Full waveform inversion

Full Waveform Inversion (FWI) estimates subsurface model attributes by minimizing the difference between observed and simulated data as well as incorporating structural prior information. It relies on the consistent implementation of acoustic and visco/poro-elastic wave propagation equations.

As exascale high-performance computing continues to evolve, FWI is becoming increasingly feasible.

Owing to its reliance upon substantiated theoretical grounds, the method promises to provide superior insights into the course of subsurface resource characterization.

## Natural resources: Industry challenges

## Design, plan, build, optimize

## Find, produce, optimize

## Maximize recovery

As shown in the chart at right, cumulative oil reserves in shallow-water resources (blue line) are reaching a plateau, i.e. are becoming depleted.

Conversely, deep-water resources show much more potential (black line).

The costs of drilling a deep-sea well are about 300 M$. Unfortunately, off-shore drilling is associated with a miss rate of 60-70%, i.e. two out of three drilling attempts fail, yielding billion-dollar losses.

The oil, gas and mining industries have long been properous, but now that the "easy" resources have been depleted, and they must resort to more trustworthy methodologies for understanding the subsurface constituents.

Whereas wave phenomena physics is well understood, it has not been computationally feasible to account for the true physics.

Further challenges include pricing, lower fidelity and limited depth inferences of the subsurface.

## Imaging: Seismic inversion

Seismic inversion is the process of transforming seismic data into a quantitative rock-property description of the subsurface.

This has been performed hitherto primarily by means of approximated physics or heuristic methods.

## Industry trends: Depth imaging challenges roadmap

The entire industry is moving towards imaging methods of higher fidelity, implying higher (exascale) computing.

## From data to image

The process of inference involves several stages.

First, the goal is to find model parameters (e.g. acoustic, where κ = bulk modulus, ρ = density, or elastic (Lame coefficients)), for which the simulated data is in good agreement with the true data.

The discrepancy between the two is considered in the misfit term.

Owing to the ill-posed nature of the problem, additional *a priori* information is incorporated into the solution in the form of regularization and/or constraints.

## Robust seismic imaging

**Big Data and even bigger model for complex processing**

Seismic surveys take months to complete, and yield petabytes of data. Inverting these data requires solving millions of PDEs on large 3D meshes.

Conventional noise models amplify data noise. Hence, they may introduce many undesirable artifacts in the data, which are a major impediment to imaging. Instead, robust statistics measures, which may be non-convex, can be utilized to obtain high-fidelity images.

Full wave inversion is non-convex regardless of penalty choice because observation is nonlinear. Heavy-tailed models are less conservative with regard to outlier distributions. Subsurface images drive key acquisition and development decisions. Many major oil companies are making significant efforts to take their imaging capabilities to the next level.

The image (top right) shows full wave inversion recovery based on the conventional approach. The image (bottom right) demonstrates image recovery of the same noise-corrupted data when robust measures are considered.

## Forward/adjoint modelling

A first outcome:

- Output: Pressure files generated every 40th timestep (∼25 GB of data) for a very small subsurface model interpolated from the SEG/EAGE salt dome model.

## Source estimation

Source estimation is a fundamental ingredient of Full Waveform Inversion (FWI). In such seismic inversion methods, wavelet intensity and phase spectra are usually estimated statistically, although for the FWI formulation as a nonlinear least-squares optimization problem it can naturally be incorporated to the workflow.

Modern approaches for source estimation consider robust misfit functions leading to the well-known robust FWI method. The present work uses synthetic data generated from a high-order spectral element forward solver to produce observed data, which in turn are used to estimate the intensity and the location of the point seismic source term of the original elastic wave PDE. A min-max filter approach is used to convert the original source estimation problem into a state problem conditioned to the observations and a non-standard uncertainty description.

The resulting numerical scheme uses an implicit midpoint method to solve, in parallel, the chosen 2D and 3D numerical examples running on an IBM Blue Gene/Q using a grid defined by approximately 16,000 5th-order elements, resulting in a total of approximately 6.5 million degrees of freedom. Further details are available in [2].

## List of references

[1] A Spectral Element Solution of the Wave Equation in 3D

Sergiy Zhuk, Steve Moore, James Korte, Andrew Rawlinson, Tigran Tchrakian, Lior Horesh, Aleksandr Aravkin

SEG workshop on FWI, 2014.

[2] Source Estimation for Wave Equations with Uncertain Parameters

Sergiy Zhuk, Stephen Moore, Alberto Costa Nogueira Junior, Andrew Rawlinson, Tigran Tchrakian, Lior Horesh, Aleksandr Aravkin and Albert Akhriev

European Control Conference, 2015.