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...@en

Scattering and inverse scattering theories are based on the Lippmann-Schwinger equation and its linearised version, the so-called Born equation. In this chapter, we seek alternative representations that can be the basis for different methods of model- ing and inversion. One representation transforms the Lippmann-Schwinger equation into one that represents directional wave-fields, while another invokes operators in- stead model parameters. We investigate and derive different forms of the latter. At the end, we come to a classification of the different representations described in this chapter. We classify them as those that invoke solely model parameters and those that invoke operators. Each of these families can be further grouped into those that use directional wavefields and those that use non-directional wave-fields. This chapter derives formulas that can be the basis for methods such as a directional inverse scattering theory, full wavefield modeling, model-independent joint migration inversion and operator-based non-directional method that invokes contrasts of the so-called Helmholtz operator.@en

We propose in this paper a theorem that combines together operator retrieval from model-independent joint migration inversion and homogeneous Green’s function re- construction. The resulting method overcomes the under-determination problem that we face in retrieving those operators and hence we turn this problem from undeter- mined inverse problem to over-determined inverse problem and hence that results in a more stable (well-posed) inverse problem. @en

The kinematical aspects of wave propagation are often analysed using slowness curves, which are well-known not only for isotropic media but also for anisotropic ones. The dynamical aspects are often analysed using reflection coefficient curves which are also well-known for both isotopic and anisotropic media. However, those curves are known for either fully or laterally homogenous rather than heterogeneous media. We generate and analyse those curves in a heterogeneous medium and find that the velocity-normalised slowness curves or cosine-sine curves deviate from the classical circular shape in the homogeneous situation. The deviation is dependent on frequency and hence those curves are dispersive, unlike their counterparts for homogenous media. The reflection coefficient curves also exhibit such deviation from the classical situation involving two homogeneous half spaces. Such deviation in reflection coefficient curves would have an impact on AVO/AVA analysis.

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We extend in this paper the full wave-field modeling method to the elastodynamic situation. While, the acoustic case encompasses directional decomposition - where up-going and downgoing wavefields are modeled in addition to modal decomposition, through which heterogeneity is properly handled - the elastodynamic case encompasses one additional type of decomposition; the one related to polarisations, where the wavefield is separated into its constituent polarisations: quasi-P, quasi-SV, and quasi-SH waves. All three types of decomposition are present in the equations given in this paper. Although we begin with the most general form of equations for anisotropic inhomogeneous media, we reduce those equations to the isotropic case, and we consider only P-SV waves for simplicity. We implement only the squared wavenumber operator, whose implementation closely resembles that of the acoustic case.@en

We extend in this chapter the full wave-field modeling method to the elastodynamic situation. While, the acoustic case encompasses directional decomposition — where up-going and downgoing wavefields are modeled in addition to modal decomposi- tion, through which heterogeneity is properly handled — the elastodynamic case en- compasses one additional type of decomposition; the one related to polarizations, where the wavefield is separated into its constituent polarizations: quasi-P, quasi-SV, and quasi-SH waves. All three types of decomposition are present in the equations given in this chapter. Although we begin with the most general form of equations for anisotropic inhomogeneous media, we reduce those equations to the isotropic case, and we consider only P-SV waves for simplicity. We implement only the squared wavenumber operator, whose implementation closely resembles that of the acoustic case. @en

Instead of attempting to retrieve model parameters (velocity and density in the acoustic situation), Model-Independent Joint Migration Inversion tries to obtain operators, reflection and augmented transmission, the sum of transmission and wavenumber perturbation operators. However, one does not know whether the retrieved operators are correct or not without first modeling them. Thus, we model those operators for a simple model with vertical and horizontal dips only. The modeled results show that all the lateral heterogeneity is contained in the augmented transmission operator, while the reflection operator contains none of the lateral heterogeneity— only the vertical one. We then compute the critical ingredients of the inversion process, the gradients of the misfit functional. The gradients exhibit overall resemblance to those ideal operators, but less resemblance was exhibited by the augmented transmission operators and their ideal counterparts. While the resulting gradients seem promising, most of the remaining challenge would be in producing effective regularization of the augmented transmission operator. Read More: @en

