## Using the Full Wavefield

The PGS Full Waveform Inversion (FWI) solution uses diving waves, refractions, and reflections together to deliver accurate full record length model updates in a stable manner without relying on very long offsets.

PGS uses a smart imaging condition to avoid the high wavenumber artifacts (perturbation) that conventional FWI suffers from.

Additionally, PGS has modified the variable density acoustic wave-equation to one using vector-reflectivity. This enables accurate modeling of reflections using a term directly determined from the data. The objective function is constrained by an elegant regularization scheme within the inversion. The entire workflow allows accurate background models to be built, avoiding inversion results that resemble seismic images.

#### Key PGS FWI Benefits

- Uses diving waves, refractions, and reflections together to reduce dependence upon very long offsets and provide high-fidelity velocity models
- Avoids cycle-skipping artifacts and delivers models with correct long-wavelength information
- Removes migration isochron artifacts resulting in accurate deep velocity models
- High-resolution deep velocity updates translate to improved subsurface images and robust reservoir characterization
- No need to construct a density model or have an accurate velocity model with hard boundaries

#### Smart Imaging Condition for Artifact-Free FWI Using the Full Wavefield

The figure below has the technical name of being the 'sensitivity kernel' for FWI, and describes PGS' smart imaging condition. Written formally, it shows the sensitivity of the FWI misfit function to the perturbation of a specific parameter, in this case the velocity being derived from field shot gathers. In simpler terms, it shows where the velocity model is updated for a particular source-receiver offset (5 km here) using two types of FWI solution: 'Transmission FWI' which uses diving waves and refractions to provide velocity model updates, and 'Reflection FWI' which uses reflections to provide velocity model updates.

The left and right versions of this figure appear similar but they are not. First, let us examine what is different for transmission FWI and reflection FWI. Heuristically, it is obvious why transmission FWI requires extremely long offsets to provide reasonably deep velocity updates: The 'banana' transmission FWI kernel only affects the velocity model in this example for quite shallow depths. Velocity updates at larger depths will require much longer offsets. In contrast, reflection FWI clearly provides much deeper velocity updates for the same offset.

Unfortunately, FWI has historically been more challenging for most people to implement. One challenge has been the high wavenumber 'migration isochron' that dominates the left-side sensitivity kernel based upon the conventional cross-correlation imaging condition. In contrast, the right-side sensitivity kernel uses a new Inverse Scattering Imaging Condition (ISIC) developed by PGS and is free of this artifact. For more details refer to the paper by Ramos-Martinez et al. presented at the 2016 SEG conference.

The figure below compares the results of a velocity model derived using the historical implementation of reflection FWI using the cross-correlation imaging condition (upper) versus the new implementation of reflection FWI using the new PGS imaging condition (lower). The difference is remarkable. Whilst it is inviting to view the upper result as containing spectacular resolution, and thereby 'fit for direct interpretation' in the manner wherein reflectivity seismic images are interpreted, nothing could be further from the truth. Most of the high wavenumber detail in the upper velocity-model update is false: Artifacts of the migration isochron problem in the left-side FWI sensitivity kernel above. The lower velocity model update is the correct representation of the geology in this location.

#### Efficiently Incorporating the Accurate Modeling of Reflections

There are challenges with incorporating reflections in FWI, primarily but not exclusively within the modeling engine. For example, how do we model the reflections that are necessary for a full wavefield FWI? They are present in the acquired seismic data, so their accurate modeling is needed to avoid building bad models for deeper data. Seismic modeling may overcome this with hard boundaries in the velocity model or with an accurate density model, but only if we have access to them.

Recorded seismic data does not directly measure density, and in under-explored regions density models may be difficult to access or generate. In these areas highly evolved velocity models probably do not exist.

Acoustic impedance is a function of spatially varying density and velocity and enables a reformulation of the density wave-equation to one defined by reflectivity, where vector-reflectivity is defined as the normalized rate of acoustic impedance change in each vector direction. Using this reformulation, scattering is primarily produced by the reflectivity term. No approximations are made, and only one wave-equation needs to be solved, comparing favorably with the standard industry approach of two required by Born modeling. This approach enables a new and efficient modeling engine for an FWI workflow for the full wavefield, using both transmission and reflections.

The entire optimized workflow uses a reflection modeling engine and PGS’ inverse scattering imaging condition. The reflectivity term is derived directly from the seismic data and the velocity model may come from either tomography or FWI. The inversion scheme jointly updates both the velocity and the reflectivity terms in each pass; the regeneration of the reflectivity term, based on the updated velocity model, is needed to maintain accurate reflection modeling for each successive iteration, which continues until the objective function criterion is reached.

The benefits of PGS’ innovative workflow are illustrated on field seismic data from the Orphan basin, located in the north of the Grand Banks, North East of St. John's, Canada. The figure below shows a depth slice at approximately 4.5 km, which is beyond the limits of transmission for this data and acquisition geometry. At this depth, the update relies on accurate reflection modeling and an optimized imaging condition. The slice of the model after deep reflection-inclusive FWI shows an improvement in resolution, and the inclusion of a slowdown that adjusts the under-corrected common image point gathers so that when the data is migrated with the updated model, the gathers are flatter.

It is also worth noting that reflection FWI does not only contribute high-resolution, deep velocity model updates. Some areas such as those affected by seafloor rugosity or very shallow geological heterogeneities may derive more benefit to shallow velocity model building using reflections rather than diving waves and refractions.