Seismic P-wave modelling in an inhomogeneous transversely isotropic medium with a tilted symmetry axis

A method for seismic P-wave modelling to generate a seismic image of a subsurface formation that is represented in a Cartesian coordinate frame as an inhomogeneous transversely isotropic (TI) acoustic medium with a tilted symmetry axis of variable non-vertical direction, the method comprising the steps of:

• a) measuring seismic P-waves, excited by a seismic source, and propagated through the subsurface formation;
• b) generating a pseudo-acoustic stress-strain relationship of a stress tensor and a strain tensor of modelled P-waves, in the Cartesian coordinate frame by means of a TI elastic tensor, wherein each of the tensors is expressed in a rotated local Cartesian coordinate frame which rotated local Cartesian coordinate frame is rotated relative to the Cartesian coordinate frame in which the subsurface formation is imaged and aligned with said tilted symmetry axis of the variable non-vertical direction of the TI medium, in which stress-strain relationship a shear velocity along the axis of symmetry is set to zero, and which stress-strain relationship comprises an axial scalar stress component that is co-axial to the tilted symmetry axis and a lateral scalar stress component in a plane perpendicular to the tilted symmetry axis;
• c) combining the pseudo-acoustic stress-strain relationship of step b with an equation of motion to generate a coupled wave equation for the axial and lateral scalar stress components, which equation contains mixed and non-mixed second-order spatial derivatives;
• d) discretizing first-order spatial derivatives and non-mixed second-order spatial derivatives by centered finite-differences, with dedicated selection of coefficients;
• e) using bi-directional combinations of the discretized first-order derivatives for the mixed second-order derivatives in the coupled wave equation of step c, and using discretized second-order derivatives for the non-mixed second-order derivatives, while stability of an explicit time-stepping method is established by weighing down the mixed second-order derivatives;
• and
• f) forward propagating a simulated shot, and backward propagating seismic P-waves measured in step a), through an anisotropic migration model in accordance with steps a-e to generate the seismic image of the subsurface formation.