Improved Full oment Tensor Inversions Sarah E. inson and Douglas S. Dreger bstract The seismic moment tensors for certain types of sources, such as volcanic earthquakes and nuclear explosions, are expected to contain tropic components. The best method for determining moment tensors is linear moment tensor inversions. We have corrected a regional distance formulism, and this correction improves the recovery of non-double-couple moment tensors which include an tropic component. Tests with synthetic data show that our new methodology is more stable than the existing inversion approach. pplying these methods to both volcanic earthquakes and nuclear explosions, we have been able to produce full moment tensor mechanisms which fit the data very well. These inversions reveal that the nuclear explosions have strikingly anomalous source mechanisms which contain very large tropic components, making it evident that these events are not tectonic in origin. This indicates that moment tensor inversions could be an important tool for nuclear monitoring. Introduction Earthquake source mechanisms are routinely determined through moment tensor inversions. This process requires that the synthetic seismograms be represented as the linear combination of fundamental Green s functions where the weights on these Green s functions are the individual moment tensor elements. n analytical representation of this system for a general moment tensor,, was derived in Jost and Herrmann (989), based on the work of Langston (98) for a deviatoric (zero trace) moment tensor. However, an error in the Jost and Herrmann (989) derivation precludes their moment tensor inversion scheme from correctly recovering source mechanisms which include tropic components. In this paper, we present an improved moment tensor methodology which we have used to calculate more accurate moment tensors for several volcanic and nuclear explosion sources. Improved inversion design deviatoric point source can be represented by using Green s functions for three fundamental faults: a vertical strike-slip fault, a vertical dip-slip fault, and a dip-slip fault with a dip of 45 (Langston, 98). But for a complete moment tensor, we must also include the explosion Green s functions, so that Equation u u u Z R T = ZSS + ZDS + ZDD = RSS + RDS + RDD = TSS + TDS 4 5 ZEP REP where u is the surface displacement, SS is the vertical strike-slip Green s function, DS is the vertical dip-slip Green s function, DD is the 45 dip-slip Green s function, and EP is the explosion Green s function. Z, R, and T refer to the vertical, radial, and tangential components, respectively; and Equation tr( ) =
The i coefficients for a deviatoric source are given by, Equation = ( )cos(az) cos( az) sin( az) = ( ) 4 = ( )sin(az) = cos( az) sin( az) 5 sin(az) cos(az) where ij are the elements of the moment tensor, and az is the source-receiver azimuth. It should be noted that since the seismic moment tensor is symmetric, it has only six independent elements. Equation is the same as Equation 5. in Jost and Herrmann (989), which in turn is Equation in Langston (98) transformed into a coordinate system where z is positive up and the positive tangential direction is measured clockwise from north. Since any tensor can be described as the sum of deviatoric and tropic tensors, we describe the synthetic seismograms as the sum of deviatoric and tropic synthetic seismograms. For the deviatoric synthetic seismograms, we retain the Langston (98) formulas (Equation ). However, we replace the i coefficients with i, which are the i coefficients calculated from the elements of the deviatoric part of the full moment tensor, ij where, Equation 4 Substituting ij into Equation, we obtain
Equation 5 4 5 = = = ( 6 = 4 = 5 ) Note that for a deviatoric moment tensor, for which Equation must hold, Equation 5 is identical to Equation. Substituting Equation 5 into Equation and rearranging, we find Equation 6 u z Equation 7 u r ZSS ZDD ZEP cos(az) + 6 ZSS ZDD ZEP cos(az) + 6 ZDD ZEP + [ ZSS sin(az) [ ZDS cos( az) [ ZDS sin( az) RSS RDD REP cos(az) + 6 RSS RDD REP cos(az) + 6 RDD REP + [ RSS sin(az) [ RDS cos( az) [ RDS sin( az)
