PARTIAL DIFFERENTIAL EQUATIONS FOR THE PSPM OPERATOR

The PSPM operator, and any other migration operator with the same reasoning, represents a surface when the migration operator is defined in 2-D. The solutions to partial differential equations (PDE) in two variables are families of well behaved surfaces (piecewise smooth surfaces), so naturally the question arises if our migration operators cannot be represented as solutions to certain PDE. Fundamentally a migration operator is defined kinematically by the migration curves, and dynamically by an amplitude function along the migration curve. The zero-offset migration operator can be viewed as a surface. For a constant velocity you have circles and radial lines as potential characteristics. A common amplitude scheme assumes constant amplitudes on a circle (migration circle) equal to 1 / t and the characteristics are the family of curves t=p*x, which are the rays. Obviously, finding just the surface without the migration curve is of little use for the geophysicist, as often the correct kinematics offer better imaging even when the amplitude function is not well defined. The characteristic curves which are encountered in the solution of a broad family of partial differential equations may offer a solution to this problem.

The amplitudes of the operator can be discussed separately, as they don't influence the form of the characteristics. The characteristics are determined by the first two terms in a linear or quasi-linear partial differential equation. While there is a consensus in the kinematics of the migration operators, there are as many opinions in the distribution of amplitudes along the operators as there are geophysical schools. However to examine the validity of the PDE, the amplitude definition can be expressed as a general function f(x₀,t₀) without affecting our analysis.

The motivation for finding the PDE that describe the operator in a constant velocity medium is the assumption that the PDE in a variable velocity medium should have a similar form, or close enough, which can be found using analogous reasoning.

The PSPM operator is defined by the migration curves shown in Figure 1 and by the cutoff curves of constant parameter $\bf{p}$.It is natural to think about one of these families of curves as characteristic equations and try to find the partial differential equation which can describe the PSPM operator in a constant velocity medium. The simplest family of partial differential equations is the one of linear (and quasi-linear) partial differential equations, for which the solution is found along the characteristic curves.

The equations for the curves which appear in Figure 1 are  

$$t_0 = {p {{(h^2-x^2_0)} \over {x_0}} }$$ (5)

 

$$t_0 = {\tau {\sqrt {h^2-x_0^2} \over h} }$$ (6)

Equation (5) represents the location of the points which are DMO migrated with the same parameter $\bf{p}$.Equation (6) represents the DMO migration curves, a family of ellipses with horizontal semi-axis h.

The corresponding PDEs whose characteristic curves are the curves presented in Figure 1 are  

$$-{u_{x_0}}{x_0(h^2-x_0^2) \over {h^2+x_0^2}}+t_0u_{t_0}={f_1}(x_0,t_0)$$ (7)

 

$$-{u_{x_0}}{{h^2-x_0^2} \over x_0}+t_0u_{t_0}={f_2}(x_0,t_0)$$ (8)

where f₁ and f₂ are two arbitrary functions which are defined by the variation of the amplitudes of the operator along the characteristics. While studying the kinematics of the operators, these functions can be ignored.

The solution to each equation as defined in the Appendix, will provide two families of curves, as two parameters are used in solving the equation using the method of characteristics. One is the parameter which varies along the characteristic curve, the other varies along the initial condition curve.

In Figure 4 the characteristic curves are the curves of constant $\bf{p}$ parameter shown in Figure 1. The initial condition is given along an ellipse. While the characteristic curves are represented correctly in the solution, the associated curves do not follow the migration ellipses.

In Figure 5 the characteristic curves are the curves of constant $\tau$ parameter shown in Figure 1. The initial condition is given along a vertical curve of constant parameter $\bf{p}$.While the characteristic curves are represented correctly in the solution, the associated curves do not follow the cutoff curves in Figure 1.

We are faced with the problem that while the amplitudes of the operator are represented correctly in the domain considered, the pair of family curves are not.