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<div class="section" id="fixedhypocenter">
<span id="global-fixedhypocenter"></span><h1>FixedHypocenter<a class="headerlink" href="#fixedhypocenter" title="Permalink to this headline"></a></h1>
<p>Locator for re-computing source time with fixed hypocenter</p>
<div class="section" id="description">
<h2>Description<a class="headerlink" href="#description" title="Permalink to this headline"></a></h2>
<p>Mining-related events are useful as ground truth events (<span id="id1">Bondár and McLaughlin [<a class="reference internal" href="../base/references.html#id12" title="I. Bondár and K.L. McLaughlin. A new ground truth data set for seismic studies. Seismol. Res. Lett., 3:465 - 472, 2009. doi:10.1785/gssrl.80.3.465.">31</a>]</span>)
because the epicentre and depth can be constrained by physical inspection.
Unless a local seismograph network with accurate timing also locates the event,
and that information is available, the origin time must be estimated in order
for the event to be useful as ground truth. Existing location algorithms in
<cite>SeisComP</cite>, including <a class="reference internal" href="global_locsat.html#global-locsat"><span class="std std-ref">Hypo71</span></a> and <a class="reference internal" href="global_locsat.html#global-locsat"><span class="std std-ref">LOCSAT</span></a>
do not allow the determination of origin time given a set of arrivals and a
fixed hypocentre. There is a need, then, for a method of fixed hypocentre
origin time determination.</p>
<p>Objectives of this locator are:</p>
<ul class="simple">
<li><p>Inversion of arrival times of phase picks for source time fixing hypocenter location.</p></li>
<li><p>Compatibility of the method of fixed-hypocentre origin time determination with
the practise of the Comprehensive Test Ban Treaty Organization (CTBTO).</p></li>
<li><p>Adaptation of a procedure which is compatible with the other locators supported by <cite>SeisComP</cite>.</p></li>
<li><p>Adaptation of a procedure which can reproduce results of legacy locators currently
in use, such as GENLOC <span id="id2">Pavlis <em>et al.</em> [<a class="reference internal" href="../base/references.html#id57" title="G.L. Pavlis, F. Vernon, D. Harvey, and D. Quinlan. Lsqr: an algorithm for sparse linear equations and sparse least squares. ACM Transactions on Mathematical Software, 1:43 - 71, 1982. doi:10.1145/355984.355989.">52</a>]</span> and GRL, a
grid-based locator developed at the Canadian Hazards Information Service (CHIS).</p></li>
</ul>
<p>The implementation of this locator by <a class="reference internal" href="../base/glossary.html#term-gempa-gmbh"><span class="xref std std-term">gempa GmbH</span></a> was initiated and has received
initial funding from <span id="id3"><em>Natural Resources Canada (NRCAN), Earthquakes Canada</em> [<a class="reference internal" href="../base/references.html#id162" title="Natural Resources Canada (NRCAN), Earthquakes Canada. URL: https://earthquakescanada.nrcan.gc.ca/index-en.php.">16</a>]</span>.</p>
</div>
<div class="section" id="methodology">
<h2>Methodology<a class="headerlink" href="#methodology" title="Permalink to this headline"></a></h2>
<p>Given the measured arrival times <img class="math" src="../_images/math/ef0cc8a9abcd91d4c81e277fda15f02f61283b8a.png" alt="t_i^k"/> of phase <img class="math" src="../_images/math/9630132210b904754c9ab272b61cb527d12263ca.png" alt="k"/> at
station <img class="math" src="../_images/math/5aa339d4daf45a810dda332e3c80a0698e526e04.png" alt="i"/>, most methods of earthquake hypocentre location involve
minimization of the weighted squared sum of the residuals. That is,
minimization of:</p>
<div class="math">
<p><img src="../_images/math/66bfccc625b603ecc69ebbaf5e79849a6f90985a.png" alt="|r_w|^2 = \sum_{i=1}^N {w_i^2 [ t_i^k - \tau - T_{model}^k(r_i,x) ]^2}"/></p>
</div><p>The residuals are computed by subtracting the expected arrival times
<img class="math" src="../_images/math/4cd887dff87ad1c4fe54e94c441379b895433dc5.png" alt="\tau - T_{model}^k(r_i,x)"/> based on a velocity model applied at the
coordinates of each station
<img class="math" src="../_images/math/dbc4c429b48dd94e41ee866e1edbf0abededae3a.png" alt="r_i"/>.</p>
<p>Typically the weights can be a combination of the inverse of the
estimated pick uncertainty <img class="math" src="../_images/math/68d8fad26c73ee49f100be287592043b3d020379.png" alt="1/{\sigma}_i"/>, a distance term
<img class="math" src="../_images/math/1bdf84b56ea19c568b70e6dc45392b954d63a0ea.png" alt="d^k(\Delta)"/> and/or a residual weight term <img class="math" src="../_images/math/cf90d710bc5e654f04f2197992b5658a393a8c93.png" alt="p(r_i)"/>.
