Reflection angle formula

This appendix shows the derivation of freq, which serves as the basis of the frequency-domain angle gather construction (adcig).

In the process of downward continuation, the wavefield appears as a function of four variables: time t, depth z, source lateral position s, and receiver lateral position r. Both the source and receiver assume positions at depth z, where the wavefield is continued (24). It is often convenient to replace the variables s and r with the midpoint position x = (s+r)/2 and the half-offset h = (r-s)/2.

Assuming that the reflection event in the continued wavefield is described by the function t = t(z,s,r), we find from the Snell's law the following derivatives:  

$${{\partial t} \over {\partial s}} \,=\, { {\sin{(\alpha-\gamma)}} \over {v}}\;,$$ (85)

 

$${{\partial t} \over {\partial r}} \,=\, {{\sin{(\alpha+\gamma)}} \over {v}}\;,$$ (86)

where v is the wave velocity, $\alpha$ is the dip angle, and $\gamma$ is the reflection angle (local). The traveltime derivative with respect to the depth of the observation surface z has contributions from the two branches of the reflected ray, as follows:  

$${{\partial t} \over {\partial z}} \,=\, {{\cos{(\alpha-\gamma)}} \over {v}} + {{\cos{(\alpha+\gamma)}} \over {v}}\;.$$ (87)

snell3 corresponds to the well-known double-square-root equation (24). This equation simply reflects the fact that the traveltime increases with increasing depth of the reflector.

Transforming snell1-snell3 to the midpoint and half-offset coordinates, we obtain  

$${{\partial t} \over {\partial x}} \,=\, {{\partial t} \over... ...rtial r}} \,=\, { {2 \sin{\alpha}\,\cos{\gamma}} \over {v}}\;,$$ (88)

 

$${{\partial t} \over {\partial h}} \,=\, {{\partial t} \over ... ...tial s}} \,=\, { {2 \cos{\alpha}\,\sin{\gamma}} \over {v}} \;,$$ (89)

 

$${{\partial t} \over {\partial z}} \,=\, {{2 \cos{\alpha} \cos{\gamma}} \over {v}}\;.$$ (90)

At a fixed image location x, we can transform the derivatives of t(z,x,h) to the derivatives of z(t,x,h) by applying the implicit function theorem. Using snells2-snells3, we obtain  

$${{\partial z} \over {\partial h}} \,=\,- {{\partial t} \over... ...h}} / {{\partial t} \over {\partial z}} \,=\,- \tan{\gamma}\;.$$ (91)

snellz2 corresponds to freq in the adcig. It is important to note that this equation is only suitable for angle gathers constructed on images obtained by wavefield continuation methods (local), when h does not represent the surface offset, but half the distance between the downward continued sources and receivers.

In deriving snellz2, I assumed that the velocity v does not change laterally between the source and receiver positions. While this may not be true in general, snellz2 is always satisfied in the vicinity of zero half-offset after migration (h=0). Therefore, this formula is applicable near the focusing points of the downward-continued wavefield.