Solving the Cauchy problem

 To obtain an explicit solution of the Cauchy problem (-) for equation (), it is convenient to apply the following simple transform of the wavefield P:  

$$P(t_n,h,y)=Q(t_n,h,y)\,t_n\,H(t_n)\;.$$ (22)

Here the Heavyside function H is included to take into account the causality of the reflection seismic gathers (note that the time t_n=0 corresponds to the direct wave arrival). We can extrapolate Q as an even function to negative times, writing the reverse of (A-22) as follows:  

$$Q(t_n,h,y)=Q(-t_n,h,y)=P(\vert t_n\vert,h,y)/\vert t_n\vert\;.$$ (23)

With the change of function (A-22), equation () transforms to  

$$h \, {\partial^2 Q \over \partial y^2} = h\, {\partial^2 Q \... ...artial h} + t_n \, {\partial Q \over {\partial t_n}}\right) \;.$$ (24)

Applying the change of variables  

$$\rho={t_n^2 \over 2}\;,\;\nu={h^2 \over {2\,t_n^2}}$$ (25)

and Fourier transform in the midpoint coordinate y  

$$\widetilde{Q}(\rho,\nu,k)=\int\,Q(\rho,\nu,y)\,\exp (-iky)\,dy\;,$$ (26)

I further transform equation (A-24) to the canonical form of a hyperbolic-type partial differential equation with two variables:  

$${\partial^2 \widetilde{Q} \over {\partial \rho \, \partial \nu}} + k^2\,\widetilde{Q} = 0\;.$$ (27)

 

rim Figure 1 Domain of dependence of a point in the transformed coordinate system.

The initial value conditions () and () in the $\{\rho,\nu\}$ space are defined on a hyperbola of the form $\rho\,\nu=\left(h_1 \over 2\right)^2=\mbox{constant}$. Now the solution of the Cauchy problem follows directly from Riemann's method (). According to this method, the domain of dependence of each point $\{\rho,\nu\}$ is a part of the hyperbola between the points $\{\rho,{h_1^2 \over {4\,\rho}}\}$ and $\{{h_1^2 \over {4\,\nu}},\nu\}$ (Figure A-1). If we let $\Sigma$denote this curve, the solution takes an explicit integral form:

$$\begin{eqnarray} \widetilde{Q}(\rho,\nu) & = & {1 \over 2}\, \widetilde{Q}(\rho,... ...\rho_1,\nu_1;\rho,\nu) \over {\partial \nu_1}} \right)\,d \nu_1\;.\end{eqnarray}$$ (28)

Here R is the Riemann's function of equation (A-27), which has the known explicit analytical expression  

$$R(\rho_1,\nu_1;\rho,\nu)= J_0\left(2k\,\sqrt{\left(\rho_1-\rho\right)\, \left(\nu_1-\nu\right)}\right)\;,$$ (29)

where J₀ is the Bessel function of zeroth order. Integrating by parts and taking into account the connection of the variables on the curve $\Sigma$, we can simplify equation (A-28) to the form  

$$\widetilde{Q}(\rho,\nu)= \widetilde{Q}_0(\rho,\nu)+ \widetilde{Q}_1(\rho,\nu)\;,$$ (30)

where

$$\begin{eqnarray} \widetilde{Q}_0(\rho,\nu) & = & {\partial \over {\partial \rho}... ...l \widetilde{Q}(\rho_1,\nu_1) \over {\partial \nu_1}} \,d \nu_1\;.\end{eqnarray}$$ (31) (32)

Applying the explicit expression for the Riemann function R (A-29) and performing the inverse transform of both the function and the variables allows us to rewrite equations (A-30), (A-31), and (A-32) in the original coordinate system. This yields the integral offset continuation operators in the $\{t_n,h,k\}$ domain  

$$\widetilde{P}(t_n,h,k)= H(t_n)\,\left(\widetilde{P}_0(t_n,h,k) + t_n\,\widetilde{P}_1(t_n,h,k)\right)\;,$$ (33)

where

$$\begin{eqnarray} \widetilde{P}_0 & = & {\partial \over {\partial t_n}}\, \int_{... ...}\right)\, \left(t_n^2-t_1^2\right)}\right)\,{dt_1 \over t_1^2}\;,\end{eqnarray}$$ (34) (35)

$$\begin{eqnarray} \widetilde{P}^{(j)}_1(t_1,k) & = & \int\,P\,^{(j)}_1(t_1,y_1)\... ...lde{P}(t_n,h,k) & = & \int\,P(t_n,h,y)\exp (-iky)\,dy\;(j=0,1)\;.\end{eqnarray}$$ (36) (37)

The inverse Fourier transforms of equations (A-34) and (A-35) are reduced to analytically evaluated integrals () to produce explicit integral operators in the time-and-space domain  

$$P(t_n,h,y)=\mbox{sign}(h-h_1)\, {H(t_n) \over \pi}\,\left(P_0(t_n,h,y) + t_n\,P_1(t_n,h,y)\right)\;,$$ (38)

where

$$\begin{eqnarray} P_0(t_n,h,y) & = & {\partial \over {\partial t_n}}\, \iint_{\S... ..._1^2 \over t_1^2}\right)\, \left(t_n^2-t_1^2\right)-(y-y_1)^2}}\;.\end{eqnarray}$$ (39) (40)

The range of integration $\Sigma$ in (A-39) and (A-40) is defined by the inequality  

$$\left({h^2 \over t_n^2}-{h_1^2 \over t_1^2}\right)\, \left(t_n^2-t_1^2\right)-(y-y_1)^2 \gt 0\;.$$ (41)

Equations (A-38), (A-39), and (A-40) coincide with (), (), and () in the main text.