Equation 8 TSS ut sin(az) TSS sin(az) [ TSS cos(az) [ TDS sin( az) [ TDS cos( az) Note that these equations differ from Equations 5.4-5.6 in Jost and Herrmann (989). To illustrate why Equation 6-Equation 8 are preferable consider a purely tropic source, Equation 9 = = 0 Intuitively, the synthetic seismograms for this source should be, Equation 0 uz ur u = 0 t ZEP REP which is exactly obtained by Equation 6-Equation 8. However, the Jost and Herrmann (989) equations produce, Equation u z ur u = 0 t [ ZEP ZDD [ REP RDD Thus, if a non-zero trace moment tensor is used in the Jost and Herrmann (989) equations, then the tropic component is not only used to weight the tropic Green s functions ZEP and REP, but also the ZDD and RDD deviatoric Green s functions, and a full moment tensor inversion is incapable of accurately recovering an tropic component. Our new inversion system eliminates that problem. Tests with synthetic data To test the efficacy of these new equations for moment tensor inversions, and to compare the results of such inversions to those done implementing the original Jost and Herrmann (989) equations, we inverted synthetic data for many different source mechanisms. We used Green s functions calculated with frequency-wavenumber integration (Saikia, 994) for a -D velocity structure, and we bandpass filtered these Green s functions between 0 and 50 s period. The synthetic data were constructed using the same Green s functions and filtering, and the synthetic data were inverted using linear time domain moment tensor inversions (Pasyanos et al., 996; Fukuyama and Dreger, 000). 4
Figure is a sample comparn of inversions of deviatoric and non-deviatoric synthetic data. For deviatoric source mechanisms, the inversion system of Jost and Herrmann (989) and this paper result in the same mechanisms, as is expected theoretically. When we add an tropic component to those same synthetic data, the Jost and Herrmann (989) method yields a very different mechanism from the deviatoric inversion. In contrast, our method produces the same deviatoric mechanism plus an tropic component, as is predicted theoretically. pplication to real data Using this same inversion methodology for actual data from several real non-double-couple sources, we have compared the results of full moment tensor inversions based on Jost and Herrmann (989) to those implementing the equations introduced in this paper. In particular, we have focused on three case studies: three Nevada Test Site (NTS) nuclear explosions (BEXR, ONTELLO, and JUNCTION) in 99 and 99, the 997 ammoth earthquake swarm in the Long Valley caldera, and the earthquake swarm associated with the iyakejima, Japan volcanic eruption in 000 (see Table ). These NTS explosions were previously studied by Dreger and Woods (00), who found that they had very anomalous mechanisms which could be used to help distinguish them from naturally occurring earthquakes. The Dreger and Woods (00) mechanisms did not have significant tropic components, but instead had large vertical negative compensated-linearvector-dipole (CLVD) components (Knopoff and Randall, 970), which is consistent with the nuclear explosion mechanisms reported in Patton (988). This inability to recover the explosive nature of the source could be due to free-surface effects; and other studies have found that tropic components of shallow events are difficult to constrain (e.g. Patton, 998; Dufumier and Rivera, 997). However, after applying our new equations (Equation 6-Equation 8) it appears that these events actually contain large resolvable tropic components (Figure, Figure ). These new mechanisms clearly identify BEXR, ONTELLO, and JUNCTION as being principally explosive and non-tectonic in origin. In addition to the NTS explosions, we studied volcanic earthquakes from the Long Valley caldera, California (OTH in Figure ; EVT in Dreger et al., 000) and from iyakejima, Japan (IYKEJI in Figure ). Using the old moment tensor inversion method, both of these events have mechanisms which are combinations of normal faulting, CLVD components, and tropic components. Using our new methodology, the OTH mechanism behaves very similarly to the NTS explosion mechanisms: the new mechanism is comprised mostly of double-couple and tropic (DC+ISO) components with a much smaller CLVD component. This is consistent with other studies which found that regional data for events in the Long Valley caldera are best fit by DC+ISO mechanisms (Templeton and Dreger, 005). The iyakejima event seems somewhat different from the other events studied. The new mechanism is very complicated, and the CLVD component actually increases. However, the moment does decrease relative to the deviatoric mechanism. Deleted: We evaluate how well our source mechanisms fit the data