Analytical solutions of the VTI wave equation

Equations (7) and (11) have four solutions corresponding to the fourth-order nature of these two equations. Two of these four solutions are just perturbations of the isotropic wavefield solutions, and they reduce to these isotropic solutions when $\eta$ is set to zero; one solution for outgoing waves and the other for incoming waves. The initial condition will resolve whether the wave is outgoing or incoming. Equations (7) and (11) have another set of outgoing and incoming wave solutions. In this section, I solve equation (7) analytically for the case of a homogeneous medium, in hope of providing a better understanding of the two new solutions.

To solve equation (7), we use a plane wave,

F(x,z,t)= A(t) e^{i(k_x x+k_z z)},

as a trial solution. For simplicity, let us consider the two-dimensional problem, and as a result, drop all terms containing y (or k_y). Substituting the trial solution into the 2-D version of the partial differential equation (7), we obtain the following linear ordinary differential equation in A:

   $$\begin{eqnarray} \frac{d^4 A}{dt^4}+ ((1+2\,\eta) {v^2}\,{{{k_x}}^2} + {{{k_z}}^... ...2} + 2\,{v^2}\,\eta \,{{{k_x}}^2}\,{{{k_z}}^2}\,{{{v_v}}^2} A =0.\end{eqnarray}$$

The fact that equation (12) includes only even-order derivates of A implies that we have two sets of complex-conjugate solutions. These solutions are

   $$\begin{eqnarray} A_1(t) = e^{\pm{{\sqrt{\frac{a_1}{2}} }\, t }},\end{eqnarray}$$

where

$$a_1=- (1+2\,\eta){v^2}\,{{{k_x}}^2} - {{{k_z}}^2}\,{{{v_v}}^... ...eta \,{{{k_x}}^2} + {{{k_z}}^2}\,{{{v_v}}^2} \right) }^2}}},$$

and

   $$\begin{eqnarray} A_2(t) = e^{\pm{{\sqrt{\frac{a_2}{2}} }\, t }},\end{eqnarray}$$

where

$$a_2 = - (1+2\,\eta){v^2}\,{{{k_x}}^2} - {{{k_z}}^2}\,{{{v_v}... ...\eta \,{{{k_x}}^2} + {{{k_z}}^2}\,{{{v_v}}^2} \right) }^2}}}.$$

The two sets of solutions differ in the sign in front of the square root in functions a₁ and a₂. Solution (13) reduces to the isotropic medium solution when $\eta$=0. Solution (14) represents an additional wave that reduces in the isotropic limit ($\eta \rightarrow 0$) to an evanescent wave and, with proper initial conditions, its coefficient reduces to zero. However, this wave will prove to be harmful in the VTI case.

 

sol2 Figure 1 Plots of a1 (solid curve), and a2 (dashed curve), from equations (13) and (14), as a function of $\eta$. Positive a1 or a2 values result in real solutions of the wave equation that either grow or decay exponentially, depending on the sign of the exponent.

The main concern in equations 13 and 14 is the sign of a₁ and a₂. A negative sign results in an imaginary exponential term that describes wave propagation behavior. A positive sign results in a real exponential that either decays or grows depending on the sign of the exponent. Considering that we have conjugate solutions, at least one of the solutions will grow exponentially and cause serious instability problems in the numerical implementation regardless of the initial condition. Moreover, numerical noise will insure the instability of the problem when we have a solution that grows exponentially.

Figure 1 shows plots of a₁ (the solid curve), and a₂ (the dashed curve) as a function of $\eta$. The value of a₁ is negative independently of the value of $\eta$ (for practical $\eta$ values), which corresponds to the P-wave solution. If $\eta=0$,a₂=0 and the exponential term equals 1. However, for negative $\eta$,a₂ is positive and the exponential term grows rapidly with time, a clear area of instability in the problem. Positive $\eta$ gives negative values of a₂, which results in a wave that is different from the P-wave, propagating in the medium. A numerical example of this phenomenon is given in Figure 2.

Again, two of the solutions [equations (13)] are just perturbations of familiar isotropic medium solutions. Therefore, using the method of perturbation (described in Appendix B), we can obtain these two desired solutions by perturbing the isotropic ones. Doing so should provide accurate results for small $\eta$. This procedure is similar to the Born approximation Cohen and Bleistein (1979) for small perturbations in velocity.

The unwanted additional wave, which I refer to as an artifact, satisfies the following dispersion relation from a₂:

$$k_z^2 = {\frac{\left( {{\omega }^2} - {v^2}\,{{{k_x}}^2} \ri... ... + \eta \right) \,{{{k_x}}^4}-{{\omega }^4} }}}{{{{v_v}}^2}}}.$$ (15)

It has evanescent features (k_z is complex) for $\eta=0$ (isotropic medium), and therefore exponentially decays as a function of z. Detailed study of this artifact is beyond the scope of this paper. However, putting enough vertical distance between the source and the anisotropic layer will help eliminate this wave.