Filtering Effect in Waveform Data Inversion for Seismic Source Process
Yuji Yagi (University of Tsukuba)
Zisin, 66(4), p. 147-149, 2014, https://doi.org/10.4294/zisin.66.147

The Japanese article was machine translated using DeepL. I hope that this article will help
you understand the meaning of applying filters when performing waveform inversion.
Yuji Yagi§1. Introduction
Since the waveform inversion for determining the source process was invented in the 1980s,
many researchers have developed analytical methods [e.g., Hartzell and Heaton (1983),
Beroza and Spudich (1988), Sekiguchi et al. (2000), Ji et al. (2002), Yagi and Fukahata (2011)].
Currently, program sources for classical analysis methods are distributed [e.g., Kikuchi andKanamori (2006)], and these methods have been applied to many earthquakes [e.g., Lay et al.
(2011)].
In many source process analyses, filters are often applied to the observed waveforms andGreen's functions to reduce modeling errors in the Green's functions or to remove noise.
However, it is known that the applied filters can change the solution [e.g., Cho and Nakanishi(2000)]. In this section, we discuss the meaning of applying filters in waveform inversion,using a simple case.

§2. source process analysis§2.1 Considerations in the time domain
Consider the case of fault slip on a fault plane Σ. For simplicity, the discussion will proceedwith the equation for data from a single point and a single component, but the same conclusion
can be reached for the case of multiple observation points and multiple components. The
observed seismic waveform u can be written as follows.

1u ( t ) = ∫ G ( ξ,t ) *D! ( ξ,t ) dΣ + e ( t ) .

(1)Σwhere G and D! are the Green's functions and the slip rate function on the fault plane Σ , ξis the coordinates on the fault plane, e is the observation error, and * is the convolution withrespect to time. The modeling error is assumed to be negligible. Applying the filter B ( t ) tothe above equation, we obtain

B ( t ) *u ( t ) = ∫ B ( t ) *G ( ξ,t ) *D! ( ξ,t ) dΣ + B ( t ) * e ( t ) .
Σ(2)The result is as follows. Now, discretizing the slip distribution of the fault in time and spaceand describing it vectorially, we obtain

Bu = BGm + Be .

(3)

B is the matrix in which the filter is run, an N × N matrix where N is the number of data, u
is the N-row data vector, G is the N × M kernel matrix where M is the number of model
parameters, m is the M-row model vector, and e is the N-row error vector. The least-squares
solution m̂ isT T−1m̂ = ( GT BT C −1d BG ) G B C d Bu ,−1(4)where C d is the covariance matrix of the filtered data Bu . Assuming that the observation
error is uncorrelated and Gaussian with mean zero, the data covariance matrix is

C d = σ 2 BBT ,(5)2where σ 2 is the variance of the observation error. Since the filter matrix is an N × N squarematrix, the least-squares solution ism̂ = ( GT G ) GT u ,−1(6)The least-squares solution is obtained independent of the filter employed. In other words, theleast-squares solution is obtained independently of the filter employed. This result holds evenwhen constraint conditions are imposed on the solution.
Many analyses treat the data covariance matrix as a diagonal matrix. This assumption
corresponds to the assumption that the data covariance matrix after applying the filter can beapproximated by a Gaussian distribution with mean zero and no correlation. In other words,
the data covariance matrix after applying the filter can be approximated by

C d = BeeT BT ≈ σ 2 I ,(7)where I is the identity matrix. If the number of data and the number of model parameters
are the same, then G is a square matrix, and if the inverse matrix exists, thenm̂ = ( GT BT BG ) GT BT Bu = G −1u .
−1(8)Even when the covariance of the errors is approximated as in equation (7), the result of theanalysis is independent of the filter employed.
If the filter is applied only to the observed waveform and not to the Green's function, weobtain a filtered slip time function [Nakano et al. (2008)]. The solution in this case is3m̂ = ( GT G ) GT Bu .
−1(9)§2.2 Considerations in the frequency domain
To simplify the problem, we assume a point epicenter. The observation equation when thefilter is applied at this time isB ( t ) *u ( t ) = B ( t ) *G ( ξ C ,t ) * M! 0 ( t ) + B ( t ) * e ( t ) .