with the variance, Equation σ = ( i j k data ijk N and the variance reduction (VR), Equation VR = i j k synth ( data ijk i j k s da where N is the number of data points (0 per trace), is the number of free parameters, and i, j, and k refer to station, seismogram component, and time sample, respectively. For each event, we examine multiple moment tensor inversions with different source depths. Because of this, we consider the depth of the source to be a variable in addition to the elements of the moment tensor. So the number of free parameters is six for a deviatoric inversion, and seven for a full moment tensor inversion. Discussion and conclusions In order to study the source mechanisms of complex sources such as volcanic earthquakes and nuclear explosions, we must be able to determine all six elements of the seismic moment tensor. We have derived a system of equations (Equation 6-Equation 8) in which the deviatoric and tropic parts of the moment tensor are separated. Inversions of synthetic data (Figure ) show that this system of equations is capable of accurately recovering tropic components. These equations supercede those for the proposed inversion scheme in Jost and Herrmann (989). 5
pplying these new equations to several volcanic earthquakes and nuclear explosions, we find that these new equations lead to mechanisms with larger tropic components, smaller CLVD components, and smaller total moments, than the mechanisms determined using the old inversion scheme (Figure ). In general, the new mechanisms appear to be predominantly DC+ISO. One concern with full moment tensor inversions is that they can have much larger moments than deviatoric moment tensor inversions for the same event. Such increases in moment can be caused by trade-offs between elements of the moment tensor, and thus indicate that the full moment tensor solution may be invalid (Dreger and Woods, 00). For all of the events discussed here, our new inversion system results in moments which are the same as or smaller than the moments from the old full moment tensor inversions (Figure ). Remarkably, the full moment tensor inversions for the three NTS explosions result in smaller moment magnitudes than those derived from deviatoric moment tensor inversions (Table ), thereby increasing the L : 0 differential measure, which has been proposed as a regional discriminant (Woods et al., 99). For the three explosions in questions the L - W differential ranges from.0 to.6 magnitude units. Our inversions of the NTS explosions (BEXR, ONTELLO, and JUNCTION) have very large tropic components. Unlike the findings from some previous studies of nuclear explosions, the source mechanisms of these events truly look like explosions. Dreger and Woods (00) note that the NTS explosions were ideally located since they were at regional distances from large seismic networks and had well-studied travel paths. oment tensor inversions of nuclear explosions in regions with poorly understood travel paths and low quality or scarce data may not be able to resolve the tropic part of the source mechanism. But the results presented in this study indicate that full moment tensor inversions could be extremely useful in nuclear monitoring. Regardless of its value for nuclear monitoring, the inversion scheme presented in this paper results in improved full moment tensor inversions, and this, in turn, will lead to a better understanding of complex source mechanisms. cknowledgements We used data from broadband stations in the TERRscope network and the Berkeley Digital Seismic Network for events located in California and Nevada, and data from the F-net Broadband Seismograph Network for the iyakejima earthquake. References ki, K., and P. G. Richards, Quantitative Seismology, second edition, University Science Books, Sausalito, 00. Dreger, D., and B. Woods, Regional distance seismic moment tensors of nuclear explosions, Tectonophysics, 56, 9-56, 00. Dreger, D. S., H. Tkalcic, and. Johnston, Dilational processes accompanying earthquakes in the Long Valley Caldera, Science, 88, -5, 000. Dufumier, H., and L. Rivera, On the resolution of the tropic component in moment tensor inversion,, 595-606, 997. Fukuyama, E., and D. S. Dreger, Performance test of an automated moment tensor determination system for the future Tokai earthquake, Earth Planets Space, 5, 8-9, 000. Helmberger, D. V., Theory and application of synthetic seismograms, Earthquakes: observation, theory and interpretation, edited by H. Kanamori and E. Boschi, pp. 74-, North- Holland, 98. Formatted: Font: Italic 6
