Frozen Gaussian approximation for 3D seismic tomography

Abstract

Three-dimensional (3D) wave-equation-based seismic tomography is computationally challenging in large scales and high-frequency regime. In this paper, we apply the frozen Gaussian approximation (FGA) method to compute 3D sensitivity kernels and seismic tomography of high-frequency. Rather than standard ray theory used in seismic inversion (e.g. Kirchhoff migration and Gaussian beam migration), FGA is used to compute the 3D high-frequency sensitivity kernels for travel-time or full waveform inversions. Specifically, we reformulate the equations of the forward and adjoint wavefields for the purpose of convenience to apply FGA, and with this reformulation, one can efficiently compute the Green's functions whose convolutions with source time function produce wavefields needed for the construction of 3D kernels. Moreover, a fast summation method is proposed based on local fast Fourier transform which greatly improves the speed of reconstruction as the last step of FGA algorithm. We apply FGA to both the travel-time adjoint tomography and full waveform inversion (FWI) on synthetic crosswell seismic data with dominant frequencies as high as those of real crosswell data, and confirm again that FWI requires a more sophisticated initial velocity model for the convergence than travel-time adjoint tomography. We also numerically test the accuracy of applying FGA to local earthquake tomography. This study paves the way to directly apply wave-equation-based seismic tomography methods into real data around their dominant frequencies.

Frozen Gaussian approximation for 3-D elastic wave equation and seismic tomography

“…then use FGA to compute the forward and adjoint wavefields and construct 3-D kernels of different phases by the methods of wave-equation-based traveltime tomography Liu & Gu 2012) and FWI (Pratt & Shipp 1999;Virieux & Operto 2009). Since FWI requires a more sophisticated initial velocity model for the convergence than wave-equation-based traveltime tomography, we use a hierarchical strategy as in our previous work (Chai et al 2018), which first uses waveequation-based traveltime tomography to create a macro-scale model and then adopts FWI to generate a high-resolution micro-scale model. For the signals received at the top station (Fig.…”

“…We will follow the strategy described in Chai et al (2018); Wei & Yang (2012), and consider the case illustrated in Fig. 7, where the level set functions φ re,tr for the transmitted P-waves satisfy the same evolution as φ in eq.…”

Method of Multi-Angle Transmission Radiowave Tomography of Dielectric Objects

“…However, methods of this kind do not take into account the wave nature of radiation, but ray approximation is applied. Similar approaches are used in seismic tomography [33,34], where the seismic wave velocity distribution in the medium is reconstructed by time delays. The use of such methods is widespread in the method of ultrasonic nondestructive testing [35], in which, similarly to electromagnetic waves, in the scalar approximation, ultrasonic waves are described by the wave equation.…”

Seismic Tomography with Random Batch Gradient Reconstruction

“…To use RBM in seismic tomography, natural choices for computing synthetic seismograms are numerical methods of particle type, e.g., generalized ray theory [11,32], Kirchhoff migration [8,15], Gaussian beam migration [12,13,21,9,7,22], and frozen Gaussian approximation (FGA) [4,5,10,3]. Here for the sake of convenience, we use FGA to compute wave equations, which do not need to solve ray paths by shooting to reach the receivers, and can provide accurate solutions in the presence of caustics and multipathing, with no requirement on tuning beam width parameters to achieve a good resolution [2,12,6,23,19,33].…”

“…The above computation can also performed to wave-equation-based travel-time inversion, yielding a similar formulations except that the adjoint source function becomes (c.f. [5])…”