Wave-equation traveltime inversion (WTI) is a useful tool for background velocity model building. It is generally formulated and implemented in the time domain, in which the gradient is calculated by temporally cross-correlating the source- and receiver-side wavefields. The time-domain source-side snapshots are either stored in memory or reconstructed through back propagation. The memory requirements or the computational cost of WTI are thus prohibitively expensive, especially for 3D applications. To partially alleviate this problem, we provide an implementation of WTI in the frequency domain with a monofrequency component. Since only one frequency is used, it is affordable to directly store the source- and receiver-side wavefields in memory. There is no need for wavefield reconstruction during gradient calculation. In such a way, we have dramatically reduced the memory requirements and computational cost compared with the traditional time-domain WTI realization. For practical implementation, the frequency-domain wavefield is calculated by time-domain finite-difference forward modeling and is transformed to frequency domain by on-the-fly Discrete Fourier Transform (DFT). Numerical examples on a simple lateral periodic velocity model and the Marmousi model demonstrate that the proposed method can obtain accurate background velocity models comparable to those from time-domain WTI and frequency-domain WTI with multiple frequencies. A field data set test shows that the proposed method obtains a background velocity model that well predicts the seismic wave traveltime.
Wave-equation traveltime inversion (WTI) can be used to automatically obtain a background near-surface velocity model (NSM), which overcomes the high-frequency approximation in ray theory. It is generally implemented in the time domain. However, the commonly used gradient-based optimisation methods (such as the steepestdescent method) in WTI have a low convergence rate and may yield less accurate results within limited iterations in geologically complex regions. To increase the convergence rate and improve the inversion accuracy, we propose a frequency-domain truncated Gauss-Newton first-arrival wave-equation traveltime inversion (GN-WTI) method to retrieve the background NSM. As only a few frequencies are used for inversion, the proposed frequency-domain WTI method significantly reduces the computational memory requirements by more than two orders of magnitude in comparison with the conventional time-domain WTI method. Therefore, the proposed method is especially advantageous for the building of large three-dimensional models. In this GN-WTI method, according to the derived explicit traveltime residual kernel, the gradient and Hessian vector products can be computed efficiently using an elegant and improved scattering integral approach as long as the source-side wavefields and nonredundant receiver-side Green's functions are computed and stored in advance. The conjugate gradient approach is used to solve the Gauss-Newton normal equation to obtain the Gauss-Newton direction in inner loops. Here, the Gauss-Newton Hessians of the ray-based traveltime inversion and WTI are compared to demonstrate the advantages of WTI. The trial runs with a simple periodic velocity model example showed that the proposed GN-WTI method outperforms the WTI method when using the steepest-descent and limited-memory Broyden-Fletcher-Goldfarb-Shanno approaches in terms of the convergence rate and inversion accuracy. A complex Marmousi model was further used to illustrate the effectiveness of GN-WTI. The proposed method should be beneficial in near-surface velocity model building.
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