from
0.1 (the rightmost red circle in each panel) to 2.0 (the leftmost red circle)
with an interval of <inline-formula><mml:math display="inline"><mml:mn>0.02</mml:mn></mml:math></inline-formula>. Panels <bold>(a–c)</bold> show the trade-off curves
of <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave checkerboard resolution tests for models <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">4</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> from the
1st, 2nd and 3rd iteration. Panels <bold>(d–f)</bold> are for <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave
checkerboard tests. The blue star in each panel represents the values of
model variance and data variance for the optimal damping parameter <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">ϵ</mml:mi></mml:math></inline-formula>
(values indicated in the same panel) for <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave or <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave at each
iteration. The value of the unitless model variance <inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi mathvariant="italic">σ</mml:mi><mml:mi>m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula> is at the
scale of <inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">4</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>.</p></caption>
          <?xmltex \igopts{width=241.848425pt}?><graphic xlink:href="https://www.solid-earth.net/5/1169/2014/se-5-1169-2014-f07.png"/>

        </fig>

</sec>
</sec>
<sec id="Ch1.S3">
  <title>Checkerboard resolution tests</title>
      <p>We are ready to conduct wave-equation-based travel-time seismic
tomography (WETST) based on the selected data, model parameterization,
and inversion scheme laid out in previous sections. Prior to showing
the tomographic results, we first examine the validity and reliability
of this tomographic inversion based on checkerboard resolution
tests. The checkerboard model is composed of alternating positive and
negative velocity anomalies of <inline-formula><mml:math display="inline"><mml:mrow><mml:mn mathvariant="normal">5</mml:mn><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mi mathvariant="italic">%</mml:mi></mml:mrow></mml:math></inline-formula> on the 3-D inversion grid
nodes. Synthetic data are calculated for the checkerboard model based
on 2-D finite-difference modeling. The starting velocity model is <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula>,
i.e., the 1-D layered model with an undulated Moho as introduced in
Sect. 2.2. The checkerboard patterns for both <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> and
<inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave velocity structures will be recovered through
iterative procedures based on WETST.</p>
<sec id="Ch1.S3.SS1">
  <title>Data variance vs. model variance trade-off analysis</title>
      <p>In order to obtain the discrete velocity perturbation <inline-formula><mml:math display="inline"><mml:mi mathvariant="bold-italic">X</mml:mi></mml:math></inline-formula> in
Eq. (<xref ref-type="disp-formula" rid="Ch1.E8"/>) at each iteration, the damping parameter
<inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">ϵ</mml:mi></mml:math></inline-formula> and the smoothing parameter <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">η</mml:mi></mml:math></inline-formula> should be determined
beforehand. In practice, these two parameters can be chosen via
a trade-off analysis of data variance <inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula> and model
variance <inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula> <xref ref-type="bibr" rid="bib1.bibx50" id="paren.40"/>. For the sake of
computational efficiency, the unbiased data variance is approximated
by
            <disp-formula content-type="numbered" id="Ch1.E11"><mml:math display="block"><mml:mrow><mml:msubsup><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup><mml:mo>≈</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mn mathvariant="normal">1</mml:mn><mml:mrow><mml:mi>M</mml:mi><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:mfrac></mml:mstyle><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mi>M</mml:mi></mml:munderover><mml:msup><mml:mfenced close=")" open="("><mml:msubsup><mml:mi>T</mml:mi><mml:mi>i</mml:mi><mml:mtext>obs</mml:mtext></mml:msubsup><mml:mo>-</mml:mo><mml:msubsup><mml:mi>T</mml:mi><mml:mi>i</mml:mi><mml:mtext>syn</mml:mtext></mml:msubsup><mml:mo>-</mml:mo><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mi>N</mml:mi></mml:munderover><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>X</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:mover accent="true"><mml:mi>d</mml:mi><mml:mo mathvariant="normal">¯</mml:mo></mml:mover></mml:mfenced><mml:mn mathvariant="normal">2</mml:mn></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>
          where the data average <inline-formula><mml:math display="inline"><mml:mover accent="true"><mml:mi>d</mml:mi><mml:mo mathvariant="normal">¯</mml:mo></mml:mover></mml:math></inline-formula> is estimated as
            <disp-formula content-type="numbered" id="Ch1.E12"><mml:math display="block"><mml:mrow><mml:mover accent="true"><mml:mi>d</mml:mi><mml:mo mathvariant="normal">¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mn mathvariant="normal">1</mml:mn><mml:mi>M</mml:mi></mml:mfrac></mml:mstyle><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mi>M</mml:mi></mml:munderover><mml:mfenced open="(" close=")"><mml:msubsup><mml:mi>T</mml:mi><mml:mi>i</mml:mi><mml:mtext>obs</mml:mtext></mml:msubsup><mml:mo>-</mml:mo><mml:msubsup><mml:mi>T</mml:mi><mml:mi>i</mml:mi><mml:mtext>syn</mml:mtext></mml:msubsup><mml:mo>-</mml:mo><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mi>N</mml:mi></mml:munderover><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>X</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula>
          The unbiased model variance <inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula> is calculated using
the formula
            <disp-formula content-type="numbered" id="Ch1.E13"><mml:math display="block"><mml:mrow><mml:msubsup><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mn mathvariant="normal">1</mml:mn><mml:mrow><mml:mi>N</mml:mi><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:mfrac></mml:mstyle><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mi>N</mml:mi></mml:munderover><mml:mo>(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:mover accent="true"><mml:mi>X</mml:mi><mml:mo mathvariant="normal">¯</mml:mo></mml:mover><mml:msup><mml:mo>)</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>
          with <inline-formula><mml:math display="inline"><mml:mrow><mml:mover accent="true"><mml:mi>X</mml:mi><mml:mo mathvariant="normal">¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mi>N</mml:mi></mml:msubsup><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>/</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:math></inline-formula> is the mean of <inline-formula><mml:math display="inline"><mml:mi mathvariant="bold-italic">X</mml:mi></mml:math></inline-formula>. The
trade-off analysis tries to find optimal damping and smoothing
parameters that reduce most of the data variance without giving rise
to too large model variance <xref ref-type="bibr" rid="bib1.bibx15 bib1.bibx50" id="paren.41"/>. For the
checkerboard resolution tests, we search the damping parameter
<inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">ϵ</mml:mi></mml:math></inline-formula> in the range <inline-formula><mml:math display="inline"><mml:mrow><mml:mo>[</mml:mo><mml:mn>0.1</mml:mn><mml:mo>,</mml:mo><mml:mn>2.0</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula> with a step of <inline-formula><mml:math display="inline"><mml:mn>0.02</mml:mn></mml:math></inline-formula>, but set
the smoothing parameter as <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="italic">η</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:mrow></mml:math></inline-formula> at each iteration to reflect the
knowledge that the inverted structures are not smooth and have
perturbations of opposite signs at neighboring
nodes. Figure <xref ref-type="fig" rid="Ch1.F7"/> shows the trade-off curves for both
<inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave checkerboard resolution tests at
the first three iterations. Based on the <inline-formula><mml:math display="inline"><mml:mi>L</mml:mi></mml:math></inline-formula> curve method
<xref ref-type="bibr" rid="bib1.bibx3 bib1.bibx44 bib1.bibx50" id="paren.42"><named-content content-type="pre">e.g.,</named-content></xref>, we choose the
optimal damping parameter <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">ϵ</mml:mi></mml:math></inline-formula> for <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave or
<inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave test at each iteration near the corner of the
corresponding trade-off curve. For example, to obtain the
<inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave velocity model <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> from the starting model
<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula>, the optimal damping parameter in Eq. (<xref ref-type="disp-formula" rid="Ch1.E8"/>)
is chosen as <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="italic">ϵ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.42</mml:mn></mml:mrow></mml:math></inline-formula> which gives the data variance
<inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>1.571</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">4</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow></mml:math></inline-formula> and model
variance <inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>11.59</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">4</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>
(Fig. <xref ref-type="fig" rid="Ch1.F7"/>a). Note that the model variance is calculated
with respect to the model in the previous iteration. Since the data
variance is significantly reduced from model <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> to
<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">4</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> and the value of the data variance in model
<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">4</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> is very small for either <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave or
<inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave checkerboard test (Fig. <xref ref-type="fig" rid="Ch1.F7"/>), we stop
the iteration procedure at the fourth model <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">4</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula>.</p>

