Operator-Based Modeling and Inversion: An Operator Approach to the Forward and Inverse Scattering Problems

The seismic method has many applications. It is important in the critical sector of energy. Besides being used in imaging oil and gas reservoirs, it is also utilized in other sectors of energy such as geothermal energy exploration and development. It also plays a role in extracting other resources such as minerals or in the process of monitoring CO2 sequestration to reduce the carbon footprint of humankind. While seismic waves can occur naturally, their study gives insight in analysing the occurrence of and mitigating risks related to earthquakes. As far as active-source seismic is concerned: seismic images make it possible to see what is in the subsurface with minimal expensive and invasive operations such as drilling unnecessary holes in the subsurface — similar to what medical professionals use ultrasound or X-ray images for. Several methods have been proposed to analyze seismic data. A popular method nowadays is full waveform inversion (FWI), for instance, which attempts to fit all the recorded waveformwith amodel. This process solves, in fact, a very complicated highly non-linear inverse problem. Another process that uses such inversion process, but which tries to separate classes of parameters to reduce non-linearity, is joint migration inversion (JMI), in which scattering properties of the subsurface are separated from the propagation properties of seismicwaves. Currently those two methods, FWI and JMI, are generally model-dependent — that is they have been formulated to fit specific physics model such as isotropic acoustic media, transversally isotropic media with or without absorption. Hence, they would tend to have biases towards those particular models. Another paradigmis the so-called data-driven paradigm, or data-adaptive paradigm, and since it is formulated in terms of operators, one could also refer to it as operatorbased. Since it contains less biases towards a particular physics model or require no detailed knowledge of model parameters, beforehand, some also refer to it as modelindependent, as it does not need to force the data to fit a specific model, rather the process adapts to the model contained within the data. A process such as surface-related multiple elimination follows this paradigm. Another process, which is also shown in this dissertation, separates the surface multiples scattering-order-by-scattering-order without the need to assume a specific physics model. The process is referred to as scattering order decomposition. So, this dissertation looks into the problem of extending the inversion process to the model-independent or the operator-based paradigm. This dissertation looks first into the theoretical underpinning of this problem, where integral representations are used to study it. These representations are divided into four categories: first model-based representations are derived and presented as directional and non-directional. So, it places in context those theories. Next, the operator-based representations are also divided into directional and non-directional. Finally, four representations are derived, in this dissertation, which have the potential for applications in modeling, inversion and various seismic data analysis processes. Modeling is needed before any inversion since the inverse problem is ill-posed or illconditioned and hence no unique solutions exist but rather preconditioned or regularized solutions to these problems are normally used. Moreover, the inverse problem uses modeling iteratively and also back-projects the data residuals with the forward modeling mechanism. Therefore, the next chapters study operator generation and the subsequent modeling of wavefields with these derived operators...