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

• Main Authors: P. Tong, D. Zhao, D. Yang, X. Yang, J. Chen, Q. Liu
• Format: Article
• Language: English
• Published: Copernicus Publications 2014-11-01
• Series: Solid Earth
• Online Access: http://www.solid-earth.net/5/1151/2014/se-5-1151-2014.pdf
• ISSN: 1869-9510; 1869-9529

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 &Delta;<i>t</i> = <i>T</i>^obs&ndash;T^syn and the relative velocity perturbation &delta; c(<b><i>x</i></b>)/c(<b><i>x</i></b>) connected by a finite-frequency travel-time sensitivity kernel <i>K</i>(<b><i>x</i></b>) is theoretically derived using the adjoint method. To accurately calculate the travel-time residual &Delta;<i>t</i>, two automatic arrival-time 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 <i>T</i>^obs of observed seismograms are usually determined by manual hand picking in real applications. Travel-time sensitivity kernel <i>K</i>(<b><i>x</i></b>) is constructed by convolving a~forward wavefield <i>u</i>(<i>t</i>,<b><i>x</i></b>) with an adjoint wavefield <i>q</i>(<i>t</i>,<b><i>x</i></b>). 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.