<?xmltex \floatpos{t}?><table-wrap id="Ch1.T2" specific-use="star"><caption><p>Structural similarity indices (SSIM) <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">ζ</mml:mi></mml:math></inline-formula> between the
checkerboard models and the iteratively updated inversion results (<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> to
<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">4</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula>, Figs. <xref ref-type="fig" rid="Ch1.F8"/> and <xref ref-type="fig" rid="Ch1.F9"/>) at seven different depths
for <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave checkerboard resolution tests.</p></caption><oasis:table frame="topbot"><oasis:tgroup cols="8">
     <oasis:colspec colnum="1" colname="col1" align="left"/>
     <oasis:colspec colnum="2" colname="col2" align="left"/>
     <oasis:colspec colnum="3" colname="col3" align="left"/>
     <oasis:colspec colnum="4" colname="col4" align="left"/>
     <oasis:colspec colnum="5" colname="col5" align="left"/>
     <oasis:colspec colnum="6" colname="col6" align="left"/>
     <oasis:colspec colnum="7" colname="col7" align="left"/>
     <oasis:colspec colnum="8" colname="col8" align="left"/>
     <oasis:thead>
       <oasis:row rowsep="1">  
         <oasis:entry colname="col1">Depth</oasis:entry>  
         <oasis:entry colname="col2">1.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula></oasis:entry>  
         <oasis:entry colname="col3">5.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula></oasis:entry>  
         <oasis:entry colname="col4">10.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula></oasis:entry>  
         <oasis:entry colname="col5">15.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula></oasis:entry>  
         <oasis:entry colname="col6">21.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula></oasis:entry>  
         <oasis:entry colname="col7">28.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula></oasis:entry>  
         <oasis:entry colname="col8">40.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula></oasis:entry>
       </oasis:row>
     </oasis:thead>
     <oasis:tbody>
       <oasis:row>  
         <oasis:entry colname="col1"><inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave: model 2</oasis:entry>  
         <oasis:entry colname="col2">0.8711</oasis:entry>  
         <oasis:entry colname="col3">0.9175</oasis:entry>  
         <oasis:entry colname="col4">0.9205</oasis:entry>  
         <oasis:entry colname="col5">0.7437</oasis:entry>  
         <oasis:entry colname="col6">0.6321</oasis:entry>  
         <oasis:entry colname="col7">0.4186</oasis:entry>  
         <oasis:entry colname="col8">0.5026</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1"><inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave: model 3</oasis:entry>  
         <oasis:entry colname="col2">0.8569</oasis:entry>  
         <oasis:entry colname="col3">0.9285</oasis:entry>  
         <oasis:entry colname="col4">0.9300</oasis:entry>  
         <oasis:entry colname="col5">0.8941</oasis:entry>  
         <oasis:entry colname="col6">0.7343</oasis:entry>  
         <oasis:entry colname="col7">0.4430</oasis:entry>  
         <oasis:entry colname="col8">0.5013</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1"><inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave: model 4</oasis:entry>  
         <oasis:entry colname="col2">0.9044</oasis:entry>  
         <oasis:entry colname="col3">0.9402</oasis:entry>  
         <oasis:entry colname="col4">0.9407</oasis:entry>  
         <oasis:entry colname="col5">0.9225</oasis:entry>  
         <oasis:entry colname="col6">0.7882</oasis:entry>  
         <oasis:entry colname="col7">0.4674</oasis:entry>  
         <oasis:entry colname="col8">0.5013</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1"><inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave: model 2</oasis:entry>  
         <oasis:entry colname="col2">0.8831</oasis:entry>  
         <oasis:entry colname="col3">0.9206</oasis:entry>  
         <oasis:entry colname="col4">0.9206</oasis:entry>  
         <oasis:entry colname="col5">0.7767</oasis:entry>  
         <oasis:entry colname="col6">0.6652</oasis:entry>  
         <oasis:entry colname="col7">0.3998</oasis:entry>  
         <oasis:entry colname="col8">0.5052</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1"><inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave: model 3</oasis:entry>  
         <oasis:entry colname="col2">0.8541</oasis:entry>  
         <oasis:entry colname="col3">0.9245</oasis:entry>  
         <oasis:entry colname="col4">0.9279</oasis:entry>  
         <oasis:entry colname="col5">0.8901</oasis:entry>  
         <oasis:entry colname="col6">0.7764</oasis:entry>  
         <oasis:entry colname="col7">0.4347</oasis:entry>  
         <oasis:entry colname="col8">0.5068</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1"><inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave: model 4</oasis:entry>  
         <oasis:entry colname="col2">0.9047</oasis:entry>  
         <oasis:entry colname="col3">0.9389</oasis:entry>  
         <oasis:entry colname="col4">0.9416</oasis:entry>  
         <oasis:entry colname="col5">0.9133</oasis:entry>  
         <oasis:entry colname="col6">0.8199</oasis:entry>  
         <oasis:entry colname="col7">0.4614</oasis:entry>  
         <oasis:entry colname="col8">0.5060</oasis:entry>
       </oasis:row>
     </oasis:tbody>
   </oasis:tgroup>

</oasis:table></table-wrap>

<?xmltex \floatpos{t}?><table-wrap id="Ch1.T3"><caption><p>The root mean square (rms) values of <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave
travel-time residuals in iteratively updated models.</p></caption><oasis:table frame="topbot"><oasis:tgroup cols="5">
     <oasis:colspec colnum="1" colname="col1" align="left"/>
     <oasis:colspec colnum="2" colname="col2" align="left"/>
     <oasis:colspec colnum="3" colname="col3" align="left"/>
     <oasis:colspec colnum="4" colname="col4" align="left"/>
     <oasis:colspec colnum="5" colname="col5" align="left"/>
     <oasis:thead>
       <oasis:row rowsep="1">  
         <oasis:entry colname="col1">rms</oasis:entry>  
         <oasis:entry colname="col2">Model 1</oasis:entry>  
         <oasis:entry colname="col3">Model 2</oasis:entry>  
         <oasis:entry colname="col4">Model 3</oasis:entry>  
         <oasis:entry colname="col5">Model 4</oasis:entry>
       </oasis:row>
     </oasis:thead>
     <oasis:tbody>
       <oasis:row>  
         <oasis:entry colname="col1"><inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave</oasis:entry>  
         <oasis:entry colname="col2">0.2540</oasis:entry>  
         <oasis:entry colname="col3">0.1928</oasis:entry>  
         <oasis:entry colname="col4">0.1754</oasis:entry>  
         <oasis:entry colname="col5">0.1661</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1"><inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave</oasis:entry>  
         <oasis:entry colname="col2">0.4724</oasis:entry>  
         <oasis:entry colname="col3">0.3543</oasis:entry>  
         <oasis:entry colname="col4">0.3196</oasis:entry>  
         <oasis:entry colname="col5">0.3043</oasis:entry>
       </oasis:row>
     </oasis:tbody>
   </oasis:tgroup>

