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<div class="section" id="fixedhypocenter">
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<span id="global-fixedhypocenter"></span><h1>FixedHypocenter<a class="headerlink" href="#fixedhypocenter" title="Permalink to this headline">¶</a></h1>
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<p>Locator for re-computing source time with fixed hypocenter</p>
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<div class="section" id="description">
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<h2>Description<a class="headerlink" href="#description" title="Permalink to this headline">¶</a></h2>
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<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>)
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because the epicentre and depth can be constrained by physical inspection.
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Unless a local seismograph network with accurate timing also locates the event,
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and that information is available, the origin time must be estimated in order
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for the event to be useful as ground truth. Existing location algorithms in
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<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>
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do not allow the determination of origin time given a set of arrivals and a
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fixed hypocentre. There is a need, then, for a method of fixed hypocentre
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origin time determination.</p>
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<p>Objectives of this locator are:</p>
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<ul class="simple">
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<li><p>Inversion of arrival times of phase picks for source time fixing hypocenter location.</p></li>
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<li><p>Compatibility of the method of fixed-hypocentre origin time determination with
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the practise of the Comprehensive Test Ban Treaty Organization (CTBTO).</p></li>
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<li><p>Adaptation of a procedure which is compatible with the other locators supported by <cite>SeisComP</cite>.</p></li>
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<li><p>Adaptation of a procedure which can reproduce results of legacy locators currently
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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
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grid-based locator developed at the Canadian Hazards Information Service (CHIS).</p></li>
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</ul>
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<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
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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>
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</div>
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<div class="section" id="methodology">
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<h2>Methodology<a class="headerlink" href="#methodology" title="Permalink to this headline">¶</a></h2>
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<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
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station <img class="math" src="../_images/math/5aa339d4daf45a810dda332e3c80a0698e526e04.png" alt="i"/>, most methods of earthquake hypocentre location involve
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minimization of the weighted squared sum of the residuals. That is,
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minimization of:</p>
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<div class="math">
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<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>
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</div><p>The residuals are computed by subtracting the expected arrival times
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<img class="math" src="../_images/math/4cd887dff87ad1c4fe54e94c441379b895433dc5.png" alt="\tau - T_{model}^k(r_i,x)"/> based on a velocity model applied at the
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coordinates of each station
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<img class="math" src="../_images/math/dbc4c429b48dd94e41ee866e1edbf0abededae3a.png" alt="r_i"/>.</p>
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<p>Typically the weights can be a combination of the inverse of the
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estimated pick uncertainty <img class="math" src="../_images/math/68d8fad26c73ee49f100be287592043b3d020379.png" alt="1/{\sigma}_i"/>, a distance term
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<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)"/>.
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Alternative weighting schemes can be applied but in this
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implementation we weight by pick uncertainty alone: <img class="math" src="../_images/math/eb6982d524ed71f7530ce3eb31b4814c1aef31b1.png" alt="w_i=\frac{1}{{\sigma}_i}"/>.</p>
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<p>In the general case, the model is a nonlinear function of its inputs, and there
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is no analytic solution for the origin time and hypocenter that minimize the
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norm. Typically, the solution is found iteratively, based on an initial guess
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for the origin time and hypocenter. This is the normal procedure for an earthquake
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without an a priori estimate of the hypocentral location.
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When the hypocenter is in fact accurately constrained, the modeled travel time
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is a constant, so we can project each phase arrival back to an equivalent origin
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time</p>
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<div class="math">
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<p><img src="../_images/math/e07cf5ca46edb86938c9b742583b78cf5dfd5a2c.png" alt="\tau_i^k = t_i^k - T_{model}^k (r_i,x)"/></p>
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</div><p>so that we only have to find which minimizes:</p>
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<div class="math">
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<p><img src="../_images/math/ce91df5771b45d3d22d0246721cb9eda678dc1ea.png" alt="|r_w|^2 = \sum_{i=1}^{N}w_i^2 [\tau_i^k - \tau]^2"/></p>
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</div><p>The residuals are minimized by:</p>
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<div class="math">
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<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>
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</div><p>Thus, the origin time is simply the weighted mean of the equivalent origin
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times, according to the velocity model, associated with the arrivals.</p>
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<p>The standard error of this estimate is:</p>
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<div class="math">
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<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>
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</div><p>The methodology for estimating error intervals and ellipses recommended for
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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
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<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
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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>).
