Theory of differential offset continuation

Proof of amplitude equivalence

Let us now consider the connection between the laws of traveltime transformation and the laws of the corresponding amplitude transformation. The change of the wave amplitudes in the OC process is described by the first-order partial differential transport equation (5). We can find the general solution of this equation by applying the method of characteristics. The solution takes the explicit integral form

$$A_n\left(t_n\right)=A_0\left(t_0\right)\,\exp{\left(\int_{t_... ...u_n \over \partial h} \right)^{-1}\right]\, d\tau_n\right)}\;.$$ (42)

The integral in equation (42) is defined on a curved time ray, and $A_n(t_n)$ stands for the amplitude transported along this ray. In the case of a plane dipping reflector, the ray amplitude can be immediately evaluated by substituting the explicit traveltime and time ray equations from the preceding section into (42). The amplitude expression in this case takes the simple form

$$A_n\left(t_n\right)=A_0\left(t_0\right)\,\exp{\left(-\int_{t... ...u_n}{\tau_n}\right)} = A_0\left(t_0\right)\,{t_0 \over t_n}\;.$$ (43)

In order to consider the more general case of a curvilinear reflector, we need to take into account the connection between the traveltime derivatives in (42) and the geometry of the reflector. As follows directly from the trigonometry of the incident and reflected rays triangle (Figure 1),

$$h = {r-s \over 2}= D\,{{\cos{\alpha}\,\sin{\gamma}\,\cos{\gamma}} \over {\cos^2{\alpha}-\sin^2{\gamma}}}\;,$$ (44)

$$y = {r+s \over 2}= x+D\,{{\cos^2{\alpha}\,\sin{\alpha}} \over {\cos^2{\alpha}-\sin^2{\gamma}}}\;,$$ (45)

$$y_0 = x+D\,\sin{\alpha}\;,$$ (46)

where $D$ is the length of the normal ray. Let $\tau_0=2\,D/v$ be the zero-offset reflection traveltime. Combining equations (44) and (46) with (9), we can get the following relationship:

$$a={\tau_n\over\tau_0}={{\cos{\alpha}\,\cos{\gamma}}\over \le... ...}}}\right)^{1/2}= {h\,\over\sqrt{h^2-\left(y-y_0\right)^2}}\;,$$ (47)

which describes the ``DMO smile'' (40) found by Deregowski and Rocca (1981) in geometric terms. Equation (47) allows for a convenient change of variables in equation (42). Let the reflection angle $\gamma$ be a parameter monotonically increasing along a time ray. In this case, each time ray is uniquely determined by the position of the reflection point, which in turn is defined by the values of $D$ and $\alpha$. According to this change of variables, we can differentiate (47) along a time ray to get

$${{d\tau_n}\over\tau_n}=-{{\sin^2{\alpha}}\over {2\,\cos^2{\g... ...mma}-\sin^2{\alpha}\right)}}\, d\left(\cos^2{\gamma}\right)\;.$$ (48)

Note also that the quantity $h\,\left(\tau_n\,{\partial \tau_n \over \partial h} \right)^{-1}$ in equation (42) coincides exactly with the time ray invariant $C_3$ found in equation (27). Therefore its value is constant along each time ray and equals

$$h\,\left(\tau_n\,{\partial \tau_n \over \partial h}\right)^{-1}= -{v^2 \over 4\, \sin^2{\alpha}}\;.$$ (49)

Finally, as shown in Appendix ,

$$\tau_n\,\left({\partial^2 \tau_n \over \partial y^2}- {\part... ...\,\left({\sin^2{\alpha}+DK}\over {\cos^2{\gamma}+DK}\right)\;,$$ (50)

where $K$ is the reflector curvature at the reflection point. Substituting (48), (49), and (50) into (42) transforms the integral to the form

$$\begin{eqnarray*} \int_{t_o}^{t_n}\left[ h\,\left({\partial^2 \tau_n \over \par... ...artial \tau_n \over \partial h} \right)^{-1}\right]\, d\tau_n = \end{eqnarray*}$$

$$= -{1 \over 2}\,\int_{\cos^2{\gamma_0}}^{\cos^2{\gamma}} \l... ...r {\cos^2{\gamma'}+DK}}\right)\,d\left(\cos^2{\gamma'}\right)$$ (51)

which we can evaluate analytically. The final equation for the amplitude transformation is

$$A_n = A_0\,{\sqrt{\cos^2{\gamma}-\sin^2{\alpha}}\over \sqrt{\cos^2{\gam... ...2{\alpha}}}\,\left( {\cos^2{\gamma_0}+DK}\over{\cos^2{\gamma}+DK}\right)^{1/2}=$$

$$= A_0\,{{\tau_0\,\cos{\gamma}}\over{\tau_n\,\cos{\gamma_0}}}\, \left( {\cos^2{\gamma_0}+DK}\over{\cos^2{\gamma}+DK}\right)^{1/2}\;.$$ (52)

