Identifying Isotropic Events Using an Improved Regional Moment Tensor Inversion Technique

Sean R. Ford, Douglas S. Dreger, William R. Walter (Lawrence Livermore National Laboratory)


Identification of events with demonstrably significant non-double-couple components can aid in understanding the source processes of seismic events in volcanic and geothermal areas (e.g. Dreger et al., 2000), and in nuclear event screening and possibly discrimination (Dreger and Woods, 2002). We implement the time-domain full regional waveform inversion for the complete moment tensor devised by Minson and Dreger (2006) after Herrman and Hutchensen (1993) based on the work of Langston (1981). The complete moment tensor allows for a characterization of the relative amounts of deviatoric and isotropic source components, the similarity of those components with prior events in the source region, and a constraint on the source depth. This information can aid in the discrimination of events.

Data and Methods

In general, synthetic seismograms are represented as the linear combination of fundamental Greenís functions where the weights on these Greenís functions are the individual moment tensor elements. Synthetic displacement seismograms are calculated with a frequency-wavenumber integration method (Saikia, 1994) for a one-dimensional (1-D) velocity model of eastern California and western Nevada (Song et al., 1996). The synthetic data is filtered with a 4-pole acausal Butterworth filter between 0.02 and 0.05 Hz. At these frequencies, where the dominant wavelengths are approximately 100 km, we assume a point source for the low-magnitude ($M_W \leq 5.6$) regional events investigated in this study. Data are collected from the TERRAscope network stations, ISA, PAS, and PFO. We remove the instrument response, rotate to the great-circle frame, integrate to obtain displacement, and filter similarly the synthetic seismograms.

We calibrate the algorithm by calculating the full and deviatoric moment tensor for the 1992 Little Skull Mountain event (Figure 10.1a). The deviatoric solution is obtained by constraining the trace of the moment tensor to be zero. Our result fits the data very well and is highly similar to the double-couple solution of Walter (1993), the deviatoric solution of Ichinose et al. (2003), and the full solution of Dreger and Woods (2002), where we assume a source depth of 9 km. The deviatoric component of the full moment tensor is decomposed to a double-couple and compensated linear vector dipole (CLVD) that shares the orientation of the major axis. The 1992 Little Skull Mountain event is almost purely double-couple and there is little change between the full and deviatoric solutions. The best-fit double-couple mechanism produces source parameters of strike 35$^{\circ }$and 196$^{\circ }$, rake -78$^{\circ }$and -104$^{\circ }$, and dip 50$^{\circ }$and 42$^{\circ }$, for the two focal planes, respectively. The total scalar moment ($M_0$) is $2.92\times10^{24}$ dyne-cm, which results in an $M_W$ of 5.58.

Figure 10.1: Moment tensor analysis of the (a) 1992 Little Skull Mt. earthquake and (b) 1992 BEXAR Nevada Test Site explosion. Data (solid line) and synthetics (dashed grey line) produced by inversion in the 20-50 s passband and resulting full and deviatoric (zero trace) focal spheres with best-fit double-couple planes (black lines), where the radius of the deviatoric sphere is relative to the total scalar moment contribution.
\epsfig{, width=7.7608cm}\end{center}\end{figure}

With the same algorithm we calculate the full and deviatoric moment tensor for the 1991 Nevada nuclear test site explosion, BEXAR ($m_b=5.6$ and $M_S=4.2$, NEIC; Figure 10.1b). The solution fits the data well and is similar to the full solution of Minson and Dreger (2006), where we assume a source depth of 1 km. The moment tensor has a large isotropic component, and the ratio of deviatoric moment ($M_{DEV}$) to isotropic moment ($M_{ISO}$) is 0.65, where the $M_0$ is $4.79 \times 10^{22}$ dyne-cm ($M_W$ of 4.39). $M_{ISO}$ and $M_{DEV}$ are defined according to Bowers and Hudson (1999) and $M_0 = M_{ISO} + M_{DEV}$.

It is difficult to grasp the source-type from the standard focal mechanism plot. Following the source-type analysis described in Hudson et al. (1989) we calculate $2\sigma$ and $k$, which are given by

\begin{displaymath}\sigma = \frac{-m_1}{\vert m_3\vert} \textrm{,} \end{displaymath}


\begin{displaymath}k = \frac{M_{ISO}}{\vert M_{ISO}\vert + \vert m_3\vert} \textrm{,} \end{displaymath}

where $m_1$ and $m_3$ are the deviatoric principal moments which are ordered $\vert m_1\vert \leq \vert m_2\vert \leq \vert m_3\vert$. $\sigma $ is a measurement of the departure of the deviatoric component from a pure double-couple mechanism, and is $0$ for a pure double-couple and $\pm0.5$ for a pure CLVD. $k$ is a measure of the volume change, where $+1$ would be a full explosion and $-1$ a full implosion. $\sigma $ and $k$ for the Little Skull Mountain earthquake and NTS explosion, BEXAR are given in Figure 10.2. Error in the values is derived from the standard error in the moment tensor elements given by the estimated covariance matrix obtained in the weighted least-squares inversion. Figure 10.2 shows that the Little Skull Mountain earthquake is within the error of being a perfect double-couple event ($2\sigma = 0$) with no volume change ($k = 0$). The BEXAR test, on the other hand, has a large volume increase with a large variance in the deviatoric source.