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Uncertainty is represented by a set of points <img class="math" src="../_images/math/c24babb8b2b84f2c211a040b48c4e2e04bb74871.png" alt="x_e"/> around the final estimate
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<img class="math" src="../_images/math/4d932942be8cf79d3cda089d58de1a45ca6f8597.png" alt="x_f"/> satisfying:</p>
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<div class="math">
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<p><img src="../_images/math/b5359fb7461e8422b3d0ea043241d3902ed8c878.png" alt="\kappa_p^2 &= (x_e - x_f)^TC_m(x_e-x_f), \\
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\kappa_p^2 &= Ms^2F_p(M,K+N-M), \\
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s^2 &= \frac{Ks_K^2+|r_w|^2}{K+N-M}"/></p>
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</div><p>where:</p>
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<ul>
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<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>
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<li><p><img class="math" src="../_images/math/8ab4b82ece5a67cb801cf339bca8ff343a570c2e.png" alt="s^2"/>: Ratio of actual to assumed.</p></li>
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<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>
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<li><p><img class="math" src="../_images/math/86991529ca92ae0a8532508be15f15400052d4a4.png" alt="F_p(m,n)"/>: Fisher-Snedecor quantile function (inverse cumulative F-distribution)
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for and degrees of freedom of numerator and denominator sum of squares,
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respectively, and probability.</p></li>
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<li><p><img class="math" src="../_images/math/141bbefb74014fc5e43499901bf78607ae335583.png" alt="p"/>: Confidence level: the desired probability that the true epicentre should
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fall within the uncertainty bounds.</p></li>
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<li><p><img class="math" src="../_images/math/3bfb3a64189a14b2704f4610827762d5e3145114.png" alt="N"/>: Sum of all arrival time, azimuth or slowness estimates. Here, only
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arrival times are considered for inversion.</p></li>
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<li><p><img class="math" src="../_images/math/4abba779877abb276b98ccb2b4ba9bf2e41947ab.png" alt="M"/>: Number of fitted parameters:</p>
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<ul class="simple">
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<li><p>3: error ellipsoid</p></li>
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<li><p>2: error ellipse</p></li>
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<li><p>1: depth or time error bounds.</p></li>
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</ul>
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<p>Here, <img class="math" src="../_images/math/dabb61ecc4da8df35c00e765a39fd383714b573a.png" alt="M = 1"/> as we only invert for the time.</p>
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</li>
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<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>
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<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"/>.
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<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>
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<li><p><img class="math" src="../_images/math/4fa53fbce202dca5a7572c6eb0dd0b33bd9e601b.png" alt="r_w"/>: Vector of weighted residuals.</p></li>
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</ul>
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<p>Although this formulation is complex it is useful it because allows the analyst to
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balance a priori and a posteriori estimates of the ratio of actual to assumed
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data variances.</p>
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<p>The covariance matrix in the general case is computed from the weighted sensitivity
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matrix <img class="math" src="../_images/math/93b581f18e6ddf1ae1ab7475c44b41837e648994.png" alt="A_w"/>, the row-weighted matrix of partial derivatives of arrival
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time with respect to the solution coordinates.</p>
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<div class="math">
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<p><img src="../_images/math/746b9634c2f299dd2005c85460e58fae6d08a49e.png" alt="C_m = A^T_wA_w"/></p>
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</div><p>However, when origin time is the only coordinate, the partial derivatives with
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respect to origin time are unity, the weighted sensitivity matrix is simply a
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row vector of weights, and the time-time covariance
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<img class="math" src="../_images/math/95c69b0bc6dae384ad6ff06f0ec8ecaaf23987e4.png" alt="c_{tt}"/> is simply the sum of the squares of these weights.</p>
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<div class="math">
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<p><img src="../_images/math/3a34284a39272ffe88e206a0c9e216e716d08a39.png" alt="c_{tt} = \sum_{i=1}^{N}w_i^2"/></p>
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</div><p>It is recommended that fixed-hypocentre origin time confidence intervals be
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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,
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that is, that the time error bounds be represented using</p>
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<div class="math">
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<p><img src="../_images/math/6d03a1b7af5b2b4b70c5442100e5e890ecc841ae.png" alt="\Delta t_p &= \sqrt{ \frac{\kappa_p^2}{c_{tt}} } \\
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&= \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>
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</div><p>In addition to recording arrival weights and residuals, distances and azimuths,
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and other details of origin quality, the details of a ground-truth-level (GT1)
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fixed-hypocentre origin time estimate are recorded as:</p>
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<ul class="simple">
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<li><p>origin.time = <img class="math" src="../_images/math/914b2d4b6659b86d3153d5510839dfb254dfc8a3.png" alt="\tau"/></p></li>
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<li><p>origin.time_errors.uncertainty = <img class="math" src="../_images/math/c65793dc2e246814b66c7cdd4e3a1c6dc504bbfc.png" alt="\Delta t_p"/></p></li>
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<li><p>origin.time_errors.confidence_level = <img class="math" src="../_images/math/13f30937182d0536a97f9f716e6024b4c3951b86.png" alt="100p"/></p></li>
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<li><p>origin.quality.standard_error = <img class="math" src="../_images/math/b52df27bfb0b1e3af0c2c68a7b9da459178c2a7d.png" alt="\sigma"/></p></li>
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<li><p>origin.quality.ground_truth_level = GT1</p></li>
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</ul>
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<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>
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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>
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</div>
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<div class="section" id="application">
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<h2>Application<a class="headerlink" href="#application" title="Permalink to this headline">¶</a></h2>
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<ol class="arabic">
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<li><p>Configure the parameters in the section <em>FixedHypocenter</em> of the global configuration.</p></li>
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<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
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from the available locators.</p>
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<div class="figure align-center" id="id8">
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<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>
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<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>
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</div>
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</li>
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<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>
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<ul class="simple">
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<li><p>set the locator type / interface: “FixedHypocenter”</p></li>
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<li><p>if requested, set the profile as [interface]/[model], e.g.: LOCSAT/iasp91 or libtau/ak135</p></li>
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</ul>
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</li>
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<li><p>Run the module with FixedHypocenter</p>
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<p>Origins created by the FixedHypocenter locator can be identified by the methodID
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and the <em>confidence/description</em> comment of the origin paramters, e.g.:</p>
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<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><origin publicID="Origin/20200102030459.123456.8222">
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...
