DERIVING THE KINEMATIC EQUATIONS

The main goal of this appendix is to derive the partial differential equation describing the image surface in a depth-midpoint-offset-velocity space.

 

vlcray Figure 11 Reflection rays in a constant velocity medium (a scheme).

The derivation starts with observing a simple geometry of reflection in a constant-velocity medium, shown in Figure 11. The well-known equations for the apparent slowness  

$${{\partial t} \over {\partial s}} \,=\, { {\sin{\alpha_1}} \over {v}}\;,$$ (64)

 

$${{\partial t} \over {\partial r}} \,=\, {{\sin{\alpha_2}} \over {v}}$$ (65)

relate the first-order traveltime derivatives for the reflected waves to the emergency angles of the incident and reflected rays. Here s stands for the source location at the surface, r is the receiver location, t is the reflection traveltime, v is the constant velocity, and $\alpha_1$ and $\alpha_2$ are the angles shown in Figure 11. Considering the traveltime derivative with respect to the depth of the observation surface z, we can see that the contributions of the two branches of the reflected ray, added together, form the equation  

$$- {{\partial t} \over {\partial z}} \,=\, {{\cos{\alpha_1}} \over {v}} + {{\cos{\alpha_2}} \over {v}}\;.$$ (66)

It is worth mentioning that the elimination of angles from equations (64), (65), and (66) leads to the famous double-square-root equation,  

$$- v\,{{\partial t} \over {\partial z}} \,=\, \sqrt{1 - v^2\,... ...t{1 - v^2\,\left({{\partial t} \over {\partial r}}\right)^2}\;,$$ (67)

published in the Russian literature by Belonosova and Alekseev 1967 and commonly used in the form of a pseudo-differential dispersion relation Claerbout (1985); Clayton (1978) for prestack migration Popovici (1995); Yilmaz (1979). Considered locally, equation (67) is independent of the constant velocity assumption and enables prestack downward continuation of reflected waves in heterogeneous media.

Introducing midpoint coordinate $x = {{s+ r} \over 2}$ and half-offset $h = {{r - s} \over 2}$, we can apply the chain rule and elementary trigonometric equalities to formulas (64) and (65) and transform these formulas to  

$${{\partial t} \over {\partial x}} \,=\, {{\partial t} \over... ...rtial r}} \,=\, { {2 \sin{\alpha}\,\cos{\gamma}} \over {v}}\;,$$ (68)

 

$${{\partial t} \over {\partial h}} \,=\, {{\partial t} \over ... ...tial s}} \,=\, { {2 \cos{\alpha}\,\sin{\gamma}} \over {v}} \;,$$ (69)

where $\alpha = {{\alpha_1 + \alpha_2} \over 2}$ is the dip angle, and $\gamma = {{\alpha_2 - \alpha_1} \over 2}$ is the reflection angle Claerbout (1985); Clayton (1978). Equation (66) transforms analogously to  

$$- {{\partial t} \over {\partial z}} \,=\, {{2 \cos{\alpha} \cos{\gamma}} \over {v}}\;.$$ (70)

This form of equation (66) is used to describe the stretching factor of the waveform distortion in depth migration Tygel et al. (1994).

Dividing (68) and (69) by (70), we obtain  

$${{\partial z} \over {\partial x}} \,=\, - \tan{\alpha}\;,$$ (71)

 

$${{\partial z} \over {\partial h}} \,=\, - \tan{\gamma}\;.$$ (72)

Substituting formulas (71) and (72) into equation (70) yields yet another form of the double-square-root equation:  

$$- {{\partial t} \over {\partial z}} \,=\, {2 \over {v}}\, \s... ...\, \sqrt{1 + \left({\partial z} \over {\partial h}\right)^2}\;,$$ (73)

which is analogous to the dispersion relationship of Stolt prestack migration Stolt (1978).

The law of sines in the triangle formed by the incident and reflected ray leads to the explicit relationship between the traveltime and the offset:  

$$v\,t = 2\,h\, {{\cos{\alpha_1}+ \cos{\alpha_2}} \over \sin{\... ...\alpha_1\right)}} = 2\,h\,{\cos{\alpha} \over \sin{\gamma}} \;.$$ (74)

The combination of formulas (74), (68), and (69) forms the basic kinematic equation of the offset continuation theory Fomel (1995):  

$${{\partial t} \over {\partial h}} \, \left(t^2 + {{4\,h^2} \... ...2\,- \left({{\partial t} \over {\partial x}}\right)^2\right)\;.$$ (75)

Differentiating (74) with respect to the velocity v yields  

$$- v^2\,{{\partial t} \over {\partial v}} = 2\,h\,{\cos{\alpha} \over \sin{\gamma}}\;.$$ (76)

Finally, dividing (76) by (70), we get  

$$v\,{{\partial z} \over {\partial v}} = {h \over {\cos{\gamma}\,\sin{\gamma}}}\;.$$ (77)

Equation (77) can be written in a variety of ways with the help of an explicit geometric relationship between the half-offset h and the depth z,  

$$h = z\, {{\sin{\gamma}\,\cos{\gamma}} \over {\cos^2{\alpha}-\sin^2{\gamma}}}\;,$$ (78)

which follows directly from the trigonometry of the triangle in Figure 11 Fomel (1995). For example, equation (77) can be transformed to the form obtained recently by Liu and Bleistein 1995:  

$$v\,{{\partial z} \over {\partial v}} = {z \over{\cos^2{\alp... ...\sin^2{\gamma}}} = {z \over{\cos{\alpha_1}\,\cos{\alpha_2}}}\;.$$ (79)

In order to separate different factors contributing to the velocity continuation process, we can transform this equation to the form  

$$= z\,\left(1 + \left({{\partial z} \over {\partial x}}\right... ...^2\, \left({{\partial z} \over {\partial h}}\right)^2\right)\;.$$ (80)

Rewritten in terms of the vertical traveltime, it further transforms to equation (1) in the main text. Yet another form of the kinematic velocity continuation equation follows from eliminating the reflection angle $\gamma$ from equations (77) and (78). The resultant expression takes the following form:  

$$v\,{{\partial z} \over {\partial v}} = {{2\,(z^2 + h^2)} \o... ... + {{2\,h^2} \over {\sqrt{z^2 + h^2 \sin^2{2\,\alpha}} + z}}\;.$$ (81)

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