</oasis:table></table-wrap>

</sec>
<sec id="Ch1.S3.SS2">
  <title>Resolution results</title>
      <p>Figures <xref ref-type="fig" rid="Ch1.F8"/> and <xref ref-type="fig" rid="Ch1.F9"/> show the iterative results of
checkerboard tests at five representative layers in the crust for the
<inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave velocity (<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>) and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave velocity (<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>) structures,
respectively. Generally speaking, the checkerboard patterns are well resolved
by WETST in the source area of the Landers earthquake. This indicates that
both <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave data coverages are adequate enough, and the
tomographic results inverted based on these data are reliable and can be used
for further interpretation. More specifically, the checkerboard patterns at
the five layers are almost recovered even at the first iteration
(Fig. <xref ref-type="fig" rid="Ch1.F8"/>a–e and <xref ref-type="fig" rid="Ch1.F9"/>a–e), and the subsequent
iterations only slightly refine the models (Fig. <xref ref-type="fig" rid="Ch1.F8"/>f–o
and <xref ref-type="fig" rid="Ch1.F9"/>f–o). For both <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave tests, WETST has
higher resolution in the upper crust (0–5.5 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>) and middle crust
(5.5–16.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>) than that in the lower crust (<inline-formula><mml:math display="inline"><mml:mrow><mml:mo>&gt;</mml:mo><mml:mn>16.0</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>).
This may be due to two main reasons. First, as most of the 2041 earthquakes
used in this study are located above <inline-formula><mml:math display="inline"><mml:mn>20.0</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>
(Fig. <xref ref-type="fig" rid="Ch1.F2"/>a), the inversion grid nodes in the upper and middle
crust are sampled by more data than those in the lower crust, which provides
better constraints to the anomalies in the upper and middle crust. Secondly,
the inversion grid nodes in the lower crust are mainly covered by travel-time
sensitivity kernels for first arrivals at long epicentral distances as shown
in Fig. <xref ref-type="fig" rid="Ch1.F6"/>. The resolving ability of the travel-time data is
proportional to the width of the first Fresnel zone proportional to
<inline-formula><mml:math display="inline"><mml:msqrt><mml:mrow><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>L</mml:mi></mml:mrow></mml:msqrt></mml:math></inline-formula>, where <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">λ</mml:mi></mml:math></inline-formula> is the wavelength and <inline-formula><mml:math display="inline"><mml:mi>L</mml:mi></mml:math></inline-formula> is the
traveling distance <xref ref-type="bibr" rid="bib1.bibx48 bib1.bibx45" id="paren.43"><named-content content-type="pre">e.g.,</named-content></xref>. A long traveling
distance would result in relatively low resolution. In addition, the edges of
the model range are likely to have poor resolution due to the lack of well
crisscrossed kernels therein.</p>
      <p>To further investigate the recovery ability of our tomographic method
WETST, we calculate the structural similarity (SSIM) index <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">ζ</mml:mi></mml:math></inline-formula>
between the inverted model and the input checkerboard model
<xref ref-type="bibr" rid="bib1.bibx36 bib1.bibx37" id="paren.44"/>. The SSIM index <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">ζ</mml:mi></mml:math></inline-formula> between two velocity
(or other positive physical parameter) models A and B is defined as
            <disp-formula content-type="numbered" id="Ch1.E14"><mml:math display="block"><mml:mrow><mml:mi mathvariant="italic">ζ</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="normal">A</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">B</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mn mathvariant="normal">2</mml:mn><mml:msub><mml:mi mathvariant="italic">μ</mml:mi><mml:mi mathvariant="normal">A</mml:mi></mml:msub><mml:msub><mml:mi mathvariant="italic">μ</mml:mi><mml:mi mathvariant="normal">B</mml:mi></mml:msub><mml:msub><mml:mi mathvariant="italic">σ</mml:mi><mml:mtext>AB</mml:mtext></mml:msub></mml:mrow><mml:mrow><mml:mfenced close=")" open="("><mml:msubsup><mml:mi mathvariant="italic">μ</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mi mathvariant="italic">μ</mml:mi><mml:mi mathvariant="normal">B</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mfenced><mml:mfenced close=")" open="("><mml:msubsup><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">A</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">B</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mfenced></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>+</mml:mo><mml:mn>0.5</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>
          where <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="italic">μ</mml:mi><mml:mi mathvariant="normal">A</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="italic">μ</mml:mi><mml:mi mathvariant="normal">B</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">A</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>,
<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">B</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, and <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="italic">σ</mml:mi><mml:mtext>AB</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> are the average of A, average of
B, variance of A, variance of B, and covariance of A and B, respectively. The
<inline-formula><mml:math display="inline"><mml:mn>0.5</mml:mn></mml:math></inline-formula> is added to ensure that SSIM index <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">ζ</mml:mi></mml:math></inline-formula> is in the range <inline-formula><mml:math display="inline"><mml:mrow><mml:mo>[</mml:mo><mml:mn>0.0</mml:mn><mml:mo>,</mml:mo><mml:mn>1.0</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula>,
and it is 1.0 only when A and B are identical <xref ref-type="bibr" rid="bib1.bibx36" id="paren.45"/>.
Table <xref ref-type="table" rid="Ch1.T2"/> shows the SSIM indices between the iteratively recovered
<inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave velocity models (<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> to <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">4</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula>,
Figs. <xref ref-type="fig" rid="Ch1.F8"/> and <xref ref-type="fig" rid="Ch1.F9"/>) and the input checkerboard models
at seven vertical layers. It can be observed that the SSIM indices at depths less than <inline-formula><mml:math display="inline"><mml:mn>21.0</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> generally approach 1.0 through the
iterations for both <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave tests. The recovery rates of the
final <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave velocity and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave velocity models <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">4</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> are above 0.9 at
the depths less than <inline-formula><mml:math display="inline"><mml:mn>21.0</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> and greater than <inline-formula><mml:math display="inline"><mml:mn>0.78</mml:mn></mml:math></inline-formula> at the depth
of <inline-formula><mml:math display="inline"><mml:mn>21.0</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>, again indicating that the heterogeneities from the
surface to the depth of <inline-formula><mml:math display="inline"><mml:mn>21.0</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> can be well resolved in this study. However, the SSIM indices at the depths 28 and 40 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> are only around 0.5,
implying decreased resolution in the lowermost crust and the uppermost
mantle. It is worth noting that the SSIM indices at 1.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> are
