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.

S U M M A R YWave-equation traveltime tomography tries to obtain a subsurface velocity model from seismic data, either passive or active, that explains their traveltimes. A key step is the extraction of traveltime differences, or relative phase shifts, between observed and modelled finite-frequency waveforms. A standard approach involves a correlation of the observed and measured waveforms. When the amplitude spectra of the waveforms are identical, the maximum of the correlation is indicative of the relative phase shift. When the amplitude spectra are not identical, however, this argument is no longer valid. We propose an alternative criterion to measure the relative phase shift. This misfit criterion is a weighted norm of the correlation and is less sensitive to differences in the amplitude spectra. For practical application it is important to use a sensitivity kernel that is consistent with the way the misfit is measured. We derive this sensitivity kernel and show how it differs from the standard banana-doughnut sensitivity kernel. We illustrate the approach on a cross-well data set.In ray-based tomography, the aim is to construct a subsurface velocity model that explains the picked traveltimes of the measured data. Such a model can be obtained in an iterative manner by back projecting the traveltime differences along rays in the current velocity model. This procedure will lead to satisfactory results when the wave propagation is sufficiently well approximated by ray theory. To extract more information from the data than just the traveltimes of a few selected arrivals, seismologists are moving towards full-waveform processing and inversion of all available data. This trend is driven by the availability of high-quality broad-band data (earthquake data from USArray, for example) and a need to incorporate finite-frequency effects to process data from geologically complex areas (sub-salt exploration for the detection of hydrocarbons, for example). Also, the computing resources needed to routinely model 3-D wave-propagation in complex media are becoming a commodity. There have been two major developments regarding finite-frequency or full-waveform inversion in the last decades. The first is least-squares inversion-or waveform tomography-of seismic data pioneered by Tarantola & Valette (1982) that was aimed originally at inversion of active seismic reflection data. The second is the development of a finite-frequency analogue of ray-based traveltime tomography by Luo & Schuster (1991) (see also TrompRe-use of this article is permitted in accordance with the Terms and Conditions set out at

Many inverse and parameter estimation problems can be written as PDE-constrained optimization problems. The goal is to infer the parameters, typically coefficients of the PDE, from partial measurements of the solutions of the PDE for several right-hand-sides. Such PDE-constrained problems can be solved by finding a stationary point of the Lagrangian, which entails simultaneously updating the parameters and the (adjoint) state variables. For large-scale problems, such an all-at-once approach is not feasible as it requires storing all the state variables. In this case one usually resorts to a reduced approach where the constraints are explicitly eliminated (at each iteration) by solving the PDEs. These two approaches, and variations thereof, are the main workhorses for solving PDE-constrained optimization problems arising from inverse problems. In this paper, we present an alternative method that aims to combine the advantages of both approaches. Our method is based on a quadratic penalty formulation of the constrained optimization problem. By eliminating the state variable, we develop an efficient algorithm that has roughly the same computational complexity as the conventional reduced approach while exploiting a larger search space. Numerical results show that this method indeed reduces some of the non-linearity of the problem and is less sensitive to the initial iterate.

We consider a class of inverse problems in which the forward model is the solution operator to linear ODEs or PDEs. This class admits several dimensionalityreduction techniques based on data averaging or sampling, which are especially useful for large-scale problems. We survey these approaches and their connection to stochastic optimization. The data-averaging approach is only viable, however, for a leastsquares misfit, which is sensitive to outliers in the data and artifacts unexplained by the forward model. This motivates us to propose a robust formulation based on the Student's t-distribution of the error. We demonstrate how the corresponding penalty function, together with the sampling approach, can obtain good results for a large-scale seismic inverse problem with 50 % corrupted data.