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<timeFixed>false</timeFixed>
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<epicenterFixed>true</epicenterFixed>
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<methodID>FixedHypocenter</methodID>
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<earthModelID>iasp91</earthModelID>
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...
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<comment>
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<text>Confidence coefficient: K-weighted ($K$=8, $s_K$=1 s), $\kappa_p$ = 1.6, $n_{eff}$ = 5.0</text>
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<id>confidence/description</id>
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</comment>
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...
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</origin>
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</pre></div>
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</div>
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</li>
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</ol>
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</div>
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<div class="section" id="module-configuration">
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<span id="global-fixedhypocenter-configuration"></span><h2>Module Configuration<a class="headerlink" href="#module-configuration" title="Permalink to this headline">¶</a></h2>
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<div class="admonition note">
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<p class="admonition-title">Note</p>
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<p><strong>FixedHypocenter.*</strong>
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<em>Locator parameters: FixedHypocenter</em></p>
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</div>
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<dl class="confval">
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<dt id="confval-FixedHypocenter.profiles">
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<code class="sig-name descname">FixedHypocenter.profiles</code><a class="headerlink" href="#confval-FixedHypocenter.profiles" title="Permalink to this definition">¶</a></dt>
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<dd><p>Default: <code class="docutils literal notranslate"><span class="pre">LOCSAT/iasp91,LOCSAT/tab</span></code></p>
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<p>Type: <em>list:string</em></p>
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|
<p>Defines a list of available travel time tables. Each item
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|
is a tuple separated by a slash with format “[interface]/[model]”.
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|
Built-in interfaces are “LOCSAT” and “libtau”.
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|
Other interfaces might be added via plugins. Please check their
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documentation for the required interface name.</p>
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</dd></dl>
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|
|
<dl class="confval">
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<dt id="confval-FixedHypocenter.usePickUncertainties">
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<code class="sig-name descname">FixedHypocenter.usePickUncertainties</code><a class="headerlink" href="#confval-FixedHypocenter.usePickUncertainties" title="Permalink to this definition">¶</a></dt>
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<dd><p>Default: <code class="docutils literal notranslate"><span class="pre">false</span></code></p>
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<p>Type: <em>boolean</em></p>
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|
|
<p>Whether to use pick time uncertainties rather than a fixed
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|
|
time error. If true, then the uncertainties are retrieved from
|
|
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each individual pick object. If they are not defined, then the
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|
|
default pick time uncertainty as defined by defaultTimeError
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will be used instead.</p>
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|
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</dd></dl>
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|
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|
|
|
<dl class="confval">
|
|
|
<dt id="confval-FixedHypocenter.defaultTimeError">
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|
|
<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>
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|
|
<p>Type: <em>double</em></p>
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|
<p>Unit: <em>s</em></p>
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|
|
<p>The default pick time uncertainty if pick uncertainties are
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|
|
not going to be used or if they are absent.</p>
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|
</dd></dl>
|
|
|
|
|
|
<dl class="confval">
|
|
|
<dt id="confval-FixedHypocenter.degreesOfFreedom">
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|
|
<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>
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|
<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>
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|
</dd></dl>
|
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</div>
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<script>$('#searchbox').show(0);</script>
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<h3><a href="../index.html">Table of Contents</a></h3>
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<ul>
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<li><a class="reference internal" href="#">FixedHypocenter</a><ul>
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<li><a class="reference internal" href="#description">Description</a></li>
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<li><a class="reference internal" href="#methodology">Methodology</a></li>
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<li><a class="reference internal" href="#application">Application</a></li>
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<li><a class="reference internal" href="#module-configuration">Module Configuration</a></li>
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</ul>
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</li>
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<h4>Previous topic</h4>
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<p class="topless"><a href="global_locsat.html"
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title="previous chapter">LOCSAT</a></p>
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<h4>Next topic</h4>
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<p class="topless"><a href="../toc/extensions/scevent.html"
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title="next chapter">scevent</a></p>
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