(10)

The moment-rate function is where ξ C is the location of the point source and M! 0 is themoment rate function. When discretizing the moment rate function, the number of model
parameters and the sampling interval are the same as the number of data and the samplinginterval. Fourier transforming equation (10) yields

B! ( f ) u! ( f ) = B! ( f ) G! ( ξ C , f ) M!" 0 ( f ) + B! ( f ) e! ( f ) .

(11)

The following is an example of the Fourier transform. where f is the frequency and thesuperscript ~ denotes the Fourier transform. Since the moment rate function to be obtained isobtained by division, it can be understood more directly that the solution obtained isindependent of frequency.
In actual analysis, it is known that the solution varies depending on the filter applied. Why
do the solutions change depending on the filter applied? Many studies employ a condition toconstrain the solution by approximating the observation error after applying a filter with anuncorrelated Gaussian distribution with mean zero. Consider the Dumped Least Square
Solution (DLSS), which introduces the simplest constraint condition. If the seismic
waveforms at times outside the range of analysis are set to zero, the DLSS is obtained byminimizing:4

∫ { B (t ) * ⎡⎣G (ξ ,t ) ∗ M! (t ) − u (t )⎤⎦} dt + α ∫ M! (t ) dt .
∞−∞

2C

20∞

−∞

20(12)

where α 2 is a hyperparameter that controls the strength of the constraint. Using Perceval's
theorem, eq. (12) becomes

2∞∞!" ( f ) 2 df .
! ( f ) ⎡G! ( ξ C , f ) M!" 0 ( f ) − u! ( f ) ⎤ df + α 2BM∫−∞∫−∞ 0⎣⎦(13)

The solution can be obtained independently for each frequency. Therefore, the solution
corresponding to the DLSS isM"! 0 ( f ) =B" * ( f ) B" ( f ) G" * ( ξ C , f ) u" ( f ),B" * ( f ) B" ( f ) G" * ( ξ C , f ) G" ( ξ C , f ) + α 2(14)where the superscript * means complex conjugate. The DLSS solution corresponds to thesolution with water level correction [Gubbins (2004)]. In the frequency band removed by thefilter, the DLSS solution is more complex than B! ( f ) " 0M"! 0 ( f ) #0=0
0 +α2(15)Therefore, we obtain a solution that reflects only the DLSS constraint condition that the model
weights are zero. In other words, in the frequency band removed by the filter, the solution
obtained is determined by the constraint condition. As a result, the analytical results will varydepending on the filter employed. If we return to the time domain, we obtain a solution forthe moment rate function with filters similar to those applied to the observed waveform andthe Green's function. The same conclusion is obtained when a finite fault model is employed.
Also, the solution corresponding to the DLSS when the observation error before applying

5the filter is an uncorrelated Gaussian distribution with mean zero isM"! 0 ( f ) =G" * ( ξ C , f ) u" ( f )G" * ( ξ C , f ) G" ( ξ C , f ) + α 2(16)and the solution obtained is independent of the filter. In other words, the solution changes
depending on how the error model is assumed. The setting of the error model is one of themost important items when performing inverse analysis [e.g., Yagi and Fukahata (2011)].

§3. Conclusion
The result that the analytical results varied with the filter employed was caused by (1)the assumption that the observation error after applying the filter could be approximated byan uncorrelated Gaussian distribution with mean zero, and (2) the application of constraints.
Considering that the source image in the filtered frequency band is determined by theconstraints, it is desirable to use waves with the widest possible bandwidth to obtain an
appropriate source image. On the other hand, if a narrow bandwidth filter is applied to theDLSS, the moment rate function will have a filtered source image similar to the filter applied
to the observed waveform and the Green's function.
Although not discussed in this paper, further discussion of the properties of the momentrate functions obtained is warranted if smoothing and nonnegative constraints are applied anda simplified error model is assumed.

ACKNOWLEDGEMENTS
Discussions with Dr. Chen Ji of the University of California, Santa Barbara and Dr.
Kimiyuki Asano of the Disaster Prevention Research Institute, Kyoto University were very
helpful in preparing this manuscript. Dr. Haruko Sekiguchi, an anonymous reviewer, and Dr.
Takanori Matsuzawa, a member of the editorial board, provided useful comments that
contributed to the improvement of the paper. I would like to express my gratitude to them.

6This research was supported by a Grant-in-Aid for Scientific Research from the Ministry ofEducation, Culture, Sports, Science and Technology of Japan (Grant-in-Aid for Scientific
Research No. 24540450, 24101012).

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