Full Waveform Inversion
Full Waveform Inversion (FWI) is a methodology that seeks to find high-resolution, high-fidelity velocity models of the subsurface capable of matching individual synthetic seismic waveforms with an original raw field dataset. This is achieved iteratively by determining and minimizing a residual; the difference between modeled and recorded data.
PGS FWI utilizes the full wavefield, so both diving waves (wavefronts continuously refracted upwards through the earth due to the presence of a vertical velocity gradient) and reflections. FWI is successful in resolving small scale features; in particular, in shallow-water environments where reflection based tomographic inversion methods are limited.
As FWI operates directly on shot gathers, it can be deployed early in the model building process and it is an efficient velocity model building tool. FWI can be used as part of any Prestack Depth Migration (PSDM) velocity model building flow with virtually no time impact, as the input data requires minimum pre-processing and any free-surface effects can be left in the data.
The method begins from an initial starting model which is then iteratively improved using a sequence of linearized inversions, to solve the full non-linear 3D FWI problem.
FWI and the Inverse Scattering Imaging Condition
In conventional Reverse Time Migration (RTM), there is a forward and reverse time propagation of the source and receiver wavefield. In FWI, the back projection step basically constitutes a RTM run where the receiver wavefield is replaced by the residual between the modeled and recorded data. To rely less on refractions and to better utilize reflections in the FWI updates, the PGS implementation of FWI uses a variation of the Inverse Scattering Imaging Condition (ISIC) found in PGS RTM.
While in RTM, ISIC is used to remove backscattered and diving wave energy to produce a clean migration operator, the opposite is the case for FWI; we want only the backscattered and the diving wave energy. The ISIC formulation for RTM is based on a sum of two imaging kernels (to remove the low wavenumbers), whereas for FWI it is based on the difference of the same two kernels, to remove the high wavenumbers.
This is illustrated in the figure below where we using a simple v(z) model with a horizontal reflector and have plotted a conventional FWI gradient and the ISIC gradient. For an offset of 5km, the diving wave only samples the upper parts of the model, while the deeper parts of the conventional FWI gradient are dominated by the migration operator artifact. It is obvious why 'transmission FWI' requires long offsets to provide a deeper update, and lower frequencies will increase the width of the kernel, allowing for the update of larger areas in the model. In contrast, using the ISIC gradient of PGS' FWI, we can suppress the migration operator artifact, and hence reveal the underlying backscattered energy. This can then be used to provide model updates much deeper in the section than with conventional FWI.
A comparison of a conventional FWI gradient (left) and the PGS ISIC FWI gradient (right). The PGS FWI kernel (right) removes the migration isochron resulting in accurate velocity models.
The next figure shows the recovered velocity perturbations showing this effect. Conventional FWI only recovers the interfaces as there are no diving waves at this depth due to the limited offsets available in the data. The incorrect velocity perturbation field has been generated by the presence of the migration operator artifact in the FWI gradient. However, with the improved gradient of PGS' FWI we are able to correctly reconstruct the perturbations solely based on the backscattered energy, providing much deeper velocity updates for the same limited offset range.
Conventional FWI and PGS ISIC FWI on a five layer synthetic model. There are no diving waves at the depth of the interfaces. The conventional FWI recovers an incorrect solution due to the presence of the migration operator in the FWI gradient. With the PGS solution we correctly reconstruct the velocity perturbations solely based on backscattered energy, providing much deeper velocity updates for the same offset.
The resulting FWI models are often used as input to ray-trace based migration methods such as Kirchhoff or Beam migrations to improve the underlying reservoir image. However to retain the structural detail of the velocity field achieved with this FWI update the use of high-fidelity migration algorithms such as RTM or WEM is recommended. Unlike conventional reflection tomography, FWI typically uses wide-angle refracted arrivals and reflections to build its model.
Avoiding Cycle Skipping
Low frequencies in the field data are essential for robust and effective inversion without cycle skipping issues. GeoStreamer® dual-sensor acquisition provides these crucial low frequencies from the deep tow of the streamer. To ensure convergence of the model and avoid cycle skipping, FWI starts with the lowest frequencies in the data which contain coherent energy. The frequency content may subsequently be increased to add spatial resolution to the updates.
Additionally, to overcome cycle skipping limitations associated with simplistic starting models, PGS FWI uses sophisticated regularization schemes like the L1 norm of total variation (TV) of the model to stabilize the inversion space.
An accurate convergence of an FWI model phase is measured at each iterative stage using an integrated quality control procedure. Metrics demonstrating an improvement in the correlation of modeled and field data are used alongside data observations in both data and image space.
3D View of SWIM image at 250 m depth with FWI velocity model overlaid.