Electronic Supplement to
Source-Type-Specific Inversion of Moment Tensors

by Avinash Nayak and Douglas S. Dreger

This electronic supplement contains a description and figures comparing inversion performance using spherical and Cartesian eigenvector parameterizations.

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Comparison between Results of Inversions Using Spherical and Cartesian Eigenvector Parameterizations

Assuming three independent trigonometric angles ${\theta }_1$, ${\theta }_2$, and ${\theta }_3$ with domains $[-\pi ,\pi ],[-\pi /2,\pi /2]$ and $[-\pi ,\pi ]$ respectively, we can define eigenvectors ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$ in spherical coordinates as $${\mathbf{e}}_1=[\cos {\theta }_1\cos {\theta }_2,\sin {\theta }_1\cos {\theta }_2,\sin {\theta }_2],$$ $${\mathbf{e}}_2=\frac{[\cos {\theta }_3\sin {\theta }_2,\sin {\theta }_3\sin {\theta }_2,-\cos {\theta }_2(\cos {\theta }_1\cos {\theta }_3+\sin {\theta }_1\sin {\theta }_3)]}{r_0},$$ $$r_0=\sqrt{(\cos {\theta }_2(\cos {\theta }_1\cos {\theta }_3+\sin {\theta }_1\sin {\theta }_3){)}^2+(\sin {\theta }_2{)}^2},$$ and $${\mathbf{e}}_3={\mathbf{e}}_1\times {\mathbf{e}}_2\mathrm{.}$$

The partial derivatives of moment tensor (MT) elements with respect to (${\theta }_1$, ${\theta }_2$, and ${\theta }_3$) can be easily derived as for the Cartesian parameters in the Appendix in the main article. Following the procedure in the Tests on Synthetic Waveforms section in the main article, synthetic waveforms were computed for a hypothetical event assuming a pure double-couple (DC) MT solution ($\mathbf{\lambda }=[0.7071,0,-0.7071]$ , $M_0=1\times {10}^{15}\mathrm{N}·\mathrm{m}$, $\varphi =325°$, $\delta =60°$, $\zeta =112°$) for the same depth and station configuration. We used the damped least squares (LS) inversion procedure established in the Inverse Problem Formulation section in the main article to compare results, assuming both Cartesian ($a_1,a_2,a_3,b_1,b_2$ ) and spherical (${\theta }_1,{\theta }_2,{\theta }_3$ ) parameterizations. We randomly generated 200 sets of orthonormal eigenvectors and $m_0$ values (between 1.0 and 10) and used them as initial values for both sets of inversions. For the spherical parameterization, the damping parameter was fixed at $1\times {10}^{-7}$ from trial and error.

Figures S1 and S2 show the evolution of all parameters from their initial to final values as a function of the number of iterations for all 200 initial models and inversions with both parameterizations. Figure S1 shows that most of the initial models (197 out of 200) converge to the final correct MT solution, as shown by the final variance reduction ($\mathrm{VR}>99.9\%$ ) for most VR trajectories. For our choice of damping parameters, the inversions with spherical parameterization required more iterations (50–200) to converge to the correct solution, as compared to inversions with Cartesian parameterization (10 –25). The final eigenvectors are one of the four combinations of ($\pm {\mathbf{e}}_1,\pm {\mathbf{e}}_2$ ), in which the eigenvectors of the correct MT are ${\mathbf{e}}_1=[0.3222,0.924,0.206]$, ${\mathbf{e}}_2=[0.868,-0.3752,0.3252]$, and ${\mathbf{e}}_3=[-0.3778,-0.074,0.9229]$. Figure S2a shows that different initial values of ($a_1,a_2,a_3,b_1,b_2$ ) can lead to different final values during the course of the iterative inversion, and multiple final combinations of ($a_1,a_2,a_3,b_1,b_2$ ) can give the same MT elements and hence the same VR. Inverting for eigenvectors using spherical parameters (Fig. S2b) also suffers from the issue of nonuniqueness in final values, because phase shifts in (${\theta }_1,{\theta }_2,{\theta }_3$ ) by $\pm \pi$ can also lead to the same MT elements.

