True‐amplitude transformation to zero offset of data from curved reflectors

Transformation to zero offset (TZO), alternatively known as migration to zero offset (MZO), or the combination of normal moveout and dip moveout (NMO/DMO), is a process that transforms data collected at finite offset between source and receiver to a pseudozero offset trace. The kinematic validity of NMO/DMO processing has been well established. The TZO integral operators proposed here differ from their NMO/DMO counterparts by a simple amplitude factor. (The form of the operator depends on how the input and output variables are chosen from among the combinations of midpoint or wavenumber with time or frequency.) With this modification in place, the dynamical validity for planar reflectors of the proposed TZO operators of this paper have been established in earlier studies. This means that the traveltime and geometrical spreading terms of the finite offset data are transformed to their counterparts for zero offset data, while the finite offset reflection coefficient is preserved. The main purpose of this study is to show that dynamical validity of the TZO operator extends to the case of curved reflectors in the 2.5-D limit. Thus, at the cost of a simple additional multiplicative factor in any standard NMO/DMO operator to produce the corresponding TZO operator, the amplitude factor attributed to curvature effects in finite offset data is transformed by this TZO processing to the corresponding curvature factor for zero offset data. This problem has also been addressed in a more general context by Tygel and associates. However, in the generality, some of the specifics and interpretations of the simpler problem are lost. Thus, we see some value in presenting this analysis where one can carry out all calculations explicitly and see specific quantities that are more familiar and accessible to users of DMO. Furthermore, in this paper, we show how processing of the input data with a second TZO operator allows for the extraction of the cosine of the preserved specular angle, a necessary piece of information for amplitude versus angle (AVA) analysis. We then discuss the possibility of using the output of our processing formalism at multiple offsets to create a table of angularly dependent reflection coefficients and attendant incidence angles as a function of offset. This is the basis of a proposed amplitude versus offset/amplitude versus angle (AVO/AVA) analysis of the pseudozero offset traces. Finally, we describe the modifications of Hale DMO and Gardner/Forel DMO to obtain true amplitude output equivalent to ours and also how to extract the cosine of the specular angle for these forms of DMO. This last does not depend on true amplitude processing, but only on processing two DMO operators with slightly different kernels and then taking the quotient of their peak amplitudes.