Generalized full wavefield modeling is a modeling process that utilizes reflection and transmission operators for inhomogeneous media in order to produce the wavefields. We extend this modeling process to the quasi-elastic anisotropic situation, where the elastodynamic generalized vertical wavenumber operator for P-waves is embedded in the acoustic formulation, assuming TIV media. Two examples are computed; one for homogeneous and the other for inhomogeneous media. The wavefields generated exhibit the typical angular distortion behavior introduced by anisotropy. While useful on its own, this extension of generalized full wavefield modeling paves the road to the inversion process, model-independent joint migration inversion. @en

Generalized Full Wavefield Modeling is a directional modeling method, which simulates wavefields such as upgoing and downgoing wavefields. The most straightforward implementation of such a method is to employ the Neumann's iterative method, which is, nonetheless, well-known not to be necessarily convergent for all situations. Thus, we use three other methods that represent a generalization of the Neumann's solution; one is preconditioned stationary overrelaxation, and the other two are preconditioned conjugate gradient and a truncated Krylov method, the so-called GMRes. We compare the convergence of all those methods, as well as, stationary and successive overrelaxation methods without preconditioning. We find that such truncated Krylov method, i.e., GMRes, is overall faster to converge and requires no preconditioning to assure convergence. We show two examples, one using a velocity model linearly increasing with depth and one using a complex salt model adapted from the SEG SEAM model. In the first model, GFWMod provides the upgoing and downing diving waves including the horizontally propagating constituents, while in the second model, it provides the evolution of the scattering process with different iterations, providing insight into the actual scattering process.@en

Generalized Full Wavefield Modeling is a directional modeling method that incorporates reflection and transmission operators. It employs Neumann's method, an iterative inversion scheme for obtaining the solutions (the wavefields), which is, nonetheless, well-known not to be necessarily convergent for all situations. Thus, we use two methods that represent a generalization of the Neumann's solution; one is stationary overrelaxation, and the other is successive overrelaxation. Both methods attempt to scale the wavefield residual such that it does not grow with increasing number of iterations. One method, stationary overrelaxation, uses a constant scale factor while the other, successive overrelaxation, varies the scale factor with iterations. A numerical example shows clearly that successive overrelaxation yields a stable solution, unlike the other two methods (Neumann's and stationary overrelaxation), since it forces the residual to be reducing in each iteration.@en

Considering the practical aspects of implementing the full wavefield modeling method is critical for analyzing this method and its theoretical, as well as, numerical limitations. We implement stretched-coordinate perfectly matched layer (SC-PML) in order to model the wavefields more accurately, where the operators are implemented using a staggered-grid finite difference scheme. Although the PML results in non-self adjoint Helmholtz operator and, hence, gives complex-valued eigen values and eigen functions when the modal decomposition is carried out, we show that its solution is kinematically consistent with that obtained independently from the eikonal equation. We also show that a particularly difficult case, diving waves, can indeed be numerically modeled. This work sets the stage for further investigation of full wavefield modeling as a potential competitor to conventional two-way wave equation modeling in terms of completeness and accuracy.@en

We derive a representation theorem for modeling directional wavefields using reciprocity theorem of the convolution-type. A Neumann series expansion of the representation yields a series that is similar to that of Bremmer. A generalized Neumann series is also derived similar to that used for solving the non-directional Lippmann-Schwinger representation. An example shows how the series can model each scattering order separately for inhomogeneous media. This could potentially be useful in imaging and inverse problems. @en

