Performance comparison of Newton and quasi-Newton methods in frequency-domain acoustic full waveform inversion for a synthetic model

Summary: Today, the significance of the velocity model estimation in seismic migration as well as time to depth conversion of seismic sections is very clear to every geophysicist. From a practical point of view, approaches such as well logging techniques or direct observations cannot provide a reliable description of regional scale physical properties of the earth. Seismic imaging is a tool to describe the earth physical properties. Today, one of the imaging techniques, which is highly welcomed by geoscientists is the full waveform inversion (FWI). The full waveform inversion - an efficient method in seismic imaging - uses all available information in the seismograms, including the amplitude, phase, and the first arrival time so as to judge the physical properties of the earth. In fact, FWI is represented as an optimization problem by defining a misfit function, which measures differences between the observed waveforms and the computed waveforms. In FWI, the size of the model space parameters will not allow us using global minimization algorithms. Therefore, we minimize the misfit function by local optimization methods. A waveform inversion problem is usually solved repetitively by the gradient-based solutions. In this paper, we have compared two gradient optimization algorithms of Gauss-Newton algorithm based on the main diagonal of the pseudo Hessian matrix (GN-DPH) and the limited memory-BFGS (LBFGS). Although the L-BFGS does not need the computation of Hessian matrix, but according to the numerical tests on synthetic models, we have found that GN-DPH algorithm results are more accurate than L-BFGS after 40 iterations.
From a general point of view, in oil and gas exploration, approaches for delineating physical properties of the earth can be divided into direct solutions and indirect solutions. In large scale seismic problems, we have to focus on indirect solutions. Consequently, we use imaging techniques to extract desired parameters of the earth. There are various approaches for the imaging and estimations of the earth elastic parameters. FWI is one of efficient imaging techniques, which has become popular recently in seismic communities.
Methodology and Approaches: We have compared one of Newton-based methods, namely Gauss-Newton based on the main diagonal of the pseudo Hessian Matrix (GN-DPH), and the limited memory-BFGS (L-BFGS). For the forward modeling, we have solved 2D acoustic wave equation using finite-difference approximation equipped with a second order absorbing boundary condition in the frequency domain. To estimate the gradient vector efficiently, the adjoint-state method (Plesseix, 2006) has been used. In the GN-DPH algorithm, a method developed by Shin et al. (2006) has been applied in order to compute the Hessian matrix, and in the L-BFGS algorithm, we have followed a preconditioned L-BFGS approach developed by Nocedal (1980). In the numerical example, the performance of the aforementioned algorithms has been examined.
Results and By comparison of the two reviewed optimization methods in terms of the gradient pre conditioners, we found that both methods succeeded to estimate the true velocity model. Analyzing the model estimation error revealed that the GN-DPH method was more efficient than the L-BFGS during the first 40 iterations performed in the study. Decreasing the normalized misfit function values showed the advantage of GN-DPH algorithm over the L BFGS. There is two justifications for this fact: First, the convergence rate for the GN-DPH is of a second order, whereas that of the L-BFGS is super-linear. Second reason is valuable information, which exists on the main diagonal of the linear part of the Hessian matrix that acts as pre-conditioner in the GN-DPH algorithm. On the other hand, it should be noted that by increasing the number of iterations, better results can be expected due to the fact that the Hessian matrix includes both linear and nonlinear parts and is more accurate than the other investigated method. It is also interesting to note that by increasing the complexity of a true model, we will definitely encounter a more nonlinear misfit function, which needs a more efficient Hessian for preconditioning the gradient vector where the importance of L BFGS algorithm will be highlighted there.