Offset continuation geometry: time rays

To study the laws of traveltime curve transformation in the OC process, it is convenient to apply the method of characteristics Courant (1962) to the eikonal-type equation (). The characteristics of equation () [ bi-characteristics with respect to equation ()] are the trajectories of the high-frequency energy propagation in the imaginary OC process. Following the formal analogy with seismic rays, I call those trajectories time rays, where the word time refers to the fact that the trajectories describe the traveltime transformation Fomel (1994). According to the theory of first-order partial differential equations, time rays are determined by a set of ordinary differential equations (characteristic equations) derived from equation () :

$$\begin{eqnarray} {{{dy} \over {dt_n}} = - {{2 h Y} \over {t_n H}}}\;,\; {{{dY} ... ...\over t_n}}}\;,\; {{{dH} \over {dt_n}} = {{Y^2} \over {t_n H}}}\;,\end{eqnarray}$$ (162)

where Y corresponds to $\partial \tau_n \over \partial y$ along a ray and H corresponds to ${\partial \tau_n} \over {\partial h}$. In this notation, equation () takes the form  

$$h\, (Y^2-H^2) = -\, t_n H$$ (163)

and serves as an additional constraint for the definition of time rays. System () can be solved by standard mathematical methods Tenenbaum and Pollard (1985). Its general solution takes the parametric form, where the time variable t_n is the parameter changing along a time ray:

$$\begin{eqnarray} y(t_n) = C_1-C_2\,t_n^2 \; & ; & \;h(t_n)=t_n \sqrt{C_2^2 t_n^2... ...t_n) = {{C_2\,t_n}\over C_3}\; & ; & \;H(t_n)={h \over {C_3\,t_n}}\end{eqnarray}$$ (164) (165)

and C₁, C₂, and C₃ are independent coefficients, constant along each time ray. To determine the values of these coefficients, we can pose an initial-value (Cauchy) problem for the system of differential equations (). The traveltime curve $\tau_n(y;h)$ for a given common offset h and the first partial derivative ${\partial \tau_n} \over {\partial h}$ along the same constant offset section provide natural initial conditions for the Cauchy problem. A particular case of those conditions is the zero-offset traveltime curve. If the first partial derivative of traveltime with respect to offset is continuous, it vanishes at zero offset according to the reciprocity principle (traveltime must be an even function of the offset):  

$$t_0\left(y_0\right)=\tau_n(y;0), \left. {\partial \tau_n \over \partial h} \right\vert _{h=0}=0\,.$$ (166)

Applying the initial-value conditions to the general solution () generates the following expressions for the ray invariants:

$$\begin{eqnarray} C_1 & = & y+h\,{Y \over H}=y_0-{t_0\left(y_0\right) \over t_0'\... ...er {\tau_n\,H}}= -{1 \over \left(t_0'\left(y_0\right)\right)^2}\;,\end{eqnarray}$$ (167)

where $t_0'\left(y_0\right)$ denotes the derivative $\frac{d\,t_0}{d\,y_0}$. Finally, substituting () into (), we obtain an explicit parametric form of the ray trajectories:  

$$\left\{ \begin{array} {rcl} y_1\left(t_1\right) & = & \displ... ...0\right)\,t_0'\left(y_0\right)\right)^2}}}\;.\end{array}\right.$$ (168)

Here y₁, h₁, and t₁ are the coordinates of the continued seismic section. The first of equations () indicates that the time ray projections to a common-offset section have a parabolic form. Time rays do not exist for $t_0'\left(y_0\right)=0$ (a locally horizontal reflector) because in this case post-NMO offset continuation transform is not required.

