Theory of differential offset continuation

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Appendix A

Second-order reflection traveltime derivatives

This appendix contains a derivation of equations connecting second-order partial derivatives of the reflection traveltime with the geometric properties of the reflector in a constant velocity medium. These equations are used in the main text of this paper to describe the amplitude behavior of offset continuation. Let $\tau(s,r)$ be the reflection traveltime from the source $s$ to the receiver $r$. Consider a formal equality

$$\tau(s,r)=\tau_1\left(s,x(s,r)\right)+\tau_2\left(x(s,r),r\right)\;,$$ (114)

where $x$ is the reflection point parameter, $\tau_1$ corresponds to the incident ray, and $\tau_2$ corresponds to the reflected ray. Differentiating (A-1) with respect to $s$ and $r$ yields

$${\partial \tau \over \partial s} = {\partial \tau_1 \over \partial s} + {\partial \tau \over \partial x}\, {\partial x \over \partial s}\;,$$ (115)

$${\partial \tau \over \partial r} = {\partial \tau_2 \over \partial r} + {\partial \tau \over \partial x}\, {\partial x \over \partial r}\;.$$ (116)

According to Fermat's principle, the two-point reflection ray path must correspond to the traveltime stationary point. Therefore

$${\partial \tau \over \partial x} \equiv 0$$ (117)

for any $s$ and $r$. Taking into account (A-4) while differentiating (A-2) and (A-3), we get

$${\partial^2 \tau \over \partial s^2} = {\partial^2 \tau_1 \over \partial s^2} + B_1\, {\partial x \over \partial s}\;,$$ (118)

$${\partial^2 \tau \over \partial r^2} = {\partial^2 \tau_2 \over \partial r^2} + B_2\, {\partial x \over \partial r}\;,$$ (119)

$${\partial^2 \tau \over \partial s \partial r} = B_1\, {\partial x \over \partial r}\;= B_2\, {\partial x \over \partial s}\;,$$ (120)

where

$$B_1={\partial^2 \tau_1 \over \partial s \partial x}\;;\; B_2={\partial^2 \tau_2 \over \partial r \partial x}\;.$$

Differentiating equation (A-4) gives us the additional pair of equations

$$C\,{\partial x \over \partial s}+B_1 = 0\;,$$ (121)

$$C\,{\partial x \over \partial r}+B_2 = 0\;,$$ (122)

where

$$C={\partial^2 \tau \over \partial x^2}= {\partial^2 \tau_1 \over \partial x^2}+ {\partial^2 \tau_2 \over \partial x^2}\;.$$

Solving the system (A-8) - (A-9) for $\partial x \over \partial s$ and $\partial x \over \partial r$ and substituting the result into (A-5) - (A-7) produces the following set of expressions:

$${\partial^2 \tau \over \partial s^2} = {\partial^2 \tau_1 \over \partial s^2} - C^{-1}\,B_1^2\;;$$ (123)

$${\partial^2 \tau \over \partial r^2} = {\partial^2 \tau_2 \over \partial r^2} - C^{-1}\,B_2^2\;;$$ (124)

$${\partial^2 \tau \over \partial s \partial r} = - C^{-1}\,B_1\,B_2\;.$$ (125)

In the case of a constant velocity medium, expressions (A-10) to (A-12) can be applied directly to the explicit equation for the two-point eikonal

$$\tau_1(y,x)=\tau_2(x,y)={\sqrt{(x-y)^2+z^2(x)}\over v}\;.$$ (126)

