RANGE OF VALIDITY FOR BOLONDI'S OC EQUATION

From the OC characteristic equation (5) 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 (reflection traveltime has to be an even function of offset). Neglecting $\partial \tau_n \over \partial h$ in (5) leads to the characteristic equation  

$$h \, {\left( \partial \tau_n \over \partial y \right)}^2 = \, - \, \tau_n \, {\partial \tau_n \over \partial h}\;,$$ (50)

which corresponds to the approximate OC equation (2) of Bolondi et al. 1982. Comparing (A-1) and (5), note that approximation (A-1) is valid only if  

$${\left( \partial \tau_n \over \partial h \right)}^2 \, \ll\, {\left( \partial \tau_n \over \partial y \right)}^2 \,\,\,.$$ (51)

To find the geometric constraints implied by inequality (A-2), we can express the traveltime derivatives in geometric terms. As follows from expressions (11) and (12),  

$${{\partial \tau} \over {\partial x}} \,=\,{{\partial \tau} \... ...ial r}} \,=\, { {2 \sin{\alpha} \cos{\gamma}} \over {v}}\,\,\,;$$ (52)

 

$${{\partial \tau} \over {\partial h}} \,=\,{{\partial \tau} \... ...al s}} \,=\, { {2 \cos{\alpha} \sin{\gamma}} \over {v}} \,\,\,.$$ (53)

Expression (10) allows transforming (A-3) and (A-4) to the form  

$$\tau_n \, {{\partial \tau_n} \over {\partial y}} = \tau \, {... ...\,{{\sin{\alpha} \cos{\alpha} \cot{\gamma}} \over {v^2}}\,\,\,;$$ (54)

 

$$\tau_n \, {{\partial \tau_n} \over {\partial h}} = \tau \, {... ...\over {v^2}} \,=\,-\, 4h\,{{\sin^2{\alpha}} \over {v^2}}\,\,\,.$$ (55)

Without loss of generality, we can assume $\alpha$ to be positive. Consider a plane tangent to the true reflector at the reflection point (Figure A-1). The traveltime of the wave, reflected from the plane, has the well-known explicit expression  

$$\tau\,=\,{2 \over v}\,\sqrt{L^2+h^2\,\cos^2{\alpha}}\,\,\,,$$ (56)

where L is the length of the normal ray from the midpoint. As follows from combining (A-7) and (10),  

$${\cos{\alpha} \cot{\gamma}} \,=\, {L \over h} \,\,\,.$$ (57)

We can then combine equalities (A-8), (A-5), (A-6) (A-3), and (A-4) to transform inequality (A-2) to the form  

$$h \ll {L \over {\sin{\alpha}}} \,=\, z\, \cot{\alpha}\,\,,$$ (58)

where z is the depth of the plane reflector under the midpoint. The proven inequality (A-9) coincides with (3) in the main text.

 

ocobol Figure 4 Reflection rays and tangent to the reflector in a constant velocity medium (a scheme).