Multiples are more complex to analyze than primaries. This is due to the fact that a multiple, as the name suggests, bounces in the subsurface more than once. We propose a method that decomposes the data by scattering order and extract each scattering order of multiples separately from the rest. The method represents an extension to SRME, where SRME is simply the first iteration. Similar to SRME, no prior knowledge of the subsurface is assumed, as the method is purely data-driven. A numerical example shows that the method is capable of separating different orders of scattering. Application to field data from the North Sea demonstrates the effectiveness of the method. @en

We formulate the seismic inverse problem in an operator-based model-independent approach where the model parameters, such as velocity and density in the acoustic situation, are not required. The output of the inversion, however, are reflection and augmented transmission operators. The augmented transmission operator is a combination of both generalized wavenumber and transmission operators. The theory employs directional reciprocity theorem and a numerical example illustrates those operators that are sought by our method. @en

Joint Migration Inversion (JMI) offers an attractive feature. It is an operator-based model-independent approach to the inverse problem, in contrast with the model-dependent conventional approach of Full Waveform Inversion, which not only uses the physical model parameters, velocity and density in the acoustic situation, but also forces the data to obey a certain model, e.g. isotropic or anisotropic. The operators sought by the proposed JMI method are reflection and augmented transmission operators (the sum of slowness and transmission operators), yet the reference/background operators are only the simpler Green’s primary-only operators. This formulation is sufficient to explain not only the primaries but also the multiples. Then, the operator-based inverse problem can be solved in a non-linear sense, where phantom sources are obtained only as an intermediary step to obtain those operators. A numerical example shows that the method is capable of distinguishing between the relatively easily-obtained vertical heterogeneity, embedded in the reflection operator, and the more difficulty-obtained lateral heterogeneity, embedded in the augmented transmission operator. This feature, among others, are expected to have a major influence on the inversion process, including its convergence properties.@en

Full wavefield modeling is a method that invokes the transfer coefficients in order to model wave propagation. However, generalized transfer coefficients, which are more general than those that are equivalent to Zoeppritz equations in acoustic media, are needed in order to model waves in such media. We compute those generalized transfer coefficients and invoke them into the full wavefield modeling method, which models multiples along with primaries. We compute those coefficients using a formulation that allows discontinuous media to be modeled. A comparison between the conventional approach of using the acoustic equivalent of Zoeppritz equations and our proposed method show a pronounced difference. In one example, it is shown that neighboring sand channels have a pronounced impact on the reflection coefficient at a single sand channel. This work sets the stage to an elastic formulation that has the potential to replace the commonly used Zoeppritz equations.@en

Joint migration inversion (JMI) has been implemented with the implicit assumption that the medium parameters are only smoothly laterally varying. We extend JMI such that its modeling algorithm is capable of properly handling strong lateral medium variations. The extension is made by computing propagation and scattering operators using modal expansion. In order to simulate wave propagation in unbounded media, a perfectly matched layer (PML) is implemented. The resulting propagation and scattering operators are not limited by angle of propgation. The extended modeling algorithm is then used for Full Wavefield Migration (FWM) and JMI, which is also extended using a preconditioned conjugate gradient scheme. A numerical example shows that the proposed JMI algorithm is capable of recovering not only the macro-model but also the micro-model. In the numerical example, a velocity anomaly, embedded in a layered model, is recovered with high resolution, along with the accompanying image.@en

Joint migration inversion (JMI) has been implemented with the assumption that the medium parameters are only locally laterally varying. We extend JMI such that its modeling algorithm is able to handle arbitrary lateral medium variations. We show numerically that the the propagation operator, which is computed by modal expansion, is not limited by propagation angle. A perfectly matched layer (PML) is implemented to simulate wave propagation in unbounded media. The modeling algorithm is then used for Full Wavefield Migration (FWM) and JMI. We also implement JMI with a preconditioned conjugate gradient scheme. A numerical example shows that starting from a homogeneous model, the modified JMI algorithm is able to recover a velocity anomaly embedded into the model, with high resolution, along with the accompanying image.

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