smaller than those at 5.0, <inline-formula><mml:math display="inline"><mml:mn>10.0</mml:mn></mml:math></inline-formula>, and <inline-formula><mml:math display="inline"><mml:mn>15.0</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>, which is probably
caused by the better crisscrossing of the travel-time sensitivity kernels of
earthquakes below 3.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> at 5.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math display="inline"><mml:mn>10.0</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>, and
<inline-formula><mml:math display="inline"><mml:mn>15.0</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> depths than that at 1.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> depth
(Figs. <xref ref-type="fig" rid="Ch1.F5"/> and <xref ref-type="fig" rid="Ch1.F6"/>). The generally well-recovered
<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> structures in our checkerboard resolution tests also imply
that reliable Poisson's ratio structures can be derived from this
tomographic study. We have also conducted other checkerboard resolution tests
for noise data, as summarized in the Supplement. All these resolution tests
give us confidence that WETST should be able to generate high-resolution
tomographic results for the source area of the Landers earthquake.</p>
</sec>
</sec>
<sec id="Ch1.S4">
  <title>Tomographic inversions</title>
<sec id="Ch1.S4.SS1">
  <title>Resolution parameters and models evaluation</title>
      <p>The optimal regularization parameters <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">ϵ</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">η</mml:mi></mml:math></inline-formula> should be
determined to update the tomographic models at each iteration, similar
to those in the checkerboard resolution tests. In this case, we search
the optimal damping parameter <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">ϵ</mml:mi></mml:math></inline-formula> in the range <inline-formula><mml:math display="inline"><mml:mrow><mml:mo>[</mml:mo><mml:mn mathvariant="normal">6</mml:mn><mml:mo>,</mml:mo><mml:mn>40</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula> with an
interval of 1 and the optimal smoothing parameter <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">η</mml:mi></mml:math></inline-formula> over <inline-formula><mml:math display="inline"><mml:mrow><mml:mo>[</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mo>,</mml:mo><mml:mn>100</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula>
at a step of 2. In the searching procedure, we first set the smoothing
parameter <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="italic">η</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:mrow></mml:math></inline-formula> and find the optimal damping parameter <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">ϵ</mml:mi></mml:math></inline-formula>
based on the <inline-formula><mml:math display="inline"><mml:mi>L</mml:mi></mml:math></inline-formula> curve method. With the optimal damping parameter
<inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">ϵ</mml:mi></mml:math></inline-formula>, we then determine the optimal smoothing parameter <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">η</mml:mi></mml:math></inline-formula>
in the searching region. For both <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave
tomographic inversions, Fig. <xref ref-type="fig" rid="Ch1.F10"/> shows the trade-off
analysis of data variance <inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula> and model variance
<inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula> along with different damping and smoothing
parameters throughout the iterations. The optimal damping and
smoothing parameters are also indicated in
Fig. <xref ref-type="fig" rid="Ch1.F10"/>. After each model update, we compute the
root mean square (rms) value of the travel-time residuals using the
formula
            <disp-formula content-type="numbered" id="Ch1.E15"><mml:math display="block"><mml:mrow><mml:mtext>rms</mml:mtext><mml:mspace width="0.25em" linebreak="nobreak"/><mml:mo>=</mml:mo><mml:msqrt><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mn mathvariant="normal">1</mml:mn><mml:mi>M</mml:mi></mml:mfrac></mml:mstyle><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mi>M</mml:mi></mml:munderover><mml:msup><mml:mfenced close=")" open="("><mml:msubsup><mml:mi>T</mml:mi><mml:mi>i</mml:mi><mml:mtext>obs</mml:mtext></mml:msubsup><mml:mo>-</mml:mo><mml:msubsup><mml:mi>T</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi></mml:mrow><mml:mtext>syn</mml:mtext></mml:msubsup></mml:mfenced><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow></mml:msqrt><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>
          where <inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi>T</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi></mml:mrow><mml:mtext>syn</mml:mtext></mml:msubsup></mml:mrow></mml:math></inline-formula> is the arrival time of the <inline-formula><mml:math display="inline"><mml:mi>i</mml:mi></mml:math></inline-formula>-th record
in model <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>. Table <xref ref-type="table" rid="Ch1.T3"/> shows the values of
rms. For both <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave results, we can find
that rms monochronically decreases from <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> to
<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">4</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> .</p>
      <p>Figure <xref ref-type="fig" rid="Ch1.F11"/> further shows the distributions of
<inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave (Fig. <xref ref-type="fig" rid="Ch1.F11"/>a–c) and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave
(Fig. <xref ref-type="fig" rid="Ch1.F11"/>d–f) travel-time residuals
<inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs</mml:mtext></mml:msup><mml:mo>-</mml:mo><mml:msup><mml:mi>T</mml:mi><mml:mtext>syn</mml:mtext></mml:msup></mml:mrow></mml:math></inline-formula> in models
<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">4</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula>. It is clear that travel-time residuals
gradually become more centered around 0.0 s over iterations,
indicating an overall reduction in total travel-time misfit. Since
there is no significant decrease in rms (Table <xref ref-type="table" rid="Ch1.T3"/>) from <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">3</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> to <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">4</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula>, we
stop our iteration at the fourth model for both <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave and
<inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave inversions, and <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">4</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> is viewed as the final
tomographic model used for interpretations in the following sections.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F8" specific-use="star"><caption><p>Iterative results <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> <bold>(a–e)</bold>, <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">3</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> <bold>(f–j)</bold>,
and <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">4</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> <bold>(k–o)</bold> of a checkerboard resolution test for <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave
velocity structure at five representative depth layers (1.0, 5.0, <inline-formula><mml:math display="inline"><mml:mn>10.0</mml:mn></mml:math></inline-formula>,
<inline-formula><mml:math display="inline"><mml:mn>15.0</mml:mn></mml:math></inline-formula>, and <inline-formula><mml:math display="inline"><mml:mn>21.0</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>). Red and blue colors denote low- and high-velocity perturbations, respectively. The velocity perturbation in percentage
scale is shown at the right hand side. The stars denote the epicentral
locations of the Landers, the Joshua Tree, and the Big Bear earthquakes.</p></caption>
          <?xmltex \igopts{width=341.433071pt}?><graphic xlink:href="https://www.solid-earth.net/5/1169/2014/se-5-1169-2014-f08.png"/>