Jost,. L., and R. B. Herrmann, student s guide to and review of moment tensors, Seismol. Res. Lett., 60(), 7-57, 989. Knopoff, L., and. J. Randall, The compensated linear-vector dipole: possible mechanism for deep earthquakes, J. Geophys. Res., 75(6), 4957-496, 970. Langston, C.., Source inversion of seismic waveforms: The Koyna, India, earthquakes of September 967, Bull. Seismol. Soc. m., 7(), -4, 98. Pasyanos,. E., D. S. Dreger, and B. Romanowicz, Toward real-time estimation of regional moment tensors, Bull. Seismol. Soc. m., 86(5), 55-69, 996. Patton, H. J., Source models of the HRZER explosion from regional observations of fundamental-mode and higher mode surface waves, Bull. Seismol. Soc. m., 78(), - 57, 988. Saikia, C. K., odified frequency-wavenumber algorithm for regional seismograms using Filon's quadrature: modelling Lg waves in eastern North merica, Geophys. J. Int., 8, 4-58, 994. Templeton, D. C., and D. S. Dreger, (005). Source Processes of Long Valley Caldera Seismicity Determined by oment Tensor Inversion, submitted to Bull. Seismol. Soc. m. Woods, B. B., S. Kedar, and D. V. Helmberger, L : 0 as a regional seismic discriminant, Bull. Seismol. Soc. m., 8(4), 67-8, 99. Seismological Laboratory California Institute of Technology Pasadena, California 95 Department of Earth and Planetary Science University of California, Berkeley Berkeley, California 9470 Table. Hypocenter information Event name Description Origin Time (UT) Latitude Longitude L a W b W BEXR Nuclear explosion 04/04/99 9:00:00.0 7.96-6. 5.6 4. c 4.0 ONTELLO Nuclear explosion 04/6/99 5:0:00.0 7.45-6.44 5.4 4.5 c 4.4 JUNCTION Nuclear explosion 0/9/99 6:0:00.0 7.7-6.60 5.5 4.4 c 4. OTH Volcanic earthquake //997 7:0:5. d 7.66 d -8.96 d 4.9 4.8 e 4.9 IYKEJI Volcanic earthquake 07/0/000 07:0:56. f 4.87 f -9.97 f 6. 6. a oment magnitudes determined from deviatoric moment tensor inversions b oment magnitudes determined from full moment tensor inversions using corrected equations c Dreger and Woods (00) d dvanced National Seismic System (NSS) composite catalog e Dreger et al. (000) f Japan eteorological gency (J) catalog 7
Tangential Radial Vertical Tangential Radial Vertical Strike= ; 7 5._0._d8.data,0 ax mp=8.88e-07 cm VR=00.0 5._45._d8.data,45 ax mp=4.50e-07 cm VR=00.0 5._90._d8.data,90 ax mp=7.8e-07 cm VR=00.0 Rake =45 ; 49 Dip =67 ; 49 5._0._d8.data,0 o =.75e+0 w =.8 Percent DC=50 Percent CLVD=50 Percent ISO=0 ax mp=8.88e-07 cm VR=00.0 5._45._d8.data,45 ax mp=4.50e-07 cm VR=00.0 Strike= ; 7 Rake =45 ; 49 Dip =67 ; 49 o =.75e+0 w =.8 Percent DC=50 Percent CLVD=50 Percent ISO=0 Variance=4.79e-6 Var. Red=.00e+0 5._90._d8.data,90 ax mp=7.8e-07 cm VR=00.0 RES/Pdc.=9.58e-8 Variance=9.05e-6 Var. Red=.00e+0 RES/Pdc.=.8e-7 5._95._d8.data,95 ax mp=9.0e-07 cm VR=00.0 5._95._d8.data,95 ax mp=9.0e-07 cm VR=00.0 T T P P B Tangential Radial Vertical Tangential 5._0._d8.data,0 ax mp=8.88e-07 cm 5._45._d8.data,45 ax mp=5.90e-07 cm VR=00.0 5._90._d8.data,90 ax mp=7.8e-07 cm VR=00.0 VR=00.0 Strike=6 ; 4 Rake =-79 ; -5 Dip =67 ; 6 o =.9e+0 w =.8 Percent DC=4 Vertical 5._0._d8.data,0 ax mp=8.88e-07 cm 5._45._d8.data,45 ax mp=5.90e-07 cm VR=00.0 Percent CLVD=4 Percent ISO=4 Variance=4.9e- Var. Red=.00e+0 Radial 5._90._d8.data,90 ax mp=7.8e-07 cm VR=00.0 VR=00.0 RES/Pdc.=.0e-4 Strike= ; 7 Rake =45 ; 49 Dip =67 ; 49 o =.75e+0 w =.8 Percent DC= Percent CLVD= Percent ISO=6 Variance=.85e- Var. Red=.00e+0 RES/Pdc.=.e-4 5._95._d8.data,95 ax mp=9.0e-07 cm VR=00.0 5._95._d8.data,95 ax mp=9.0e-07 cm VR=00.0 P T T P C D Figure. Comparn of moment tensor inversions of synthetic data using the equations from Jost and Herrmann (989) ( and C) with the ones introduced in this paper (B and D). and B are inversions of deviatoric mechanisms comprised of a double-couple component (strike, rake 45, dip 67 ) and a compensated-linear-vector-dipole component with the same principle axes as the double-couple component. Both inversions correctly recover the input mechanism (=-0.96, =0.895, =0.04, =-0.74, =-., =0.8 in 00 dyne-cm). C and D are the same as and B, except that an tropic component had been added to the synthetic data (=0.076, =0.895, =.04, =-0.74, =-.9, =.8 in 00 dyne-cm). The Jost and Herrmann (989) method (C) results in a completely different mechanism. In contrast, the method introduced in this paper (D) recovers the correct deviatoric mechanism as well as the added tropic component.
