Kinematics of Residual DMO

The partial differential equation for kinematic residual DMO is the third term in (1):  

$${{\partial \tau} \over {\partial v}} = - {{h^2 v} \over {\t... ...ght)^2\, \left({{\partial \tau} \over {\partial h}}\right)^2\;.$$ (30)

It is more convenient to consider the residual dip-moveout process coupled with residual normal moveout. Etgen 1990 describes this procedure as the cascade of inverse DMO with the initial velocity v₀, residual NMO, and DMO with the updated velocity v₁. The kinematic equation for residual NMO+DMO is the sum of the two terms in (1):  

$${{\partial \tau} \over {\partial v}} = {{h^2} \over {v^3\,\... ..., \left({{\partial \tau} \over {\partial h}}\right)^2\right)\;.$$ (31)

If the boundary data for equation (31) are on a common-offset gather, it is appropriate to rewrite this equation purely in terms of the midpoint derivative ${{\partial \tau} \over {\partial x}}$, eliminating the offset-derivative term ${{\partial \tau} \over {\partial h}}$. The resultant expression, derived in Appendix A, has the form  

$$v^3\,{{\partial \tau} \over {\partial v}} = {{2\,h^2} \over... ...left(v,{{\partial \tau} \over {\partial x}}\right)} + \tau}}\;,$$ (32)

where  

$$Q(v,\tau_x) = {{\tau_x^2} \over {\left(1 + v^2\,\tau_x^2\right)^2}}\;.$$ (33)

The direct solution of equation (32) is nontrivial. A simpler way to obtain this solution is to decompose residual NMO+DMO into three steps and to evaluate their contributions separately. Let the initial data be the zero-offset reflection event $\tau_0(x_0)$. The first step of the residual NMO+DMO is the inverse DMO operator. One can evaluate the effect of this operator by means of the offset continuation concept Fomel (1995). According to this concept, each point of the input traveltime curve $\tau_0(x_0)$travels with the change of the offset from zero to h along a special trajectory, which I call a time ray. Time rays are parabolic curves of the form  

$$x\left(\tau\right) = x_0+{{\tau^2-\tau_0^2\left(x_0\right)} \over {\tau_0\left(x_0\right)\,\tau_0'\left(x_0\right)}}\;,$$ (34)

with the final points constrained by the equation  

$$h^2 = \tau^2\,{{\tau^2-\tau_0^2\left(x_0\right)} \over {\left(\tau_0\left(x_0\right)\,\tau_0'\left(x_0\right)\right)^2}}\;.$$ (35)

The second step of the cumulative residual NMO+DMO process is the residual normal moveout. According to equation (23), residual NMO is a one-trace operation transforming the traveltime $\tau$ to $\tau_1$ as follows:  

$$\tau_1^2 = \tau^2 + h^2\,s\;,$$ (36)

where  

$$s = {1 \over v_0^2} - {1 \over v_1^2}\;.$$ (37)

The third step is dip moveout corresponding to the new velocity v₁. DMO is the offset continuation from h to zero offset along the redefined time rays Fomel (1995)  

$$x_2\left(\tau_2\right) = x + {{h\,X} \over {\tau_1^2\,H}}\,\left(\tau_1^2-\tau_2^2\right)\;,$$ (38)

where $H = {{\partial \tau_1} \over {\partial h}}$, and $X = {{\partial \tau_1} \over {\partial x}}$. The end points of the time rays (38) are defined by the equation  

$$\tau_2^2 = - \tau_1^2\,{{\tau_1\,H} \over {h\,X^2}}\;.$$ (39)

The partial derivatives of the common-offset traveltimes are constrained by the offset continuation kinematic equation  

$$h\,(H^2 - Y^2) = \tau_1\,H\;,$$ (40)

which is equivalent to equation (75) in Appendix A. Additionally, as follows from equations (36) and the ray invariant equations from Fomel (1995),  

$$\tau_1\,X = \tau\,{{\partial \tau} \over {\partial x}} = {{... ...^2\,\tau_0'\left(x_0\right)} \over {\tau_0\left(x_0\right)}}\;.$$ (41)

Substituting (34), (35), (36), (40), and (41) into equations (38) and (39) and performing the algebraic simplifications, we arrive at the parametric expressions for velocity rays of the residual NMO+DMO process:  

$$\left\{ \begin{array} {rcl} x_2(s) & = & \displaystyle{x_0 +... ...splaystyle{{{\tau_1^2(s)} \over {T_2(s)}}}\;,\end{array}\right.$$ (42)

where the function $T\left(h,\tau_0(x_0),\tau_0'\left(x_0\right)\right)$ is defined by  

$$T\left(h,\tau,\tau_x\right) = {{\tau + \sqrt{\tau^2 + 4\,h^2\,\tau_x^2}} \over 2}\;,$$ (43)

