Theory of differential offset continuation

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 (4). The characteristics of equation (4) [ bi-characteristics with respect to equation (1)] 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 (4) :

$${{{dy} \over {dt_n}} = - {{2 h Y} \over {t_n H}}}\;,\; {{{dY} \over {dt_n}} = {Y \over t_n}}\;,$$

$${{{dh} \over {dt_n}} = {-{1 \over H}+{{2 h} \over t_n}}}\;,\; {{{dH} \over {dt_n}} = {{Y^2} \over {t_n H}}}\;,$$ (23)

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 (4) takes the form

$$h\, (Y^2-H^2) = -\, t_n H$$ (24)

and serves as an additional constraint for the definition of time rays. System (23) 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:

$$y(t_n) = C_1-C_2\,t_n^2 \; ; \;h(t_n)=t_n \sqrt{C_2^2 t_n^2 + C_3}\;;$$ (25)

$$Y(t_n) = {{C_2\,t_n}\over C_3}\; ; \;H(t_n)={h \over {C_3\,t_n}}$$ (26)

and $C_1$, $C_2$, and $C_3$ are independent coefficients, constant along each time ray. To find the values of these coefficients, we can pose an initial-value problem for the system of differential equations (23). 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. 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): $t_0\left(y_0\right)=\tau_n(y;0), \left. {\partial \tau_n \over \partial h} \right\vert _{h=0}=0\,.$ Applying the initial-value conditions to the general solution (26) generates the following expressions for the ray invariants:

$$C_1 = y+h\,{Y \over H}=y_0-{t_0\left(y_0\right) \over t_0'\left(y_0\rig... ...\over {\tau_n^2\,H}}= -{1 \over t_0\left(y_0\right)\,t_0'\left(y_0\right)}\;;\;$$

$$C_3 = {h \over {\tau_n\,H}}= -{1 \over \left(t_0'\left(y_0\right)\right)^2}\;,$$ (27)

where $t_0'\left(y_0\right)$ denotes the derivative $\frac{d\,t_0}{d\,y_0}$. Finally, substituting equations (27) into (26), we obtain an explicit parametric form of the ray trajectories:

$$y_1\left(t_1\right) = \displaystyle{y+{{h\,Y} \over {t_n^2\,H}}\,\left(t_n^2-t_1^2\righ... ...2-t_0^2\left(y_0\right)} \over {t_0\left(y_0\right)\,t_0'\left(y_0\right)}}}\;;$$ (28)

$$h_1^2\left(t_1\right) = \displaystyle{{{h\,t_1^2} \over {t_n^3\,H}}\, \left(t_n^2+t_1^2\,... ...0\right)} \over {\left(t_0\left(y_0\right)\,t_0'\left(y_0\right)\right)^2}}}\;.$$ (29)

Here $y_1$, $h_1$, and $t_1$ are the coordinates of the continued seismic section. Equations (28) 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

$$t_0(x) = \displaystyle{t_v \sec{\alpha}= t_v(x)\,\sqrt{1+u^2\left(t_v'(x)\right)^2}}\;,$$ (30)

$$y_0(x) = \displaystyle{x+u t_v\tan{\alpha} =x+u^2\,t_v(x)t_v'(x)}\;,$$ (31)

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

$${{dt_0}\over{dy_0}}={{t_0'(x)}\over{y_0'(x)}}={{\sin{\alpha}}\over u}= {{t_v'(x)} \over \sqrt{1+u^2\left(t_v'(x)\right)^2}}$$ (32)

and substituting equations (30) and (31) into (28) and (29), we get

$$y_1\left(t_1\right) = \displaystyle{x+{{t_1^2-t_v^2\left(x\right)} \over {t_v\left(x\right)\,t_v'\left(x\right)}}}\;;$$ (33)

$$u^2 t^2\left(t_1\right) = \displaystyle{t_1^2\,{{t_1^2-t_v^2\left(x\right)} \over {\left(t_v\left(x\right)\,t_v'\left(x\right)\right)^2}}}\;,$$ (34)

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.

Theory of differential offset continuation