Extended Waveform Inversion Research Articles

Anderson-accelerated augmented Lagrangian for extended waveform inversion

The augmented Lagrangian (AL) method provides a flexible and efficient framework for solving extended-space full-waveform inversion (FWI), a constrained nonlinear optimization problem whereby we seek model parameters and wavefields that minimize the data residuals and satisfy the wave-equation constraint. The AL-based wavefield reconstruction inversion, also known as iteratively refined wavefield reconstruction inversion, extends the search space of FWI in the source dimension and decreases the sensitivity of the inversion to the initial model accuracy. Furthermore, it benefits from the advantages of the alternating direction method of multipliers, such as generality and decomposability for dealing with nondifferentiable regularizers, e.g., total variation regularization, and large-scale problems, respectively. In practice, any extension of the method aiming at improving its convergence and decreasing the number of wave-equation solves would have great importance. To achieve this goal, we recast the method as a general fixed-point iteration problem, which enables us to apply sophisticated acceleration strategies such as Anderson acceleration. The accelerated algorithm stores a predefined number of previous iterates and uses their linear combination together with the current iteration to predict the next iteration. We investigate the performance of our accelerated algorithm on a simple checkerboard model and the benchmark Marmousi II and 2004 BP salt models through numerical examples. These numerical results confirm the effectiveness of our algorithm in terms of convergence rate and the quality of the final estimated model.

Extended full waveform inversion with matching filter

ABSTRACTFull waveform inversion has shown its huge potentials in recovering a high‐resolution subsurface model. However, conventional full waveform inversion usually suffers from cycle skipping, resulting in an inaccurate local minimum model. Extended waveform inversion provides an effective way to mitigate cycle skipping. A matching filter between the predicted and observed data can provide an additional degree of freedom to improve the data fitting and avoid the cycle skipping. We extend the search space to treat the matching filter as an independent variable that we use to bring the compared data within a half cycle to obtain the accurate direction of velocity updates. We formulate the objective function using the penalty method by linearly combining a data‐misfit term and a penalty term. The objective function with a reasonable penalty parameter has a larger region of convergence compared to conventional full waveform inversion. We search for the optimal solution over the extended model by updating the matching filter and the velocity in a nested way. The normalization of the data can bring us an equivalent normalization to the filter and a more effective convergence. In the synthetic Marmousi model, the proposed inversion method recovers the velocity model stably and accurately starting from a linearly increasing model in the case of lack of low frequencies below 3 Hz, in which conventional full waveform inversion suffers from cycle skipping. We also use a marine field data to demonstrate the effectiveness and practicality of the proposed method.

Source-independent extended waveform inversion based on space-time source extension: Frequency-domain implementation

Source signature estimation from seismic data is a crucial ingredient for successful application of seismic migration and full-waveform inversion (FWI). If the starting velocity deviates from the target velocity, FWI method with on-the-fly source estimation may fail due to the cycle-skipping problem. We have developed a source-based extended waveform inversion method, by introducing additional parameters in the source function, to solve the FWI problem without the source signature as a priori. Specifically, we allow the point source function to be dependent on spatial and time variables. In this way, we can easily construct an extended source function to fit the recorded data by solving a source matching subproblem; hence, it is less prone to cycle skipping. A novel source focusing annihilator, defined as the distance function from the real source position, is used for penalizing the defocused energy in the extended source function. A close data fit avoiding the cycle-skipping problem effectively makes the new method less likely to suffer from local minima, which does not require extreme low-frequency signals in the data. Numerical experiments confirm that our method can mitigate cycle skipping in FWI and is robust against random noise.

Volume source-based extended waveform inversion

Full-waveform inversion (FWI) faces the persistent challenge of cycle skipping, which can result in stagnation of the iterative methods at uninformative models with poor data fit. Extended reformulations of FWI avoid cycle skipping through adding auxiliary parameters to the model so that a good data fit can be maintained throughout the inversion process. The volume-based matched source waveform inversion algorithm introduces source parameters by relaxing the location constraint of source energy: It is permitted to spread in space, while being strictly localized at time [Formula: see text]. The extent of source energy spread is penalized by weighting the source energy with distance from the survey source location. For transmission data geometry (crosswell, diving wave, etc.) and transparent (nonreflecting) acoustic models, this penalty function is stable with respect to the data-frequency content, unlike the standard FWI objective. We conjecture that the penalty function is actually convex over much larger region in model space than is the FWI objective. Several synthetic examples support this conjecture and suggest that the theoretical limitation to pure transmission is not necessary: The inversion method can converge to a solution of the inverse problem in the absence of low-frequency data from an inaccurate initial velocity model even when reflections and refractions are present in the data along with transmitted energy.

A discrepancy-based penalty method for extended waveform inversion

Extended waveform inversion globalizes the convergence of seismic waveform inversion by adding nonphysical degrees of freedom to the model, thus permitting it to fit the data well throughout the inversion process. These extra degrees of freedom must be curtailed at the solution, for example, by penalizing them as part of an optimization formulation. For separable (partly linear) models, a natural objective function combines a mean square data residual and a quadratic regularization term penalizing the nonphysical (linear) degrees of freedom. The linear variables are eliminated in an inner optimization step, leaving a function of the outer (nonlinear) variables to be optimized. This variable projection method is convenient for computation, but it requires that the penalty weight be increased as the estimated model tends to the (physical) solution. We describe an algorithm based on discrepancy, that is, maintaining the data residual at the inner optimum within a prescribed range, to control the penalty weight during the outer optimization. We evaluate this algorithm in the context of constant density acoustic waveform inversion, by recovering background model and perturbation fitting bandlimited waveform data in the Born approximation.