Asymptotic Solutions for High Frequency Helmholtz Equations in Anisotropic Media with Hankel Functions

Abstract

We present asymptotic methods for solving high frequency Helmholtz equations in anisotropic media. The methods are motivated by Babich’s expansion that uses Hankel functions of the first kind to approximate the solution of high frequency Helmholtz equation in isotropic media. Within Babich’s expansion, we can derive the anisotropic eikonal equation and a recurrent system of transport equations to determine the phase and amplitude terms of the wave, respectively. In order to reconstruct the wave with the phase and amplitude terms for any high frequencies, they must be computed with high-order accuracy, for which a high-order factorization approach based on power series expansions at the primary source is applied first to resolve the source singularities, after that high-order schemes can be implemented efficiently. Rigorous formulations are derived, and numerical examples are presented to demonstrate the methods.

Appendices

Appendix A: WKB Approximation for Eq. (1) in 2D

We derive the governing equations for the phase and amplitude terms in the WKB expansion (2) for the anisotropic Helmholtz equation (1) in 2D.

Theorem 2

In the WKB approximation (2) for the anisotropic Helmholtz equation (1) in 2D, the phase $\tau$ satisfies the anisotropic eikonal equation (8), and the amplitude terms $\{ A_k \}_{k=0}^\infty$ satisfy the following recurrent system,

$$\begin{aligned} \begin{aligned}&A_{-1} \equiv 0, \\&\qquad ( \beta \tau _x + \gamma \tau _z + a\tau _{xx}-2c\tau _{xz}+b\tau _{zz})A_k + 2 \{(a\tau _x-c\tau _z)(A_k)_x + (b\tau _z - c\tau _x)(A_k)_z \} \\&\quad = -( \beta (A_{k-1})_x + \gamma (A_{k-1})_z) - (a (A_{k-1})_{xx} - 2c (A_{k-1})_{xz} + b(A_{k-1})_{zz}), ~ k \ge 0, \end{aligned} \end{aligned}$$

(21)

where $\beta \equiv a_x - c_z$, and $\gamma \equiv b_z - c_x$.

Theorem 2 can be proved by careful calculation. We have

$$\begin{aligned} \begin{aligned}&U_x = \sum _{k=0}^\infty \left( \frac{\tau _x A_k}{(\iota \omega )^{k-1}} + \frac{(A_k)_x}{(\iota \omega )^k}\right) e^{\iota \omega \tau };\\&U_z = \sum _{k=0}^\infty \left( \frac{\tau _z A_k}{(\iota \omega )^{k-1}} + \frac{(A_k)_z}{(\iota \omega )^k}\right) e^{\iota \omega \tau };\\&U_{xx} = \sum _{k=0}^\infty \left( \frac{\tau _x^2 A_k}{(\iota \omega )^{k-2}} + \frac{2\tau _x (A_k)_x+ \tau _{xx} A_k}{(\iota \omega )^{k-1}} + \frac{(A_k)_{xx}}{(\iota \omega )^k}\right) e^{\iota \omega \tau };\\&U_{zz} = \sum _{k=0}^\infty \left( \frac{\tau _z^2 A_k}{(\iota \omega )^{k-2}} + \frac{2\tau _z (A_k)_z+ \tau _{zz} A_k}{(\iota \omega )^{k-1}} + \frac{(A_k)_{zz}}{(\iota \omega )^k}\right) e^{\iota \omega \tau };\\&U_{xz} = U_{xz} = \sum _{k=0}^\infty \left( \frac{\tau _x \tau _z A_k}{(\iota \omega )^{k-2}} + \frac{\tau _z (A_k)_x + \tau _x (A_k)_z + \tau _{xz} A_k}{(\iota \omega )^{k-1}} + \frac{(A_k)_{xz}}{(\iota \omega )^k}\right) e^{\iota \omega \tau }.\\ \end{aligned} \end{aligned}$$

By substitution the above formulas into Eq. (1), we have

$$\begin{aligned} \begin{aligned}&\sum _{k=0}^\infty \left( \frac{A_k}{(\iota \omega )^{k-2}} \{a \tau _x^2 - 2c \tau _x \tau _z + b \tau _z^2 - 1/v^2\} \right. \\&\left. \quad +\frac{1}{(\iota \omega )^{k-1}} \{ (\beta \tau _x + \gamma \tau _z + a\tau _{xx}-2c\tau _{xz}+b\tau _{zz})A_k \right. \\&\left. \quad +\,2(a\tau _x - c\tau _z) (A_k)_x + 2(b\tau _z - c \tau _x)(A_k)_z \}\right. \\&\left. \quad +\,\frac{1}{(\iota \omega )^k} \{ \beta (A_k)_x + \gamma (A_k)_z+ a (A_k)_{xx} - 2c (A_k)_{xz} + b(A_k)_{zz}\} \right) e^{\iota \omega \tau } =0. \end{aligned} \end{aligned}$$

Then collecting coefficient for $O(1/(\iota \omega )^{k-2})$ term and letting it be equal to 0 yields the anisotropic eikonal equation (8), and collecting coefficients for $O(1/(\iota \omega )^{k-1})$ term and let it be equal to 0 yields the recurrent system (21).

The factorization techniques can also be applied to resolve the source singularities for computing amplitude terms $\{ A_k \}_{k=0}^\infty$, for instance, see [22, 24, 28] for similar techniques applied for isotropic cases.

