Application of the near-optimal quadrature to 2.5D wave modeling

Numerical simulation of wave propagation in heterogeneous medium provides a fundamental tool in many inverse scattering problems. Although real data are acquired in three-dimensional (3D) world, we can still apply some approximation to simplify the model and to accelerate the computation. For example, when the data are acquired along a straight line, they are mainly sensitive to the formation in the vertical plane of the acquisition line. Therefore, we can use a two-dimensional (2D) model as an approximation while maintaining the 3D acquisition layout. This defines the settings of the two-and-a-half dimensional (2.5D) modeling schemes. Here we study a two-and-a-half dimensional (2.5D) finite-difference algorithm to model elastic wave propagation in heterogeneous media. In 2.5D problems, it is assumed that the elastic properties of models are invariant along a certain direction. Therefore, we can convert the three-dimensional (3D) problem into a set of two-dimensional (2D) problems in the spectral domain. The 3D solutions are then obtained by applying a numerical integration in the spectral domain. Usually the quadrature points used in this spectral-domain integration are sampled from the real axis in the spectral domain. However, because of the oscillating nature of the integrand, the convergence of this quadrature can be very slow especially at high frequencies. In this work, we apply the near-optimal quadrature scheme to the spectral integration. This is equivalent to transform the contour of integration from the real axis into a path in the complex plane and near-optimal quadrature is computed along the path numerically. Numerical studies show more than ten times of reduction in the number of quadrature points compared with sampling along the real axis in the spectral domain. This scheme alleviates the bottleneck of computing speed in the 2.5D elastic wave modeling. Furthermore, it can improve the computational efficiency of 2.5D elastic full waveform inversion algorithms. This technique can also be applied to solve 2.5D electromagnetic problems.