True‐amplitude imaging and dip moveout

True‐amplitude seismic imaging produces a three dimensional (3-D) migrated section in which the peak amplitude of each migrated event is proportional to the reflectivity. For a constant‐velocity medium, the standard imaging sequence consisting of spherical‐divergence correction, normal moveout (NMO), dip moveout (DMO), and zero‐offset migration produces a true‐amplitude image if the DMO step is done correctly. There are two equivalent ways to derive the correct amplitude‐preserving DMO. The first is to improve upon Hale’s derivation of F-K DMO by taking the reflection‐point smear properly into account. This yields a new Jacobian that simply replaces the Jacobian in Hale’s method. The second way is to calibrate the filter that appears in integral DMO so as to preserve the amplitude of an arbitrary 3-D dipping reflector. This latter method is based upon the 3-D acoustic wave equation with constant velocity. The resulting filter amounts to a simple modification of existing integral algorithms. The new F-K and integral DMO algorithms resulting from these two approaches turn out to be equivalent, producing identical outputs when implemented in nonaliased fashion. As dip increases, their output become progressively larger than the outputs of either Hale’s F-K method or the integral method generally associated with Deregowski and Rocca. This trend can be observed both on model data and field data. There are two additional results of this analysis, both following from the wave‐equation calibration on an arbitrary 3-D dipping reflector. The first is a proof that the entire imaging sequence (not just the DMO part) is true‐amplitude when the DMO is done correctly. The second result is a handy formula showing exactly how the zero‐phase wavelet on the final migrated image is a stretched version of the zero‐phase deconvolved source wavelet. This result quantitatively expresses the loss of vertical resolution due to dip and offset.