Numerical examples

Figure  presents a simple flat-layer model. Figures  and  show the result of modeling the two-way travel time with the conventional normal moveout equation, and with the non-hyperbolic moveout equation, using an initial velocity of 1100 m/s, a velocity gradient of 125 $\mbox{s}^{-1}$ and a constant value of $\gamma=2$.

Both results (Figures  and ) present a hyperbolic moveout at near offset or small offset-to-depth ratio. This result resembles the well known theoretical presentation of Tessmer and Behle (1988). However, the non-hyperbolic moveout is dominant for large offsets and shallow depths, as can be observed in Figure .

 

mod5 Figure 1 Reflectivity model.

 

mod5NMO Figure 2 Modeled data, using the non-hyperbolic equation with a constant value of $\gamma=2$.

 

mod5NMO1 Figure 3 Modeled data, using the the traditional hyperbolic equation.

I also apply the non-hyperbolic moveout equation to a PS CMP gather from the Alba dataset acquired on the Alba oil field in the North sea. The data set is a multicomponent 3-D Ocean Bottom Seismic experiment. Figure  shows the original CMP gather before (left), after (center) non-hyperbolic moveout, and after traditional hyperbolic moveout (right). I use an initial velocity value of 1100 m/s, and a velocity gradient of 125 $\mbox{s}^{-1}$ and a constant $\gamma=2.0$. Note that even though the events are not totally corrected, the non-hyperbolic correction gives a better result for shallow events at large offsets. These events are flatter after the non-hyperbolic moveout correction than with the hyperbolic moveout correction.

 

cmp_comp
Figure 4 PS CMP gather from the Alba dataset, original gather (left), after non-hyperbolic moveout (center), and after traditional hyperbolic moveout (right). I performed non-hyperbolic moveout and traditional moveout both with an initial velocity of 1100 m/s, a velocity gradient of 125 $\mbox{s}^{-1}$ with $\gamma=2.0$ for the non-hyperbolic case.