Alternative weighting schemes can be applied but in this
implementation we weight by pick uncertainty alone: <img class="math" src="../_images/math/eb6982d524ed71f7530ce3eb31b4814c1aef31b1.png" alt="w_i=\frac{1}{{\sigma}_i}"/>.</p>
<p>In the general case, the model is a nonlinear function of its inputs, and there
is no analytic solution for the origin time and hypocenter that minimize the
norm. Typically, the solution is found iteratively, based on an initial guess
for the origin time and hypocenter. This is the normal procedure for an earthquake
without an a priori estimate of the hypocentral location.
When the hypocenter is in fact accurately constrained, the modeled travel time
is a constant, so we can project each phase arrival back to an equivalent origin
time</p>
<div class="math">
<p><img src="../_images/math/e07cf5ca46edb86938c9b742583b78cf5dfd5a2c.png" alt="\tau_i^k = t_i^k - T_{model}^k (r_i,x)"/></p>
</div><p>so that we only have to find which minimizes:</p>
<div class="math">
<p><img src="../_images/math/ce91df5771b45d3d22d0246721cb9eda678dc1ea.png" alt="|r_w|^2 = \sum_{i=1}^{N}w_i^2 [\tau_i^k - \tau]^2"/></p>
</div><p>The residuals are minimized by:</p>
<div class="math">
<p><img src="../_images/math/a3c64e2d4f039151334ad193c4df506d7c5a4d27.png" alt="\tau = \frac{\sum_{i=1}^{N}w_i^2 (\tau_i^k)^2}{\sum_{i=1}^{N}w_i^2}."/></p>
</div><p>Thus, the origin time is simply the weighted mean of the equivalent origin
times, according to the velocity model, associated with the arrivals.</p>
<p>The standard error of this estimate is:</p>
<div class="math">
<p><img src="../_images/math/450e7d546e262c19d94b8fdfc9f5d971d36f7a1b.png" alt="\sigma = \sqrt{\frac{\sum_{i=1}^{N}w_i^2 [\tau_i^k - \tau]^2}{\sum_{i=1}^{N}w_i^2}}."/></p>
</div><p>The methodology for estimating error intervals and ellipses recommended for
standard processing at the CTBTO (<span id="id4">Lee and Lahr [<a class="reference internal" href="../base/references.html#id51" title="W.H. Lee and J.C. Lahr. Hypo71 (revised): a computer program for determining local earthquake hypocentral parameters, magnitude, and first motion pattern of local earthquakes. US Geol. Survey Open-file Report 75-311, 1975. URL: https://pubs.er.usgs.gov/publication/ofr75311, doi:10.3133/ofr75311.">48</a>]</span>) is that of
<span id="id5">Jordan and Sverdrup [<a class="reference internal" href="../base/references.html#id41" title="T.H. Jordan and K.A. Sverdrup. Teleseismic location techniques and their application to earthquake clusters in the south-central pacific. Bull. Seismol. Soc. Am., 4:1105 – 1130, 1981. doi:10.1785/BSSA0710041105.">44</a>]</span> and is implemented
in LOCSAT (<span id="id6">Bratt and Bache [<a class="reference internal" href="../base/references.html#id20" title="S.R. Bratt and T.C. Bache. Locating events with a sparse network of regional arrays. Bull. Seismol. Soc. Am., 78(2):780 - 798, 1988. URL: https://pubs.geoscienceworld.org/ssa/bssa/article-pdf/78/2/780/5334120/bssa0780020780.pdf, doi:10.1785/BSSA0780020780.">38</a>]</span>).
