Source estimation for wavefield-reconstruction inversion

Abstract

Source estimation is essential for all wave-equation-based seismic inversions, including full-waveform inversion (FWI) and the recently proposed wavefield-reconstruction inversion (WRI). When the source estimation is inaccurate, errors will propagate into the predicted data and introduce additional data misfit. As a consequence, inversion results that minimize this data misfit may become erroneous. To mitigate the errors introduced by the incorrect and preestimated sources, an embedded procedure that updates sources along with medium parameters is necessary for the inversion. So far, such a procedure is still missing in the context of WRI, a method that is, in many situations, less prone to local minima related to so-called cycle skipping, compared with FWI through exact data fitting. Although WRI indeed helps to mitigate issues related to cycle skipping by extending the search space with wavefields as auxiliary variables, it relies on having access to the correct source functions. To remove the requirement of having the accurate source functions, we have developed a source-estimation technique specifically designed for WRI. To achieve this task, we consider the source functions as unknown variables and arrive at an objective function that depends on the medium parameters, wavefields, and source functions. During each iteration, we apply the so-called variable projection method to simultaneously project out the source functions and wavefields. After the projection, we obtain a reduced objective function that only depends on the medium parameters and invert for the unknown medium parameters by minimizing this reduced objective. Numerical experiments illustrate that this approach can produce accurate estimates of the unknown medium parameters without any prior information of the source functions.

Characterization of a ground penetrating radar shielded antenna using laboratory measurements, FDTD modeling and swarm global optimization

“…Different methods have been used to estimate the source signature for FWI, such as the characterization of the antenna as an infinitesimal dipole, where the energy in TE mode is distributed uniformly for any angle [3], or the signature estimation from the acquired data using the adjoint state method [4], the variable projection method [5], [6] and gradient-based optimization methods [7]. Other proposed methods are reverse-time propagation [8], and the deconvolution of radar data with the parameters system [9].…”

Efficient extended-search space full-waveform inversion with unknown source signatures

“…Recently, Fang et al (2018) tackled the source signature estimation problem in WRI. The formulation of Fang et al (2018) groups together the wavefield and the source signature into a single optimization variable such that WRI can be cast as a separable nonlinear leastsquares problem, which can be tackled with the variable projection method.…”

“…Recently, Fang et al (2018) tackled the source signature estimation problem in WRI. The formulation of Fang et al (2018) groups together the wavefield and the source signature into a single optimization variable such that WRI can be cast as a separable nonlinear leastsquares problem, which can be tackled with the variable projection method. The issue with this approach is that the data assimilation makes the augmented wave-equation (normal) operator source dependent hence making the method expensive since the Cholesky decomposition needs to be performed for each source when a direct method is used.…”

“…Also, block-processing of multiple right-hand sides is not possible anymore with iterative methods. To bypass this issue, Fang et al (2018) partition the normal-operator matrix into source-dependent and source-independent blocks and apply the block matrix inversion formula (Press et al 2007, page 81) to the partitioned matrix. This block reformulation involves only the inverse of the source-independent normal operator, hence making this approach manageable.…”

“…The focus of this paper is to propose an alternative formulation of the source signature estimation problem to that proposed by Fang et al (2018) with the aim to reduce the computational overhead while making the source signature reconstruction potentially more robust when crude background models are used. Our approach relies on a variable projection method, where the closed-form expression of the source signature is projected in the wavefield reconstruction subproblem.…”