Radiative transfer of elastic waves in two-dimensional isotropic scattering media: Semi-analytical approach for isotropic source radiation
We formulate the radiative transfer of P- and S-wave energy from an isotropically radiating source in a two-dimensional infinite isotropic scattering medium. For a stable numerical calculation of seismogram envelopes, we take a semi-analytical approach: Energy densities of P and S waves are divided into three parts of the direct-wave terms, the single-scattering terms, and the multiple-scattering terms, and the first two terms and the last term are evaluated analytically and numerically, respectively. Concerning the single-scattering terms, the P- to-S and S-to-P single conversion scattering terms are expressed analytically with a complete elliptic integral of the first kind. The multiple-scattering terms are represented by a double integral with respect to frequency and wavenumber, and can be numerically evaluated by a discrete wavenumber summation and a Fast Fourier Transform. The results based on the numerical implementation are confirmed with an independent numerical calculation using the Monte Carlo method. Our formulation is also applied to consider the equilibration between P and S waves at larger lapse times. The equilibrated S-to-P energy ratio is reproduced and the equilibration time is first derived for two-dimensional cases. Our formulation will be a reference for the understanding of more complex cases.
Key wordsRadiative transfer two-dimensional S-to-P energy ratio equilibration
Seismogram envelopes have been used to estimate the strength of scattering and absorption in the Earth. Synthesizing realistic seismogram envelopes is needed to model observed envelopes and to invert for the strength of scattering and absorption in the Earth. Radiative transfer theory, which was originally developed in astrophysics (e.g. Chandrasekhar, 1960), is widely used in seismology, because it offers clear physical meanings and mathematical tractability. Various kinds of settings have been modeled using radiative transfer theory in seismology. For more detailed developments, readers can refer to Sato and Fehler (1998). Modeling started from the single scattering of scalar waves for an isotropic source and isotropic scattering in three-dimensional media (e.g. Aki and Chouet, 1975; Sato, 1977a), and in two-dimensional media (e.g. Kopnichev, 1975). Later, multiple scattering was included in the modeling for the three-dimensional case by Wu (1985) and Zeng et al. (1991), and for the two-dimensional case by Shan and Gao (1988). Vector waves (or P waves and S waves) were first considered for the single-scattering case by Sato (1977b) and for the multiple scattering case by Zeng (1993) and Sato (1994), all of which treated three-dimensional cases. However, concerning two-dimensional cases, we have only found a paper by Trégourès and van Tiggelen (2002) which derived a radiative transfer equation in a thin-plate bounded by two free surfaces. A canonical problem for a two-dimensional full space has not been found.
The partitioning of P waves and S wavesisanimportant subject. The S-to-P energy ratio is known to be equilibrated after multiple conversion scatterings between P waves and S waves. The equilibrated ratio was elegantly derived for three-dimensional full spaces by Weaver (1982), and, later, for two-dimensional full spaces by Sánchez-Sesma and Campillo (2006). To consider the transition process into the equilibrium, radiative transfer theory for vector waves is necessary; see Zeng (1993), Sato (1994) and Ryzhik et al. (1996). Snieder (2002) proposed a simpler ball-counting method. The S-to-P energy ratio can be used to judge whether a wavefield is in a diffuse state (multiple scattering regime) or not. Shapiro et al. (2000) conducted the first seismic observation in this context and concluded that seismic coda is in a diffuse state. Sánchez-Sesma and Campillo (2006) clarified that the S-to-P energy ratio is also important to retrieve strict Green’s tensors from cross-correlations of noise records using seismic interferometry, which is a rapidly-growing field in seismology, acoustics, exploration geophysics, etc. (e.g. Curtis et al., 2006).
In this study, we formulate the radiative transfer of elastic waves in a two-dimensional full space for the first time. Simple assumptions on the isotropy of source radiation and scattering enable us to derive semi-analytical expressions: the direct-wave terms and the single-scattering terms are analytically evaluated, and the multiple-scattering terms having an order higher than or equal to two are numerically calculated. The numerical implementation of our formulation is tested and confirmed by comparisons with the purely numerical approach of a Monte Carlo simulation. Finally, we discuss the energy partitioning of a P wave and an S wave in two-dimensional media as an application of our formulation.
