Zhang's improved DMO

Zhang (1988) observed that in Hale's DMO the reflection point in the nonzero-offset case does not coincide with the reflection point in the zero-offset case. He shows a new formula for DMO which takes into account not only a time correction but also a mid-point correction.

 

ZhangDMO
Figure 13 Geometry for a dipping reflector in a constant velocity medium. Notice that the reflection point for the nonzero-offset ray R is the same as the one for the zero-offset ray JR. The dipping angle is $\theta$.

The $NMO \cdot DMO$ correction in constant velocity media as defined in Part 1 is written

$$\left \{ \begin{array} {l} \Delta y_0 = \displaystyle{ {2h^2... ... {t_h^2-4h^2{{\cos^2\theta} \over v^2} }. }\end{array} \right.$$

We can isolate the NMO transformation from equation (12) which is only a time-shift

$$t_n^2={t_h^2-{4h^2 \over v^2}}$$

and write just for the DMO operator  

$$\left \{ \begin{array} {l} y_0 = \displaystyle{y- {{2h^2 \si... ...t_n^2+4h^2{{\sin^2\theta} \over v^2} } } }.\end{array} \right.$$ (20)

To clarify the sign convention for the spatial coordinate in equation (20) note that the y axis is increasing to the left, and the angles are positive if they dip toward the right and negative if they dip toward the left. In Figure the angle $\theta$ is negative. This can be also derived from equation (16)

$${{dt_0} \over {dy_0}} = {{ 2 \sin \theta} \over v}$$

where the sign of dy₀ determines the sign of the angle.

The next steps follow exactly Hale's reasoning by defining another field p₀(t₀,y₀,h), with the addition that not only the time variable is changed but also the common-midpoint variable. This accounts for the fact that the DMO transformation defined by Zhang moves the nonzero-offset reflection point to a zero-offset reflection point. Stacking after this transformation produces true common depth point gathers. In the transformation defined by Hale, the reflection point for the nonzero-offset is different from the reflection point in the zero-offset case.

The Fourier transform of the new field is  

$$P_0(\omega_0,k_{y_0},h)={\int_{t_0}dt_0\int_{y_0}dy_0 \: {e^{i (\omega_{0} t_0- k_{y_0} y_0) }}p_0(t_0,y_0,h)}.$$ (21)

We need to replace the variables t₀ and y₀ in equation (21) with the known variables t_n and y. Fortunately it is not necessary to express explicitly t_n=t_n(t₀,y₀) and y=y(t₀,y₀), though we assume the respective dependencies. From equation (20) we can express the differentials of the new variables  

$$\left \{ \begin{array} {l} dy_0 = \displaystyle{dy} \\ \\ dt... ...n^2\theta} \over v^2})^{3 \over 2} }}dt_n }\end{array} \right.$$ (22)

and introduce them in equation (21). After using equation (16) to express

$${{2 \sin \theta} \over v}={{k_{y_0}} \over {\omega_0}}$$

the phase becomes

$$\begin{array} {lcl} \Phi & = & \displaystyle{ {\omega_{0} t_... ...t{t_n^2+h^2{k_{y_0}^2 \over \omega_0^2}}-k_{y_0} y}\end{array}$$

which should be noted is the same phase as in Hale's equation (19).

Equation (21) becomes  

$$P_0(\omega_0,k_{y_0},h)={\int_{t_n}dt_n\int_{y}dy {{t_n(t_n^... ...0 \sqrt{t_n^2+h^2 {k_{y_0}^2 \over \omega_0^2}}- k_{y_0} y)}}}.$$ (23)

Comparing equation (23) with Hale's equation (19) we notice that surprisingly the phase term is the same and the only difference is in the amplitude term. The two Jacobians are:

$$\begin{array} {lcl} J_H & = & {t_{n} \over {\sqrt{t_n^2+h^2 ... ...^2+h^2 {k_{y_0}^2 \over \omega_0^2})^{3 \over 2}}} .\end{array}$$

The ratio between the two Jacobians is

$${J_Z \over J_H} = {{t_n^2+2h^2{k_{y_0}^2 \over \omega_0^2}} \over {t_n^2+ h^2{k_{y_0}^2 \over \omega_0^2}}}.$$ (24)