PS-DMO smile derivation

The trajectory equation of conversion points is  

 43#43 (14)

By introducing the ratio, 44#44, and by following Huub Den Rooijen's derivations, as well as work by (), we have the following standard form of elliptical equation:  

 45#45 (15)

where D is the element responsible for transforming the data from CMP coordinates to CCP coordinates. D comes from solving equation () for z². The difference among the existing PS-DMO operators is in their definition of D. In this paper we use the definition presented by ():

46#46 (16)

where t_n is the NMO-corrected time.

The spatial shift distance needed to convert CMP gathers to common-reflection point gathers is  

 47#47 (17)

The one-way normal-incidence distance R is given by  

 48#48 (18)

Using equations () and () we eliminate the z and x dependencies. This result is substituted in the relationship for the one-way normal-incidence distance R [equation ()] in order to get the equation:  

 49#49 (19)

The two-way normal-incidence time is  

 50#50 (20)

and the traveltime equation, in terms of the normal moveout time, is  

 51#51 (21)

Combining equations (), () and () produces the PS-DMO smile equation:  

 52#52 (22)

where