Equation () has the known general solution, expressed in terms of cylinder functions of complex order Watson (1952)
In the general case of offset continuation, C1 and C2 are constrained by the two initial conditions () and (). In the special case of continuation from zero offset, we can neglect the second term in () as vanishing at the zero offset. The remaining term defines the following operator of inverse DMO in the domain:
The DMO operator now can be derived as the inversion of operator (), which is a simple multiplication by . Therefore, offset continuation becomes a multiplication by (the cascade of two operators). This fact demonstrates an important advantage of moving to the log-stretch domain: both offset continuation and DMO are simple filter multiplications in the Fourier domain of the log-stretched time coordinate.
In order to compare operator () with the known versions of log-stretch DMO, we need to derive its asymptotical representation for high frequency . The required asymptotic expression follows directly from the definition of function in () and the known asymptotical representation for a Bessel function of high order Watson (1952):
Asymptotical representation () is valid for high frequency and . It can be shown that the phase function defined in () coincides precisely with the analogous term in Liner's exact log DMO Liner (1990), which was proven to provide the correct geometric properties of DMO. Similar expressions for the log-stretch phase factor were derived in different ways by Zhou et al. (1996) and Canning and Gardner (1996). However, the amplitude term differs from the previously published ones because of the difference in the amplitude preservation properties.
A number of approximate log DMO operators have been proposed in the literature. As shown by Liner (1990), all of them but exact log DMO distort the geometry of reflection effects at large offsets. The distortion is caused by the implied approximations of the true phase function . Bolondi's OC operator Bolondi et al. (1982) implies , Notfors' DMO Notfors and Godfrey (1987) implies , and the ``full DMO'' Bale and Jakubowicz (1987) has . All these approximations are valid for small (small offsets or small reflector dips) and have errors of the order of (Figure ). The range of validity of Bolondi's operator is defined in equation ().
Figure 8 Phase functions of the log DMO operators. Solid line: exact log DMO; dashed line: Bolondi's OC; dashed-dotted line: Bale's full DMO; dotted line: Notfors' DMO.
In practice, seismic data are often irregularly sampled in space but regularly sampled in time. This makes it attractive to apply offset continuation and DMO operators in the domain, where the frequency corresponds to the log-stretched time and y is the midpoint coordinate. Performing the inverse Fourier transform on the spatial frequency transforms the inverse DMO operator () to the domain, where the filter multiplication becomes a convolutional operator:
Inverting operator (), we can obtain the DMO operator in the domain.