1. Wave equation and the finite difference method - 2023.2 English

Equation (1) is the basic mathematical equation for elastic wave propagation.

where \(\nabla^2\) is the Laplace operator, \(p(\mathbf{x}, t)\) is the wave field and \(v(\mathbf{x})\) is the wave velocity. In 2D-RTM, \(\mathbf{x} = \{x, z\}\), which is the length and depth of the imaging area respectively. In 3D-RTM, \(\mathbf{x} = \{x, y, z\}\), which is the length, width and depth of the imaging area respectively.

Due to the non-uniform structures in the subsurface, there is no theoretical solution for Equation 1. Hence, RTM is based on a numerical solution called a finite difference method. All spacial dimensions and time domains are discretized properly. The partial derivative with respect to time and the Laplace operator also have corresponding transformation according to the discretization scheme.

For instance, the Pluto synthetic seismic velocity model which was used to evaluation our 2D-RTM implementation has been discretized into a regular grid both spatially and temporally. The number of partiitions are listed in Table 1.

where \(N_x\), \(N_z\), \(N_t\) denote the number of grids along \(x\), \(z\), \(t\) respectively.

Similar to the approaches presented in many research publications, we also adopted Order-2 temporal scheme and Order-8 spatial scheme. Following these discretization schemes, the 2D wave equation can be re-interpreted by Equation (2) and (3) shown below. Equation (2*) and (3) are for 3D RTM.

Where all \(c_n\) are the coeffients shown in Table 2 below.