Uncertainty is represented by a set of points <img class="math" src="../_images/math/c24babb8b2b84f2c211a040b48c4e2e04bb74871.png" alt="x_e"/> around the final estimate
<img class="math" src="../_images/math/4d932942be8cf79d3cda089d58de1a45ca6f8597.png" alt="x_f"/> satisfying:</p>
<div class="math">
<p><img src="../_images/math/b5359fb7461e8422b3d0ea043241d3902ed8c878.png" alt="\kappa_p^2 &amp;= (x_e - x_f)^TC_m(x_e-x_f), \\
\kappa_p^2 &amp;= Ms^2F_p(M,K+N-M), \\
s^2 &amp;= \frac{Ks_K^2+|r_w|^2}{K+N-M}"/></p>
</div><p>where:</p>
<ul>
<li><p><img class="math" src="../_images/math/4577806854ed7741af5ad0aa1abe2909567a3083.png" alt="C_m"/>: Covariance matrix, corresponding to the final hypocentre estimate <img class="math" src="../_images/math/4d932942be8cf79d3cda089d58de1a45ca6f8597.png" alt="x_f"/>.</p></li>
<li><p><img class="math" src="../_images/math/8ab4b82ece5a67cb801cf339bca8ff343a570c2e.png" alt="s^2"/>: Ratio of actual to assumed.</p></li>
<li><p><img class="math" src="../_images/math/2255befd84ee2fcea10694752b43e60d80a4f992.png" alt="\kappa_p^2"/>: The “confidence coefficient” at probability <img class="math" src="../_images/math/27dc86f9f1b1c3435b2403a869b5870c582facea.png" alt="\rho"/>.</p></li>
<li><p><img class="math" src="../_images/math/86991529ca92ae0a8532508be15f15400052d4a4.png" alt="F_p(m,n)"/>: Fisher-Snedecor quantile function (inverse cumulative F-distribution)
for and degrees of freedom of numerator and denominator sum of squares,
respectively, and probability.</p></li>
<li><p><img class="math" src="../_images/math/141bbefb74014fc5e43499901bf78607ae335583.png" alt="p"/>: Confidence level: the desired probability that the true epicentre should
fall within the uncertainty bounds.</p></li>
<li><p><img class="math" src="../_images/math/3bfb3a64189a14b2704f4610827762d5e3145114.png" alt="N"/>: Sum of all arrival time, azimuth or slowness estimates. Here, only
arrival times are considered for inversion.</p></li>
<li><p><img class="math" src="../_images/math/4abba779877abb276b98ccb2b4ba9bf2e41947ab.png" alt="M"/>: Number of fitted parameters:</p>
<ul class="simple">
<li><p>3: error ellipsoid</p></li>
<li><p>2: error ellipse</p></li>
<li><p>1: depth or time error bounds.</p></li>
</ul>
<p>Here, <img class="math" src="../_images/math/dabb61ecc4da8df35c00e765a39fd383714b573a.png" alt="M = 1"/> as we only invert for the time.</p>
</li>
<li><p><img class="math" src="../_images/math/8caebcef0e5d5769000618e5116d6051c25bd98e.png" alt="s_K^2"/>: A prior estimate of the ratio of actual to assumed data variances; typically set to 1.</p></li>
<li><p><img class="math" src="../_images/math/52ddc0cde6d632f631533173562fe3ca375b1f32.png" alt="K"/>: Number of degrees of freedom in prior estimate <img class="math" src="../_images/math/8caebcef0e5d5769000618e5116d6051c25bd98e.png" alt="s_K^2"/>.
<img class="math" src="../_images/math/52ddc0cde6d632f631533173562fe3ca375b1f32.png" alt="K"/> can be configured by <a class="reference internal" href="#confval-FixedHypocenter.degreesOfFreedom"><code class="xref std std-confval docutils literal notranslate"><span class="pre">FixedHypocenter.degreesOfFreedom</span></code></a>.</p></li>
<li><p><img class="math" src="../_images/math/4fa53fbce202dca5a7572c6eb0dd0b33bd9e601b.png" alt="r_w"/>: Vector of weighted residuals.</p></li>
</ul>
<p>Although this formulation is complex it is useful it because allows the analyst to
balance a priori and a posteriori estimates of the ratio of actual to assumed
data variances.</p>
<p>The covariance matrix in the general case is computed from the weighted sensitivity
matrix <img class="math" src="../_images/math/93b581f18e6ddf1ae1ab7475c44b41837e648994.png" alt="A_w"/>, the row-weighted matrix of partial derivatives of arrival
time with respect to the solution coordinates.</p>
<div class="math">
<p><img src="../_images/math/746b9634c2f299dd2005c85460e58fae6d08a49e.png" alt="C_m = A^T_wA_w"/></p>
</div><p>However, when origin time is the only coordinate, the partial derivatives with
respect to origin time are unity, the weighted sensitivity matrix is simply a
row vector of weights, and the time-time covariance