        </fig>

      <?xmltex \floatpos{t}?><fig id="Ch1.F9" specific-use="star"><caption><p>The same as Fig. <xref ref-type="fig" rid="Ch1.F8"/> but for <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave velocity
structure.</p></caption>
          <?xmltex \igopts{width=341.433071pt}?><graphic xlink:href="https://www.solid-earth.net/5/1169/2014/se-5-1169-2014-f09.png"/>

        </fig>

</sec>
<sec id="Ch1.S4.SS2">
  <title>Tomographic images</title>
      <p>We present iteratively updated map views of <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> (Fig. <xref ref-type="fig" rid="Ch1.F12"/>)
and <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> (Fig. <xref ref-type="fig" rid="Ch1.F13"/>) models at five representative depths for
the Landers earthquake area. It can be observed that the general
patterns of <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> revealed by models
<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">4</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> are almost the same, with only slight
increase in the amplitudes of velocity anomalies over iterations
(Figs. <xref ref-type="fig" rid="Ch1.F12"/> and <xref ref-type="fig" rid="Ch1.F13"/>). This is consistent with
the significant rms reduction from <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> to <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula>,
and minor reduction in the following updates, as shown in
Table <xref ref-type="table" rid="Ch1.T3"/>. However, it should be also noted that velocity
anomalies near the boundaries of the study area become more clear over
iterations, which agrees with the checkerboard resolution tests
showing increased recovery from <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> to <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">4</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula>
(Table <xref ref-type="table" rid="Ch1.T2"/>), especially in the boundary regions
(Figs. <xref ref-type="fig" rid="Ch1.F8"/> and <xref ref-type="fig" rid="Ch1.F9"/>). These results imply the
necessity of iteratively improving the velocity models, even though
the patterns of velocity anomalies could be almost recovered in the
first iteration based on WETST with the LSQR solver.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F10" specific-use="star"><caption><p>Trade-off analysis of data variance <inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula> and model
variance <inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula> for 35 damping values equally in <inline-formula><mml:math display="inline"><mml:mrow><mml:mo>[</mml:mo><mml:mn>6.0</mml:mn><mml:mo>,</mml:mo><mml:mn>40.0</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula>
and 50 smoothing values over <inline-formula><mml:math display="inline"><mml:mrow><mml:mo>[</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mo>,</mml:mo><mml:mn>100.0</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula> with an interval of 2.0 at each
iteration to obtain <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave <bold>(a–f)</bold> and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave <bold>(g–l)</bold>
models <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">4</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula>. By setting the smoothing parameter <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="italic">η</mml:mi><mml:mo>=</mml:mo><mml:mn>0.0</mml:mn></mml:mrow></mml:math></inline-formula>, the
optimal damping parameter <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">ϵ</mml:mi></mml:math></inline-formula> is first determined based on the <inline-formula><mml:math display="inline"><mml:mi>L</mml:mi></mml:math></inline-formula>
curve method as shown in <bold>(a)</bold>, <bold>(c)</bold>, <bold>(e)</bold>,
<bold>(g)</bold>, <bold>(i)</bold>, and <bold>(k)</bold> at each iteration. The purple
stars highlight the values of data variance <inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula> and model
variance <inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula> calculated with the optimal damping
parameters. The optimal smoothing parameter <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">η</mml:mi></mml:math></inline-formula> is then determined with
the corresponding optimal damping parameter also based on the <inline-formula><mml:math display="inline"><mml:mi>L</mml:mi></mml:math></inline-formula> curve
method in <bold>(b)</bold>, <bold>(d)</bold>, <bold>(f)</bold>, <bold>(h)</bold>,
<bold>(j)</bold>, and <bold>(l)</bold>. The blue stars are at the crosses determined
by the values of data variance <inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula> and model variance
<inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula> calculated with the optimal damping and smoothing
parameters. The value of the unitless model variance <inline-formula><mml:math display="inline"><mml:mrow><mml:msubsup><mml:mi mathvariant="italic">σ</mml:mi><mml:mi>m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula> is at the
scale of <inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">4</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>.</p></caption>
          <?xmltex \igopts{width=369.885827pt}?><graphic xlink:href="https://www.solid-earth.net/5/1169/2014/se-5-1169-2014-f10.png"/>

        </fig>

      <?xmltex \floatpos{t}?><fig id="Ch1.F11" specific-use="star"><caption><p><inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave <bold>(a–c)</bold> and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave <bold>(d–f)</bold> travel-time
residuals <inline-formula><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mtext>obs</mml:mtext></mml:msup><mml:mo>-</mml:mo><mml:msup><mml:mi>T</mml:mi><mml:mtext>syn</mml:mtext></mml:msup></mml:mrow></mml:math></inline-formula> in models <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">3</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> (blue
histograms) compared to those in model <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">4</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> (red histograms). The mean
<inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">μ</mml:mi></mml:math></inline-formula> and standard deviation <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">σ</mml:mi></mml:math></inline-formula> of the travel-time residuals in models
<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">3</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> (corresponding to blue histograms) are shown in each panel. In
<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">4</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula>, the mean and standard deviation values of the <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave travel-time
residuals are <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="italic">μ</mml:mi><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mn>0.0457</mml:mn></mml:mrow></mml:math></inline-formula> s and <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="italic">σ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.1596</mml:mn></mml:mrow></mml:math></inline-formula> s, and those of the
<inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave travel-time residuals are <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="italic">μ</mml:mi><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mn>0.0652</mml:mn></mml:mrow></mml:math></inline-formula> s and <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="italic">σ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.2972</mml:mn></mml:mrow></mml:math></inline-formula> s.</p></caption>
          <?xmltex \igopts{width=369.885827pt}?><graphic xlink:href="https://www.solid-earth.net/5/1169/2014/se-5-1169-2014-f11.png"/>

        </fig>

      <p>We summarize the main features of the final tomographic model
<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">4</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula>. Map views of <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> (Fig. <xref ref-type="fig" rid="Ch1.F12"/>k–o) and <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>
(Fig. <xref ref-type="fig" rid="Ch1.F13"/>k–o) reveal large velocity variations of up to
<inline-formula><mml:math display="inline"><mml:mrow><mml:mo>±</mml:mo><mml:mn mathvariant="normal">8</mml:mn></mml:mrow></mml:math></inline-formula> %, which indicate strong lateral heterogeneities in the
model region. The epicentral areas of the Landers, Big
Bear, and Joshua Tree earthquakes exhibit clear lateral
velocity contrasts from the surface to about <inline-formula><mml:math display="inline"><mml:mn>15.0</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> depth
(Figs. <xref ref-type="fig" rid="Ch1.F12"/>k–n and <xref ref-type="fig" rid="Ch1.F13"/>k–n). In the
Mojave block <xref ref-type="bibr" rid="bib1.bibx4" id="paren.46"/>, north of the San Andreas fault, high
<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> anomalies are generally visible at the shallow depth
1.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> (Figs. <xref ref-type="fig" rid="Ch1.F12"/>k and <xref ref-type="fig" rid="Ch1.F13"/>k) and negative
velocity perturbations exists in the middle crust
(Figs. <xref ref-type="fig" rid="Ch1.F12"/>m, n and <xref ref-type="fig" rid="Ch1.F13"/>m, n). Similar depth
variation of the velocity structures in this region was also reported
by <xref ref-type="bibr" rid="bib1.bibx57" id="text.47"/>. In the upper crust, low-velocity anomalies
(Figs. <xref ref-type="fig" rid="Ch1.F12"/>k, l and <xref ref-type="fig" rid="Ch1.F13"/>k, l) exist along the
San Andreas fault (SAF) and the San Jacinto fault (SJF) but only
beneath the northwestern portion of the Elsinore fault (EF)
<xref ref-type="bibr" rid="bib1.bibx10" id="paren.48"/>. Additionally, a significant high-velocity zone is visible
between the SAF and the SJF, which results in strong velocity
contrasts across the two faults near the surface <xref ref-type="bibr" rid="bib1.bibx1 bib1.bibx17" id="paren.49"/>. The high-velocity zone between the EF and the north portion
of the SJF may indicate a reversal in the velocity contrast polarity
along the SJF at around 5.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> depth (Figs. <xref ref-type="fig" rid="Ch1.F12"/>l
and <xref ref-type="fig" rid="Ch1.F13"/>l) <xref ref-type="bibr" rid="bib1.bibx1" id="paren.50"/>. In the middle crust, the SAF,
the SJF, and the EF roughly show relatively high-velocity anomalies
(Figs. <xref ref-type="fig" rid="Ch1.F12"/>m, n and <xref ref-type="fig" rid="Ch1.F13"/>m, n)
<xref ref-type="bibr" rid="bib1.bibx18" id="paren.51"/>. However, in the lower crust (<inline-formula><mml:math display="inline"><mml:mn>21.0</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>), low-velocity anomalies are generally reported along these fault systems
(Figs. <xref ref-type="fig" rid="Ch1.F12"/>o and <xref ref-type="fig" rid="Ch1.F13"/>o). We will discuss this
low-velocity feature in detail in the next section. Beneath the Salton
Trough (ST), which is a sediment-filled graben near the southern part
of the SAF <xref ref-type="bibr" rid="bib1.bibx1" id="paren.52"/>, a pronounced low <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> anomaly
exists in the upper crust (Figs. <xref ref-type="fig" rid="Ch1.F12"/>k, l
and <xref ref-type="fig" rid="Ch1.F13"/>k, l), and high <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave velocity structures
are revealed in the middle and lower crust
(Fig. <xref ref-type="fig" rid="Ch1.F12"/>m–o). This is consistent with the results of
<xref ref-type="bibr" rid="bib1.bibx1" id="text.53"/>.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F12" specific-use="star"><caption><p>Map views of the <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave tomography at five representative depths
for models <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> (left column), <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">3</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> (middle column), and <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">4</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> (right
column). The layer depth is shown just on the right hand side of each row.
Red and blue colors denote low and high velocities, respectively. The
velocity perturbation scale (in percent) is also shown. On each map, grey
lines denote active faults, and the empty stars indicate the epicentral
locations of the Landers earthquake, the Big Bear earthquake, and the Joshua
Tree earthquake (Fig. <xref ref-type="fig" rid="Ch1.F1"/>). SAF is the short form for the San
Andreas fault, SJF is the San Jacinto fault, EF is the Elsinore fault, and ST
is the Salton Trough.</p></caption>
          <?xmltex \igopts{width=341.433071pt}?><graphic xlink:href="https://www.solid-earth.net/5/1169/2014/se-5-1169-2014-f12.png"/>