We propose an extended full-waveform inversion formulation that includes general convex constraints on the model. Though the full problem is highly nonconvex, the overarching optimization scheme arrives at geologically plausible results by solving a sequence of relaxed and warm-started constrained convex subproblems. The combination of box, total-variation, and successively relaxed asymmetric total-variation constraints allows us to steer free from parasitic local minima while keeping the estimated physical parameters laterally continuous and in a physically realistic range. For accurate starting models, numerical experiments carried out on the challenging 2004 BP velocity benchmark demonstrate that bound and total-variation constraints improve the inversion result significantly by removing inversion artifacts, related to source encoding, and by clearly improved delineation of top, bottom, and flanks of a high-velocity high-contrast salt inclusion. The experiments also show that for poor starting models these two constraints by themselves are insufficient to detect the bottom of high-velocity inclusions such as salt. Inclusion of the one-sided asymmetric total-variation constraint overcomes this issue by discouraging velocity lows to buildup during the early stages of the inversion. To the author's knowledge the presented algorithm is the first to successfully remove the imprint of local minima caused by poor starting models and band-width limited finite aperture data. † John "Ernie" Esser passed away on March 8, 2015 while preparing this manuscript. The original is posted here: https://www.slim.eos.ubc.ca/content/total-variation-regularization-strategies-full-waveform-inversion-improving-robustness-noise. arXiv:1608.06159v1 [math.OC]

We develop and apply methods for resolution analysis in tomography, based on stochastic probing of the Hessian or resolution operators. Key properties of our methods are (i) low algorithmic complexity and easy implementation, (ii) applicability to any tomographic technique, including full‐waveform inversion and linearized ray tomography, (iii) applicability in any spatial dimension and to inversions with a large number of model parameters, (iv) low computational costs that are mostly a fraction of those required for synthetic recovery tests, and (v) the ability to quantify both spatial resolution and interparameter trade‐offs. Using synthetic full‐waveform inversions as benchmarks, we demonstrate that autocorrelations of random‐model applications to the Hessian yield various resolution measures, including direction‐ and position‐dependent resolution lengths and the strength of interparameter mappings. We observe that the required number of random test models is around five in one, two, and three dimensions. This means that the proposed resolution analyses are not only more meaningful than recovery tests but also computationally less expensive. We demonstrate the applicability of our method in a 3‐D real‐data full‐waveform inversion for the western Mediterranean. In addition to tomographic problems, resolution analysis by random probing may be used in other inverse methods that constrain continuously distributed properties, including electromagnetic and potential‐field inversions, as well as recently emerging geodynamic data assimilation.

We explore the use of stochastic optimization methods for seismic waveform inversion. The basic principle of such methods is to randomly draw a batch of realizations of a given misfit function and goes back to the 1950s. The ultimate goal of such an approach is to dramatically reduce the computational cost involved in evaluating the misfit. Following earlier work, we introduce the stochasticity in waveform inversion problem in a rigorous way via a technique calledrandomized trace estimation. We then review theoretical results that underlie recent developments in the use of stochastic methods for waveform inversion. We present numerical experiments to illustrate the behavior of different types of stochastic optimization methods and investigate the sensitivity to the batch size and the noise level in the data. We find that it is possible to reproduce results that are qualitatively similar to the solution of the full problem with modest batch sizes, even on noisy data. Each iteration of the corresponding stochastic methods requires an order of magnitude fewer PDE solves than a comparable deterministic method applied to the full problem, which may lead to an order of magnitude speedup for waveform inversion in practice.

Many inverse problems include nuisance parameters which, while not of direct interest, are required to recover primary parameters. Structure present in these problems allows efficient optimization strategies -a well known example is variable projection, where nonlinear least squares problems which are linear in some parameters can be very efficiently optimized. In this paper, we extend the idea of projecting out a subset over the variables to a broad class of maximum likelihood (ML) and maximum a posteriori likelihood (MAP) problems with nuisance parameters, such as variance or degrees of freedom. As a result, we are able to incorporate nuisance parameter estimation into large-scale constrained and unconstrained inverse problem formulations. We apply the approach to a variety of problems, including estimation of unknown variance parameters in the Gaussian model, degree of freedom (d.o.f.) parameter estimation in the context of robust inverse problems, automatic calibration, and optimal experimental design. Using numerical examples, we demonstrate improvement in recovery of primary parameters for several largescale inverse problems. The proposed approach is compatible with a wide variety of algorithms and formulations, and its implementation requires only minor modifications to existing algorithms.