Full Waveform Inversion Implementation Research Articles

Elastic full-waveform inversion on Caesar-Tonga — Case study

Full-waveform inversion (FWI) has been widely used on 3D data sets to build detailed velocity models over the past 10 years. Most of these projects used pressure data and an acoustic approximation, with the assumption that the field data were dominated with P-waves. This approach of FWI can determine parameters related to the acoustic wave equation. It focuses on updating velocities by minimizing the misfit between observed and model data. Acoustic FWI, using the pressure component of collected data, has shown tremendous potential in simple geologic settings. Successful FWI projects, using wide-azimuth streamer, ocean bottom, and land survey geometries, have convinced the oil industry to pursue the next step by involving more physical properties. However, questions remain on how far we can properly describe field data with the acoustic approximation and at what point we need to switch to a much more expensive elastic wave equation implementation. In a complicated geologic region such as the Gulf of Mexico (GoM), the seismic wavefield can be complex and elastic FWI is needed to achieve a better velocity model, even when using mostly pressure data alone. We demonstrate the application of elastic FWI on sparse-node ocean-bottom-node data from the GoM and show comparisons to the acoustic solution. The comparisons demonstrate the benefits of the elastic FWI implementation when applied to image steeply dipping Miocene sands beneath a complex salt canopy, despite the increased computational expense. Furthermore, we demonstrate that when elastic FWI is applied to sufficiently high frequencies, the FWI-derived reflectivity product and velocity model are reliable interpretation products.

Frequency-domain acoustic full waveform inversion with an embedded boundary method for irregular topography

In the implementation of full waveform inversion (FWI) to identify subsurface velocity distributions with land seismic data, which are often acquired in regions with irregular topography, wave equation-based modelling requires caution. In particular, when using the finite difference method (FDM), unwanted scattered waves are generated because irregular surfaces crossing a rectangular grid are discretized via a staircase approximation; hence, if the problems caused by this staircase approximation are disregarded, FDM-based FWI may fail due to the presence of undesirable wavefields. To resolve this problem, this study develops a 2D frequency-domain acoustic FWI technique using a 9-point FDM-based modelling scheme that includes an embedded boundary method (EBM). This study suggests a workflow for the whole EBM-based FWI process from the calculation of coefficients for the EBM-based 9-point FDM modelling to applying it to FWI for proper velocity updates. In numerical examples, using velocity models with a tilted surface and an arbitrarily fluctuating surface, we synthesize seismic data and verify the accuracy of EBM-based 9-point FDM modelling and its superiority over the conventional FDM by comparing it with wavefields derived from the spectral element method. Then, we show that our EBM-based FWI is able to estimate subsurface velocity distributions even though the model has irregular topography, which spoils the result of the conventional FWI.

Datum-based waveform inversion using a subsurface-scattering imaging condition

Full-waveform inversion (FWI) solves model optimization problems by fitting simulated data to the observed data. The implementation of FWI requires involving as many physical features of the subsurface medium in the simulation as possible. The computation can be extremely costly and complex because the FWI algorithm usually deploys a consistent discretization over the entire model space, whereas a high-resolution analysis (and the accompanying complex physics) is often only required in the reservoir region. As an alternative, we have developed an FWI optimization scheme based on a convolution type of modeling from the datum. The solution of such an inversion consists of the overburden, which includes the medium above a datum level, and the virtual data at the datum, which represent the underlying medium, i.e., the reservoir. We formulate the redatuming operation using a modified expression of the extended Born representation. Based on that, the virtual data can be retrieved by using the subsurface-scattering imaging condition. By measuring the data misfit at the surface acquisition, we implement a simultaneous inversion for the overburden velocity and the virtual data at that datum. This velocity inversion is crucial to the redatuming, but we can rely on fairly simple physics compared to the complex reservoir region. The redatumed data, in turn, are involved in resolving the overburden velocity. We develop a robust estimate of the velocity using low-wavenumber updates along the reflection wavepaths generated by our modeling process including the overburden scattering and those coming from the datum. Using numerical examples, we determine that our optimization is capable of mitigating the complex wave-propagation effects of the overburden medium and output redatumed data with plausible relative amplitude, which is achieved given little knowledge of the model.

Long-wavelength FWI updates in the presence of cycle skipping

Full-waveform inversion (FWI) has become the tool of choice for building high-resolution velocity models. Its success depends on producing seamless updates of the short- and long-wavelength model features while avoiding cycle skipping. Classic FWI implementations use the L2 norm to measure the data misfit in combination with a gradient computed by a crosscorrelation imaging condition of the source and residual wavefields. The algorithm risks converging to an inaccurate result if the data lack low frequencies and/or the initial model is far from the true one. Additionally, the model updates may display a reflectivity imprint before the long-wavelength features of the model are fully recovered. We propose a new solution to this fundamental challenge by combining the quadratic form of the Wasserstein distance (W2 norm) for measuring the data misfit with a robust implementation of a velocity gradient. The W2 norm reduces the risk of cycle skipping, whereas the velocity gradient effectively eliminates the reflectivity imprint and emphasizes the long-wavelength model updates. We illustrate the performance of the new solution on a field survey acquired offshore Brazil. We demonstrate how FWI successfully updates the earth model and resolves a high-velocity carbonate section missing from the initial velocity model.