Earthquake location BEXR Nuclear explosion Seismic station OTH IYKEJI Figure. Results of full moment tensor inversions of real data. These mechanisms were computed using the moment tensor inversion method presented in this paper, and they produce synthetics which fit the observed waveforms quite well. comparn of these mechanisms to those determined with the Jost and Herrmann (989) methodology is presented in Figure. We evaluate how well our source mechanisms fit the data with the variance, σ, and the variance reduction VR, which are defined as σ = (data i j k ijk syntheticijk ) N (dataijk syntheticijk ) i j k VR = 00% data ijk i j k where N is the number of data points (0 per trace), is the number of free parameters, and i, j, and k refer to station, seismogram component, and time sample, respectively. For each event, we examine multiple moment tensor inversions with different source depths. Because of this, we consider the depth of the source to be a variable in addition to the elements of the moment tensor. So the number of free parameters is six for a deviatoric inversion, and seven for a full moment tensor inversion.
ethodology of Jost and Herrmann (989) This paper Event oment tensors in dyne-cm (Deviatoric) [ethodology of Jost and Herrmann (989) <ethodology of this paper> P-wave radiation pattern W ( L ) P-wave radiation pattern W BEXR (7.8, -0.7, -96., 46.8, -59, -49.6) 0 0 [8., -05, -6.5, 908.4, -9., -48.4 0 0 <98.8, -05, -6.5, 474, -9, 40.4> 0 0 4.5 (5.6) 4.0 ONTELLO (94, -4.8, -0.5, 98., -0.7, -49.) 0 0 [040., -4.6, -7.8, 45.6, -6.7, -56. 0 0 <48.7, -4.6, -7.8, 589, -6.6, 597.> 0 0 4.6 (5.4) 4.4 JUNCTION (8.4, -6.9, -.6,., 8.4, -40.6) 0 0 [90.6, -8.4, -4.7, 988, 8.4, -6.7 0 0 <9.4, -8.4, -4.7, 478.7, 8.4, 655.9> 0 0 4.5 (5.5) 4. OTH (-406, 476.9, -8.4, 846.8, -76.4, -440.8) 0 0 [45.6, 55.8, -46., 60., -8.6, -5.8 0 0 4.9 4.9 <.7, 55.8, -46., 6., -8.6, 44.> 0 0 IYKEJI (44., 8., -585.9, 470.7, -86.6, -89495.) 0 [57., 84., -59., 400.7, -807.8, -750.4 0 6. 6. <688.4, 84., -59., 7.8, -807.8, 58646.> 0 Figure. Comparn of full moment tensor inversions using the methodology of Jost and Herrmann (989) to those computed using the new methodology proposed in this paper. oment tensors are given as,,,,, using ki convention (ki and Richards, 00). Our new inversion methodology produces mechanisms for the three nuclear explosions (BEXR, JUNCTION, and OTH) which are substantially more explosive in character than the old mechanisms. These new mechanisms also have smaller W, indicating that the L : 0 discriminant (Woods et al., 99) would be useful in this situation.