Differentiating (A-13) and taking into account the trigonometric relationships for the incident and reflected rays (Figure 1), one can evaluate all the quantities in (A-10) to (A-12) explicitly. After some heavy algebra, the resultant expressions for the traveltime derivatives take the form

$${\partial \tau \over \partial s} = {\partial \tau_1 \over \partial s} = {\sin{\alpha_1}\over v} \;;\; {\partial \tau \over \partial r} = {\partial \tau_2 \over \partial r} = {\sin{\alpha_2}\over v}\;;$$ (127)

$${\partial \tau_1 \over \partial x} = {\sin{\gamma}\over v \cos{\alpha}} \;;\; {\partial \tau_2 \over \partial x} = - {\sin{\gamma}\over v \cos{\alpha}}\;;$$ (128)

$$B_1 = {\partial^2 \tau_1 \over \partial s\,\partial x} = {\cos{\alpha_1... ...{\alpha}}}\, \left(-1-{\sin{\gamma}\over\cos{\alpha}}\,\sin{\alpha_1}\right)\;;$$ (129)

$$B_2 = {\partial^2 \tau_2 \over \partial r\,\partial x} = {\cos{\alpha_2... ...{\alpha}}}\, \left(-1+{\sin{\gamma}\over\cos{\alpha}}\,\sin{\alpha_2}\right)\;;$$ (130)

$$B_1\,B_2 = {\cos^6{\gamma}\over v^2\,D^2\,a^4}\;;\; B_1+B_2 = -2\,{\cos^3{\gamma}\over v\,D\,a^2}\,\left(2\,a^2-1\right)\;;$$ (131)

$${\partial^2 \tau_1 \over \partial x^2} = {{\cos^2{\gamma}+D\... ...2{\gamma}+D\,K}\over{v\,D\,\cos^3{\alpha}}}\,\cos{\alpha_2}\;;$$ (132)

$$C={\partial^2 \tau_1 \over \partial x^2}+{\partial^2 \tau_2 ... ...\gamma}\,{{\cos^2{\gamma}+D\,K}\over{v\,D\,\cos^3{\alpha}}}\;.$$ (133)

Here $D$ is the length of the normal (central) ray, $\alpha$ is its dip angle ( $\alpha={{\alpha_1+\alpha_2}\over 2}$, $\tan{\alpha}=z'(x)$), $\gamma$ is the reflection angle $\left(\gamma={{\alpha_2-\alpha_1}\over 2}\right)$, $K$ is the reflector curvature at the reflection point $\left(K=z''(x)\,\cos^3{\alpha}\right)$, and $a$ is the dimensionless function of $\alpha$ and $\gamma$ defined in (47).

The equations derived in this appendix were used to obtain the equation

$$\tau_n\,\left({\partial^2 \tau_n \over \partial y^2}- {\part... ...\,\left({\sin^2{\alpha}+DK}\over {\cos^2{\gamma}+DK}\right)\;,$$ (134)

which coincides with (50) in the main text.

Appendix B

The kinematics of offset continuation

This Appendix presents an alternative method to derive equation (70), which describes the summation path of the integral offset continuation operator. The method is based on the following considerations.

The summation path of an integral (stacking) operator coincides with the phase function of the impulse response of the inverse operator. Impulse response is by definition the operator reaction to an impulse in the input data. For the case of offset continuation, the input is a reflection common-offset gather. From the physical point of view, an impulse in this type of data corresponds to the special focusing reflector (elliptical isochrone) at the depth. Therefore, reflection from this reflector at a different constant offset corresponds to the impulse response of the OC operator. In other words, we can view offset continuation as the result of cascading prestack common-offset migration, which produces the elliptic surface, and common-offset modeling (inverse migration) for different offsets. This approach resemble that of Deregowski and Rocca (1981). It was also applied to a more general case of azimuth moveout (AMO) by Fomel and Biondi (1995) and fully generalized by Bleistein and Jaramillo (2000). The geometric approach implies that in order to find the summation pass of the OC operator, one should solve the kinematic problem of reflection from an elliptic reflector whose focuses are in the shot and receiver locations of the output seismic gather.

