A Nodal Integral Scheme for Acoustic Wavefield Simulation

Abstract

The nodal integral method (NIM) has been applied to a variety of both linear and non-linear problems. Here, we outline the development of an NIM scheme for the indefinite Helmholtz equation to simulate acoustic wave propagation. NIM has a simple formulation, and can yield accurate results with coarse grids. In the development of the scheme, the partial differential equation is averaged over a node by a transverse integration process (TIP). While performing the TIP, the product of the wave number and the function are incorporated in pseudo source terms, leading to a simple scheme. The ordinary differential equations resulting from the TIP are solved analytically, constituting the distinguishing step in NIM that leads to accurate solutions. We present the dispersion analysis of the NIM scheme for the very first time to quantify numerical dispersion. We then benchmark the wavefields and time domain seismograms computed from NIM with the analytical solution for a homogeneous model. We find that the NIM scheme yields second-order accurate results for the homogeneous model. Finally, we compute time domain seismograms for the Marmousi model and compare the results with the second-order central finite difference method in the time domain. The second-order scheme presented here is the very first application of NIM to the solution of the acoustic wave equation.

Appendix 1: Coefficients for the final scheme

The expressions for the coefficients in Eq. 26, 27 and 34 are given by

$$\begin{aligned} A_1= & {} \frac{-3ab^2}{-3b^2 + a^2\left( -3 + b^2k^2\right) } \end{aligned}$$

(A.1)

$$\begin{aligned} B_1= & {} \frac{-3a^2b}{-3b^2 + a^2\left( -3 + b^2k^2\right) } \end{aligned}$$

(A.2)

$$\begin{aligned} A_2= & {} \frac{9a}{-6b^2 + 2a^2\left( -3 + b^2k^2\right) } \end{aligned}$$

(A.3)

$$\begin{aligned} B_2= & {} \frac{9b}{-6b^2 + 2a^2\left( -3 + b^2k^2\right) } \end{aligned}$$

(A.4)

$$\begin{aligned} A_3= & {} \frac{-24a^2-6b^2 + 8a^2b^2k^2}{-6ab^2 + 2a^3\left( -3 + b^2k^2\right) } \end{aligned}$$

(A.5)

$$\begin{aligned} B_3= & {} \frac{-24b^2-6a^2 + 8a^2b^2k^2}{-6b^3 + 2a^2b\left( -3 + b^2k^2\right) } \end{aligned}$$

(A.6)

$$\begin{aligned} A_4= & {} \frac{-3b^2 + 2a^2\left( -3 + b^2k^2\right) }{-6ab^2 + 2a^3\left( -3 + b^2k^2\right) } \end{aligned}$$

(A.7)

$$\begin{aligned} B_4= & {} \frac{-3a^2 + 2b^2\left( -3 + a^2k^2\right) }{-6b^3 + 2a^2b\left( -3 + b^2k^2\right) } \end{aligned}$$

(A.8)

$$\begin{aligned} A_{11}= & {} \frac{3a^2b}{-3b^2 + a^2\left( -3 + b^2k^2\right) } \end{aligned}$$

(A.9)

$$\begin{aligned} A_{22}= & {} \frac{1}{2b} + \frac{9b}{2\left( -3b^2+a^2\left( -3 + b^2k^2\right) \right) } - \frac{3a^2bk^2}{2\left( -3b^2+a^2\left( -3+b^2k^2\right) \right) } \end{aligned}$$

(A.10)

$$\begin{aligned} A_{33}= & {} -\frac{1}{2b} + \frac{9b}{2\left( -3b^2+a^2\left( -3 + b^2k^2\right) \right) } - \frac{3a^2bk^2}{2\left( -3b^2+a^2\left( -3+b^2k^2\right) \right) } \end{aligned}$$

(A.11)

$$\begin{aligned} A_{44}= & {} -\frac{9b}{2\left( -3b^2+a^2\left( -3+b^2k^2\right) \right) } \end{aligned}$$

(A.12)