        </fig>

      <?xmltex \floatpos{t}?><fig id="Ch1.F13" specific-use="star"><caption><p>The same as Fig. <xref ref-type="fig" rid="Ch1.F12"/> but for <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave tomography.</p></caption>
          <?xmltex \igopts{width=341.433071pt}?><graphic xlink:href="https://www.solid-earth.net/5/1169/2014/se-5-1169-2014-f13.png"/>

        </fig>

      <p>A series of vertical cross-sectional views from the surface to 40 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>
depth for <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, and Poisson's ratio <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">σ</mml:mi></mml:math></inline-formula> structures are shown in
Figs. <xref ref-type="fig" rid="Ch1.F14"/> and <xref ref-type="fig" rid="Ch1.F15"/>. Since both <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>
structures are almost well recovered in the crust (Figs. <xref ref-type="fig" rid="Ch1.F8"/> and
<xref ref-type="fig" rid="Ch1.F9"/> and Table <xref ref-type="table" rid="Ch1.T2"/>), Poisson's ratio models in
Figs. <xref ref-type="fig" rid="Ch1.F14"/>c, f, i and <xref ref-type="fig" rid="Ch1.F15"/>c, f, i can be viewed as
being reliably determined based on Eq. (<xref ref-type="disp-formula" rid="Ch1.E9"/>). Note that the map
views of the iteratively updated Poisson's ratio <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">σ</mml:mi></mml:math></inline-formula> structure are
included in the Supplement (Figs. S3–S5). Figure <xref ref-type="fig" rid="Ch1.F14"/> shows three
cross sections along the profiles through the hypocenters of the Landers
earthquake, the Joshua Tree earthquake, the Big Bear earthquake, and the 1999
Hector Mine earthquake where profile AB is nearly parallel to the fault zone
of the Landers earthquake. It can be observed that the Landers mainshock is
located in a high-velocity, low Poisson's ratio anomaly
(Fig. <xref ref-type="fig" rid="Ch1.F14"/>a–c and g–i). Additionally, the hypocenters of the Joshua Tree, Big Bear, and 1999 Hector Mine earthquakes are
at or near high-velocity and low Poisson's ratio anomalies
(Fig. <xref ref-type="fig" rid="Ch1.F14"/>). By inverting <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave arrival times from aftershocks
of 1992 southern California earthquakes, <xref ref-type="bibr" rid="bib1.bibx14" id="text.54"/> also reported that
high <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> anomalies occur at or near nucleation sites of the Joshua Tree,
Landers, and Big Bear mainshocks. Both velocity and Poisson's ratio
structures change drastically around the source areas of the Landers
mainshock and the other three large earthquakes. Material properties of the
source areas of the four large earthquakes are consistent with those of the
brittle seismogenic layer, which is characterized by high velocity and low
Poisson's ratio <xref ref-type="bibr" rid="bib1.bibx46" id="paren.55"/>. A prominent feature of the vertical
cross sections along profiles CD and EF is the low <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, low <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, and high
Poisson's ratio structure to the west of the Big Bear mainshock hypocenter in
the lower crust (Fig. <xref ref-type="fig" rid="Ch1.F14"/>d–i), which has been interpreted as
a ductile and weak region <xref ref-type="bibr" rid="bib1.bibx52 bib1.bibx56" id="paren.56"/>. A low-velocity and high
Poisson's ratio structure is also visible in the lower crust close to the
hypocenter of the Landers mainshock <?xmltex \hack{\mbox\bgroup}?>(Fig. <xref ref-type="fig" rid="Ch1.F14"/>a–c and g–i)<?xmltex \hack{\egroup}?>. In addition, tomographic results and seismicity along the profile
AB confirm the conclusion of <xref ref-type="bibr" rid="bib1.bibx18" id="text.57"/> that shallow earthquakes mostly
occurred in high <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> regions and mid-crustal earthquakes occurred in low
<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> zones. Seismicity along profile AB is mainly the aftershocks of the
Landers earthquake (Fig. <xref ref-type="fig" rid="Ch1.F14"/>a–c), and it can be observed that to
the south of the Landers mainshock hypocenter, seismicity strikes across the
Joshua Tree aftershock zone, extends about <inline-formula><mml:math display="inline"><mml:mn>40.0</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> south of the
epicenter of the Landers mainshock, and terminates within a few kilometers of
the SAF. Immediately following that of the Salton Trough, a low-velocity and
high Poisson's ratio anomaly exists near the surface <xref ref-type="bibr" rid="bib1.bibx52" id="paren.58"/>. Additionally, to
the north of the Landers mainshock, aftershocks extend about
<inline-formula><mml:math display="inline"><mml:mn>60.0</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> to the Camp Rock fault and are surrounded by low-velocity
and high Poisson's ratio rocks beneath them <xref ref-type="bibr" rid="bib1.bibx9 bib1.bibx52" id="paren.59"/>.</p>
      <p>Figure <xref ref-type="fig" rid="Ch1.F15"/> shows the vertical cross sections along the Elsinore
fault (EF), the San Jacinto fault (SJF), and the San Andreas fault (SAF).
Beneath the northwest segment of the EF, a low-velocity and high Poisson's
ratio anomaly is visible in the upper and middle crust, underlaid by
a high-velocity and low Poisson's ratio structure (Fig. <xref ref-type="fig" rid="Ch1.F15"/>a–c).
These are contrary to the structural properties under the central and
southeast sections of the EF, which generally exhibit high velocity and low
Poisson's ratio in the upper and middle crust and low-velocity and high
Poisson's ratio beneath (Fig. <xref ref-type="fig" rid="Ch1.F15"/>a–c). Seismicity along the EF
is generally focused between 5.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mn>15.0</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>. Velocity
and Poisson's ratio models reveal complex patterns beneath both the SJF and
the SAF (Fig. <xref ref-type="fig" rid="Ch1.F15"/>d–i). Alternating high- and low-velocity
variations can be observed along the faults near the surface, which can be
interpreted as manifestations of the complex surface geological patterns.
However, along both the SJF and the SAF, we can observe a layer with low
velocity and high Poisson's ratio at the depth of about 5.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>. Right
beneath this layer, high-velocity and low Poisson's ratio structures exist in
the middle crust. Seismicity along the SJF and the SAF mainly occurred in
this high-velocity and low Poisson's ratio region. Additionally, the
seismicity along the SJF is much more active than that along the SAF for
study area <xref ref-type="bibr" rid="bib1.bibx17" id="paren.60"/>. The lower crust is generally dominated by
low-velocity and high Poisson's ratio structures. Specifically, near the
southeast sections of the SJF and the SAF which are close to the Salton
Trough, there are mainly low-velocity and high Poisson's ratio structures at
shallow depths and high-velocity and low Poisson's ratio anomalies in the
middle and lower crust. These features are consistent with the extension and
crustal thinning of the Salton Trough region <xref ref-type="bibr" rid="bib1.bibx1" id="paren.61"/>.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F14" specific-use="star"><caption><p>Vertical cross sections of <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave velocity, <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave velocity, and
Poisson's ratio images (<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn mathvariant="normal">4</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula>) along profile AB <bold>(a–c)</bold>, CD
<bold>(d–f)</bold> and EF <bold>(g–i)</bold> as indicated on the inset map
<bold>(j)</bold>. Low velocity and high Poisson's ratio are shown in red color,
while high velocity and low Poisson's ratio are represented by blue color.
The scales for the velocity and Poisson's ratio <inline-formula><mml:math display="inline"><mml:mi mathvariant="italic">σ</mml:mi></mml:math></inline-formula> perturbations (in
%) are shown on the right. Small grey dots denote events with magnitudes
greater than 1.5 between January 1992 and November 2013 that are located
within 3.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> width along each profile. The hypocenters for the
Landers mainshock (<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi mathvariant="normal">w</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> 7.3) hypocenter at 7.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> depth
and the Hector Mine earthquake at 6.0 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> are shown by the red and
brown star, respectively. The hypocenters for the Joshua Tree earthquake at
<inline-formula><mml:math display="inline"><mml:mn>12.4</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> and the Big Bear earthquake at <inline-formula><mml:math display="inline"><mml:mn>14.4</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> are
indicated by blue stars. The dashed lines represent the Moho discontinuity
obtained by <xref ref-type="bibr" rid="bib1.bibx58" id="text.62"/>. CRF is short for the Camp Rock fault, also
indicated on the inset map <bold>(j)</bold>.</p></caption>
          <?xmltex \igopts{width=341.433071pt}?><graphic xlink:href="https://www.solid-earth.net/5/1169/2014/se-5-1169-2014-f14.png"/>