In order to solve this problem , let us consider an elliptic surface of the general form

$$h(x)=\sqrt{d^2-\beta\,(x-x')^2}\;,$$ (135)

where $0 < \beta < 1$. In a constant velocity medium, the reflection ray path for a given source-receiver pair on the surface is controlled by the position of the reflection point $x$. Fermat's principle provides a required constraint for finding this position. According to Fermat's principle, the reflection ray path corresponds to a stationary value of the travel-time. Therefore, in the neighborhood of this path,

$${\partial \tau(s,r,x) \over \partial x} = 0\;,$$ (136)

where $s$ and $r$ stand for the source and receiver locations on the surface, and $\tau$ is the reflection traveltime

$$\tau(s,r,x) = { \sqrt{h^2(x)+(s-x)^2} \over v} + { \sqrt{h^2(x)+(r-x)^2} \over v}\;.$$ (137)

Substituting equations (B-3) and (B-1) into (B-2) leads to a quadratic algebraic equation on the reflection point parameter $x$. This equation has the explicit solution

$$x(s,r)= x' + {{\xi^2+H^2-h^2+\mbox{sign}(h^2-H^2)\, \sqrt{\l... ...^2-H^2-h^2\right)^2-4\,H^2\,h^2}\over {2\,\xi\,(1-\beta)}}}\;,$$ (138)

where $h=(r-s)/2$, $\xi = y-x'$, $y=(s+r)/2$, and $H^2=d^2\,\left({1 \over \beta} - 1\right)$. Replacing $x$ in equation (B-3) with its expression (B-4) solves the kinematic part of the problem, producing the explicit traveltime expression

$$\tau(s,r)=\left\{ \begin{array}{lcr}\displaystyle{ {1 \over... ... {1-\beta}}} & \mbox{for} & h^2 < H^2 \end{array} \right.\;,$$ (139)

where

$$f=\sqrt{(r-x')^2-H^2}\; , \;g=\sqrt{(s-x')^2-H^2}\;,$$

$$F=\sqrt{H^2-(r-x')^2}\; , \;G=\sqrt{H^2-(s-x')^2}\;.$$

The two branches of equation (B-5) correspond to the difference in the geometry of the reflected rays in two different situations. When a source-and-receiver pair is inside the focuses of the elliptic reflector, the midpoint $y$ and the reflection point $x$ are on the same side of the ellipse with respect to its small semi-axis. They are on different sides in the opposite case (Figure B-1).

ell Figure B-1. .

Reflections from an ellipse. The three pairs of reflected rays correspond to a common midpoint (at 0.1) and different offsets. The focuses of the ellipse are at 1 and -1.

If we apply the NMO correction, equation (B-5) is transformed to

$$\tau_n(s,r)=\left\{ \begin{array}{lcr}\displaystyle{ {1 \ov... ...^2+(F+G)^2}} & \mbox{for} & h^2 < H^2 \end{array} \right.\;.$$ (140)

Then, recalling the relationships between the parameters of the focusing ellipse $r$, $x'$ and $\beta$ and the parameters of the output seismic gather (Deregowski and Rocca, 1981)

$$r={ {v\,t_n} \over 2}\;,\;x'=y\;,\; \beta={t_n^2 \over {t_n^2+{{4\,h^2} \over v^2}}}\;,\; H=h\;,$$ (141)

and substituting expressions (B-7) into equation (B-6) yields the expression

$$t_1(s_1,r_1;s,r,t_n)=\left\{ \begin{array}{lcr}\displaystyl... ...+(F+G)^2}} & \mbox{for} & h_1^2 < h^2 \end{array} \right.\;,$$ (142)

where

$$f=\sqrt{(r_1-r)\,(r_1-s)}\;,\;g=\sqrt{(s_1-r)\,(s_1-s)}\;,$$

$$F=\sqrt{(r-r_1)\,(r_1-s)}\;,\;G=\sqrt{(s_1-r)\,(s-s_1)}\;.$$

It is easy to verify algebraically the mathematical equivalence of equation (B-8) and equation (70) in the main text. The kinematic approach described in this appendix applies equally well to different acquisition configurations of the input and output data. The source-receiver parameterization used in (B-8) is the actual definition for the summation path of the integral shot continuation operator (Bagaini and Spagnolini, 1996,1993). A family of these summation curves is shown in Figure B-2.

shc
Figure B-2. .

Summation paths of the integral shot continuation. The output source is at -0.5 km. The output receiver is at 0.5 km. The indexes of the curves correspond to the input source location.

Theory of differential offset continuation