Comparison with Bolondi's OC equation

Equation (1) and the previously published OC equation () differ only with respect to the single term $\partial^2 P \over {\partial h^2}$. However, this difference is substantial.

From the offset continuation characteristic equation (4), we can conclude that the first-order traveltime derivative with respect to offset decreases with a decrease of the offset. At zero offset the derivative equals zero, as predicted by the principle of reciprocity (the reflection traveltime has to be an even function of offset). Neglecting ${\partial \tau_n} \over {\partial h}$ in (4) leads to the characteristic equation  

$$h \, {\left( \partial \tau_n \over \partial y \right)}^2 = \, - \, \tau_n \, {\partial \tau_n \over \partial h}\;,$$ (13)

which corresponds to the approximate OC equation of (). The approximate equation has the form  

$$h \, {\partial^2 P \over \partial y^2} \, = \, t_n \, {\partial^2 P \over {\partial t_n \, \partial h}}\;.$$ (14)

Comparing (13) and (4), note that approximation (13) is valid only if  

$${\left( \partial \tau_n \over \partial h \right)}^2 \, \ll\, {\left( \partial \tau_n \over \partial y \right)}^2 \,\,\,.$$ (15)

To find the geometric constraints implied by inequality (15), we can express the traveltime derivatives in geometric terms. As follows from expressions (10) and (11),

$$\begin{eqnarray} {{\partial \tau} \over {\partial x}} & = & {{\partial \tau} \ov... ... {\partial s}} \,=\, { {2 \cos{\alpha} \sin{\gamma}} \over {v}}\;.\end{eqnarray}$$ (16) (17)

Expression (9) allows transforming equations (16) and (17) to the form

$$\begin{eqnarray} \tau_n \, {{\partial \tau_n} \over {\partial y}} & = & \tau \, ... ...h} \over {v^2}} \,=\,-\, 4h\,{{\sin^2{\alpha}} \over {v^2}}\,\,\,.\end{eqnarray}$$ (18) (19)

Without loss of generality, we can assume $\alpha$ to be positive. Consider a plane tangent to a true reflector at the reflection point (Figure 3). The traveltime of a wave, reflected from the plane, has the well-known explicit expression  

$$\tau\,=\,{2 \over v}\,\sqrt{L^2+h^2\,\cos^2{\alpha}}\,\,\,,$$ (20)

where L is the length of the normal ray from the midpoint. As follows from combining (20) and (9),  

$${\cos{\alpha} \cot{\gamma}} \,=\, {L \over h} \,\,\,.$$ (21)

We can then combine equalities (21), (18), and (19) to transform inequality (15) to the form  

$$h \ll {L \over {\sin{\alpha}}} \,=\, z\, \cot{\alpha}\,\,,$$ (22)

where z is the depth of the plane reflector under the midpoint. For example, for a dip of 45 degrees, equation (14) is satisfied only for offsets that are much smaller than the depth.

 

ocobol Figure 3 Reflection rays and tangent to the reflector in a constant velocity medium (a scheme).