Early arrival waveform inversion (EWI) is an essential approach to obtaining velocity structures in near-surface. Due to suffering from a cycle‐skipping issue, it is difficult to reach the global minima for conventional EWI with the misfit function of least-squares norm (L2‐norm). Following the optimal transportation theory, we developed an EWI solution with a new objective function based on quadratic‐Wasserstein‐metric (W2-norm) to maintain the geometric characteristics of the distribution and improve the stability and convexity of the inverse problem. First, we gave the continuous form of the adjoint source and the Frechet gradient of the Wasserstein metric for seismic early arrival, which leads to an easy and efficient way to implement in the adjoint-state method. Then, we conducted two synthetic experiments on the target model containing some velocity anomalies and hidden layers to test its effectiveness in mapping accurate and high-resolution near-surface velocity structure. The results show that the W2-normed EWI can mitigate cycle-skipping issues compared with the L2-normed EWI. In addition, it can deal with hidden layers and is robust in terms of noise. The application to a real dataset indicates that this new solution can recover more details in the shallow structure, especially in the aspect of dealing with hidden layers.
Near-surface imaging structures often plays a significant role in the field of environmental and engineering geophysics. Early-arrival waveform inversion (EWI) is state-of-the-art method to imaging near-surface structures due to its high resolution. However, the method faces with cycle-skipping issue which might lead to an unexpected local minimum. Envelope inversion (EI) could deal with this issue which contributes to the ultralow-frequency information extracted from the envelope but has a low resolution. We have developed a curvelet-based joint waveform and envelope inversion (CJWEI) method for inverting imaging near-surface velocity structures. By inverting two types of data, we are able to recover the low- and high-wavenumber structures and mitigate the cycle-skipping problem. Curvelet transform was used to decompose seismic data into different scales and provide a multiscale inversion strategy to further reduce non-uniqueness of waveform inversion efficiently. With synthetic test and real data application, we demonstrate that our method can constrain the anomalies and hidden layers in the shallow structure more efficiently as well as is robust in terms of noise. The proposed multiscale joint inversion offers a computational efficiency and high precision to imaging fine-scale shallow underground structures.
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