Investigation of Source Extension Methods, the Discrepancy Algorithm, and Noise Estimation to Overcome Cycle Skipping in Full Waveform Inversion

Abstract

Full waveform inversion (FWI) is a geophysical technique used to create highly detailed mod- els of the Earth’s subsurface which can be used to explore for hydrocarbons and to predict natural hazards such as earthquakes. Seismic waves are generated from controlled sources such as vibroseis trucks on land and airguns in marine environments. These waves propagate through the Earth’s subsurface materials, reflecting off of interfaces underground. The for- ward problem in FWI predicts the data based on a given model of the subsurface, while the inverse problem estimates subsurface parameters, such as sound velocity, by minimizing the difference between recorded data and data we predict from solving our mathematical model (the wave equation). Solving this minimization problem is computationally prohibitive, so we rely on local gradient-based optimization methods. The success of such methods depends on the accuracy of the initial guess. Otherwise, FWI tends to get stuck at a suboptimal solution, a problem known as cycle-skipping. This dissertation explores several source extension methods to overcome the cycle-skipping problem in FWI for transmitted data. These methods add additional degrees of freedom to the objective function, expanding the solution space to include models which may or may not be physical. This updated objective function is convex with appropriate penalty parameters, allowing local gradient-based optimization methods to find the correct geological model from a wider range of initial models. When close to the correct model, the physical constraints are reimposed in the problem by the penalty term. For a simple homogeneous medium experiment with single trace acoustic data, we illustrate how extended source inversion (ESI) avoids cycle skipping by relaxing the requirement that the source must be compactly supported and by adding a soft penalty to control the extent of the source. The discrepancy algorithm dynamically adjusts the penalty weight to maintain data error within a specified range, ensuring accurate model estimates. The update of the penalty parameter relies on having an accurate estimate of the noise level in the data which is generally unknown a priori. Numerical examples show that the extended method successfully overcomes cycle-skipping without the need for a good initial model, that the algorithm can dynamically update the noise level in the data (and hence the penalty parameter), leading to a reliable and accurate solution to the inverse problem. Additionally, this dissertation investigates the matched source waveform inversion (MSWI) method which extends the solution space by assuming that each data trace is a function of both receiver and source location. For single arrival data, MSWI is closely related to travel- time inversion. The MSWI objective function includes a data misfit term and a penalty term to keep an adaptive filter close to the Dirac delta function. MSWI is equivalent to the source extension method described above when the penalty parameter in MSWI approaches zero. Experiments demonstrate that for more complex heterogeneous media experiments MSWI successfully reduces cycle-skipping in single-arrival transmission data (even when moderate amounts of noise are present in the data), while FWI often fails due to cycle skipping. The inverted model resulting from MSWI can, therefore, provide a good starting model for FWI. However, the results also demonstrate that MSWI applied to multi arrival transmission data fails