Comparison with Bolondi's OC equation

Equation () and the previously published OC equation Bolondi et al. (1982) 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 (), 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 () leads to the characteristic equation  

$$h \, {\left( \partial \tau_n \over \partial y \right)}^2 = \, - \, \tau_n \, {\partial \tau_n \over \partial h}\;,$$ (152)

which corresponds to the approximate OC equation of Bolondi et al. (1982). The approximate equation has the form  

$$h \, {\partial^2 P \over \partial y^2} \, = \, t_n \, {\partial^2 P \over {\partial t_n \, \partial h}}\;.$$ (153)

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

$${\left( \partial \tau_n \over \partial h \right)}^2 \, \ll\, {\left( \partial \tau_n \over \partial y \right)}^2 \,\,\,.$$ (154)

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

$$\begin{eqnarray} {{\partial \tau} \over {\partial x}} & = & {{\partial \tau} \ov... ... {\partial s}} \,=\, { {2 \cos{\alpha} \sin{\gamma}} \over {v}}\;.\end{eqnarray}$$ (155) (156)

Expression () allows transforming equations () and () to the form

$$\begin{eqnarray} \tau_n \, {{\partial \tau_n} \over {\partial y}} & = & \tau \, ... ...h} \over {v^2}} \,=\,-\, 4h\,{{\sin^2{\alpha}} \over {v^2}}\,\,\,.\end{eqnarray}$$ (157) (158)

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

$$\tau\,=\,{2 \over v}\,\sqrt{L^2+h^2\,\cos^2{\alpha}}\,\,\,,$$ (159)

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

$${\cos{\alpha} \cot{\gamma}} \,=\, {L \over h} \,\,\,.$$ (160)

We can then combine equalities (), (), and () to transform inequality () to the form  

$$h \ll {L \over {\sin{\alpha}}} \,=\, z\, \cot{\alpha}\,\,,$$ (161)

where z is the depth of the plane reflector under the midpoint. For example, for a dip of 45 degrees, equation () 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).