In case of a plane reflector, the curvature $K$ is zero, and equation (52) coincides with (43). In the general case can be rewritten as

$$A_n={{c\,\cos{\gamma}}\over{\tau_n\,\sqrt{\cos^2{\gamma}+DK}}}\;,$$ (53)

where $c$ is constant along each time ray (it may vary with the reflection point location on the reflector but not with the offset). We should compare equation (53) with the known expression for the reflection wave amplitude of the leading ray series term in 2.5-D media (Bleistein et al., 2001):

$$A={{C_R(\gamma) \Psi}\over G}\;,$$ (54)

where $C_R$ stands for the angle-dependent reflection coefficient, $G$ is the geometric spreading

$$G=v \tau {\sqrt{\cos^2{\gamma}+DK}\over \cos{\gamma}}\;,$$ (55)

and $\Psi$ includes other possible factors (such as the source directivity) that we can either correct or neglect in the preliminary processing. It is evident that the curvature dependence of the amplitude transformation (53) coincides completely with the true geometric spreading factor (55) and that the angle dependence of the reflection coefficient is not accounted for the offset continuation process. If the wavelet shape of the reflected wave on seismic sections [$R_n$ in equation (3)] is described by the delta function, then, as follows from the known properties of this function,

$$A\,\delta\left(t-\tau(y,h)\right)=\left\vert{{dt_n} \over {d... ...ht) = {t \over t_n}\,A\,\delta\left(t_n-\tau_n(y,h)\right)\;,$$ (56)

which leads to the equality

$$A_n=A\,{t \over t_n}\;.$$ (57)

Combining equation (57) with equations (54) and (53) allows us to evaluate the amplitude after continuation from some initial offset $h_0$ to another offset $h_1$, as follows:

$$A_1={{C_R(\gamma_0) \Psi_0}\over G_1}\;.$$ (58)

According to equation (58), the OC process described by equation (1) is amplitude-preserving in the sense that corresponds to the definition of Born DMO (Bleistein, 1990; Liner, 1991). This means that the geometric spreading factor from the initial amplitudes is transformed to the true geometric spreading on the continued section, while the reflection coefficient stays the same. This remarkable dynamic property allows AVO (amplitude versus offset) analysis to be performed by a dynamic comparison between true constant-offset sections and the sections transformed by OC from different offsets. With a simple trick, the offset coordinate is transferred to the reflection angles for the AVO analysis. As follows from (47) and (9),

$${\tau_n^2 \over \tau\,\tau_0}=\cos{\gamma}\;.$$ (59)

If we include the ${t_n^2 \over t\,t_0}$ factor in the DMO operator (continuation to zero offset) and divide the result by the DMO section obtained without this factor, the resultant amplitude of the reflected events will be directly proportional to $\cos{\gamma}$, where the reflection angle $\gamma$ corresponds to the initial offset. Of course, this conclusion is rigorously valid for constant-velocity 2.5-D media only.

Black et al. (1993) suggest a definition of true-amplitude DMO different from that of Born DMO. The difference consists of two important components:

1. True-amplitude DMO addresses preserving the peak amplitude of the image wavelet instead of preserving its spectral density. In the terms of this paper, the peak amplitude corresponds to the pre-NMO amplitude $A$ from formula (54) instead of corresponding to the spectral density amplitude $A_n$. A simple correction factor $t \over t_n$ would help us take the difference between the two amplitudes into account. Multiplication by $t \over t_n$ can be easily done at the NMO stage.
2. Seismic sections are multiplied by time to correct for the geometric spreading factor prior to DMO (or, in our case, offset continuation) processing.

As follows from (55), multiplication by $t$ is a valid geometric spreading correction for plane reflectors only. It is the amplitude-preserving offset continuation based on the OC equation (1) that is able to correct for the curvature-dependent factor in the amplitude. To take into account the second aspect of Black's definition, we can consider the modified field $\hat{P}$ such that

$$\hat{P}\left(y,h,t_n\right)=t\,P\left(y,h,t_n\right)\;.$$ (60)

Substituting (60) into the OC equation (1) transforms the latter to the form

$$h \, \left( {\partial^2 \hat{P} \over \partial y^2} - {\part... ..._n \, \partial h}}\, -{\partial \hat{P} \over \partial h}\; .$$ (61)

Equations (61) and (1) differ only with respect to the first-order damping term $\partial \hat{P} \over \partial h$. This term affects the amplitude behavior but not the traveltimes, since the eikonal-type equation (4) depends on the second-order terms only. Offset continuation operators based on (61) conform to Black's definition of true-amplitude processing.

Fomel and Bleistein (2001) describe an alternative approach to confirming the kinematic and amplitude validity of the offset continuation equation. Applying equation (1) directly on the Kirchhoff model of prestack seismic data shows that the equation is satisfied to the same asymptotic order of accuracy as the Kirchhoff modeling approximation (Bleistein, 1984; Haddon and Buchen, 1981).

Theory of differential offset continuation