        </fig>

      <?xmltex \floatpos{t}?><fig id="Ch1.F15" specific-use="star"><caption><p>The same as Fig. <xref ref-type="fig" rid="Ch1.F14"/> but along the Elsinore fault (EF),
the San Jacinto fault (SJF) and the San Andreas fault (SAF), denoted by
cross sections GH <bold>(a–c)</bold>, IJ <bold>(d–f)</bold> and KLM
<bold>(g–i)</bold>, respectively.</p></caption>
          <?xmltex \igopts{width=341.433071pt}?><graphic xlink:href="https://www.solid-earth.net/5/1169/2014/se-5-1169-2014-f15.png"/>

        </fig>

</sec>
</sec>
<sec id="Ch1.S5" sec-type="conclusions">
  <title>Discussion and conclusions</title>
      <p>Our new tomographic models in general agreement with the results of previous
studies for overlapped research regions <xref ref-type="bibr" rid="bib1.bibx52 bib1.bibx57 bib1.bibx35 bib1.bibx30 bib1.bibx19 bib1.bibx1" id="paren.63"><named-content content-type="pre">e.g.,</named-content></xref>. As shown in Figs. <xref ref-type="fig" rid="Ch1.F12"/>
and <xref ref-type="fig" rid="Ch1.F13"/>, the tomographic models have mainly four typical
features. (1) Strong lateral heterogeneities (up to <inline-formula><mml:math display="inline"><mml:mrow><mml:mo>±</mml:mo><mml:mn mathvariant="normal">8</mml:mn><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi mathvariant="italic">%</mml:mi></mml:mrow></mml:math></inline-formula>) exist in
the crust <xref ref-type="bibr" rid="bib1.bibx55 bib1.bibx30" id="paren.64"><named-content content-type="pre">e.g.,</named-content></xref>, which reflects complex
compositional, structural, and petrophysical variations. Since crustal
heterogeneities undoubtedly affect seismic wave propagation <xref ref-type="bibr" rid="bib1.bibx30" id="paren.65"/>,
an accurate forward modeling technique is essential for correctly capturing
the interactions between seismic waves and heterogeneous structures. This
indicates the necessity of solving full wave equations in complex structure
imaging. (2) Significant lateral velocity contrasts can be observed in the
epicentral areas of the Landers, Big Bear, and the Joshua Tree earthquakes
from the surface to the middle crust and also across the San Jacinto fault
and the San Andreas fault near the surface <xref ref-type="bibr" rid="bib1.bibx1" id="paren.66"/>. (3) The
velocity structures in the upper crust correlate well with the surface
geological features <xref ref-type="bibr" rid="bib1.bibx55 bib1.bibx35 bib1.bibx17" id="paren.67"/>. For example, due to
the fractured rocks within the fault zones and the thick sedimentary
materials <xref ref-type="bibr" rid="bib1.bibx35" id="paren.68"/>, low-velocity anomalies are prominent along the
San Andreas fault and the San Jacinto fault, near the coast, and beneath the
Salton Trough in the upper crust (Figs. <xref ref-type="fig" rid="Ch1.F12"/>k, l
and <xref ref-type="fig" rid="Ch1.F13"/>k, l). (4) Pronounced low-velocity anomalies are recovered
along the Elsinore fault, the San Jacinto fault, and the San Andreas fault in
the lower crust. Because of their poor resolution in the lower crust, this
feature was not reported by previous crustal tomographic studies that also
used only first arrival-time data <xref ref-type="bibr" rid="bib1.bibx18 bib1.bibx34" id="paren.69"><named-content content-type="pre">e.g.,</named-content></xref>.
Contrary to that, our tomographic results have satisfactory recovery rates at
<inline-formula><mml:math display="inline"><mml:mn>21.0</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> depth and clearly reveal these low-velocity anomalies
(Figs. <xref ref-type="fig" rid="Ch1.F12"/>o and <xref ref-type="fig" rid="Ch1.F13"/>o). The adjoint tomography of the
southern California crust <xref ref-type="bibr" rid="bib1.bibx30" id="paren.70"/> shows visible but less significant
low <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave velocity anomalies along the three faults at <inline-formula><mml:math display="inline"><mml:mn>20.0</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>
depth. By combining earthquake recordings and ambient-noise cross-correlation
phase measurements, stacking of station-to-station correlations of ambient
seismic noise by incorporating receiver function analysis with gravity and
magnetic data, <xref ref-type="bibr" rid="bib1.bibx13" id="text.71"/> also discovered the low-velocity anomalies
in the lower crust of southern California with full 3-D waveform tomographic
inversions. <xref ref-type="bibr" rid="bib1.bibx11" id="text.72"/> proposed that a magmatic intrusion at a depth
of about 20 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> exists in the southwest of Salton Sea. It extends for
70 <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula> in the SW–NE direction and may imply the existence of fluids
<xref ref-type="bibr" rid="bib1.bibx11" id="paren.73"/>. Since their reported magmatic intrusion zone is
partially within our study area and appears to be covered by low-velocity
anomalies, it may be possible to associate the low-velocity anomalies in the
lower crust with the existence of crustal fluids.</p>
      <p>Seismicity in the study area mainly occurred in the regions with high
velocity and low Poisson's ratio, which can be associated with the
brittle seismogenic layers <xref ref-type="bibr" rid="bib1.bibx46" id="paren.74"/>. Particularly, the seismic
rupture zone in the upper crust around the Landers earthquake fault
zone (Fig. <xref ref-type="fig" rid="Ch1.F14"/>a–c) generally shows high <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, high <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, and
relatively low Poisson's ratio <xref ref-type="bibr" rid="bib1.bibx52" id="paren.75"/>. <xref ref-type="bibr" rid="bib1.bibx53" id="text.76"/>
suggested that high-velocity areas are generally considered to be
strong and brittle parts of the fault zone, which are capable of
generating earthquakes. In contrast, low-velocity regions may
represent the regions of either higher degree of fracture, high fluid
pressure, or higher temperatures where deformations are more likely to
be aseismic. In addition, a closer observation reveals that the
mainshocks of the Landers earthquake (<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi mathvariant="normal">w</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> 7.3) and other three strong
earthquakes with magnitudes greater than 6.0 (the Joshua Tree, Big
Bear, and Hector Mine earthquakes) occurred very close to the boundaries of
high <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, high <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and low Poisson's ratio anomalies
