PDE

The function inside the integral in equation (13) satisfies the partial differential equation  

$${2 \over v^2} {{\partial^2 p } \over {\partial t^2_0}}= {{1 ... ...er v^2} p_{tt}+p_{yy}-p_{hh})^2-{4 \over v^2} p_{tt}p_{yy}}}}.$$ (15)

Notice that for flat layers (p_yy=0) the equation is reduced to

$${1 \over v^2}{{\partial^2 p } \over {\partial t^2_0}}= {{1 \over v^2} p_{tt}-p_{hh}}$$

which is solved by doing NMO as time migration in time and offset coordinates. For the zero-offset case (p_hh=0), the equation becomes an identity. To verify the partial differential equation, I coded equation (13) for a constant velocity case and for several depth varying velocity functions. Equation (13) can be used to extrapolate the initial wavefield in t₀ using a v_rms velocity for each zero-offset time value. The results are shown in the following synthetic cases.

Figure shows the velocity and the travel-time map used to model the constant-offset sections. Since for this first case the velocity was constant the information in this figure is trivial, however I kept it just to be consistent with the layout of the other figures. The number of offsets used is 32, with a maximum half-offset h=930 meters. In all cases the structural model was a diffractor at a depth of 1250 m. Figure shows a comparison between the first zero-offset panel obtained from modeling and the output of the MZO algorithm applied to the 32 constant-offset sections. The kinematics of the initial zero-offset section in Figure a coincide with the output of the MZO algorithm in Figure b. The artifacts in Figure b are due to the Fourier domain implementation of the algorithm. The next figures use a depth variable velocity to model the constant-offset sections and the corresponding v_rms to migrate the data to zero-offset.

Figure a shows a root-mean-square velocity with a slow increase, corresponding to an interval velocity with several linearly increasing velocity layers. The travel-time map model in Figure b can be used to identify the boundaries of sharper velocity changes. Comparing Figures a and b we notice the excellent kinematic match between the zero-offset model and the output of the migration to zero-offset algorithm using the depth variable velocity.

Figure shows a velocity model with an increasing jump, while in Figure , the velocity has a decreasing jump. Again in both cases, comparing Figures a with b and Figures a with b, the kinematics of the initial zero-offset section and the output of the migration to zero-offset are extremely similar even though the velocities used show quite large variations.

 

Fig1
Figure 1 The constant velocity medium case.
a. V_rms velocity used.
b. Zero-offset travel-time map used for modeling.

 

Fig2
Figure 2 Comparison between zero-offset input and constant velocity MZO output.
a. First panel in the input cube: zero-offset.
b. MZO output after stacking.

 

Fig3
Figure 3 The depth variable velocity case.
a. V_rms velocity used.
b. Zero-offset travel-time map used for modeling.

 

Fig4
Figure 4 Comparison between zero-offset input and v(z) velocity MZO output.
a. First panel in the input cube: zero-offset.
b. MZO output after stacking.

 

Fig5
Figure 5 The depth variable velocity case.
a. V_rms velocity used.
b. Zero-offset travel-time map used for modeling.

 

Fig6
Figure 6 Comparison between zero-offset input and v(z) velocity MZO output.
a. First panel in the input cube: zero-offset.
b. MZO output after stacking.

 

Fig7
Figure 7 The depth variable velocity case.
a. V_rms velocity used.
b. Zero-offset travel-time map used for modeling.

 

Fig8
Figure 8 Comparison between zero-offset input and v(z) velocity MZO output.
a. First panel in the input cube: zero-offset.
b. MZO output after stacking.