The Perturbation method

The general procedure of the perturbation method is to identify a small parameter (i.e., $\eta$), such that when this parameter is set to zero the problem (for example, the differential equation) becomes soluble. Thematically, the approach decomposes a tough problem into an infinite number of relatively easy ones. Hence, perturbation method is most useful when the few first steps reveal the important features of the solution and the remaining ones give small corrections. Another feature of the perturbation method is that it does not add additional solutions to the solutions already obtained from the unperturbed medium (i.e., background medium). The method simply perturbes those solutions obtained for the background medium (in our case isotropic) to accommodate the perturbation in the medium.

From equation (7), one parameter that can clearly be small is $\eta$. Setting $\eta$=0 yields the acoustic wave equation for elliptically anisotropic media. For isotropic (or elliptically anisotropic) acoustic media, we have two complex-conjugate solutions; one corresponds to outgoing waves and the other to incoming waves. Perturbation from a background medium of $\eta$=0 will result in only two solutions as well.

According to the perturbation theory (Bender and Orszag, 1978), the solution of equation (7) can be represented in a power-series expansion in terms of $\eta$. Since in practical applications, this power series is truncated, we use only the first four terms. Therefore, to find the perturbed solution, we write

$$F(x,z,t)= A_0(x,z,t)+A_1(x,z,t) \eta +A_2(x,z,t) \eta^2 +A_3(x,z,t) \eta^3 ,$$

as a trial solution of equation (7) for the two-dimensional problem. In addition, we consider

$$P(x,z,t)=\frac{\partial^2 F}{\partial t^2} = B_0(x,z,t)+ B_1(x,z,t) \eta +B_2(x,z,t) \eta^2 +B_3(x,z,t) \eta^3 ,$$

as a trial solution of equation (11). Because $\eta$ is a variable, we can set the coefficients of each power of $\eta$ separately to equal zero. This will provide a recursive method of solving for the coefficients, A_i and B_i. For example, B₀ is obtained directly from setting $\eta$=0, which corresponds to the solution of the elliptically anisotropic problem. Although, $\eta$ can vary with position, its variation is taken to be much smaller than those associated with the wavefield (which is true asymptotically, where $\omega \rightarrow \infty$).

Substituting the trial solution into the partial differential equation (11), and considering only the term of zero-power in $\eta$ yields  

$$\frac{\partial^2 B_0}{\partial t^2}= v_v^2 \frac{\partial^2 B_0}{\partial z^2}+v^2 \frac{\partial^2 B_0}{\partial x^2}.$$ (32)

If a force function was present in equation (11), the force function would transport as is to this zero-order equation. Equation (32) is just the acoustic wave equation for elliptically anisotropic media, and can be solved using the finite-difference approach.

The coefficient of first power in $\eta$ yields  

$$\frac{\partial^2 B_1}{\partial t^2}= v_v^2 \frac{\partial^2 B_1}{\partial z^2}+v^2 \frac{\partial^2 B_1}{\partial x^2} +f_0,$$ (33)

where  

$$f_0 = 2 v^2 \left(\frac{\partial^2 B_0}{\partial z^2}-v_v^2 \frac{\partial^4 A_0}{\partial t^2 \partial z^2} \right),$$ (34)

and

$$B_0 = \frac{\partial^2 A_0}{\partial t^2} .$$

Equation (34) is calculated using the solution B₀ of equation (32). Note that equations (32) and (33) are the same differential equations with different forcing functions. In other words, they share the same Green's function, that of the elliptically anisotropic background medium.

The next power in $\eta$ gives  

$$\frac{\partial^2 B_2}{\partial t^2}= v_v^2 \frac{\partial^2 B_2}{\partial z^2}+v^2 \frac{\partial^2 B_2}{\partial x^2} +f_1,$$ (35)

where  

$$f_1 = 2 v^2 \left(\frac{\partial^2 B_1}{\partial z^2}-v_v^2 \frac{\partial^4 A_1}{\partial x^2 \partial z^2} \right),$$ (36)

and

$$B_1 = \frac{\partial^2 A_1}{\partial t^2},$$

which is the same as the previous power $\eta$ term. Actually, all terms that follow have the same form. A Green's function is calculated for the elliptically anisotropic background problem can be used to recursively estimate all B_i from the different forcing functions, and subsequently to estimate the wavefield.

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