Seismic Diffraction Modeling with the Tip-wave Superposition Method

The transmission-propagation operator theory provides mathematical tools for describing wave propagation in laterally inhomogeneous layered media. The theory is a hybrid of the integral representation (integral-equation) method and the spatial-frequency decomposition (slowness) method. The theory introduces a new wave-motion statement of the forward problem using two operators; the convolutional transmission operator and the feasible propagation operator. In this theory, the exact solution of the initial-boundary value problem is written as the sum of sequentially reflected and transmitted wave events. We further develop the tip-wave superposition method based on the approximations of the two operators in the seismic-frequency range. The feasible propagation operators inside the layers are approximated in the form of layer matrices. The transmission operators at the interfaces are approximated with effective reflection/transmission coefficients, which generalize plane-wave reflection/transmission coefficients to curved interfaces, non-planar wavefronts and finite frequencies. Each element of the layer matrix contains a feasible beam of the tip waves diverging from a small interface element with diffraction at concave parts of the layer boundary and reflecting/transmitting at the interface. We illustrate the potential of the tip-wave superposition method through diffraction modeling of the Green’s function in a layered medium.