William W. Symes, Huiyi Chen, Susan E. Minkoff
Study of a simple single-trace transmission example shows how an extended source formulation of full-waveform inversion can produce an optimization problem without spurious local minima ("cycle skipping"), hence efficiently solvable via Newton-like local optimization methods. The data consist of a single trace extracted from a causal pressure field, propagating in a homogeneous fluid according to linear acoustics, and recorded at a given distance from a transient point energy source. The source intensity ("wavelet") is presumed quasi-impulsive, with zero energy for time lags greater than a specified maximum lag. The inverse problem is: from the recorded trace, recover both the sound velocity or slowness and source wavelet with specified support, so that the data is fit with prescribed RMS relative error. The least-squares objective function has multiple large residual minimizers. The extended inverse problem permits source energy to spread in time, and replaces the maximum lag constraint by a weighted quadratic penalty. A companion paper shows that for proper choice of weight operator, any stationary point of the extended objective produces a good approximation of the global minimizer of the least squares objective, with slowness error bounded by a multiple of the maximum lag and the assumed noise level. This paper summarizes the theory developed in the companion paper and presents numerical experiments demonstrating the accuracy of the predictions in concrete instances. We also show how to dynamically adjust the penalty scale during iterative optimization to improve the accuracy of the slowness estimate.
William W. Symes, Huiyi Chen, Susan E. Minkoff
Iterative inversion of seismic, ultrasonic, and other wave data by local gradient-based optimization of mean-square data prediction error (Full Waveform Inversion or FWI) can fail to converge to useful model estimates if started from an initial model predicting wave arrival times in error by more than half a wavelength (a phenomenon known as cycle skipping). Matched Source Waveform Inversion (MSWI) extends the wave propagation model by a filter that shifts predicted waves to fit observed data. The MSWI objective adds a penalty for deviation of this filter from the identity to the mean-square data misfit . The extension allows the inversion to make large model adjustments while maintaining data fit and so reduces the chances of local optimization iterates stagnating at non-informative model estimates. Theory suggests that MSWI applied to acoustic transmission data with single-arrival wavefronts may produce an estimate of refractive index similar to the result of travel time inversion, but without requiring explicit identification of travel times. Numerical experiments conform to this expectation, in that MSWI applied to single arrival transmission data gives reasonable model estimates in cases where FWI fails. This MSWI model can then be used to jumpstart FWI for further refinement of the model. The addition of moderate amounts of noise (30\%) does not negatively impact MSWI's ability to converge. However, MSWI applied to data with multiple arrivals is no longer theoretically equivalent to travel-time tomography and exhibits the same tendency to cycle-skip as does FWI.
Fernando Llorente, Daniel Waxman, Sanket Jantre, Nathan M. Urban, Susan E. Minkoff
Gaussian processes (GPs) offer a flexible, uncertainty-aware framework for modeling complex signals, but scale cubically with data, assume static targets, and are brittle to outliers, limiting their applicability in large-scale problems with dynamic and noisy environments. Recent work introduced decentralized random Fourier feature Gaussian processes (DRFGP), an online and distributed algorithm that casts GPs in an information-filter form, enabling exact sequential inference and fully distributed computation without reliance on a fusion center. In this paper, we extend DRFGP along two key directions: first, by introducing a robust-filtering update that downweights the impact of atypical observations; and second, by incorporating a dynamic adaptation mechanism that adapts to time-varying functions. The resulting algorithm retains the recursive information-filter structure while enhancing stability and accuracy. We demonstrate its effectiveness on a large-scale Earth system application, underscoring its potential for in-situ modeling.