Mitigating local minima in full-waveform inversion by expanding the search space

Abstract

Wave equation based inversions, such as full-waveform inversion and reverse-time migration, are challenging because of their computational costs, memory requirements and reliance on accurate initial models. To confront these issues, we propose a novel formulation of wave equation based inversion based on a penalty method. In this formulation, the objective function consists of a data-misfit term and a penalty term, which measures how accurately the wavefields satisfy the wave equation. This new approach is a major departure from current formulations where forward and adjoint wavefields, which both satisfy the wave equation, are correlated to compute updates for the unknown model parameters. Instead, we carry out the inversions over two alternating steps during which we first estimate the wavefield everywhere, given the current model parameters, source and observed data, followed by a second step during which we update the model parameters, given the estimate for the wavefield everywhere and the source. Because the inversion involves both the synthetic wavefields and the medium parameters, its search space is enlarged so that it suffers less from local minima. Compared to other formulations that extend the search space of wave equation based inversion, our method differs in several aspects, namely (i) it avoids storage and updates of the synthetic wavefields because we calculate these explicitly by finding solutions that obey the wave equation and fit the observed data and (ii) no adjoint wavefields are required to update the model, instead our updates are calculated from these solutions directly, which leads to significant computational savings. We demonstrate the validity of our approach by carefully selected examples and discuss possible extensions and future research.

Adaptive waveform inversion: Theory

“…van Herrmann (2013) andvan Leeuwen et al (2014) introduce wavefield reconstruction inversion (WRI) as an alternative to FWI that is able to circumvent local minima in the conventional FWI objective function. Like AWI, this method introduces a nonphysical model extension, and then attempts to recover an earth model by collapsing this extended model onto a physical model that is still able to explain the observed data.…”

“…These methods currently appear to fall into one of two groups: either the methods are efficient and affordable on commercial data sets in three dimensions, but fail to converge when the data and/or the model are realistically complicated, or the methods work well on realistic data, but have not so far been proven to be affordably achievable on 3D field data. Methods in the first group include those of van Mulder (2008, 2010), Luo and Sava (2011), and perhaps Ma and Hale (2013), and methods in the second group include those of van Leeuwen and Herrmann (2013) and Almomin (2012, 2015). Mathematically, our approach is most closely related to the first two methods, but it takes two important concepts from the other methods, and it is these concepts that seem to be central to the success of AWI on realistic data sets.…”

“…We expect this to be a common feature of any successful, broadly applicable, waveform-inversion method. Conventional FWI achieves this by using low frequencies and a good starting model; AWI and Ma and Hale's (2013) approach achieve this by using matching filters; and AWI, van Leeuwen and Herrmann (2013), and Biondi and Almomin (2015) all achieve it by reformulating the underlying problem so that the predicted data need no longer obey a physical wave equation.…”

“…The second concept, explored in depth by Symes (2008), is that the dimensionality of the unknown model space is hugely expandedby the Wiener-filter coefficients in AWI, by adding an additional dimension to the subsurface model in Biondi and Almomin's (2015) approach, and by treating the entire wavefield as part of the unknown model space in the approach of van Leeuwen and Herrmann (2013). It is not clear if this is an essential feature for a successful method, but we note that it is common to all the methods that currently appear able to overcome cycle skipping during waveform inversion in complicated models.…”

Combining global optimization and boundary integral methods to robustly estimate subsurface velocity models

“…Researchers also make use of different objective functions and regularization techniques (Burstedde and Ghattas, 2009;van Leeuwen and Herrmann, 2013) to make the inversion more convex, mitigating the need for global solvers.…”