Seismic inversion by Newtonian machine learning

Abstract

<jats:p> We present a wave-equation inversion method that inverts skeletonized seismic data for the subsurface velocity model. The skeletonized representation of the seismic traces consists of the low-rank latent-space variables predicted by a well-trained autoencoder neural network. The input to the autoencoder consists of seismic traces, and the implicit function theorem is used to determine the Fréchet derivative, i.e., the perturbation of the skeletonized data with respect to the velocity perturbation. The gradient is computed by migrating the shifted observed traces weighted by the skeletonized data residual, and the final velocity model is the one that best predicts the observed latent-space parameters. We denote this as inversion by Newtonian machine learning (NML) because it inverts for the model parameters by combining the forward and backward modeling of Newtonian wave propagation with the dimensional reduction capability of machine learning. Empirical results suggest that inversion by NML can sometimes mitigate the cycle-skipping problem of conventional full-waveform inversion (FWI). Numerical tests with synthetic and field data demonstrate the success of NML inversion in recovering a low-wavenumber approximation to the subsurface velocity model. The advantage of this method over other skeletonized data methods is that no manual picking of important features is required because the skeletal data are automatically selected by the autoencoder. The disadvantage is that the inverted velocity model has less resolution compared with the FWI result, but it can serve as a good initial model for FWI. Our most significant contribution is that we provide a general framework for using wave-equation inversion to invert skeletal data generated by any type of neural network. In other words, we have combined the deterministic modeling of Newtonian physics and the pattern matching capabilities of machine learning to invert seismic data by NML. </jats:p>

Skeletonized Wave-Equation Refraction Inversion With Autoencoded Waveforms

“…Inserting (10)- (14) to (4), we have the ultimate gradient expression. In this study, the gradient of refraction AE inversion for the latentspace misfit function is derived using our notation similar to that in [23]. Its final form is given as…”

“…If the dimension of the latent space is set to two as z = (z 1 , z 2 ), then the degree of freedom to express the envelopes is increasing. Therefore, the corresponding inversion theory in [23] needs to be modified with a few changes to the misfit function. The misfit function associated with the 2-D latent space autoencoder can be defined as…”

“…To generalize the skeletonized inversion procedure to any type of skeletal data, Chen and Schuster [23] developed the Newtonian machine learning (NML) method. Here, they used the compressed latent space variables of an autoencoder [24]- [26] as the skeletonized data.…”

“…Here, we refer to NML as autoencoded (AE) inversion. The synthetic and field data are windowed about the first arrivals, and some 1 (or 2)×1 vectors of latent-space coefficients are used for skeletonizing traces, not only the 1-D latent space vector in [23]. Therefore, all the formulas in the theoretical section are redeveloped in this study.…”

Seismic inversion by multi-dimensional Newtonian machine learning

“…The conventional skeletonized data, such as traveltime, often requires manual picking, which can be labor-intensive for large datasets. To solve this problem, Chen and Schuster (2019) proposed a Newtonian machine learning (NML) method suggesting that the skeletonized data can be automatically learned by an autoencoder network. For a well-trained autoencoder, the encoder and decoder network contains the common parts among all the training examples, however, the information preserved in the latent space, also denoted as the skeletonized representation, indicates their differences.…”