THE INTEGRAL OPERATOR FOR OFFSET CONTINUATION

The integral solution of problem (1)-(3) is obtained in Appendix A with the help of the classic methods of mathematical physics. It takes the explicit form

$$\begin{eqnarray} P(t_n,h,y) & = & \int\!\!\int P^{(0)}_1(t_1,y_1)\,G_0(t_1,h_1,y... ...\!\int P^{(1)}_1(t_1,y_1)\,G_1(t_1,h_1,y_1;t_n,h,y)\,dt_1\,dy_1\;,\end{eqnarray}$$ (4)

where the ``Green's functions'' G₀ and G₁ are expressed as

$$\begin{eqnarray} G_0(t_1,h_1,y_1;t_n,h,y) & = & \mbox{sign}(h-h_1)\,{H(t_n) \ove... ..._n \over t_1^2}\,\left\{ H(\Theta) \over \sqrt{\Theta}\right\}\;,\end{eqnarray}$$ (5) (6)

and the parameter $\Theta$ is  

$$\Theta(t_1,h_1,y_1;t_n,h,y) = \left(h_1^2/t_1^2-h^2/t_n^2\right)\,\left(t_1^2-t_n^2\right)- \left(y_1-y\right)^2\;.$$ (7)

H stands for the Heavyside step-function.

From formulas (5) and (6) one can see that the impulse response of the offset continuation operator is discontinuous in the time-offset-midpoint space on a surface defined by the equality

 

$$\Theta(t_1,h_1,y_1;t_n,h,y) = 0$$ (8)

that describes the ``wavefronts'' of the offset continuation process. In terms of the theory of characteristics Courant (1962), the surface $\Theta=0$ corresponds to the characteristic conoid formed by bi-characteristics of equation (1) - ``time rays'' Fomel (1995) emerging from the point $\{t_n,h,y\}=\{t_1,h_1,y_1\}$(Figure 1).

 

offcon
Figure 1 Constant-offset sections of the characteristic conoid - ``offset continuation fronts'' (left), and branches of the conoid used in the integral OC operator (right). The upper part of the plots (small times) corresponds to continuation to smaller offsets; the lower part (large times) corresponds to larger offsets.

As a second-order differential equation of the hyperbolic type, equation (1) describes two different processes. The first process is ``forward'' continuation from smaller to larger offsets; the second one is ``reverse'' continuation in the opposite direction. These two processes are clearly separated in the high-frequency asymptotics of operator (4). To obtain the asymptotic representation, it is sufficient to note that ${1 \over \sqrt{\pi}}\, {H(t) \over \sqrt{t}}$ is the impulse response of the causal half-order integration operator, and that $H(t^2-a^2) \over \sqrt{t^2-a^2}$ is asymptotically equivalent to $H(t-a) \over {\sqrt{2a}\,\sqrt{t-a}}$(t, a >0). Thus, the asymptotic form of the integral offset continuation operator becomes

$$\begin{eqnarray} P^{(\pm)}(t_n,h,y) & = & {\bf D}^{1/2}_{\pm\,t_n}\,\int w^{(\pm... ...h,t_n)\, P^{(1)}_1(\theta^{(\pm)}(\xi;h_1,h,t_n),y_1-\xi)\,d\xi\;.\end{eqnarray}$$ (9)

Here the signs ``+'' and ``-'' correspond to the type of continuation (the sign of h-h₁); ${\bf D}^{1/2}_{\pm\,t_n}$ and ${\bf I}^{1/2}_{\pm\,t_n}$ stand for the operators of causal and anticausal half-order differentiation and integration applied with respect to the time variable t_n; the summation paths $\theta^{(\pm)}(\xi;h_1,h,t_n)$ correspond to the two non-negative sections of the characteristic conoid (8) (Figure 1):

 

$$t_1=\theta^{(\pm)}(\xi;h_1,h,t_n)= {t_n \over h}\,\sqrt{{U \pm V} \over 2 }\;,$$ (10)

where $U=h^2+h_1^2-\xi^2$, and $V=\sqrt{U^2-4\,h^2\,h_1^2}$; $\xi$ is the midpoint separation (the integration parameter); and $w^{(\pm)}_0$and $w^{(\pm)}_1$ are the following weighting functions:

$$\begin{eqnarray} w^{(\pm)}_0 & = & {1 \over \sqrt{2\,\pi}}\, {\theta^{(\pm)}(\xi... ...rt{t_n}\, h_1} \over {\sqrt{V}\,\theta^{(\pm)}(\xi;h_1,h,t_n)}}\;.\end{eqnarray}$$ (11) (12)

Expression (10) for the summation path of the OC operator was obtained previously by Stovas and Fomel 1993 and Biondi and Chemingui . A somewhat different form of it is proposed by Bagaini et al. 1994. I describe the kinematic interpretation of formula (10) in Appendix B.

The limit of expression (10) for the output offset h approaching zero can be evaluated by L'Hospitale's rule. As one would expect, it coincides with the well-known expression for the summation path of the integral DMO operator Deregowski and Rocca (1981)

 

$$t_1=\theta^{(-)}(\xi;h_1,0,t_n)= \lim_{h \rightarrow 0} {{t_... ...rt{{U - V} \over 2 }}= {{t_n\,h_1} \over \sqrt{h_1^2-\xi^2}}\;.$$ (13)