Figure 10.2: Source type plot for the 1992 Little Skull Mt. earthquake (black) and 1991 BEXAR NTS explosion (grey), with standard error (bars).
\epsfig{, width=5.8375cm}\end{center}\end{figure}

Error in the principal axes is analyzed by plotting the best-fit and scatter density of the axes of minimum compression (T), maximum compression (P) and null (N). The scatter density plot is obtained by randomly selecting moment tensor elements assuming a normal distribution for each element described by the standard error (given by the estimated covariance matrix), and diagonalizing the resulting moment tensor to obtain the principal axes. Principal axes plots for the Little Skull Mountain earthquake and NTS explosion, BEXAR are given in Figure 10.3. The axes for the Little Skull Mountain event are well constrained, while those for the BEXAR test are more variable. However, the BEXAR test axes do not deviate greatly from the axes of the Little Skull Mountain event, which is likely due to the similar tectonic stresses.

Figure 10.3: Scatter density of principal axes with best fit axes marked by the T, P, and N for the (a) Little Skull Mt. earthquake and (b) BEXAR explosion.
\epsfig{, width=7.81cm}\end{center}\end{figure}

In an effort to better characterize the source significance we adopt the source convention described in Riedesel and Jordan (1989). Vectors are defined describing the general,

\begin{displaymath}\textrm{MT} = \sum_{i=1}^3 M_i \mathbf{\hat{M}_i}\textrm{,} \end{displaymath}


\begin{displaymath}\textrm{DC} = \mathbf{\hat{M}_1} - \mathbf{\hat{M}_3}\textrm{,} \end{displaymath}


\begin{displaymath}\textrm{ISO} = \sum_{i=1}^3 \mathbf{\hat{M}_i}\textrm{,} \end{displaymath}

and CLVD sources,

\begin{displaymath}\textrm{CLVD1} = \mathbf{\hat{M}_1} - \frac{\mathbf{\hat{M}_2...
... + \frac{\mathbf{\hat{M}_2}}{2} - \mathbf{\hat{M}_3}\textrm{,} \end{displaymath}

where $\mathbf{\hat{M}_1}$, $\mathbf{\hat{M}_2}$, and $\mathbf{\hat{M}_3}$ are the T, N, and P axes, respectively, and $M_1$, $M_2$, and $M_3$ are the principal moments. The T, N, and P axes are chosen as in the double couple case, so that $M_1 \geq M_2 \geq M_3$. The source vectors are subspaces of the space defined by the principle axes of the moment tensor. The vectors are plotted on the focal sphere (similar to the T, N, and P axes) for the Little Skull Mountain earthquake and NTS explosion, BEXAR in Figure 10.4. The general source vector, MT, for the Little Skull Mountain event lies on the great-circle connecting the double-couple and CLVD sources. This great-circle defines the subspace on which MT must lie if the source is purely deviatoric. The MT vector is also collinear with the DC vector, which is to say that the source is almost purely double-couple. The MT vector for the BEXAR test lies well off the line defining the deviatoric solution space. The scatter density of possible MT vectors is also plotted and none of them intersect the deviatoric solution space, which is to say that the solution has a significant isotropic component.

Figure 10.4: Source vector plot with density plot of general source vector, MT, for the (a) Little Skull Mt. earthquake and (b) BEXAR explosion. See text for definition of vectors. The great-circle line connecting the CLVD1, DC, and CLVD2 vectors defines the purely deviatoric solution space.
\epsfig{, width=7.81cm}\end{center}\end{figure}

Discussion and Conclusions

The 1992 Little Skull Mountain event is a well-constrained, highly double-couple earthquake with an $M_W$ of 5.6. The 1991 NTS nuclear test, BEXAR ($m_b$=5.6 and $M_S$=4.2, NEIC), has a significant positive isotropic component with an $M_W$ of 4.4. The deviatoric components of both events may be responding to the same general Basin and Range stress field of NW-SE extension. Analysis of $\sigma $ versus $k$ and the source vectors described above allows for an interpretation of the source with error. There are several sources of error in the moment tensor inversion, and the probabilistic method used in this study has the ability to incorporate those sources and produce empirical probability densities of the analyzed parameters (i.e., $\sigma $, $k$, and the source vectors). For example, several velocity models could be used to create the Greenís functions for the linear inversion. Each of the moment tensor solutions and their associated scatter density could then be plotted as in Figures 4-6. These types of plots would aid in the understanding of how parameterization choice nonlinearly affects the moment tensor solutions, and help map the solution space of Ďbest-fití moment tensors.

The analysis presented here shows that high quality solutions can be obtained for sparsely-recorded events at regional distances, and that these solutions have the potential to discriminate between volume changing (explosions) and double-couple (earthquakes) sources. In the future, we will test the sensitivity of the inversion to noise and non-ideal station spacing. We will also increase the population of moment tensors for man-made and natural events that deviate from the well recorded, large magnitude, small tectonic release cases presented here. Only an analysis of a wide range of events in different environments will allow for a true comparison of explosion and earthquake moment tensor populations.


Figures were made with Generic Mapping Tools (Wessell and Smith, 1998). This research is sponsored by the Department of Energy through the National Nuclear Security Administration, Office of Nonproliferation Research and Development, Office of Defense Nuclear Nonproliferation.


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