Full waveform inversion (FWI) is a powerful method for providing a high-resolution description of the subsurface. However, the misfit function of the conventional FWI method (metric llt;subgt;2lt;/subgt;-norm) is usually dominated by spurious local minima owing to its nonlinearity and ill-posedness. In addition, FWI requires intensive wavefield computation to evaluate the gradient and step length. We consider a general inversion method using a deep neural network for the FWI problem. This deep-learning inversion method reparameterizes physical parameters using the weights of a deep neural network, such that the inversion amounts to reconstructing these weights. One advantage of this deep-learning inversion method is that it can serve as an iterative regularization method, benefiting from the representation of the network. Thus, it is suitable to solve ill-posed nonlinear inverse problems. Furthermore, this method possesses good computational efficiency because it only requires first-order derivatives. In addition, it can be easily accelerated by using multiple graphics processing units (GPUs) and central processing units (CPUs), for weight updating and forward modeling. Synthetic experiments, based on the Marmousi2, 2004 BP, and a metal ore model are used to show the numerical performances of the deep-learning inversion method. Comprehensive comparisons with a total-variation regularized FWI are presented to show the ability of our method to recover sharp boundaries. Our numerical results indicate that this deep-learning inversion approach is effective, efficient, and can capture salient features of the model.

In this work, we extend the finite-difference contrast source inversion (FD-CSI) method to the frequency-domain elastic wave equations, where the parameters describing the subsurface structure are simultaneously reconstructed. The FD-CSI method is an iterative nonlinear inversion method, which exhibits several strengths. First, the finite-difference operator only relies on the background media and the given angular frequency, both of which are unchanged during inversion. Therefore, the matrix decomposition is performed only once at the beginning of the iteration if a direct solver is employed. This makes the inversion process relatively efficient in terms of the computational cost. In addition, the FD-CSI method automatically normalizes different parameters, which could avoid the numerical problems arising from the difference of the parameter magnitude. We exploit a parallel implementation of the FD-CSI method based on the domain decomposition method, ensuring a satisfactory scalability for large-scale problems. A simple numerical example with a homogeneous background medium is used to investigate the convergence of the elastic FD-CSI method. Moreover, the Marmousi II model proposed as a benchmark for testing seismic imaging methods is presented to demonstrate the performance of the elastic FD-CSI method in an inhomogeneous background medium.

This paper extends the finite-difference contrast source inversion method to reconstruct the mass density for two-dimensional elastic wave inversion in the framework of the full-waveform inversion. The contrast source inversion method is a nonlinear iterative method that alternatively reconstructs contrast sources and contrast function. One of the most outstanding advantages of this inversion method is the highly computational efficiency, since it does not need to simulate a full forward problem for each inversion iteration. Another attractive feature of the inversion method is that it is of strong capability in dealing with nonlinear inverse problems in an inhomogeneous background medium, because a finite-difference operator is used to represent the differential operator governing the two-dimensional elastic wave propagation. Additionally, the techniques of a multiplicative regularization and a sequential multifrequency inversion are employed to enhance the quality of reconstructions for this inversion method. Numerical reconstruction results show that the inversion method has an excellent performance for reconstructing the objects embedded inside a homogeneous or an inhomogeneous background medium.

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