Synthetic example

Figure  displays a simple synthetic model. The model consists of four horizontal layers embedded in a vertical gradient velocity model. The initial P-velocity is 2 km/s with a vertical-gradient of 0.4 $\mbox{s}^{-1}$, and the initial S-velocity is 1 km/s with a vertical gradient of 0.2 $\mbox{s}^{-1}$; therefore, this example has a constant P-to-S velocity ratio of $\gamma=2$.

 

model
Figure 3 Simple synthetic model, The model consists on four horizontal layers with a vertical gradient velocity model,

Figure  shows the results for this experiment. Panel (a) exhibits the synthetic CMP gather. Panel (b) shows the same CMP gather as in the previous panel after hyperbolic NMO, that is using two terms in the total traveltime equation. Panel (c) shows the same CMP gather as in panel (a) after non-hyperbolic NMO, that is using three terms in the total traveltime equation.

I use a Kirchhoff modeling scheme to obtain the synthetic data in panel (a). Observe the change in the polarity along the four reflections at zero offset. This polarity flip is characteristics of converted-wave data, since the P-to-S velocity ratio is constant then the polarity flip happens at zero offset.

Compare the results in panels (b) and (c). The hyperbolic normal moveout result, panel (b), presents an overcorrection for the the first reflection, that is visible at large offset values. This overcorrection is less severe after using non-hyperbolic moveout correction. However, the overcorrection caused by hyperbolic solution disappears after the third reflector, the hyperbolic moveout correction and the non-hyperbolic moveout equation achieve the same result for deep reflectors and short offset.

This simple example shows that for large offset-to-depth ratio more terms in the total traveltime equation yield better flat results are achieved. Also, for small offset-to-depth ratio both approximations achieved the same result; therefore, it is possible to still hyperbolic moveout equation to obtain a simple model of the Earth's subsurface.

 

synt-nmo
Figure 4 (a) Synthetic CMP gather. (b) Two term NMO result. (c) Three term NMO result.