The actual parameter that determines a particular time ray is the reflection point location. This important conclusion follows from the known parametric equations  

$$\left\{ \begin{array} {rcl} t_0(x) & = & \displaystyle{t_v \... ...x+u t_v\tan{\alpha} =x+u^2\,t_v(x)t_v'(x)}\;,\end{array}\right.$$ (169)

where x is the reflection point, u is half of the wave velocity (u=v/2), t_v is the vertical time (reflector depth divided by u), and $\alpha$ is the local reflector dip. Taking into account that the derivative of the zero-offset traveltime curve is  

$${{dt_0}\over{dy_0}}={{t_0'(x)}\over{y_0'(x)}}={{\sin{\alpha}}\over u}= {{t_v'(x)} \over \sqrt{1+u^2\left(t_v'(x)\right)^2}}$$ (170)

and substituting () into (), we get  

$$\left\{ \begin{array} {rcl} y_1\left(t_1\right) & = & \displ... ...t(x\right)\,t_v'\left(x\right)\right)^2}}}\;,\end{array}\right.$$ (171)

where $t^2\left(t_1\right)=t_1^2+h_1^2\left(t_1\right)/u^2$.

To visualize the concept of time rays, let us consider some simple analytic examples of its application to geometric analysis of the offset-continuation process.

The simplest and most important example is the case of a plane dipping reflector. Putting the origin of the y axis at the intersection of the reflector plane with the surface, we can express the reflection traveltime after NMO in the form  

$$\tau_n(y,h)=p\,\sqrt{y^2-h^2}\;,$$ (172)

where $p=2\,{ \sin{\alpha} \over v}$, and $\alpha$ is the dip angle. The zero-offset traveltime in this case is a straight line:  

$$t_0\left(y_0\right)=p\,y_0\;.$$ (173)

According to equations (), the time rays in this case are defined by  

$$y_1\left(t_1\right)={t_1^2 \over {p^2\,y_0}}\;;\; h_1^2\left... ...p^2\,y_0^2} \over {p^4\,y_0^2}}\;;\; y_0={{y^2-h^2} \over y}\;.$$ (174)

The geometry of the OC transformation is shown in Figure .

 

ocopln
Figure 4 Transformation of the reflection traveltime curves in the OC process: the case of a plane dipping reflector. Left: Time coordinate before the NMO correction. Right: Time coordinate after NMO. The solid lines indicate traveltime curves at different common-offset sections; the dashed lines indicate time rays.

The second example is the case of a point diffractor (the left side of Figure ). Without loss of generality, the origin of the midpoint axis can be put above the diffraction point. In this case the zero-offset reflection traveltime curve has the well-known hyperbolic form  

$$t_0\left(y_0\right)={\sqrt{z^2+y_0^2} \over u}\;,$$ (175)

where z is the depth of the diffractor and u=v/2 is half of the wave velocity. Time rays are defined according to equations (), as follows:  

$$y_1\left(t_1\right)={{u^2\,t_1^2-z^2} \over y_0}\;;\; u^2\,t... ...\left(t_1\right)= u^2\,t_1^2\,{{u^2\,t_1^2-z^2} \over y_0^2}\;.$$ (176)

 

ococrv
Figure 5 Transformation of the reflection traveltime curves in the OC process. Left: the case of a diffraction point. Right: the case of an elliptic reflector. Solid lines indicate traveltime curves at different common-offset sections, dashed lines indicate time rays.

The third example (the right side of Figure ) is the curious case of a focusing elliptic reflector. Let y be the center of the ellipse and h be half the distance between the foci of the ellipse. If both foci are on the surface, the zero-offset traveltime curve is defined by the so-called ``DMO smile'' Deregowski and Rocca (1981):  

$$t_0\left(y_0\right)={t_n \over h}\,\sqrt{h^2-\left(y-y_0\right)^2}\;,$$ (177)

where $t_n=2\,z/v$, and z is the small semi-axis of the ellipse. The time-ray equations are  

$$y_1\left(t_1\right)=y+{h^2\over {y-y_0}}\,{{t_1^2-t_n^2} \ov... ...r \left(y-y_0\right)^2}\,{{t_1^2-t_n^2} \over t_n^2} \right)\;.$$ (178)

When y₁ coincides with y, and h₁ coincides with h, the source and the receiver are in the foci of the elliptic reflector, and the traveltime curve degenerates to a point t₁=t_n. This remarkable fact is the actual basis of the geometric theory of dip moveout Deregowski and Rocca (1981).