Avoiding Cycle-Skipping

Low frequencies in the field data are essential for robust and effective inversion without cycle-skipping issues. GeoStreamer multisensor acquisition provides these crucial low frequencies from the deep tow of the streamer. 

Full Waveform Inversion (FWI) inverts for the velocities that explain the observed data. This process is non-linear and the inversion can easily lead to the wrong solution. To mitigate this, conventional inversions require rich low-frequency data and an initial model close to the true model. This article By using:

  • 'Dynamic Time Warping', which computes a shift field (ms) that aligns the field seismic data to the modeled data. The shifts are used to build a more reliable initial FWI model than using conventional FWI.

These methods relax the requirement to have an accurate initial model and to have low-frequency recordings. It could enable faster computation of a velocity model with less geological input or a detailed starting model.

Cycle-Skipping and FWI

When performing FWI, a data residual is minimized to update the velocity model. The residual is the difference between the acquired data and synthetic modeled data using the velocity model. Cycle-skipping can occur when the phase match between the two data sets is greater than half a wavelength, causing erroneous model updates, which, in turn, can lead to incorrectly imaged seismic data. Cycle-skipping mitigation remains the primary challenge for FWI’s implementation in all velocity regimes. Traditional least-squares solutions are performed on the residual of the waveforms, and thus know nothing about the requirements for phase alignment.

Using the Wasserstein Distance

PGS is implementing a method that uses a minimization of the so-called ‘Wasserstein distance’, which is the distance required to map one statistical representation to another through the use of distribution functions. This method reduces the half-cycle requirement of traditional FWI schemes. Relaxing this criterion enables the starting model for FWI to be less accurate, and therefore could conceivably reduce turnaround by by-passing the need to prepare a starting model for FWI that meets the half-cycle criterion.

Warping to Mitigate Cycle-Skipping

Dynamic time-warping (DTW) is a method that aligns two or more time-samples sequences, for example, two seismic data sets. The resulting time shifts can vary dynamically in both space and time. DTW is a method using global optimization and therefore can avoid cycle-skipping challenges that affect localized approaches. 

In FWI, DTW is used to align the field seismic data to the synthetic modeled data. In conventional FWI the residual between the field and modeled data is backpropagated to form the gradient. The update to the model is computed as a scaled version of this gradient. In DTW FWI, the residual between the warped field data and the synthetic modeled data is used to update the model.

Cycle-skipping in a data set may be endemic or localized. PGS FWI only uses DTW where cycle-skipping occurs between the two sequences. Where none is present, the solution reverts to a conventional FWI approach.

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 will converge to the correct global solution.

With these methods, there is less emphasis on an accurate starting model or needing rich low-frequency data to avoid cycle-skipping issues in conventional FWI. Therefore, PGS FWI will create a more reliable image, in an accelerated timeframe.

New W2 Norm FWI Applied to Data from the Ceará Basin, Brazil

The performance of one of the new solutions is demonstrated on a GeoStreamer survey acquired offshore Brazil and illustrates how FWI successfully updates the earth model and resolves a high-velocity carbonate section missing from the initial velocity model.

The new FWI algorithm was applied to a GeoStreamer survey acquired in the Ceará Basin, offshore Fortaleza, Brazil. The starting (a) and final (b) FWI models are shown for a line in the proximity of a well. The final FWI model matches the well trend capturing the spatial variability of the carbonates.
The new FWI algorithm was applied to a GeoStreamer survey acquired in the Ceará Basin, offshore Fortaleza, Brazil. The starting (a) and final (b) FWI models are shown for a line in the proximity of a well. The final FWI model matches the well trend capturing the spatial variability of the carbonates.