- Mitigating cycle-skipping reduces the need for accuracy in the starting model and shortens turnaround time.
- Using reflections in FWI enables updates beyond the maximum depth of penetration of refracted energy.
- Combining both produces more accurate models and reduces structural uncertainty in the imaged data.

Since the commercialization of Full waveform inversion (FWI) after 2010, there has been a continual evolution to overcome the dual challenges of using reflections and mitigating cycle-skipping, as successful implementations would save both time and money. Using reflections allows a greater diversity of applications while mitigating cycle skipping simplifies the starting point and produces more accurate models and reliable subsurface images.

#### Overcoming Challenges to Increase FWI Accuracy and Efficiency

FWI minimizes the difference between acquired seismic data and a synthetic equivalent. The synthetic data is modeled using a wavelet that replicates the seismic source, and earth models. If the models used in FWI do not accurately represent real subsurface velocities, there will be a kinematic misalignment between the synthetic and acquired data. FWI uses this difference, called a residual, to update the models used to generate the synthetic data in the inversion. If the misalignment is greater than half a waveform period, then ‘cycle-skipping’ may occur. The minimization scheme in the inversion may result in the wrong waveforms aligning, leading to an erroneous update, and uncertainty in the imaging when using the model.

Using reflections in FWI is more challenging than using refractions or diving waves, but if reflections are used then the updates are no longer limited to the depth of penetration of the refracted energy.

Reflectivity isochrones form an image in a wave-equation migration, but contaminate the velocity sensitivity kernels in FWI and force the velocity updates to occur at discrete layer boundaries. PGS FWI implementation solves this problem by using an inverse scattering imaging condition, which removes the reflectivity isochrones, allowing backscattered energy to drive the velocity update.

Another hurdle for reflection FWI is to minimize the kinematic difference of the same reflection events. To generate all the needed reflections in the synthetic data, we need contrasts in the earth models. The reflectivity seen in the seismic data may relate to contrast in either velocity or density. The velocity model may not be well resolved when FWI is started, and it is often difficult to create an accurate survey-wide density model. Therefore, modeling all the necessary reflections for FWI can be a problem.

PGS has developed solutions to both challenges. Cycle-skipping mitigation uses dynamic time-warping (DTW), and reflections are used by reformulating the acoustic variable density wave-equation in terms of vector reflectivity.

#### A New Formulation for Reflectivity FWI

Including reflections in FWI requires hard boundaries in either the input velocity or the density models. These simulate backscattered energy and generate the appropriate velocity sensitivity kernels. Imposing hard boundaries in either the velocity or the density models may be difficult, especially if dense and accurate measures of density are not available, or the velocity model is immature or inaccurate.

A costly and less accurate alternative is to use first-order approximations to decompose the seismic wavefields into background and perturbations in the wave-equation. Rather than use this approach, PGS uses the wave equation, formulated in terms of vector reflectivity, to produce reflections in the modeled data. The vector reflectivity wave-equation is derived by parametrizing the variable density wave-equation. As the reflectivity is derived from the seismic data, PGS FWI no longer needs a speculative density function.

#### Reflectivity FWI Case Study from Canada

Data from the PGS / TGS Tablelands survey in Canada’s Orphan Basin showcases the benefits of PGS’ new reflection-driven FWI. Refracted waves dominate the shallow FWI update. The basin has a two-km water-column and PGS acquired the seismic data with a maximum offset of eight km. Refraction-driven FWI is difficult beyond the top Cretaceous, which occurs at approximately four-km depth.

Vector reflectivity FWI accurately resolved the model deeper than the top Cretaceous. Using reflections with frequencies up to 25 Hz produces a high-resolution FWI model, three km beyond the maximum penetration depth of the refracted waves.

The first image displays the FWI difference co-rendered on the reflectivity model. The under-corrected input migrated gathers (bottom left) show a slower velocity is required. After reflection-driven FWI, the gathers are much flatter (bottom right – yellow arrows).

The second image shows a depth slice at 5.6 km, with the initial velocity model (B) and the final FWI model (C). The final reflectivity-driven FWI velocity model has enhanced spatial resolution, conformable with the structure.

#### Eliminating Cycle skipping with Dynamic Time-Warping

PGS’ new FWI code stops the inversion creating an inaccurate model, caused by cycle-skipping. The code computes a shift field that dynamically aligns or ‘warps’ the seismic data to match the synthetic modeled data, satisfying the cycle-skipping assumption of a conventional FWI. The residual between the warped seismic data and the modeled data is backpropagated to construct the gradient in the usual way, and this is used to update the model. Like conventional FWI, the process is iteratively applied. As the model is updated, the amount of warping required for subsequent passes decreases, and the result converges to the correct global solution.

#### A Successful Elimination of Cycle-Skipping in Brazil

We can demonstrate an application of this approach using a dataset from the Ceara Basin, offshore Brazil. PGS acquired the survey with 14 multisensor streamers, separated by 100 m, each with a maximum offset of eight km. Shallow carbonate layers complicate the setting and are poorly resolved in the legacy model. The water column is very short (50 m), preventing the use of conventional tomographic methods, as no shallow reflections are recorded before their critical angle. Additionally, the velocity contrast of the shallow carbonate layer limits the depth of penetration of refracted waves for FWI.

The initial pass using dynamic time-warping needs large shifts to match equivalent waveforms in the field and the modeled data. The first image shows that far-offset data require transporting up to 500 ms (below right), so cycle-skipping prevents the use of a conventional approach.

Using conventional FWI, the inversion isolates the top of the carbonate layers (below left). Underneath this initial increase in velocity, is a velocity inversion in the shallow sediments (orange arrows). Well profiles from the area suggest this is unexpected. Using the proposed split-optimization workflow, the deeper carbonate velocities are captured (below right – blue arrows), in agreement with well information reported near to the area of study.

#### A More Automated Approach Applicable to Wider Use

Solving the challenge of cycle skipping with dynamic time-warping reduces pressure on the starting criterion of the initial FWI model; therefore, we can start FWI earlier and reduce its cycle-time. Creating high-resolution models with FWI is on the critical path of seismic processing projects, so reducing the time to build a model with FWI naturally shortens the entire project turnaround.

Using FWI with reflections can be challenging, as reflections need to be present in the modeled data, from contrasts in either the density or velocity models. If both are immature, then arbitrary augmentation of one or the other is required to generate the reflectivity. Using a vector reflectivity wave-equation, PGS employs reflections in the inversion without the expense of solving two different propagation steps. This results in a more cost-effective implementation that is free of model manipulation and applicable to a much wider range of geological settings. Our research is continuing, and we look forward to sharing updates on further automation of the use and application of FWI.

By mitigating cycle-skipping and using reflection data PGS is able to build FWI models faster, with greater accuracy, for a wider range of data, and in a more cost-conscious manner. PGS is continuing with FWI research to automate its use and application further.

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