2.1 Basic equations
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.
2.3 Total energy density of P waves and S waves
2.4 Self-consistency of the formulation
3. Numerical Calculation
3.1 Numerical implementation of the formulation
3.2 Monte Carlo simulation to confirm the numerical implementation
It is necessary to check the numerical implementation of our formulation. For the calculation of seismogram envelopes, a Monte Carlo simulation (e.g. Gusev and Abubakirov, 1987; Hoshiba, 1991) is versatile enough to be applicable to any type of random media and source radiation at the cost of computation time. In this study, we use a two-dimensional version of Yoshimoto (2000), which is equivalent to Przybilla et al. (2006). For the calculation, we emanate 1,000,000 particles from the source. On the way to receivers, scatterings can take place probabilistically according to the assumed medium parameters. The time step is assumed to be 0.0167 s from Open image in new window. We count the number of particles arriving at a receiver and at a time. We can trace each particle from the source to the receiver by keeping track of the number of scatterings. So we can extract the single-scattering terms and the multiple-scattering terms separately even with the Monte Carlo method. The resultant envelopes are also normalized according to Eq. (31) for comparison with the envelopes based on our formulation.
We have succeeded in formulating the elastic radiative transfer for a two-dimensional random full space for the first time. This is a canonical problem. It is not easy to find practical applications. However, our formulation is useful from the viewpoint of the physics of wave propagation in random media. Our formulation makes it possible to understand directly the effect of dimensionality on the radiative transfer of elastic waves by comparison with formulations for three-dimensional cases. Another benefit of our formulation concerns the validity of seismic interferometry as discussed in this section.
Recently, the partitioning of P-wave and S-wave energy draws attention to the applicability of seismic interferometry, as shown by Sánchez-Sesma and Campillo (2006). This is also of importance in investigating which is predominant between single scattering and multiple scattering in seismic coda (e.g. Shapiro et al., 2000). Intensive studies have been carried out so far in three-dimensional scattering media (e.g. Margerin et al., 2000). Our formulation enables us to quantitatively investigate the partitioning problem in two-dimensional scattering media, which has scarcely been considered.
We have formulated the radiative transfer of P-wave and S-wave energy from an isotropic source in a two-dimensional isotropic scattering media. For a stable numerical calculation of seismogram envelopes, we divide energy densities of P waves and S waves into three parts: the direct-wave terms, the single-scattering terms, and the multiple-scattering terms having an order of higher than or equal to 2. The direct-wave terms and the single-scattering terms can be evaluated analytically. Analytical expressions for the single scatterings are obtained in the spacetime domain using elliptic coordinates. Especially, the single conversion scatterings of P-to-S and S-to-P have been discovered to be represented by a complete elliptic integral of the first kind. Multiple scattering terms can be expressed by double integration with respect to frequency and wavenumber. Discrete wavenumber summation with respect to wavenumber, and FFT with respect to frequency, is adopted. The results based on the numerical implementation of our formulation have been validated by an independent numerical calculation using the Monte Carlo method. Using our formulation, we investigated the equilibration between P-wave and S-wave energy in a two-dimensional random full space. The equilibrated value of the S-to-P energy ratio has been reproduced and the equilibration time has been derived, for the first time, for two-dimensional media. Though our formulation is made for a canonical problem imposing simple assumptions of an isotropic source radiation and an isotropic scattering pattern, it enables us to see the effect of dimensionality on the radiative transfer of elastic waves. It is also helpful in understanding the mathematical background of radiative transfer theory.
Constructive comments from the editor Dr. Tatsuhiko Hara and an anonymous reviewer are greatly acknowledged. We are very grateful to the other reviewer, Dr. Jun Kawahara, for pointing out many typing errors left in the original manuscript. We used Generic Mapping Tools (Wessel and Smith, 1998). This study was supported by Grant-in-Aid for Scientific Research (C) (20540413) and for Young Scientists (B) (20740248) from the Japan Society for the Promotion of Science (JSPS) and the Ministry of Education, Culture, Sports, Science and Technology (MEXT). We also thank JSPS and Deutsche Forschungsgemeinschaft (DFG) for support under the Japan-Germany Research Cooperative Program.
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