Finite Boundary Perturbation Theory for the Elastic Equation of Motion

Nozomu Takeuchi


The crust is the most heterogeneous region of the Earth, and accurate crustal correction (i.e., accurate computation of the perturbation of synthetic seismograms caused by the crustal heterogeneities) is critical for obtaining accurate mantle structure models. The heterogeneities are usually represented by the summation of (i) the perturbation of physical properties (such as density and elastic constants) in the internal regions and (ii) the perturbation of the location of the boundaries (such as Moho and the surface), and the principal difficulty is how to compute the effect of the latter perturbation.

The computational method applied to the actual waveform inversion studies for global 3-D Earth structure thus far has been either the modal summation method (e.g., Li and Tanimoto, 1993) or the Direct Solution Method (e.g., Takeuchi et al., 2000). Both methods solve the weak form equation of motion (or its equivalence) and use vector spherical harmonics as the laterally dependent part of the trial functions. For those global trial functions, severe limitations still exist in computing the perturbation of synthetic seismograms caused by the perturbation in the location of the boundaries, because previous solutions rely on the first order perturbation theory of the free oscillation (hereafter referred as 1DT; Woodhouse, 1980). Thus, this method breaks down for strongly heterogeneous medium or for higher frequencies.

In this study, we derive the exact weak form equation of motion for the medium with finite boundary perturbations. This method can be applied to arbitrary trial functions; that is, to both global and local trial functions. We can solve the derived equation of motion by either direct solution or higher order perturbation approximations, which allows highly accurate synthetic seismograms. Hereafter we refer this solution as the finite boundary perturbation theory or FPT.

Numerical Examples

Figure: (a) The path on which the SH wave propagations are simulated (thick solid line). The star shows the location of the source. Surface (or bathymetry) topography of Crust 2.0 model (Bassin et al., 2000) are overlapped by black-and-white color scale. (b) The crustal topography model used as the perturbed model in this simulation to represent the Crust 2.0 model on the thick solid line in Figure 31.1a. The star shows the location of the source used with this perturbed model.
\epsfig{file=takeuchi_1.eps, width=16cm}\end{center}\end{figure*}

We show that in the simulations of realistic problems, a breakdown can be observed in 1DT synthetic seismograms. We consider a plane of a great circle including the path shown by the solid line in Figure 31.1a, and simulate SH waves propagating on this plane.

We compute synthetic seismograms for the initial model (isotropic PREM) and the perturbed model (Figure 31.1b), and show the record sections in Figure 31.2. For the perturbed model, we compute by using FPT and 1DT, respectively. We apply a band pass filter with the corner frequencies of $1/1000$ and $1/50$ Hz. In the record section for the initial model (Figure 31.2 left), as well as the body waves observed at the first motion part, we observe the Love waves traveling at a speed of about $4.4$ km/s (aligning almost straight in the record section). Their waveforms are almost one wave packet and do not clearly show dispersion. On the other hand, in the record section computed for the perturbed model by using FPT (Figure 31.2 middle), we observe Love waves with clear dispersion. This is a well-known feature of Love waves traveling through a continent. However, in the record section computed for the perturbed model by using 1DT (Figure 31.2 right), we cannot clearly see dispersion, an indication that 1DT breaks down for this frequency range.


In the numerical examples we showed that 1DT breaks down for surface waves with a period of $50$ seconds in a realistic problem. In recent waveform inversion studies (e.g., Mégnin and Romanowicz, 2000; Takeuchi and Kobayashi, 2004), the body waves for this frequency range are used as a data set, but the surface waves for this frequency range are excluded. This is mainly due to the insufficient accuracy of 1DT for computing the effect of crustal heterogeneities. Our method can compute accurate synthetic seismograms for arbitrary frequency ranges, and should be better able to retrieve the information in the surface waves of higher frequencies.


This research was partly supported by grants from the Japanese Ministry of Education, Culture, Sports, Science, and Technology (No. 16740249).


Bassin, C., G. Laske, and G. Masters, The current limits of resolution for surface wave tomography in North America, EOS Trans. Amer. Geophys. Un., 81, F897, 2000.

Li, X.D. and T. Tanimoto, Waveforms of long-period body waves in a slightly aspherical Earth model, Geophys. J. Int., 112, 92-102, 1993.

Mégnin, C. and B. Romanowicz, The three-dimensional shear velocity structure of the mantle from the inversion of body, surface and higher-mode waveforms, Geophys. J. Int., 143, 709-728, 2000.

Takeuchi, N., R.J. Geller and P.R. Cummins, Complete synthetic seismograms for 3-D heterogeneous Earth models computed using modified DSM operators and their applicability to inversion for Earth structure, Phys. Earth and Plan. Int., 119, 25-36, 2000.

Takeuchi, N. and M. Kobayashi, Improvement of Seismological Earth Models by Using Data Weighting in Waveform Inversion, Geophys. J. Int., 158, 681-694, 2004.

Woodhouse, J.H., The coupling and attenuation of nearly resonant multiplets in the Earth's free oscillation spectrum, Geophys. J. R. Astr. Soc., 61, 261-283, 1980.

Figure 31.2: The record section computed for the initial model (left), for the perturbed model by using FPT (middle), and for the perturbed model by using 1DT (right). The vertical axis shows the distance from the epicenter, and the horizontal axis shows the reduced time by $4.4^{-1}$ s/km. The amplitude is normalized by the maximum amplitude of each trace.
\epsfig{file=takeuchi_2.eps, width=8cm}\end{center}\end{figure}

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