Adaptive waveform inversion: algebraic interpretation, convergence analysis and alternative formulations

Abstract

Full waveform inversion (FWI) seeks to recover a high resolution, high fidelity subsurface model by iteratively updating the initial model to match the predicted data to the observed. It has been recognised as one of the most important techniques in oil and gas industrial practises as successful FWI produces accurate velocity models at high resolutions which cannot be achieved by conventional seismic techniques.

However, due to the fact that FWI exploits local optimisation methods, it can suffer from local minima. Unless an accurate initial model is provided, FWI tends to converge towards such a local minimum which could deviate far from the global minimum. Cycle-skipping is the most significant cause of the local minimum that FWI suffers from in practice, which occurs when the timeshift between the predicted data and the observed exceeds half a cycle at the lowest inversion frequencies.

Warner and Guasch (2014) proposed Adaptive waveform inversion (AWI) as a new waveform inversion scheme, which is robust against cycle-skipping. In this thesis, I have interpreted AWI from a new algebraic perspective, and performed both synthetic and real data tests to examine the convergence properties of both methods.

In the Marmousi synthetic test, I have applied both methods to 1275 initial velocity models with different degrees of smoothness and overall velocity shifts compared with the true velocity model, to examine how each method performs as the data become increasingly cycle-skipped. In the 3D OBC Tommeliten test, I have created 27 starting models with a similar method by smoothing and velocity shifting a velocity model acquired from reflection traveltime tomography. In both the synthetic and the real data tests, AWI is successful for a wider range of inaccurate initial models from which conventional FWI is unable to converge towards the global minimum.

Based on the algebraic interpretation, I have developed 5 variations of AWI, namely the non-negative AWI, the truncated AWI, the conjugate gradient AWI, the projection subspace reduction method, and the orthogonalised AWI, in order to improve its convergence efficiency and/or to mitigate its algebraic deficiencies.

The non-negative AWI shows improvements over AWI in terms of displaying broader convergence regions in both synthetic and real data tests. The truncated AWI shows faster convergence in constant velocity synthetic model tests and the conjugate gradient AWI improves the convergence efficiency of AWI without reducing the convergence regions in the Marmousi synthetic tests. The projection subspace reduction method and the orthogonalised AWI are able to mitigate the algebraic deficiencies of AWI but are not practically applicable yet, due to the complexity of seismic data and the unaffordable computational requirements, respectively.

The wider applicability of the newly developed methods needs to be demonstrated on other field datasets, and my future research project will focus on achieving this goal.