Resolving subsurface velocities with FWI

Full-waveform inversion (FWI) was introduced to the field of exploration seismology in the early ’80s by Albert Tarantola. However, it is only recently that FWI has entered the mainstream of seismic imaging and become the preferred tool for estimating complex velocity distributions. In his groundbreaking work, Tarantola laid out the theoretical foundation for FWI, and now, almost 30 years later, little of this has changed except for what one could call a revolution around the practical implementation aspects of the technology.

The key idea behind FWI is a data-space matching, where the objective is to match synthetic/modeled data to the data recorded in the field. As FWI inherently is a nonlinear problem, the solution is pursued in an iterative manner, where the data misfit or data residual is added (in a smart way) to the starting velocity model that was used to create the synthetics, creating an updated velocity model that yields synthetic data that more closely match the field recordings. This data-driven approach holds the promise of being a highly automated way of estimating velocities where, given the right data, one could directly recover an estimate of the velocity distribution in the subsurface. This velocity model could then be used to create accurate images of the subsurface, even in regions with very complex geology and velocity distributions that have normally challenged seismic imaging technology.

So why has it taken this long for the technology to mature? The key factors have been access to sufficient compute power and appropriate seismic data that support the assumptions behind FWI. Access to faster central processing units and larger compute clusters have enabled 3-D implementation of FWI that was needed to make it relevant for oil and gas exploration. Further, for 3-D marine seismic, the introduction of longer offsets (in excess of 8,000 m [26,247 ft]) and more recently the introduction of broadband seismic have enabled the recording of data with long offsets and low frequencies—data that are much better suited for FWI than the typical legacy marine dataset.

FIGURE 1. This figure illustrates the data misfit used in FWI. The modeled data on the left can be compared to the actual field data on the right, something that in practice is done inside the FWI objective function. (Source: PGS)

With access to modern marine seismic data and large, fast computers, FWI is now finally starting to live up to its promise as introduced in the ’80s. Today it does indeed offer a highly automated path to derive accurate velocity models for seismic imaging. The velocity models provided by FWI are improving the resolution and accuracy of seismic imaging and are often even serving as a useful attribute on their own.

FWI methodology

FWI is applied in an iterative manner to simplify the treatment of what is inherently a problem of wave propagation as a nonlinear function of earth parameters. The details for application of FWI by reverse time migration are well known throughout the industry; they basically involve shot records that are modeled by the two-way wave equation and data residuals (the difference between the modeled and recorded data, as shown in Figure 1) that are back-propagated to form a subsurface image. The cumulative image from all shots is then mapped into a spatial distribution for the velocity perturbation that is used to update the starting model. The above process is repeated until the data residual for all the shots (objective function) satisfies a convergence criterion.

The objective function for nonlinear problems typically has many minima, although only one minimum corresponds to the desired (global) solution. Practical inversion strategies thus incorporate procedures for inverting successive subsets of the data to guide the solution as closely as possible to the global minimum. Because a given starting model represents a smoothed representation of the desired solution, these procedures attempt to inject the longest possible wavelengths (smallest wavenumbers) in the early stages of an inversion and successively larger wavenumbers in the later stages.

FIGURE 2. This illustration shows the high-fidelity velocity models that can be derived by FWI in a shallow-water setting when sufficient offsets and low-frequency data are available. This example from the Johan Sverdrup Field in the southern North Sea shows how both shallow channels and shallow gas accumulations are recovered using a GeoStreamer dataset acquired in 2009. The maximum offset in the data was 6,000 m (19,685 ft). (Source: PGS)

Diving waves, refractions and reflections each possess different capabilities for improving the resolution of a model. Diving waves and refractions can update the velocity model anywhere between the water bottom and their turning point in the subsurface. In a given velocity regime, the depth sampled by the refractions is typically increasing as a function of longer offsets, hence the need for long offsets in the data. As a natural consequence, shallow water provides the best environment for exploiting such data for FWI. In deepwater, where diving waves and refractions may not be present, there is no alternative but to use reflections. Under these circumstances, it can be fruitful to treat the image of the residuals as a perturbation in reflectivity.

In summary, band-limited data are inverted with the lowest frequencies first because the lowest frequencies provide the most linear behavior. The lower the frequencies present in the data, the better the inversion result will be. This is where broadband seismic is making key contributions to the field of FWI. The lowest wavenumbers are selectively inverted first by picking a mute above the first breaks and then applying a time taper to window just the leading edge of the first arrivals. The length of the taper is extended for later inversion stages, which has the effect of accommodating higher wavenumbers.

Examples

An application of FWI following the strategy outlined above is shown in Figure 2. In this North Sea example, shallow velocity anomalies are beautifully recovered and help improve the overall quality of the seismic image. Here, the presence of long offsets and low frequencies in the data help provide an FWI velocity model that solved an outstanding issue of accurately determining the near-surface velocities. The area just beneath the seafloor is complicated due to the presence of channels, shale bodies and gas accumulations, all of which are resolved spatially and in magnitude by FWI.

FIGURE 3. FWI velocities with seismic overlay show how the FWI velocity recovers the velocity variations associated with shallow channels. (Source: PGS)

To assess the accuracy of lateral resolution exhibited in the FWI model, a depth slice from the PSDM image volume is displayed with the inverted velocity model overlain in Figure 3. The high-resolution velocity updates introduced by FWI correlate very well with the channels and other features shown in the PSDM image. From this depth slice the fast and slow channels in the sediments can be identified.

For inversion depths below the deepest turning point of any available diving waves and refractions, the availability of low frequencies becomes the key to successful use of FWI. Without the presence of low frequencies, low wavenumbers will be missed, resulting in an inversion result with a “ringy” appearance. In Figure 4, broadband data provide a rich low-end data spectrum that is well suited for FWI. The velocity model resulting from applying FWI to these data is geologically consistent and lacks the “ringy” and erroneous features shown in the model when FWI was applied to data lacking in low frequency.

The power of FWI

Powered by recent advances in compute power and seismic data acquisition, FWI is now emerging as a practical and important technology for determining seismic velocities. Through advanced 3-D modeling and aided by modern broadband data such as those recorded with PGS GeoStreamer technology, the power of FWI is finally available for application to the most challenging exploration problems being faced today.

FIGURE 4. The importance of low frequencies in the seismic data used for FWI is illustrated in this deepwater example. When the low frequencies are not present in the field data, as shown in the top image, the resulting FWI update gets degraded and ‘ringy.’ With proper broadband data the velocity variations can be recovered with depth as shown in the bottom part of the figure. (Source: PGS)