Theory of differential offset continuation

Offset continuation in the log-stretch domain

The log-stretch transform, proposed by Bolondi et al. (1982) and further developed by many other researchers, is a useful tool in DMO and OC processing. Applying a log-stretch transform of the form

$$\sigma = \ln\left\vert t_n \over t_* \right\vert\;,$$ (102)

where $t_*$ 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 $y$ and to the transformed (log-stretched) time coordinate $\sigma$, the partial differential equation (1) takes the form of an ordinary differential equation,

$$h\,\left({{d^2 \widehat{\widehat{P}}} \over {dh^2}} + k^2\,... ...}}\right) = i\Omega\,{{d \widehat{\widehat{P}}} \over {dh}}\;,$$ (103)

where

$$\widehat{\widehat{P}}(h) = \int\!\int P(t_n=t_*\,\exp(\sigma),h,y)\, \exp(i\Omega\sigma - iky)\,d\sigma\,dy\;.$$ (104)

Equation (104) has the known general solution, expressed in terms of cylinder functions of complex order $\lambda = {{1+i\Omega} \over 2}$ (Watson, 1952)

$$\widehat{\widehat{P}}(h) = C_1(\lambda)\,(kh)^{\lambda}\,J_{-\lambda}(kh)+ C_2(\lambda)\,(kh)^{\lambda}\,J_{\lambda}(kh)\;,$$ (105)

where $J_{-\lambda}$ and $J_{\lambda}$ are Bessel functions, and $C_1$ and $C_2$ stand for some arbitrary functions of $\lambda$ that do not depend on $k$ and $h$.

In the general case of offset continuation, $C_1$ and $C_2$ 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 ${\Omega,k}$ domain:

$$\widehat{\widehat{P}}(h) = \widehat{\widehat{P}}(0)\,Z_{\lambda}(kh)\;,$$ (106)

where $Z_{\lambda}$ is the analytic function

$$Z_{\lambda}(x) = \Gamma(1-\lambda)\,\left(x \over 2\right)^{\lambda}\, J_{-\lambda}(x)= {}_0F_1\left(;1-\lambda;-\frac{x^2}{4}\right)$$

$$= \sum_{n=0}^{\infty} {(-1)^n \over n!}\, {\Gamma(1-\lambda) \over \Gamma(n+1-\lambda)}\, \left(x \over 2\right)^{2n}\;,$$ (107)

$\Gamma$ is the gamma function and ${}_0F_1$ 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 $1/Z_{\lambda}(kh)$. Therefore, offset continuation becomes a multiplication by $Z_{\lambda}(kh_2)/Z_{\lambda}(kh_1)$ (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 $\Omega$. The required asymptotic expression follows directly from the definition of function $Z_{\lambda}$ in equation (108) and the known asymptotic representation for a Bessel function of high order (Watson, 1952):

$$J_{\lambda}(\lambda z) \stackrel{\lambda \rightarrow \infty}... ...z^2)^{1/4}\, \left\{1+\sqrt{1-z^2}\right\}^{\sqrt{1-z^2}}}}\;.$$ (108)

Substituting approximation (109) into (108) and considering the high-frequency limit of the resultant expression yields

$$Z_{\lambda}(kh) \approx \left\{{1+\sqrt{1-\left(kh \over \l... ...)^{1/4}} \approx {F(\epsilon)\,e^{i\Omega\,\psi(\epsilon)}}\;,$$ (109)

where $\epsilon$ denotes the ratio ${2\,k\,h} \over \Omega$,

$$F(\epsilon)=\sqrt{{1+\sqrt{1+\epsilon^2}} \over {2\,\sqrt{1+... ...lon^2}}}\, \exp\left({1-\sqrt{1+\epsilon^2}} \over 2\right)\;,$$ (110)

and

$$\psi(\epsilon)={1 \over 2}\,\left(1 - \sqrt{1+\epsilon^2} + \ln\left({1 + \sqrt{1+\epsilon^2}} \over 2\right)\right)\;.$$ (111)

The asymptotic representation (110) is valid for high frequency $\Omega$ and $\vert\epsilon\vert \leq 1$. The phase function $\psi$ 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 $\psi$ were derived in different ways by Zhou et al. (1996) and Canning and Gardner (1996). However, the amplitude term $F(\epsilon)$ 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 $\psi$. Bolondi's OC operator (Bolondi et al., 1982) implies $\psi(\epsilon) \approx -{\epsilon^2 \over 8}$, Notfors' DMO (Notfors and Godfrey, 1987) implies $\psi(\epsilon) \approx 1 - \sqrt{1+(\epsilon /2)^2}$, and the ``full DMO'' (Bale and Jakubowicz, 1987) has $\psi(\epsilon) \approx {1 \over 2} \ln\left[1-(\epsilon / 2)^2\right]$. All these approximations are valid for small $\epsilon$ (small offsets or small reflector dips) and have errors of the order of $\epsilon^4$ (Figure 6). The range of validity of Bolondi's operator is defined in equation (22).

pha Figure 6. 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 $\{\Omega,y\}$ domain, where the frequency $\Omega$ 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 (107) to the $\{\Omega,y\}$ domain, where the filter multiplication becomes a convolutional operator:

$$\widehat{P}(\Omega,h,y) = {\widehat{F}(\Omega) \over \sqrt{2... ...er 2}\,\ln\left(1-{\xi^2 \over h_1^2}\right)\right) \,d\xi\;.$$ (112)

Here $\widehat{F}(\Omega)$ is a high-pass frequency filter:

$$\widehat{F}(\Omega)={{\Gamma(1/2-i\Omega/2)} \over {\sqrt{1/2}\,\Gamma(-i\Omega/ 2)}}\;.$$ (113)

At high frequencies $\widehat{F}(\Omega)$ is approximately equal to $(- i \Omega)^{1/2}$, which corresponds to the half-derivative operator $\left(\partial \over \partial \sigma \right)^{1/2}$, which, in turn, is equal to the $\left(t_n {\partial \over \partial t_n} \right)^{1/2}$ term of the asymptotic OC operator (69). The difference between the exact filter $\widehat{F}$ 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
Figure 7. 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 $\Omega$.

Inverting operator (113), we can obtain the DMO operator in the $\{\Omega,y\}$ domain.

Theory of differential offset continuation