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Next: Offset continuation and DMO Up: Theory of differential offset Previous: Proof of amplitude equivalence

Integral offset continuation operator

Equation (1) describes a continuous process of reflected wavefield continuation in the time-offset-midpoint domain. In order to find an integral-type operator that performs the one-step offset continuation, I consider the following initial-value problem for equation (1):

Given a post-NMO constant-offset section at half-offset $h_1$

\left.P(t_n,h,y)\right\vert _{h=h_1}=P^{(0)}_1(t_n,y)
\end{displaymath} (62)

and its first-order derivative with respect to offset
\left.\partial P(t_n,h,y)\over \partial h\right\vert _{h=h_1}=P^{(1)}_1(t_n,y)\;,
\end{displaymath} (63)

find the corresponding section $P^{(0)}(t_n,y)$ at offset $h$.

Equation (1) belongs to the hyperbolic type, with the offset coordinate $h$ being a ``time-like'' variable and the midpoint coordinate $y$ and the time $t_n$ being ``space-like'' variables. The last condition (63) is required for the initial value problem to be well-posed (Courant, 1962). From a physical point of view, its role is to separate the two different wave-like processes embedded in equation (1), which are analogous to inward and outward wave propagation. We will associate the first process with continuation to a larger offset and the second one with continuation to a smaller offset. Though the offset derivatives of data are not measured in practice, they can be estimated from the data at neighboring offsets by a finite-difference approximation. Selecting a propagation branch explicitly, for example by considering the high-frequency asymptotics of the continuation operators, can allow us to eliminate the need for condition (63). In this section, I discuss the exact integral solution of the OC equation and analyze its asymptotics.

The integral solution of problem (62-63) for equation (1) is obtained in with the help of the classic methods of mathematical physics (Fomel, 2001,1994). It takes the explicit form

$\displaystyle P(t_n,h,y)$ $\textstyle =$ $\displaystyle \int\!\!\int P^{(0)}_1(t_1,y_1)\,G_0(t_1,h_1,y_1;t_n,h,y)\,dt_1\,dy_1$  
$\displaystyle +$   $\displaystyle \int\!\!\int P^{(1)}_1(t_1,y_1)\,G_1(t_1,h_1,y_1;t_n,h,y)\,dt_1\,dy_1\;,$ (64)

where the Green's functions $G_0$ and $G_1$ are expressed as
$\displaystyle G_0(t_1,h_1,y_1;t_n,h,y)$ $\textstyle =$ $\displaystyle \mbox{sign}(h-h_1)\,{H(t_n) \over \pi}\,
{\partial \over \partial t_n}\,\left\{
H(\Theta) \over
\sqrt{\Theta}\right\}\;,$ (65)
$\displaystyle G_1(t_1,h_1,y_1;t_n,h,y)$ $\textstyle =$ $\displaystyle \mbox{sign}(h-h_1)\,
{H(t_n) \over \pi}\,h\,
{t_n \over t_1^2}\,\left\{
H(\Theta) \over
\sqrt{\Theta}\right\}\;,$ (66)

and the parameter $\Theta$ is
\Theta(t_1,h_1,y_1;t_n,h,y) =
\end{displaymath} (67)

$H$ stands for the Heaviside step-function.

From equations (65) and (66) 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\;,
\end{displaymath} (68)

which 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 the bi-characteristics of equation (1) - time rays emerging from the point $\{t_n,h,y\}=\{t_1,h_1,y_1\}$. The common-offset slices of the characteristic conoid are shown in the left plot of Figure 5.

Figure 5.
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.
[pdf] [png] [sage]

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 (64). 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 asymptotical form of the integral offset-continuation operator becomes

$\displaystyle P^{(\pm)}(t_n,h,y)$ $\textstyle =$ $\displaystyle {\bf D}^{1/2}_{\pm\,t_n}\,\int w^{(\pm)}_0(\xi;h_1,h,t_n)\,
  $\textstyle \pm$ $\displaystyle {\bf I}^{1/2}_{\pm\,t_n}\,\int w^{(\pm)}_1(\xi;h_1,h,t_n)\,
P^{(1)}_1(\theta^{(\pm)}(\xi;h_1,h,t_n),y_1-\xi)\,d\xi\;.$ (69)

Here the signs ``$+$'' and ``$-$'' correspond to the type of continuation (the sign of ${h-h_1}$), ${\bf D}^{1/2}_{\pm\,t_n}$ 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 (68) (Figure 5):
{t_n \over h}\,\sqrt{{U \pm V} \over 2 }\;,
\end{displaymath} (70)

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:
$\displaystyle w^{(\pm)}_0$ $\textstyle =$ $\displaystyle {1 \over \sqrt{2\,\pi}}\,
{\theta^{(\pm)}(\xi;h_1,h,t_n) \over \sqrt{t_n\,V}}\;,$ (71)
$\displaystyle w^{(\pm)}_1$ $\textstyle =$ $\displaystyle {1 \over \sqrt{2\,\pi}}\,
{{\sqrt{t_n}\, h_1} \over {\sqrt{V}\,\theta^{(\pm)}(\xi;h_1,h,t_n)}}\;.$ (72)

Expression (70) for the summation path of the OC operator was obtained previously by Stovas and Fomel (1996) and Biondi and Chemingui (1994). A somewhat different form of it is proposed by Bagaini and Spagnolini (1996). I describe the kinematic interpretation of formula (70) in Appendix B.

In the high-frequency asymptotics, it is possible to replace the two terms in equation (69) with a single term (Fomel, 2003a). The single-term expression is

P^{(\pm)}(t_n,h,y) =
{\bf D}^{1/2}_{\pm\,t_n}\,\int w^{(\pm...
\end{displaymath} (73)

$\displaystyle w^{(+)}$ $\textstyle =$ $\displaystyle \sqrt{\theta^{(+)}(\xi;h_1,h,t_n) \over {2\,\pi}}\;
{{h^2-h_1^2-\xi^2} \over {V^{3/2}}}\;,$ (74)
$\displaystyle w^{(-)}$ $\textstyle =$ $\displaystyle {{\theta^{(-)}(\xi;h_1,h,t_n)} \over \sqrt{2\,\pi t_n}}\;
{{h_1^2-h^2 +\xi^2} \over {V^{3/2}}}\;.$ (75)

A more general approach to true-amplitude asymptotic offset continuation is developed by ().

The limit of expression (70) 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)

\lim_{h \rightarrow 0} {{t_...
...t{{U - V} \over 2 }}=
{{t_n\,h_1} \over \sqrt{h_1^2-\xi^2}}\;.
\end{displaymath} (76)

I discuss the connection between offset continuation and DMO in the next section.

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Next: Offset continuation and DMO Up: Theory of differential offset Previous: Proof of amplitude equivalence