Global Optimization for Full Waveform Inversion : Understanding Trade-offs and Parameter Choices

In this paper, we use a fast Helmholtz solver with global optimization methods to estimate an initial velocity model for full-waveform inversion based on raw recorded waveforms. More specifically, we combine the field expansion method for solving the Helmholtz equation with a reduced parameterization of the velocity model allowing for extremely fast forward solves of realistic velocity models. Unlike conventional Full Waveform Inversion that uses gradient based inversion techniques, global optimization methods are less sensitive to local minima and initial starting models. However, these global optimization methods are stochastic and are not guaranteed to converge to the global minima. Because our adaptation of the field expansion method is so fast, we can study the convergence of these algorithms across parameter choice and starting model. In this paper we compare two commonly used global optimization methods, particle swarm optimization, and simulated annealing, and examine the limitations of these algorithms and their dependence on choice of parameters. We find that PSO outperforms SA and that PSO converges reliably for noisy data provided the number of parameters remains small. In addition, using this methodology we are able to estimate reasonable FWI starting models with higher frequency data. INTRODUCTION Most seismic techniques such as imaging, rely on a velocity model inverted from noisy data through a non-linear inverse problem. These inverted velocity models may be inaccurate and lead to incorrect interpretations of the subsurface. For example, an erroneously fast or slow section of a velocity model could cause a syncline structure to appear as an anticline and be incorrectly interpreted as a potential trap. Full Waveform Inversion (FWI) and other gradient based methods will converge to local minima (Warner et al., 2013) and will lead to inaccurate velocity models if initiated with an inaccurate starting model (Virieux and Operto, 2009). In recent years Uncertainty Quantification (UQ) methods have been applied to the FWI problem in order to mitigate the impact of local minima and provide a more probabilistic interpretation of the sub surface (Stuart et al., 2016; Zhu and Gibson, 2016; Ray et al., 2016). See Osypov et al. (2013) for a discussion of importance of risk assessment and UQ in seismic imaging. These methods allow for an informed decision about the reliability of the subsurface image and aid in risk estimates for drilling a potential well. However, these methods either require hundreds of thousands of wave equation forward solves to converge to accurate error estimates of subsurface parameters (Ely et al., 2018), or use local approximations of a Gaussian to approximate the uncertainty (Fang et al., 2018). The number of forward solves required for MCMC is likely infeasible for many seismic imaging problems and forward models. In addition, in low noise scenarios there many only Ely, Malcolm, Nicholls 2 Global Optimization methods be a single region of high probability and MCMC techniques may be unnecessary. Instead, global optimization methods are an attractive alternative as they require far fewer forward solves and provide an estimate of the best fitting model (maximum Likelihood) at a fraction of the number of forward solves needed for UQ and yet can still avoid being trapped by local minima. In recent years global optimization methods such as Simulated Annealing (SA) (Datta and Sen, 2016; Galuzzi et al., 2017; Sajeva et al., 2016) and Particle Swarm Optimization (PSO) (Ely et al., 2015; Shaw and Srivastava, 2007) have become popular methods of calculating an initial velocity model. These methods can be an alternative or supplement to conventional methods of building velocity models such as tomography, NMO velocity analysis or very low frequency FWI (Woodward et al., 2008). Conventional initial velocity building methods typically rely on gradient based minimization and often require the manual picking of arrival times or picks in semblance space. By contrast, global optimization methods, although not guaranteed to converge to the global minima, are far more robust to local minima and frequently do not require the calculation of a gradient, reducing memory constraints and algorithmic complexity. However, starting models, frequency, and other parameter choice can have a significant impact on the convergence of these algorithms. Although these algorithms require significantly fewer iterations than UQ algorithms, rigorous comparison requires numerous global optimization runs and thousands or tens of thousands of forward solves. In addition, due to the stochastic nature of these algorithms, two different runs with the same starting model can converge to different finals models, requiring even more simulations to understand performance. Due to the computational cost of most forward solvers, previous work has been unable to accurately characterize