We provide the full MT solution and waveform fits of event TE1 (Fig. S3) of the Napoleonville salt dome seismic sequence (Nayak and Dreger, 2014a) for comparison with its MT solutions constrained to specific source types (Fig. 2 in main article). Since the publication of Nayak and Dreger (2014a), the analysis and results have been revised with updated 1D velocity models, and a broader frequency range of the waveforms (0.1–0.3 Hz used for four out of five stations instead of 0.1–0.2 Hz used for all stations previously) has been applied for the MT inversion (Nayak and Dreger, 2014b).

Figure S4 shows the network sensitivity solution (NSS) for the three events in this study, computed from waveform inversion using spherical parameterization for describing the eigenvectors. They are similar to those in Figure 4 of the main article, computed using Cartesian parameterization, demonstrating that either parameterization can be used.

Figure S5 shows a comparison of first-motion (FM) NSS, computed using the inversion scheme in this study and by forward-modeling polarities with 80 million randomly generated MTs, for two small subsets of event TE3 P-wave FM polarity data. These datasets contained 30 and 7 randomly selected polarities out of the original dataset containing 173 polarities. The results show that for both data subsets, our inversion scheme using approximation of the sign function can compute an NSS equivalent to the one estimated by the forward-modeling approach.

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Figures

Figure S1. Comparison of values of various quantities as they change during the course of the iterative inversions, assuming (left) Cartesian and (right) spherical eigenvector parameterization. e_ij is the jth component of eigenvector e_i, m₀² is the moment scale factor, and VR is variance reduction as defined in the main article. Each gray line is a parameter trajectory that shows the path of that parameter from initial value at the beginning of the inversion (at iteration 0) to the final value at the end of that inversion (ending in a black diamond). Because there are 200 separate initial models, there are 200 parameter trajectories (i.e., 200 gray curves) in each subplot. Dashed black lines indicate the theoretical values of the parameter at convergence to the correct MT solution. If an inversion has converged correctly to the true MT solution, its corresponding gray curve should meet a dashed black line at some iteration number (black diamond).

Figure S2. Similar to Figure S1 but showing change in the eigenvector model parameters for (a) Cartesian (a₁,a₂,a₃,b₁,b₂) and (b) spherical (θ₁,θ₂,θ₃) parameterization. The final solutions in (a) are nonunique and widely varying but give the same MT solution (Fig. S1).

Figure S3. Observed (solid lines) and synthetic (dotted lines) displacement waveforms and full MT solution for event TE1. (R = epicentral distance; Az = azimuth; D_max = maximum displacement amplitude at a station.) The focal mechanism plot shows the P-wave radiation pattern. Waveforms were filtered in the pass band 0.1–0.3 Hz for stations LA01, LA02, LA03, and LA09 and in 0.1–0.2 Hz for station LA08.

Figure S4. NSS of the three events in this study computed from inversion of low-frequency waveforms using spherical parameterization for describing eigenvectors: (a) TE1, (b) TE2, and (c) TE3. The contours and colors represent absolute values of VR (%). Black crosses are positions of major theoretical source types. For each event, the white star is the position of the best-fitting full MT solution from a time-domain full MT inversion of waveforms. In each plot, the white circle is the source type corresponding to the maximum VR recovered by each NSS (VR_MAX in the upper left corner). These plots are the same as for the NSS in the left panels of Figure 4 in the main article, which were computed using Cartesian parameterization of eigenvectors.

Figure S5. NSS of event TE3 using (a) 30 and (b) 7 P-wave FM polarities randomly selected from the original dataset of 173 polarities. The left panels (NSS Inversion) show the NSS computed using the inversion approach in this study. The right panels (NSS Random) show the NSS computed using randomly generated 80 million MTs. The contours and colors represent absolute values of VR (%). The VR scale is the same for both NSS plots (left and right) to enable better comparison. In each plot, the white circle is the source type corresponding to the maximum VR recovered by each NSS (VR_MAX in the lower left corner). Black crosses are positions of the major theoretical source types. “+Dipole” and “+Crack” are abbreviated to “+D” and “+C,” respectively, in the left panel in (a). The focal mechanism plots show the P-wave radiation pattern predicted by the MT solution corresponding to the maximum VR recovered by each NSS. Plotted against the predicted radiation patterns, black crosses and circles represent observed positive and negative P-wave FM polarities, respectively; the size of the polarity symbols is scaled by their quality weight (1, 2, or 3). P and T indicate pressure and tension axes, respectively.

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