<img class="math" src="../_images/math/95c69b0bc6dae384ad6ff06f0ec8ecaaf23987e4.png" alt="c_{tt}"/> is simply the sum of the squares of these weights.</p>
<div class="math">
<p><img src="../_images/math/3a34284a39272ffe88e206a0c9e216e716d08a39.png" alt="c_{tt} = \sum_{i=1}^{N}w_i^2"/></p>
</div><p>It is recommended that fixed-hypocentre origin time confidence intervals be
estimated using the method of <span id="id7">Jordan and Sverdrup [<a class="reference internal" href="../base/references.html#id41" title="T.H. Jordan and K.A. Sverdrup. Teleseismic location techniques and their application to earthquake clusters in the south-central pacific. Bull. Seismol. Soc. Am., 4:1105 – 1130, 1981. doi:10.1785/BSSA0710041105.">44</a>]</span> for error ellipsoids,
that is, that the time error bounds be represented using</p>
<div class="math">
<p><img src="../_images/math/6d03a1b7af5b2b4b70c5442100e5e890ecc841ae.png" alt="\Delta t_p &amp;= \sqrt{ \frac{\kappa_p^2}{c_{tt}} } \\
&amp;= \sqrt{ \frac{F_p(1,K+N-1)}{K+N-1} \frac{Ks_K^2 + \sum_{i=1}^{N}w_i^2 [\tau_i^k-\tau]^2}{\sum_{i=1}^{N}w_i^2}}."/></p>
</div><p>In addition to recording arrival weights and residuals, distances and azimuths,
and other details of origin quality, the details of a ground-truth-level (GT1)
fixed-hypocentre origin time estimate are recorded as:</p>
<ul class="simple">
<li><p>origin.time = <img class="math" src="../_images/math/914b2d4b6659b86d3153d5510839dfb254dfc8a3.png" alt="\tau"/></p></li>
<li><p>origin.time_errors.uncertainty = <img class="math" src="../_images/math/c65793dc2e246814b66c7cdd4e3a1c6dc504bbfc.png" alt="\Delta t_p"/></p></li>
<li><p>origin.time_errors.confidence_level = <img class="math" src="../_images/math/13f30937182d0536a97f9f716e6024b4c3951b86.png" alt="100p"/></p></li>
<li><p>origin.quality.standard_error = <img class="math" src="../_images/math/b52df27bfb0b1e3af0c2c68a7b9da459178c2a7d.png" alt="\sigma"/></p></li>
<li><p>origin.quality.ground_truth_level = GT1</p></li>
</ul>
<p>For the sake of reproducibility, a comment is added to every new <a class="reference internal" href="../base/glossary.html#term-origin"><span class="xref std std-term">origin</span></a>
reporting <img class="math" src="../_images/math/52ddc0cde6d632f631533173562fe3ca375b1f32.png" alt="K"/>, <img class="math" src="../_images/math/93ad30f90f5091591a2dc0895c6171abda508e06.png" alt="s_K"/> and <img class="math" src="../_images/math/e2e0165baaf165355d3e4661b14d1558d9bde02b.png" alt="\kappa_p"/>.</p>
</div>
<div class="section" id="application">
<h2>Application<a class="headerlink" href="#application" title="Permalink to this headline"></a></h2>
<ol class="arabic">
<li><p>Configure the parameters in the section <em>FixedHypocenter</em> of the global configuration.</p></li>
<li><p>When using in <a class="reference internal" href="scolv.html#scolv"><span class="std std-ref">scolv</span></a> the FixedHypocenter locator can be chose right away
from the available locators.</p>
<div class="figure align-center" id="id8">
<a class="reference internal image-reference" href="../_images/scolv-fixedhypocenter.png"><img alt="../_images/scolv-fixedhypocenter.png" src="../_images/scolv-fixedhypocenter.png" style="width: 18cm;" /></a>
<p class="caption"><span class="caption-text">scolv Location tab with FixHypocenter selected for relocating.</span><a class="headerlink" href="#id8" title="Permalink to this image"></a></p>
</div>
</li>
<li><p>Configure the module, e.g. <a class="reference internal" href="screloc.html#screloc"><span class="std std-ref">screloc</span></a> or <a class="reference internal" href="scolv.html#scolv"><span class="std std-ref">scolv</span></a>, which is to use FixedHypocenter:</p>
<ul class="simple">
<li><p>set the locator type / interface: “FixedHypocenter”</p></li>
<li><p>if requested, set the profile as [interface]/[model], e.g.: LOCSAT/iasp91 or libtau/ak135</p></li>
</ul>
</li>
<li><p>Run the module with FixedHypocenter</p>
<p>Origins created by the FixedHypocenter locator can be identified by the methodID
and the <em>confidence/description</em> comment of the origin paramters, e.g.:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>&lt;origin publicID=&quot;Origin/20200102030459.123456.8222&quot;&gt;
...
