### 2.1 Basic equations

We assume that *P*-wave energy *W*^{P} and *S*-wave energy *W*^{S} are impulsively radiated from an isotropic source located at the origin in a two-dimensional isotropic scattering medium (Fig. 1). Strictly speaking, assumptions of isotropic source radiation and an isotropic scattering pattern are too simple. But they greatly help to understand the mathematical background and to provide a reference for more realistic (or complex) cases. On the way to receivers, energy is scattered and converted from *P*-to-*S* and *S*-to-*P*. Energy densities for *P* waves and *S* waves at location **x** and time *t*, which are denoted as *E*^{P}(**x**, *t*) and *E*^{S}(**x**, *t*), satisfy the following integral equation (e.g. Sato, 1994):

where *a*_{o}, *ß*_{0} are, respectively, the *P*-wave and *S*-wave velocity, and are total scattering coefficients for *P*-to-*P*, *P*-to-*S*, *S*-to-*P*, S-to-S scattering. *G*^{P}(**x**, *t*) is a propagator for *P*-wave energy which is characterized by the geometrical spreading factor of a reciprocal of *r =***|x|** and propagation with *P*-wave velocity (e.g. Shang and Gao, 1988; Sato, 1993) as:

Similarly, *G*^{S} (**x**, *t*) is a propagator for *S*-wave energy as:

where *b* is an intrinsic absorption coefficient which is assumed to be equal for both *P* and *S* waves here. Operating a Fourier transform with respect to space and a Laplace transform with respect to time on Eqs. (3) and (4), we can get the following two relations:

where ^ means Laplace transform, ~ means Fourier transform, and *k***=|k|**.

The Fourier-Laplace transform of Eqs. (1) and (2) leads to the following two equations:

These are the formal solutions in the wavenumber and frequency domain. However, the direct inverse Fourier-Laplace transform of Eqs. (7) and (8) asks for careful numerical treatments, as pointed out by Zeng (1993). So we take the different approach proposed by Sato (1994). We divide the total energy density into a direct wave part, a single-scattering part, and a multiple-scattering part having an order of higher than or equal to 2 as shown in Eq. (9) for *P* waves:

The direct *P* wave is radiated from the source as a *P* wave and reaches the receiver without any scattering:

The single scattering term is composed of two parts:

On the right-hand side, the first term is a *P* wave which is radiated from the source as a *P* wave, scattered once into a *P* wave, and reaches the receiver. The second term is a *P* wave which is radiated from the source as an *S* wave, scattered once into a *P* wave, and reaches the receiver.

The last term is the summation of energy which is radiated from the source as a *P* wave or an *S* wave and scattered more than twice and reaches the receiver as a *P* wave:

The corresponding expressions for *S* waves are as follows:

We assume the same attenuation for both *P* and *S* waves as:

This is a requirement for the analytical derivation of the single-scattering terms as shown in the next subsection and Appendix. The observed ratio of the attenuation of a *P* wave to that of an *S* wave for frequencies higher than 1 Hz mostly ranges between 0.7 and 2 in the lithosphere as shown in figure 5.3 of Sato and Fehler (1998). Therefore, Eq. (17) might be the first-order approximation. However, when this assumption does not hold in a strict sense, we need to make an inverse Fourier-Laplace transform of Eqs. (7) and (8) directly.

### 2.2 Analytical representation of the single-scattering terms

In principle, it is possible to estimate the single-scattering terms both by integration in the space-time domain and the inverse Fourier-Laplace transform in the wavenumber-frequency domain. We can obtain *P*-to-*P* and *S*-to-*S* single scatterings by both methods (e.g. Sato, 1993). However, concerning *P*-to-*S* and *S*-to-*P* single conversion scatterings, we have so far only succeeded in deriving the analytical expressions by integration in the space-time domain using elliptical coordinates (e.g. page 46 in Sato and Fehler, 1998; page 1195 in Morse and Feshbach, 1953). Here, we briefly summarize the results. The detailed derivation is shown in Appendix.

Four terms of *P*-to-*P*, *P*-to-*S*, *S*-to-*P*, *S*-to-*S* single-scattering energy densities are denoted as *E*^{PP}, ^{1}(**x**, *t*), *E*^{PS}, ^{1}(**x**, *t*), *E*^{SP}, ^{1}(**x**, *t*), *E*^{SS}, ^{1}(**x**, *t*) respectively. The *P*-to-*P* single-scattering term is expressed as:

This single-scattering term is equal to the scalar wave case with -wave velocity (e.g. Sato, 1993). The *S*-to-*S* single-scattering term can be derived similarly as:

Conversion scatteringterms of *P*-to-*S* and *S*-to-*P* aremore complex:

where

and

and *K* (*x*) is the complete elliptic integral of the first kind (e.g. page 590 in Abramowitz and Stegun, 1970). It is noted that this function also appears in the single scattering of scalar waves on a spherical surface (Maeda *et al.*, 2003).

We find that the *S*-to-*P* single-scattering term has the same space-time dependence as the *P*-to-*S* single-scattering term:

### 2.3 Total energy density of *P* waves and *S* waves

Evaluating the direct-wave terms and the single-scattering terms in the space-time domain, the total energy density of *P* waves is expressed as follows:

The third term in the right-hand side corresponds to the multiple-scattering *P* wave having an order higher than or equal to 2. This term depends only on *k*. The Bessel function of the first kind *J*_{0}(*kr*) originates from a two-dimensional Fourier transform. The term can be evaluated numerically using a discrete wavenumber summation (e.g. Bouchon, 1981; Zeng, 1993), equivalent to the trapezoidal rule, with respect to wavenumber, and the Fast Fourier Transform (FFT) with respect to frequency. A similar expression for *S* waves is as follows:

### 2.4 Self-consistency of the formulation

In order to check the self-consistency of the formulation, we consider the energy conservation of a *P* wave and an *S* wave. Substituting *k =* 0 into Eq. (7) leads to:

The inverse Laplace transform of this equation leads to:

Similarly,

We find that

When *b =* 0, the sum of the *P*-wave and *S*-wave energy is conserved, which means that our formulation is self-consistent.