Wave-equation-based travel-time seismic tomography – Part 1: Method

Abstract

Abstract. In this paper, we propose a wave-equation-based travel-time seismic tomography method with a detailed description of its step-by-step process. First, a linear relationship between the travel-time residual t = T obs − T syn and the relative velocity perturbation δc(x)/c(x) connected by a finite-frequency travel-time sensitivity kernel K(x) is theoretically derived using the adjoint method. To accurately calculate the travel-time residual t, two automatic arrivaltime picking techniques including the envelop energy ratio method and the combined ray and cross-correlation method are then developed to compute the arrival times T syn for synthetic seismograms. The arrival times T obs of observed seismograms are usually determined by manual hand picking in real applications. Travel-time sensitivity kernel K(x) is constructed by convolving a forward wavefield u(t, x) with an adjoint wavefield q(t, x). The calculations of synthetic seismograms and sensitivity kernels rely on forward modeling. To make it computationally feasible for tomographic problems involving a large number of seismic records, the forward problem is solved in the two-dimensional (2-D) vertical plane passing through the source and the receiver by a high-order central difference method. The final model is parameterized on 3-D regular grid (inversion) nodes with variable spacings, while model values on each 2-D forward modeling node are linearly interpolated by the values at its eight surrounding 3-D inversion grid nodes. Finally, the tomographic inverse problem is formulated as a regularized optimization problem, which can be iteratively solved by either the LSQR solver or a nonlinear conjugate-gradient method. To provide some insights into future 3-D tomographic inversions, Fréchet kernels for different seismic phases are also demonstrated in this study.

Teleseismic Traveltime Tomography of Northern Sumatra

“…A detailed discussion of using the FMM for teleseismic traveltime tomography can be found in Rawlinson et al (). Here we use the conjugate gradient method to solve the damped least‐square problem (Tong et al, ): where A is the sensitivity matrix, X is the discrete model perturbation vector, is the a priori data covariance matrix, is the a priori model covariance matrix, η is the damping factor, and b is the relative traveltime difference vector, which is obtained by subtracting the calculated traveltime difference from the observed traveltime difference. We adopt the ak135 stratified Earth model (Kennett et al, ) as the initial model and the CRUST1.0 model (Laske et al, ) for crustal corrections.…”

“…A detailed discussion of using the FMM for teleseismic traveltime tomography can be found in Rawlinson et al (2006). Here we use the conjugate gradient method to solve the damped least-square problem (Tong et al, 2014):…”

Wave equation‐based reflection tomography of the 1992 Landers earthquake area

“…The development of seismic imaging techniques based upon solving full wave equations has become a topic of great research interest during the past several decades [e.g., Luo and Schuster , ; Tromp et al , ; Fichtner et al , ; Chen et al , ; Tape et al , ; Tong et al , ]. As one of the attempts, Tong et al [] developed a wave equation‐based traveltime seismic tomography (WETST) method, which naturally takes into account the finite‐frequency effects of seismic waves and accurately captures the interaction between seismic waves and complex velocity structure. Since only the traveltime information is considered, WETST assumes that acoustic seismic waves propagate in a 2‐D vertical plane passing through the source and receiver with P or S velocity for numerical efficiency but inverts for 3‐D velocity model of the study region.…”

A New Earthquake Location Method Based on the Waveform Inversion

“…Obviously, the global solution exists and is unique (Nocedal & Wright 1999). In the following part, the sensitivity kernel (Liu & Gu 2012;Rawlinson et al 2010;Tong et al 2014b) will be derived to solve this inversion problem iteratively.…”

“…As it was discussed in Section 2.1, the first-order Born approximation in the adjoint method requires an infinitesimal perturbation assumption of wave function δs(x, t) s(x, t) 1, see also (Liu & Gu 2012;Rawlinson et al 2010;Tong et al 2014b;Tromp et al 2005). However, as…”