Theory of differential offset continuation |

where is an arbitrarily chosen time constant, eliminates the time dependence of the coefficients in equation (1) and therefore makes this equation invariant to time shifts. After the double Fourier transform with respect to the midpoint coordinate and to the transformed (log-stretched) time coordinate , the partial differential equation (1) takes the form of an ordinary differential equation,

where

Equation (104) has the known general solution,
expressed in terms of cylinder functions of complex order
(Watson, 1952)

In the general case of offset continuation, and are
constrained by the two initial conditions (62) and
(63). In the special case of continuation from zero offset, we
can neglect the second term in (106) as vanishing at the zero
offset. The remaining term defines the following operator of inverse
DMO in the domain:

is the gamma function and is the confluent hypergeometric limit function (Petkovsek et al., 1996).

The DMO operator now can be derived as the inversion of operator (107), 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 (107) with the known versions
of log-stretch DMO, we need to derive its asymptotic representation
for high frequency . The required asymptotic expression
follows directly from the definition of function in
equation (108) and the known asymptotic representation for a Bessel
function of high order (Watson, 1952):

where denotes the ratio ,

and

The asymptotic representation (110) is valid for high
frequency and
. The
phase function defined in (112) coincides precisely
with the analogous term in Liner's *exact log DMO*
(Liner, 1990), which provides 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 6). The range of validity of Bolondi's operator is defined in equation (22).

pha
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.
Figure 6. | |
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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
is the midpoint coordinate. Performing the inverse Fourier
transform on the spatial frequency transforms the inverse DMO
operator (107) to the domain, where the
filter multiplication becomes a convolutional operator:

At high frequencies is approximately equal to , which corresponds to the half-derivative operator , which, in turn, is equal to the term of the asymptotic OC operator (69). The difference between the exact filter and its approximation by the half-order derivative operator is shown in Figure 7. This difference is a measure of the validity of asymptotic OC operators.

flt
Amplitude (left) and phase (right) of
the time filter in the log-stretch domain. The solid line is for the
exact filter; the dashed line for its approximation by the
half-order derivative filter. The horizontal axis corresponds to the
dimensionless log-stretch frequency .
Figure 7. |
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Inverting operator (113), we can obtain the DMO operator in the domain.

Theory of differential offset continuation |

2014-03-26