The promise of full waveform inversion

One of the most popular phrases in the geophysical community today is “full waveform inversion” (FWI). FWI was initially developed almost three decades ago in an attempt to obtain from seismic data quantitative information about subsurface rock properties on a very detailed scale. Over the last few years there have been some encouraging results employing FWI in seismic processing marine and land data. FWI iteratively updates an estimated subsurface model and computes corresponding synthetic data to reduce the difference (the data misfit) between the synthetic and recorded data. The objective of FWI is to match the synthetic and recorded data in a comprehensive way such that all information in waveforms (e.g., traveltimes, amplitudes, converted waves, multiples, etc.) is accounted for in the data misfit (Figure 1).

FIGURE 1. This general workflow for FWI shows the initial model containing information from legacy velocities, well logs, and non-seismic measurements for velocity analysis. (Data courtesy of Kapoor et al, 2012; images courtesy of Rocky Roden)

The FWI technique is attractive in its potential capability to estimate a subsurface model that generally produces higher resolution results than conventional processing methods and algorithms used today. Another advantage of FWI over conventional inversion techniques is that FWI can estimate multiple parameters such as velocity, density, and attenuation. This method aims to recover the true model by iteratively minimizing the difference between observed and modeled data and can be formulated in either the time or frequency domain.

Remaining challenges

There are drawbacks to FWI. This approach is very compute-intensive, requiring numerous iterations of the model to converge on an acceptable solution, and often the methods used to converge to an acceptable solution can produce spurious results. In addition, FWI is an underdetermined inverse problem with many solutions, most of which make no geologic sense. These problems are related to a typically large number of model parameters and to the absence of low frequencies in the recorded seismic data. FWI has primarily been used in defining velocity models for prestack depth migration and imaging (Figure 1). Another use for FWI is to determine reservoir properties from this inversion process, but the industry is in the early stages of this application. In current commercial practice, both approaches simplify the physics of wave propagation, emphasize only some parts of the total recorded wavefields, and seek to match only some of the properties of those wavefields. These approximations and compromises are made both to reduce the total compute cost of FWI and to circumvent the necessity to invert for multiple parameters that may be ill-constrained by the available data.

When building velocity models for seismic processing, one of the difficulties with FWI is accurately determining the misfit between the data and the model. Classical FWI involves the minimization of a least-squares misfit function between the calculated and observed data. Common approaches employ nonlinear gradient-based optimizations where complex strategies for regularizing the process (filtering, weighting, and mute of the data, etc.) are used to mitigate the nonlinearity inherent in the entire process.

FIGURE 2. A data misfit results after several iterations, producing local and global minima depending on the starting models. (Data courtesy of Ma, 2010).

One of the challenges with FWI using gradient or gradient-descent methods is the convergence to the local minima, which makes the technique very sensitive to the starting velocity model, especially when 3-D is considered. This is illustrated in Figure 2, where an initial starting model was input into the FWI process and, after several iterations, a local minimum of the misfit was determined. However, because of the nonlinearity of the forward modeling process, this local minimum as well as several other spurious local minima may be determined. Therefore, in this approach a starting model too far from the true model can produce erroneous results that may induce an improper interpretation.

Acquisition parameters

Optimally, an FWI with a starting model close to the true model can converge to a global minimum and the true model. To lessen the sensitivity of the initial velocity field, low frequencies and long offsets are required, enabling FWI to update the low-frequency component of the velocity model. This demonstrates that FWI can be used for velocity updates if the acquired data have enough low frequencies and long offsets. Particularly, the shallow part of the model could be significantly enhanced by use of FWI and can result in a more improved depth image overall. In practice, a velocity macromodel generated by traditional approaches from traveltime tomography or migration velocity analysis may serve as an initial model for FWI.

Therefore, geophysicists have to make decisions when applying FWI because of the large amount of data to be simultaneously involved in the process and the numerous unknown parameters. These issues related to FWI have led to different strategies such as stochastic methods like the genetic algorithm and simulated annealing to address them. In contrast to deterministic gradient-based methods where it is often unclear whether the final result is near the global minimum and the true model, stochastic methods search for the global minimum of the misfit function even in the absence of a good starting model. Stochastic methods do not require the calculation of gradient-descent values after several iterations. Only forward modeling is needed to evaluate the desired results.

Unlike deterministic methods that generate a single “best” model, stochastic methods yield statistical information about the range of acceptable models. One significant drawback to this stochastic approach toward FWI is that the more forward model evaluations that are generated, the higher the computational cost. The convergence to acceptable results is at times beyond practical existing computing capabilities.

FIGURE 3. These general inversion approaches are employed today. FWI could be the inversion approach of choice to determine reservoir properties in the future.

The promise

From an interpretive perspective the desired outputs from FWI are reservoir properties such as porosity, fluid type, and permeability. Seismic data responses are not necessarily a direct result of variations in reservoir properties but are due to variations of the elastic properties of the earth. It is up to the interpreter to unravel these relationships between reservoir properties and elastic properties from seismic data in a reliable manner. This is where FWI holds such promise. As Figure 3 indicates, FWI is theoretically the most advanced inversion approach of the methods employed in the industry today. Perhaps in the not-too-distant future FWI will enable reservoir property maps to be routine outputs from the seismic interpretation process.