Transmission-Propagation Operator Theory and Tip-Wave Superposition Method for sub-salt shadow wavefield description

TPOT&TWSM. Synthetic wave modeling in media with interfaces of complex geometrical shape is one of the main problems in the mathematical wave theory and its applications. Oil companies are concerned with increasing the resolution capability of seismic data for complex oil-and-gas deposits associated with salt domes, basalt traps, reefs, lenses, etc. Specialists and engineers traditionally apply numerical or approximate analytical methods to search for a compromise between the modeling speed and its correctness. In inhomogeneous block media with complex shaped intefraces, there is a problem of describing separate wave fragments (for example, primary waves), not only describing the total wavefield. This separate description of any wavefield fragments has triggered this study. We therefore propose applying the rigorous analytical Transmission-Propagation Operator Theory (TPOT) in terms of operators of propagation in blocks and transmission (reflection/refraction) at curved interfaces between the blocks. This theoretical approach allows the solution of different seismic problems in inhomogeneous media with ‘shadow’ zones of different complexity. The term shadow means zones where the rays penetrate according to the generalized Fermat’s principle, not the conventional Fermat’s principle. In addition to TPOT, we have modified the Tip-Wave Superposition Method (TWSM) on a Graphics Processing Unit (GPU) cluster in the mid-frequency range accounting for shadow zones. Publications demonstrate that there is good comparison between the TWSM results and the laboratory observations, numerical solvers and other analytical solutions. The investigation of TPOT&TWSM is so far on the canonical models level. We further plan to consider real models as well, but this is not discussed in the present thesis.

TPOT is based on two main theoretical principles: 1) rigorous explicit description of the propagation operators in domains/layers; the propagation in shadow zones is handled by the generalization of the conventional Fermat’s and Huygens’ principles for an arbitrary boundary case; and 2) rigorous explicit representation of the transmission (reflection/refraction) operators at curved interfaces; the transmission at the curved interface is handled by the generalization of the conventional Snell’s law and the conventional plane wave transmission (reflection/refraction) coefficients. TPOT is a universal solution for wave problems in complex media because it solves the problem rigorously; this solution describes the total wavefield and its separate wave fragments.
 Feasible fundamental solution (FFS) in shadow. In all problems with curved interfaces, shadow zones will be obtained because the concave parts of the interfaces create shadows behind. In TPOT&TWSM, shadow is handled as follows. All the interface points are connected to each other by a straight segment. If the segment intersects the interface, we consider that these two points do not ‘see’ each other, otherwise they do ‘see’ each other. After this procedure, propagation is ‘allowed’ only between those points which ‘see’ each other. A shadow function is responsible for the removal of the propagation between those points which do not ‘see’ each other. This shadow function is added in the kernel of the conventional propagation Kirchhoff-type operator and, therefore, corrects for the Green’s function in the kernel according to the shadow zones present. Consequently this new kernel is feasible and handles shadow zones. We call it the ‘feasible fundamental solution’ (or the feasible Green’s function). Having this feasible kernel, the propagation operator also becomes feasible and is used as a propagation computational tool in shadow.

Generalized plane waves are an analog of the conventional transmission (reflection/refraction) plane waves for the curved interface case. This generalization is obtained by introducing a local coordinate system which is fixed at the reference interface point, and leads to a space-spectral form of the boundary conditions. The new kernel of the transmission operator is the transmission coefficient based on the generalized plane wave.

TWSM computes the TPOT analytical solution in the mid-frequency range on a GPU cluster and visualizes it on a seismogram. Earlier, TWSM was run on conventional parallel systems, but we now have improved the execution time by implementing this program on the GPU system. It approximates the operators of propagation in blocks and transmission at curved interfaces in the mid-frequency (seismic frequency) range. TPOT principle 1 leads to the application of TWSM to forward and inverse seismic problems by separate wavefield description; it is done by TWSM description of the wavefield in the form of tip-wave beams, connecting the elements of the seismic model. The TWSM description of the wavefield in domains/layers with geometrical shadow zones is done by accounting for shadow by correcting the propagation operator kernel. This is a generalization of such cases as edge and tip waves from sharp edges and vertices; and cascade diffraction, for example creeping waves and ‘whispering galleries’ bending along the concave parts of interfaces. TPOT principle 2 leads to TWSM evaluation of the transmitted tip-wave beams accounting for head waves at curved interfaces. The transmission operators at curved interfaces are approximated by the effective (integrated) transmission (reflection/refraction) coefficients accounting for both curvatures of the interface. If it is necessary to account for surface waves, TWSM can reproduce them on a seismogram. This is not an area that is studied in this thesis. 
 Comparisons. Publications prove that TWSM decrease the relative AVO inversion error from 20 to 4 percent. The comparison with laboratory data demonstrates an error from 1 to 4 percent. The comparison with the finite difference method gave an 3 percent error approximately. The comparison with the theoretical approaches gave an error of 2 to 3 percent approximately.

Advantages of TPOT&TWSM. TPOT&TWSM conceptually differ from the numerical methods being exploited to solve forward and inverse seismic problems. The numerical methods represent the total solution of the equation systems, while TPOT provides not only the total wavefield but also its wave structure expressed by separate waves. Each separate wave can be represented on a seismogram without representation of the rest of the wavefield. Moreover, the solution is derived in analytical form before using TWSM programming software. TWSM just visualizes each wave fragment or group of them given by TPOT in the mid-frequency range. The method is strictly speaking valid for hmax 1,99 d, where is the dominant wave length and hmax is the maximum depth of the model. The relative error is independent of the amplitude of all the wave fragments. Therefore, all the waves on the seismogram are represented equally accurately. Moreover, TWSM gives the wave-transfer matrix description in each block/layer independently of the other blocks/layers and sources/receivers definitions.  

Applications of TPOT&TWSM. TPOT&TWSM have been applied to primary extraction (multiple removal); subsalt shadow wavefield description; wavefield description for 3D inhomogeneous media with curved interfaces/reflectors. TWSM programming software can be used for different forward problems, such as the planning of acquisition systems, wave description of physical/laboratory modeling, the description of individual waves. It also can be used for inverse problems, such as imaging in the case of laterally inhomogeneous overburden and AVO inversion.

Thesis results. The thesis contains the two main results: a theoretical description of the feasible fundamental solution choice (Chapter 2) and the comparison of TWSM with the edge wave theory for V-, U- and W-models (Chapters 3, 4, and 5)