We review the equations for correlation-based redatuming methods. A correlation-based redatuming method uses natural-phase information in the data to time shift the weighted traces so they appear to be generated by sources ͑or recorded by geophones͒ shifted to a new location. This compares to model-based redatuming, which effectively time shifts the traces using traveltimes computed from a prior velocity model. For wavefield redatuming, the daylight imaging, interferometric imaging, reverse-time acoustics ͑RTA͒, and virtualsource methods all require weighted correlation of the traces with one another, followed by summation over all sources ͑and sometimes receivers͒. These methods differ from one another by their choice of weights. The least-squares interferometry and virtual-source imaging methods are potentially the most powerful because they account for the limited source and receiver aperture of the recording geometry. Interferometry, on the other hand, has the flexibility to select imaging conditions that target almost any type of event. Stationary-phase principles lead to a Fermat-based redatuming method known as redatuming by a seminatural Green's function. No crosscorrelation is needed, so it is less expensive than the other methods. Finally, Fermat's principle can be used to redatum traveltimes.

We describe how vertical seismic profile ͑VSP͒ interferometric imaging of transmitted P-to-S ͑PS͒ waves can be used to delineate the flanks of salt bodies. Unlike traditional migration methods, interferometric PS imaging does not require the migration velocity model of the salt and/or upper sediments in order to image the salt flank. Synthetic elastic examples show that PS interferometric imaging can clearly delineate the upper and lower boundaries of a realistic salt-body model. Results also show that PS interferometric imaging is noticeably more accurate than conventional migration methods in the presence of static shifts and/or migration velocity errors. However, the illumination area of the PS transmitted waves is limited by the width of the shot and geophone aperture, which means wide shot offsets and deeper receiver wells are needed for comprehensive salt-flank imaging. Interferometric imaging results for VSP data from the Gulf of Mexico demonstrate its superiority over the traditional migration method. We also discuss other arrivals that can be used for interferometric imaging of salt flanks. For comparison, reduced-time migration results are presented, which are similar in quality to those obtained for interferometric imaging. We conclude that PS interferometric imaging of VSP data provides the geophysicist with a new tool by which salt flanks can be viewed from both above and below VSP geophone locations.

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One of the difficulties in seeing beneath salt is that the migration velocity in the salt and above it is not well known. This can lead to defocusing of migration images beneath the salt. In this paper, we show that reduced-time migration ͑RTM͒ and interferometric migration ͑IM͒ can partly mitigate this problem. RTM time-shifts the traces with the time difference between the calculated arrival time sg ref and the natural arrival time sg ref of a reference reflection, where s and g denote the source and receiver locations on the surface, respectively. We use the terms natural and calculated to represent, respectively, the arrival times that are velocity-independent ͑traveltimes directly extracted from the data without knowledge of the velocity model͒ and velocity-dependent ͑traveltimes calculated by ray tracing through a given velocity model͒. The benefit of RTM is a significant reduction of defocusing errors caused by errors in the migration velocity. IM, on the other hand, requires extrapolation of the surface data below salt using the natural arrival times sg ref of the subsalt reference reflector, and migration of the extrapolated data below the salt. The benefit with IM is that no salt velocity model is needed, so the model-based defocusing errors are, in theory, eliminated. To reduce computational time, we implement IM with a seminatural Green's function ͑combination of model-based calculated and picked natural traveltimes͒. Because no explicit data extrapolation is needed, IM with seminatural Green's functions is more cost-efficient than the standard IM. In this paper, we tested both RTM and IM with seminatural Green's functions on a synthetic and a field common-depth-point ͑CDP͒ data set, the latter from the Gulf of Mexico ͑GOM͒. Results show that both RTM and IM can remove the significant kinematic distortions caused by the overburden without knowledge of the overburden velocity.

The timing and amplitudes of arrivals recorded in seismic traces are influenced by velocity variations all along the associated raypaths. Consequently, velocity errors far from the target can lead to blurred imaging of the target body. To partly remedy this problem, we comprehensively reviewed inverting differential traveltimes that satisfied the closurephase condition. The result is that the source and receiver statics are completely eliminated in the data and velocities far from the target do not need to be known. We successfully used the phase closure equation for traveltime tomography, refraction statics, migration, refraction tomography, and earthquake location, all of which demonstrated the higher resolution achievable by processing data with differential traveltimes rather than absolute traveltimes. More generally, the stationary version of the closure-phase equation is equivalent to Fermat's principle and can be derived from the equations of seismic interferometry. In summary, the general closure-phase equation is the mathematical foundation for approximately redatuming sources and/or receivers to the target of interest without the need to accurately know the statics or the velocity model away from the target.