these impacts and benchmark these algorithms as significant simplifications were necessary to control this computational cost. For example, Sajeva et al. (2017) use analytical solutions to test problems as well as the 1D wave equation to compare global optimization algorithms. In another attempt to make the problem computationally tractable, previous work has typically shown performance with one or only a few starting models when tens or hundreds of runs are necessary to generate a statistical model of convergence and accurately compare one algorithm to another (Rios and Sahinidis, 2013). Studies in the global optimization literature use test functions that are extremely fast to evaluate but are not representative of local minima found in FWI and seismic imaging. Other work combining global optimization methods and FWI have lacked a sufficiently fast test function that accurately accounts for the physics of seismic inversion in 2D. In this paper, we use the reduced model parameterization of velocity models introduced in Frazer and Sen (1985); Zelt and Smith (1992) and Datta and Sen (2016) combined with two global optimization methods to estimate an initial velocity model. Due to the rapid speed of our forward model and we can quantitatively characterize the (1) limits on the number of degrees of freedom, (2) importance of initial guesses, (3) choice of tuning parameters, (4) comparison of different global optimization methods and (5) impact of source frequency. These comparisons, would be impossible with slower conventional finite difference solvers. From the numerical studies presented in this paper we are able to draw several conclusions about the performance of global optimization methods for FWI and what may impact the likelihood of finding a good initial model. Although the FWI problem is known to be more prone to local minima at higher frequency (Warner et al., 2013), we find that source Ely, Malcolm, Nicholls 3 Global Optimization methods frequency does not significantly decrease the likelihood of convergence and a good starting model can be found with a relatively high frequency of 5 or 8 Hz from raw waveform data. This suggests that we could build initial background velocity models with much higher frequencies than are typically used. We also find that the accuracy of the initial starting models weakly impacted the likelihood of converging to the correct model. From our results we find that only having an extremely accurate initial model improved the likelihood of convergence. This suggests that only if the initial model is in the basin of attraction of the global minima are we guaranteed to converge to the true solution and outside of this range the global optimization methods are equally likely to find an accurate solution independent of starting model. In addition, we find that our implementation of Particle Swarm Optimization (PSO) out preforms Simulated Annealing (SA) and is able to find a more accurate initial model with the same number of forward solves. The remainder of the paper is organized as follows. We first describe a reduced parameterization of the velocity model that is compatible with the field expansion forward solver. We then demonstrate that this first forward model and parametrization can yield comparable results to finite difference simulations if modifications are made to the forward solver. Second, we describe the two global optimization methods used in the paper: Particle Swarm Optimization (PSO) and Simulated Annealing (SA). Finally, we demonstrate our inversion on several synthetic models of varying complexity to determine the limitations on number of degrees of freedom, choice of initial model, and iterations needed. The field expansion method achieves significant computational saving through restricting the velocity model to consist of a series of non-overlapping layers. Although this parameterization severely restricts the velocity models we can simulate, we find that the model is sufficiently accurate to estimate initial models for FWI as demonstrated later in our paper. FORWARD MODEL Unlike gradient based minimization methods, global optimization and uncertainty quantification methods require far fewer degrees of freedom, tens versus thousands. In this section we briefly describe the reduced parametrization previously used by the authors in Ely et al. (2018) for uncertainty quantification. The reduced parameterization is compatible with the field expansion method (Malcolm and Nicholls, 2011a), allowing for extremely rapid forward modeling of the Helmholtz equation. We briefly describe the field expansion method and the modifications we make to the forward solver to mitigate several of the artifacts inherent to the field expansion method. Reduced parameterization The field expansion method is able to achieve rapid forward solves by heavily restricting the velocity model to consists of a number of non-overlapping piecewise constant layers as shown in Figure 1. Although the Earth has often been approximated by serie