Born DMO Revisited

Abstract

Born DMO (BDMO) is a method of transformation of finite offset data to zero offset data The method cascades an inversion of offset data to produce a subsurface model with a generation of zero offset data from the resulting model. Both the inversion and the modeling steps are based on a Born approximation for modeling wave prop% g&ion, hence the name, Born DMO. A method for processing spatial-temoporal domain common offset data to produce zero offset data was proposed by Liner (1989). That method required a fractional derivative on the given offset data, thus making it difficult to compare the final operator with Hale’s wavenumber-temporal domain DMO operator. Here, we derive a BDMO operator for common offset data that does not require thii fractional derivative and allows for a direct comparison. We find that the operator differs from Hale’s in only one additional simple amplitude factor. We test the operator by applying it to Kirchhoff approximate data for a tilted plane. The operator produces an output with finite offset traveltime and geometrical spreading transformed to zero offset traveltime and geometrical spreading. Furthermore, the operator preserves the angularly dependent reflection coefficient of the original data and also includes the cosine of the specular angle as a multiplicative factor. This is the fully nonlinear (in earth parameters) geometrical optics reflection coefficient, thus, indicating a more robust result than its basii in the Born approximation might lead one to expect. A second BDMO operator, including an addional scale factor in the operator, producea the same output as the first, multiplied by an additional factor of cosine-squared of the specular angle. The ratio of the outputs provides an estimate of the cosine-square. The output from Hale’s DMO produces the same result as our first BDMO operator with an addition scale factor that depends on offset, dip and zero offset diitance to the rdector. We have also compared thii result to Zhang (1988). We find that Zhaug’e result agrees with ours, except that the zero offset geometrical spreading is replaced by normal incidence geometrical spreading from the midpoint of the source/receiver pair having the same specular reflection point as the zero o&et response.