To make the l1-norm frequency-domain elastic full waveform inversion (FWI) less sensitive to initial guesses, we propose the weighting method that incorporates weighting functions to gradients at each frequency. The weighting function should be designed so that the final gradient can properly describe the differences between assumed (initial or inverted) and true models. In the l2-norm FWI, the backpropagated wavefields incited by source wavelet-deconvolved residuals, which can be easily obtained during the inversion process, can be used for the weighting function. However, in the l1-norm FWI, since the residuals normalized by their magnitudes are back-propagated to compute gradients, the normalized residuals do not reflect the differences between assumed and true models. For this reason, we approximate the residuals of the l2-norm objective function to define the weighting function for the l1-norm FWI. We demonstrate the weighting function for synthetic data with outliers obtained for the 2D section of the SEG/EAGE salt model. Numerical examples show that while the conventional l1-norm waveform inversion does not provide stable solutions for the salt model without good initial guesses, the weighting method gives inversion results comparable to true models even with the poorly estimated initial guesses.
Although full waveform inversion (FWI) is a promising method to provide subsurface material properties and has been extensively studied, there are still needs to improve FWI particularly in case of multi-parameters. In this study, we propose a multi-parameter acoustic FWI strategy that can recover both velocity and density. The strategy is developed analyzing the characteristics of acoustic FWI based on a couple of parameterizations in acoustic wave equation for heterogeneous media. Our strategy consists of two stages. In the first stage, density is fixed at an arbitrary value, and velocity is only restored updating the bulk modulus based on the wave equation parameterized by bulk modulus and density. In this case, although the bulk modulus can be wrong, the inferred velocity can be reasonable. In the second stage, both velocity and density are inverted based on the wave equation parameterized by velocity and density. For the second stage, bulk modulus rather than velocity itself is updated, which is performed by using a chain rule to compute the gradient of bulk modulus. Our FWI uses the finite-element method and the back-propagation method. Numerical examples for synthetic data of the SEG/EAGE overthrust model show that our hierarchical strategy for acoustic FWI yields better results than the conventional method.
In this paper, we propose a new objective function that incorporates both simultaneous-source technique and robust norm in the frequency domain. The proposed objective function is defined to measure the residual of the super-shot consisting of encoded shot gathers. Although the objective function does not exactly simulate the ordinary l1-norm objective function without source encoding, it has the same robustness as the least-absolute criteria. To confirm the robustness of our algorithm, we performed the full waveform inversion (FWI) using data with outliers and random noise. Based on the inversion results, we confirmed that our algorithm possesses the robust characteristics of the l1-norm objective function as well as the efficiency of the simultaneous source inversion.
Since seismic modeling is conducted repeatedly in seismic inversion, efficiency and accuracy of inversion are affected by seismic modeling algorithm used in it. One of the main factors influencing accuracy of seismic modeling can be the boundary condition used to remove edge reflections arising from finite-size models. For elastic modeling and inversion algorithms free from edge reflections, we propose using the logarithmic grid set. In the logarithmic grid set, the origin is located in the middle of surface of a given model and grid interval increases logarithmically with distance from the origin. The logarithmic grid set enables us to incorporate huge boundary areas with fewer grid points than in the conventional grid set, so that edge reflections are not recorded within the recording duration. For elastic modeling and inversion in the logarithmic grid set, wave equations and source position are first transformed to the logarithm-scaled coordintate. To convert field data from the conventional grid set to the logarithmic grid set and transform inversion results from the logarithmic grid set to the conventional grid set, interpolation algorithms are needed. In the elastic modeling algorithm, the cell-based finite-difference method, which properly describes the free-surface boundary conditions, is used. For inversion process, we apply the gradient method based on the back-propagation method, the pseudo-Hessian matrix, and the conjugate-gradient method. We also employ the frequency-marching method. Numerical examples generated for the modified version of the Marmousi-2 model showed that the elastic modeling and inversion algorithms designed in the logarithmic grid set yield reasonable results.
For a robust elastic waveform inversion algorithm, we propose incorporating a denoise function into gradients in the l1-norm waveform inversion. The denoise function is designed by the ratio of modeled data to field data summed over shots and receivers at each frequency, based on the fact that while field data are noisy, modeled data are noise-free. As a result, the denoise function is inversely proportional to the degree of noises and acts like filters. Using the denoise function, we can keep the noise-contaminated gradients from affecting model parameter updates. The denoise function is applied to synthetic data with three types of noises for the modified version of Marmousi-2 model: discontinuous monochromatic random noises, general random noises and outliers. Numerical examples show that the denoise function effectively filters out the noise-contaminated gradients during the inversion process and thus yields better inversion results than the conventional l1-norm waveform inversion.