To obtain an explicit solution of the Cauchy problem (1)-(3), it is convenient to apply the following simple transform of the function P:
Here the Heavyside function H is included to take into account the causality of the reflection seismic gathers (note that the time tn=0 corresponds to the direct wave arrival). We can evenly extrapolate the function Q to negative times, writing the reverse of (47) as follows:
The initial value conditions (2) and (3) in the space are defined on a hyperbola of the form . Now the solution of the Cauchy problem follows directly from Riemann's method Courant (1962). According to this method, the domain of dependence of each point is a part of the hyperbola between the points and (Figure 4). If we let denote this curve, the solution takes an explicit integral form:
Figure 4 Domain of dependence of a point in the transformed coordinate system.
Applying the explicit expression for the Riemann's function R (56) and performing the inverse transform of both the function and the variables allows us to rewrite equations (57), (58), and (59) in the original coordinate system. This yields the integral offset continuation operators in the domain
The inverse Fourier transforms of formulas (61) and (62) are reduced to analytically evaluated integrals Gradshtein and Ryzhik (1994) to produce explicit integral operators in the time-and-space domain