 

$$T_2(s) = \sqrt{T\left(h,\tau_1^2(s),\tau_0'\left(x_0\right)\, T\left(h,\tau_0(x_0),\tau_0'\left(x_0\right)\right)\right)}\;,$$ (44)

and  

$$\tau_1^2(s) = \tau_0\,T + s\,h^2\;.$$ (45)

The last step of the cascade of inverse DMO, residual NMO, and DMO is illustrated in Figure 5. The three plots in the figure show the offset continuation to zero offset of the inverse DMO impulse response shifted by the residual NMO operator. The middle plot corresponds to zero NMO shift, for which the DMO step collapses the wavefront back to a point. Both positive (top plot) and negative (bottom plot) NMO shifts result in the formation of the specific triangular impulse response of the residual NMO+DMO operator. As noticed by Etgen 1990, the size of the ``triangle'' operators dramatically decreases with the time increase. For large times (pseudo-depths) of the initial impulses, the operator collapses to a point corresponding to the pure NMO shift. This fact agrees with the conclusions of the preceding subsection about the comparative importance of the residual DMO term. It is illustrated in Figure 6 with the theoretical impulse response curves, and in Figure 7 with the result of an actual cascade of the inverse DMO, residual NMO, and DMO operators.

 

vlcvoc Figure 5 Kinematic residual NMO+DMO operators constructed by the cascade of inverse DMO, residual NMO, and DMO. The impulse response of inverse DMO is shifted by the residual DMO procedure. Offset continuation back to zero offset forms the impulse response of the residual NMO+DMO operator. Solid lines denote traveltime curves; dashed lines denote the offset continuation trajectories (time rays). Top plot: v1/v0 = 1.2. Middle plot: v1/v0 = 1; the inverse DMO impulse response collapses back to the initial impulse. Bottom plot: v1/v0 = 0.8. The half-offset h in all three plots is 1 km.

 

vlcvcp
Figure 6 Theoretical kinematics of the residual NMO+DMO impulse responses for three impulses. Left plot: the velocity ratio v₁/v₀ is 1.333. Right plot: the velocity ratio v₁/v₀ is 0.833. In both cases the half-offset h is 1 km.

 

vlccps
Figure 7 The result of residual NMO+DMO (cascading inverse DMO, residual NMO, and DMO) for three impulses. Left plot: the velocity ratio v₁/v₀ is 1.333. Right plot: the velocity ratio v₁/v₀ is 0.833. In both cases the half-offset h is 1 km.

Figure 8 illustrates the residual NMO+DMO velocity continuation for two particularly interesting cases. The left plot shows the continuation for a point diffractor. One can see that when the velocity error is large, focusing of the velocity rays forms a specific loop on the zero-offset hyperbola. The right plot illustrates the case of a plane dipping reflector. The image of the reflector shifts both vertically and laterally with the change in NMO velocity.

 

vlcvrd
Figure 8 Kinematic velocity continuation for residual NMO+DMO. Solid lines denote wavefronts: zero-offset traveltime curves; dashed lines denote velocity rays. Left plot: the case of a point diffractor; the velocity ratio v₁/v₀ changes from 0.9 to 1.1. Right plot: the case of a dipping plane reflector; the velocity ratio v₁/v₀ changes from 0.8 to 1.2. In both cases, the half-offset h is 2 km.

The full residual migration operator is the result of cascading residual zero-offset migration and residual NMO+DMO. I illustrate the kinematics of this operator in Figures 9 and 10, which are designed to match Etgen's Figures 2.4 and 2.5 Etgen (1990). A comparison with Figures 3 and 4 shows that including the residual DMO term affects the images of shallow objects (with the depth smaller than the offset h) and complicates the residual migration operator with cusps.

 

vlcve3 Figure 9 Summation paths of prestack residual migration for a series of depth diffractors. Residual slowness v/vd is 1.2; offset h is 1 km. This figure reproduces Etgen's Figure 2.4.

 

vlcve4 Figure 10 Summation paths of prestack residual migration for a series of depth diffractors. Residual slowness v/vd is 0.8; offset h is 1 km. This figure reproduces Etgen's Figure 2.5.