The three‐parameter equation: An efficient tool to enhance the stack

Abstract

We compare the three‐term equation to the normal moveout (NMO) equation for several synthetic data sets to analyze whether or not it is worth making the additional computational effort in the stacking process within various exploration contexts. In our evaluation we have selected two criteria: 1)The quality of the stacked image. 2) The reliability of the stacking parameters and their usefulness for further computation such as interval velocity estimation. We have simulated the stacking process very precisely, despite using only the traveltimes and not the full waveform data. The procedure searches for maximum coherency along the traveltime curve rather than a least‐square regression to it. This technique, which we call the Gaussian‐weighted least square, avoids most of the shortcomings of the least‐square method. The following are our conclusions: 1) The three term equation gives a better stack than the regular NMO. The increase in stacking energy can be more than 30 percent. 2)The calculation of interval velocities using a DIX formula rewritten for the three‐parameter equation is much more stable and accurate than the standard DIX formula. 3) The search for the three parameters is feasible in an efficient way since the shifted hyperbola requires only static corrections rather than dy namic ones. 4) Noise alters the parameters of the maximum energy stack in a way that depends on the noise type. The estimates obtained remain accurate enough for interval velocity estimation (where only two parameters are needed), but the use of the three parameters in direct inversion may be hazardous because of noise corruption. These conclusions should, however, be verified on real data examples.