(Fig. <xref ref-type="fig" rid="Ch1.F14"/>). Indeed, many large crustal earthquakes
occurred in regions with significant seismic property variations, such
as the 2008 <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi mathvariant="normal">w</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> 7.2 Iwate–Miyagi earthquake <xref ref-type="bibr" rid="bib1.bibx5" id="paren.77"/> and the
2011 <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi mathvariant="normal">w</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>=</mml:mo></mml:math></inline-formula> 7.0 Iwaki earthquake <xref ref-type="bibr" rid="bib1.bibx37" id="paren.78"/>. While the Iwate–Miyagi
earthquake and the Iwaki earthquake have been hypothesized to be
caused by fluid dehydration from the subducting Pacific plate
<xref ref-type="bibr" rid="bib1.bibx46 bib1.bibx5 bib1.bibx37" id="paren.79"><named-content content-type="pre">e.g.,</named-content></xref>, <xref ref-type="bibr" rid="bib1.bibx35" id="text.80"/>
concluded that fluids from long-term infiltration of surface water may
have triggered large earthquakes in the Landers source area.</p>
      <p>Seismic properties along the San Andreas fault, the San Jacinto fault
and the Elsinore fault are also explored in this study. Velocity and
Poisson's ratio structures in the upper crust show very complex
patterns along the three faults. These near surface features are
associated with key fault properties such as rheology, brittle–ductile
transition, pore pressure, stress, geotherm, and rupture energy
<xref ref-type="bibr" rid="bib1.bibx16 bib1.bibx10" id="paren.81"><named-content content-type="pre">e.g.,</named-content></xref>. High-velocity and low Poisson's
ratio structures are generally observed in the middle crust along the
three faults. Additionally, seismicity also mainly distributes in this
region. In the lower crust, we generally observe low-velocity and high
Poisson's ratio structures except around the area near the Salton
Trough. Since the width of fault zones ranges from tens to hundreds
meters while the lateral inversion grid spacing is about
<inline-formula><mml:math display="inline"><mml:mn>10.0</mml:mn></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">km</mml:mi></mml:math></inline-formula>, it is difficult to obtain detailed fault structures
in this regional tomographic study. A detailed discussion on the
structures of the San Jacinto and the Elsinore fault zones can be
found in <xref ref-type="bibr" rid="bib1.bibx10" id="text.82"/> which used local seismic records for
clustered fault-zone earthquakes for imaging.</p>
      <p>Based on the above discussions, we conclude that the crustal structures
beneath the 1992 Landers earthquake (<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi mathvariant="normal">w</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> 7.3) source area have
been successfully imaged based on the wave-equation-based travel-time seismic
tomography (WETST) technique. The recovered strong crustal heterogeneities
advocate the use of more subtle full wave-equation solvers in tomographic
imaging to accurately simulate seismic wave propagation in complex media. As
our forward modeling is restricted in a 2-D plane and based on an efficient
high-order central difference method, WETST only requires moderate
computational resources even when individual kernels for each
source–receiver pair are constructed. For example, a total of about 10 000
CPU hours are used to generate the <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> wave and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> wave tomographic results
in this work, much fewer than 0.8 million hours used by the adjoint
tomography of the southern California crust in <xref ref-type="bibr" rid="bib1.bibx30" id="text.83"/>. These
properties suggest that WETST can be used to reveal the structures of the
Earth's interior quickly when large data sets are involved for further
applications. Of course, the underlying 2-D acoustic wave-equation
approximation for the forward modeling ignores the effect of off-plane
structures. To what extent is this kind of approximation valid should be
further investigated and will be part of our future work. However, as it is
still <?xmltex \hack{\mbox\bgroup}?>computationally<?xmltex \hack{\egroup}?><?xmltex \hack{\mbox\bgroup}?>expensive<?xmltex \hack{\egroup}?> to <?xmltex \hack{\mbox\bgroup}?>calculate<?xmltex \hack{\egroup}?><?xmltex \hack{\mbox\bgroup}?>individual<?xmltex \hack{\egroup}?> kernels for the “3-D–3-D” tomographic method
<xref ref-type="bibr" rid="bib1.bibx31 bib1.bibx39 bib1.bibx41" id="paren.84"><named-content content-type="pre">e.g.,</named-content></xref>, WETST may serve as a bridge
between the conventional but the most widely used ray-based tomographic
methods and the promising “3-D–3-D” adjoint tomography based upon full 3-D
numerical solvers of the seismic wave equation <xref ref-type="bibr" rid="bib1.bibx20" id="paren.85"/>.</p><?xmltex \hack{\clearpage}?>
</sec>

      
      </body>
    <back><app-group>
        <supplementary-material position="anchor"><p><bold>The Supplement related to this article is available online at <inline-supplementary-material xlink:href="http://dx.doi.org/10.5194/se-14-1169-2014-supplement" xlink:title="pdf">doi:10.5194/se-14-1169-2014-supplement</inline-supplementary-material>.</bold></p></supplementary-material>
        </app-group><ack><title>Acknowledgements</title><p>We thank the Southern California Earthquake Data Center for providing the
high-quality arrival-time data used in this study. This work is supported by
NSERC through the G8 Research Councils Initiative on Multilateral Research
Grant and the Discovery Grant (no. 487237), Japan Society for the Promotion
of Science (Kiban-S 11050123), and National Natural Science Foundation of
China (grant no. 41230210). X. Yang was partially supported by the Regents
Junior Faculty Fellowship of University of California, Santa Barbara.
Numerical simulations and inversions are performed on workstations acquired
through combined funding of Canada Foundation for Innovation (CFI), Ontario
Research Fund (ORF), and University of Toronto Startup. All figures are made
with the Generic Mapping Tool (GMT) <xref ref-type="bibr" rid="bib1.bibx47" id="paren.86"/>. We thank K. Liu and
two anonymous reviewers for providing constructive comments and suggestions
that improved the manuscript. <?xmltex \hack{\newline}?><?xmltex \hack{\newline}?> Edited by:
K. Liu</p></ack><ref-list>
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