The purpose of velocity analysis is to extract the normal moveout velocity as a function of the zero-offset travel time at selected CDP locations along the seismic line. Since results of velocity analysis depend on coherency estimator, an estimator that provides a high velocity resolution is essential. Even though the conventional semblance method which is the most popular coherency estimator (Tanner and Kohler, 1969) provides a robust velocity spectrum, the tendency to smear the velocity peaks as the time increases makes the estimation of accurate velocity difficult. This estimator, however, has some resolution limits that cause problems in some cases. It fails to distinguish interfering events in a short time window and in cases of thin bedding (Lerner and Cellis, 2007).
We propose here two new coherency estimators that resolve these limitations at a minor extra-cost. The estimators are based on a differential semblance (DS) coefficient (Symes and Carazzone, 1991) that is weighted by the semblance estimator. High-resolution is introduced by sorting the traces in the data in a way that highlights the time shifts between adjacent traces within a time gate. The new estimators exploit the redundancy of seismic data in the common mid-point (CMP) to bootstrap the seismic traces in a manner that nicely brings time shifts between adjacent traces to discriminate time gates built using parameters that are close to the true stacking parameters. Bootstrapping is a statistical technique used to infer estimates of standard errors and confidence interval from data samples for which the statistical properties are unattainable via simple means. The first proposed estimator is deterministic bootstrapped differential semblance (BDS) that is based on a deterministic sorting of original offset traces by alternating near and far offsets to achieve maximized time shifts between adjacent traces. Deterministic sorting that alternates near- and far-offset traces in the time window has higher resolution than does simple bootstrapping applied to the data traces. The second was the product of several BDS terms, with the first term being the deterministic BDS defined above. The other terms were generated by random sorting of traces that alternated between near and far offsets in an unpredictable manner. The proposed estimators help in discriminating several trial parameters which produce a good guess of the flattening parameters and have direct implications in retrieving velocity information from time gathers. The suggested estimators are tested on synthetic and real data examples to show the gain in resolution they yield when applied, and they are compared with coefficient semblance. Results show that deterministic BDS coefficient provides an increased resolution with no extra computing effort compared to the BDS coefficient. Further resolution can be achieved by involving several controlled bootstrapping outcomes in the estimator, but this comes at a computing cost nearly proportional to the number of terms in the high resolution estimator.
The high-resolution BDS proves to be an efficient tool in building velocity spectra for time-domain velocity analysis and it provides more resolution with respect to conventional semblance estimator. The proposed estimators could be a good substitute for the semblance coefficient, and an economic alternative to other high resolution estimators such as eigenvalue methods that are expensive for the dense parameter tracking in high fold data sets.