&lt;timeFixed&gt;false&lt;/timeFixed&gt;
&lt;epicenterFixed&gt;true&lt;/epicenterFixed&gt;
&lt;methodID&gt;FixedHypocenter&lt;/methodID&gt;
&lt;earthModelID&gt;iasp91&lt;/earthModelID&gt;
...
&lt;comment&gt;
&lt;text&gt;Confidence coefficient: K-weighted ($K$=8, $s_K$=1 s), $\kappa_p$ = 1.6, $n_{eff}$ = 5.0&lt;/text&gt;
&lt;id&gt;confidence/description&lt;/id&gt;
&lt;/comment&gt;
...
&lt;/origin&gt;
</pre></div>
</div>
</li>
</ol>
</div>
<div class="section" id="module-configuration">
<span id="global-fixedhypocenter-configuration"></span><h2>Module Configuration<a class="headerlink" href="#module-configuration" title="Permalink to this headline"></a></h2>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p><strong>FixedHypocenter.*</strong>
<em>Locator parameters: FixedHypocenter</em></p>
</div>
<dl class="confval">
<dt id="confval-FixedHypocenter.profiles">
<code class="sig-name descname">FixedHypocenter.profiles</code><a class="headerlink" href="#confval-FixedHypocenter.profiles" title="Permalink to this definition"></a></dt>
<dd><p>Default: <code class="docutils literal notranslate"><span class="pre">LOCSAT/iasp91,LOCSAT/tab</span></code></p>
<p>Type: <em>list:string</em></p>
<p>Defines a list of available travel time tables. Each item
is a tuple separated by a slash with format “[interface]/[model]”.
Built-in interfaces are “LOCSAT” and “libtau”.
Other interfaces might be added via plugins. Please check their
documentation for the required interface name.</p>
</dd></dl>
<dl class="confval">
<dt id="confval-FixedHypocenter.usePickUncertainties">
<code class="sig-name descname">FixedHypocenter.usePickUncertainties</code><a class="headerlink" href="#confval-FixedHypocenter.usePickUncertainties" title="Permalink to this definition"></a></dt>
<dd><p>Default: <code class="docutils literal notranslate"><span class="pre">false</span></code></p>
<p>Type: <em>boolean</em></p>
<p>Whether to use pick time uncertainties rather than a fixed
time error. If true, then the uncertainties are retrieved from
each individual pick object. If they are not defined, then the
default pick time uncertainty as defined by defaultTimeError
will be used instead.</p>
</dd></dl>
<dl class="confval">
<dt id="confval-FixedHypocenter.defaultTimeError">
<code class="sig-name descname">FixedHypocenter.defaultTimeError</code><a class="headerlink" href="#confval-FixedHypocenter.defaultTimeError" title="Permalink to this definition"></a></dt>
<dd><p>Default: <code class="docutils literal notranslate"><span class="pre">1.0</span></code></p>
<p>Type: <em>double</em></p>
<p>Unit: <em>s</em></p>
<p>The default pick time uncertainty if pick uncertainties are
not going to be used or if they are absent.</p>
</dd></dl>
<dl class="confval">
<dt id="confval-FixedHypocenter.degreesOfFreedom">
<code class="sig-name descname">FixedHypocenter.degreesOfFreedom</code><a class="headerlink" href="#confval-FixedHypocenter.degreesOfFreedom" title="Permalink to this definition"></a></dt>
<dd><p>Default: <code class="docutils literal notranslate"><span class="pre">8</span></code></p>
<p>Type: <em>int</em></p>
<p>Number of degrees of freedom used for error estimate.</p>
</dd></dl>
<dl class="confval">
<dt id="confval-FixedHypocenter.confLevel">
<code class="sig-name descname">FixedHypocenter.confLevel</code><a class="headerlink" href="#confval-FixedHypocenter.confLevel" title="Permalink to this definition"></a></dt>
<dd><p>Default: <code class="docutils literal notranslate"><span class="pre">0.9</span></code></p>
<p>Type: <em>double</em></p>
<p>Confidence level between 0.5 and 1.</p>
</dd></dl>
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<h3><a href="../index.html">Table of Contents</a></h3>
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<li><a class="reference internal" href="#">FixedHypocenter</a><ul>
<li><a class="reference internal" href="#description">Description</a></li>
<li><a class="reference internal" href="#methodology">Methodology</a></li>
<li><a class="reference internal" href="#application">Application</a></li>
<li><a class="reference internal" href="#module-configuration">Module Configuration</a></li>
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