Appendix B: WKB Approximation and Babich’s Expansion for Eq. (1) in 3D

We also include the formulations of the WKB approximation and Babich’s expansion for the anisotropic Helmholtz equation (1) in three-dimensional (3D) spaces. We assume the anisotropy tensor ${\mathbf{A }}$ is given as

$$\begin{aligned} {\mathbf{A }}(\mathbf{r }) = \left( \begin{array}{ccc} a(\mathbf{r }) &{}-d(\mathbf{r }) &{} -e(\mathbf{r })\\ -d(\mathbf{r }) &{}b(\mathbf{r }) &{} -f(\mathbf{r })\\ -e(\mathbf{r }) &{}-f(\mathbf{r }) &{} c(\mathbf{r })\\ \end{array} \right) . \end{aligned}$$

Theorem 3

In the WKB approximation (2) for the anisotropic Helmholtz equation (1) in 3D, the phase $\tau$ satisfies the anisotropic eikonal equation (8), and the amplitude terms $\{ A_k \}_{k=0}^\infty$ satisfy the following recurrent system,

$$\begin{aligned} \begin{aligned}&A_{-1} \equiv 0, \\&\qquad (\beta \tau _x + \gamma \tau _y + \zeta \tau _z + a \tau _{xx} + b\tau _{yy}+c\tau _{zz} -2d\tau _{xy}-2e\tau _{xz}-2f\tau _{yz}) A_k\\&\qquad + 2 \{(a\tau _x-d\tau _y-e\tau _z)(A_k)_x \\&\qquad + (b\tau _y-d\tau _x - f\tau _z)(A_k)_y + (c\tau _z-e\tau _x - f\tau _y)(A_k)_z \} \\&\quad = -( \beta (A_{k-1})_x + \gamma (A_{k-1})_y + \zeta (A_{k-1})_z) \\&\qquad - (a (A_{k-1})_{xx} + b(A_{k-1})_{yy}+ c(A_{k-1})_{zz} \\&\qquad - 2d(A_{k-1})_{xy} - 2e (A_{k-1})_{xz} - 2f (A_{k-1})_{yz}), ~ k \ge 0, \end{aligned} \end{aligned}$$

(22)

where $\beta \equiv a_x -d_y- e_z$, $\gamma \equiv b_y -d_x - f_z$, and $\zeta = c_z-e_x-f_y$.

Theorem 4

In the Babich’s expansion (5) for the anisotropic Helmholtz equation (1) in 3D, the phase $\tau$ satisfies the anisotropic eikonal equation (8), and the amplitude terms $\{ v_k \}_{k=0}^\infty$ satisfy the following recurrent system,

$$\begin{aligned} \begin{aligned}&v_{-1} \equiv 0, \\&\qquad ( \beta T_x + \gamma T_y + \zeta T_z + (4k-6) N + aT_{xx} + bT_{yy} + cT_{zz}- 2dT_{xy}- 2eT_{xz} - 2fT_{yz} )v_k \\&\qquad + 2 \{(aT_x-dT_y-eT_z)(v_k)_x + (bT_y - dT_x-fT_z)(v_k)_y \\&\qquad + (cT_z - eT_x-fT_y)(v_k)_z \} \\&\quad = ( \beta (v_{k-1})_x + \gamma (v_{k-1})_y+ \zeta (v_{k-1})_z) \\&\qquad + (a (v_{k-1})_{xx} + b(v_{k-1})_{yy} + c(v_{k-1})_{zz}\\&\qquad - 2d (v_{k-1})_{xy}- 2e (v_{k-1})_{xz}- 2f (v_{k-1})_{yz} ), ~ k \ge 0, \end{aligned} \end{aligned}$$

(23)

where $\beta \equiv a_x -d_y- e_z$, $\gamma \equiv b_y -d_x - f_z$, $\zeta = c_z-e_x-f_y$, $N \equiv 1/v^2$, and $T\equiv \tau ^2$.

Theorems 3 and 4 can be proved similarly as in 2D cases. And the governing equations for $\tau$ and $\{v_k\}$ can be solved by the same schemes numerically. Figures  15 and 16 show plots of a 3D model on computational domain $[0,~0.5]^3$ (km) with

$$\begin{aligned} v(\mathbf{r }) = 3-1.75 e^{-((x-0.25)^2+(y-0.25)^2+(z-0.25)^2)/0.64} (\mathrm{km/s}), \end{aligned}$$

and

$$\begin{aligned} {\mathbf{A }}(\mathbf{r }) = \left( \begin{array}{ccc} 1 &{}-0.5 &{} -0.3\\ -0.5 &{}3 &{} -0.1\\ -0.3 &{}-0.1 &{} 2\\ \end{array} \right) \text{ or } \left( \begin{array}{ccc} 1+0.3\sin ^2(\pi x) &{}-0.5+0.1\cos ^2(\pi x) &{} -0.3+0.2\cos ^2(\pi y)\\ -0.5+0.1\cos ^2(\pi x) &{}3+0.2\sin ^2(\pi y) &{} -0.1-0.1\cos ^2(\pi z)\\ -0.3+0.2\cos ^2(\pi y) &{}-0.1-0.1\cos ^2(\pi z) &{} 2+0.1\sin ^2(\pi z)\\ \end{array} \right) . \end{aligned}$$

The source is $\mathbf{r }_0 = (0.25,0.25,0.25)$ (km).

Fig. 15

3D model. Case 1: slices of the wave with $\omega = 32\pi$ at $x=0.25$ (km), $y=0.25$ (km) and $z=0.25$ (km), respectively. Top: real part; Bottom: imaginary part. Left: $U_1$; Right: $U_2$ (Color figure online)

Fig. 16

3D model. Case 2: slices of the wave with $\omega = 32\pi$ at $x=0.25$ (km), $y=0.25$ (km) and $z=0.25$ (km), respectively. Top: real part; Bottom: imaginary part. Left: $U_1$; Right: $U_2$ (Color figure online)