High frequency full waveform inversion as an interpretation solution

The amount of available data to help us characterise the subsurface is ever increasing. Large seismic surveys, long offsets, multi- and full-azimuth datasets, including 3D and 4D, marine, ocean-bottom nodes and extremely high fold land surveys, are now common. In parallel, computing power is also increasing and, in combination with better data, this enables us to develop better tools and to use better physics to build models of the subsurface. Wave-equation based techniques, such as full waveform inversion (FWI), have therefore become a lot more practical. FWI uses the entire wavefield, including refractions and reflections, primaries and multiples, to generate a refined, high resolution Earth model. This technique is now commonly used at lower frequencies (up to 12 Hz) to derive more accurate models for improved seismic imaging and reduced depth conversion uncertainty. By including higher frequencies in FWI, we can attempt to resolve for finer and finer details. FWI models using the entire bandwidth of the seismic data constitute an interpretation product in itself, with applications in both structural interpretation and reservoir characterisation. Incorporating more physics within the FWI implementation, combined with modern supercomputer facilities, promises to increase the focus on very high frequency FWI in the coming years. In this paper, through a series of field examples, we illustrate the applications and rewards of high frequency FWI: from improved imaging, improved quantitative interpretation and depth conversion to a direct interpretation of the FWI models.

Pseudo-analytical finite-difference elastic-wave extrapolation based on the k-space method

Cost-effective elastic-wave modeling is the key to practical elastic reverse time migration and full-waveform inversion implementations. We have developed an efficient elastic pseudo-analytical finite-difference (PAFD) scheme for elastic-wave extrapolation. The elastic PAFD scheme is based on a modified pseudo-spectral method, k-space method, in which a pseudo-analytical operator is used to ensure the high accuracy of elastic-wave extrapolation. However, the k-space method is motivated for a pure wave mode, and thus its application in coupled first-order elastic-wave equations may cause the elastic pseudo-analytical operators to suffer from crosstalk between the P- and S-wavefields. The approaches presented attempt to overcome these shortcomings by introducing two improvements to achieve the goal. This is done, first, by performing a predictor-corrector strategy in first-order elastic-wave equations to eliminate those errors during wave extrapolation. Considering the massive computational cost in the spectral domain, we have developed an efficient elastic PAFD implementation, in which an innovative model-adaptive finite-difference coefficient-predicted scheme is provided to reduce the computational cost of elastic pseudo-analytical operator differencing. Dispersion analysis demonstrates the flexibility with varying velocity and superior performance of our PAFD scheme for spatial and temporal dispersion suppression than the existing Taylor-expansion-based scheme. Under the same simulation parameters, several numerical examples prove that the elastic PAFD scheme can provide more accurate simulation results, whereas the conventional scheme suffers from spatial or temporal dispersion errors, even in complex heterogeneous media.

Estimating a starting model for full-waveform inversion using a global optimization method

Full-waveform inversion (FWI) has become a popular method to estimate elastic earth properties from seismograms. It is formulated as a data-fitting least-squares minimization problem that iteratively updates an initial velocity model with the scaled gradient of the misfit until a satisfactory match between the real and synthetic data is obtained. However, such a local optimization approach can converge to a local minimum if the starting model used is not close enough to an optimal model. We have developed a two-step process in which we first estimate a starting model using a global optimization method. Unlike local optimization methods, a global optimization method starts with a random starting model and is not generally susceptible to be trapped in a local minimum. The starting model for FWI that we aim to estimate is sparsely parameterized and contains a set of interfaces and velocities that are used to represent the entire velocity model. We have obtained the depth of the interfaces and the velocities by minimizing the data misfit in the least-squares sense using a global optimization method called very fast simulated annealing (VFSA). Once the sparse velocity model was obtained from VFSA, we used that as a starting model in a conventional gradient-based FWI to obtain the final model. We have applied the proposed method to one synthetic data set and two field data sets from offshore India. The proposed method was able to estimate a velocity model that was not cycle skipped for realistic frequency bands. We have demonstrated that with the proper choice of model parameterization and optimization parameters, the global and gradient optimization algorithms converge in a finite number of iterations. We have determined that the resulting algorithm is computationally feasible in two dimensions and accurate for practical implementation of FWI.