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Auxiliary Material
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Determination of smoothing parameter for line source CMT and finite source model
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To determine the line location and azimuth and the critical smoothing parameter λ in (1) we
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invert for multiple combinations of line geometries by varying the center of the line source, its
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azimuth and depth. The line source geometry that minimizes misfit in the L1 sense is the best
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solution. Because this inversion is computationally simple we can perform thousands of such
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inversions without a large computational overhead and without any assumptions required about
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the geometry of the source. A single line inversion takes ~0.4s on a single 2.5 GHz processor so
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the number of line sources computed in this process must be determined a priori based on
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computational resources available for a particular monitoring network. The smoothing parameter
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λ determines the roughness of the model [Hansen, 2010] and thus controls the complexity of the
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inversion results. It is critical to have an objective way of determining λ so that no user
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interaction is required during an event. For this purpose we compute the tradeoff curves of data
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misfit ||WGm-Wd|| versus model semi-norm ||Lm|| (L-curves) for all values of λ (Figure fs1a) and
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their optimal corners from their point of maximum curvature [Hansen et al., 2007; Hansen,
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2010] for all possible source geometries. We then compute the non-parametric kernel smoothing
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density estimate of the probability density function (pdf) [Bowman and Azzalini, 1997] for all
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values (Figure fs1a). We select the mode of the resulting pdf as the preferred smoothing
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parameter λ*. Then, we compare the misfit of all inversions at the λ* optimum smoothing level
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and, as in the point source fastCMT method, select the one with the smallest misfit.
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For the finite slip inversion we use Laplacian smoothness as L in Equation (1). An optimal
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smoothing parameter λ* is determined from the maximum curvature of the L-curve (Figure fs2),
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as in the line source CMT solution.
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Finite slip inversion of only GPS data
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Figure fs3, referred to in the main manuscript, shows the line-source fastCMT approach and
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slip inversion just using the GPS data for the same stations to investigate any improvements with
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seismogeodetic data.
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References
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Bowman, A. W., and A. Azzalini (1997), Applied Smoothing Techniques for Data Analysis. pp.
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31
25-45, Oxford University Press, New York., NY.
Hansen, P.C., T.K. Jensen and G. Rodriguez (2007), An adaptive pruning algorithm for the
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discrete
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10.1016/j.cam.2005.09.026
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35
L-curve
criterion,
J.
Comput.
Appl.
Math,
198,
483-492
doi:
Hansen, P.C., (2010), Discrete Inverse Problems: Insights and Algorithms, pp. 85-105, SIAM,
Philadelphia, PA.
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Figure Captions
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Figure fs1. (a) 3800 L-curves for line source fastCMT inversions performed at different
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locations and with different orientations. Orange crosses depict the corners of each curve
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computed from the maximum curvature. (b) Probability density functions of the smoothing
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parameter that corresponds to the L-curve corners. The optimum smoothing parameter λ* is
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selected from the mode of the pdf.
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Figure fs2. (a) L-curve for the seismogeodetic slip inversion plotting the model seminorm
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||Lm||2 versus model misfit ||WGm-Wd||2 with solutions at selected values of the smoothing
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parameter λ plotted as red stars. Also plotted is the curvature κ of the L-curve, the optimum
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smoothing parameter λ* is selected as the value that corresponds to the maximum curvature
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(blue star). (b) L-curve and its curvature for the GPS-only slip inversion.
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Figure fs3. fastCMT and slip inversion results for the GPS-only displacements. Green circles are
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the point sources superimposed to compute the line source, the final averaged solution shown as
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fastCMT, and the Global CMT solution shown for comparison. The inset shows the moment
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release from the line source of CMTs as a function of distance along fault. Shown along the fault
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interface with 10 km depth contours from the Slab 1.0 model (Hayes et al., 2012) is the result of
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the slip inversion, the blue lines represent the direction of slip. The triangles indicate the
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locations of all the GPS stations used for computing the CMT solution and slip inversion. The
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two large triangles represent the fixed GPS station (0848) and the GPS/accelerometer pair
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(0914/MYG003).