Theory of differential offset continuation

Offset continuation and DMO

Dip moveout represents a particular case of offset continuation for the output offset equal to zero. In this section, I consider the DMO case separately in order to compare the solutions of equation (1) with the Fourier-domain DMO operators, which have been the standard for DMO processing since Hale's outstanding work (Hale, 1983,1984).

Equation (64) transforms to the time-wavenumber domain with the help of integral tables:

$$\widetilde{P}(t_n,h,k)= H(t_n)\,\left(\widetilde{P}_0(t_n,h,k) + t_n\,\widetilde{P}_1(t_n,h,k)\right)\;,$$ (77)

where

$$\widetilde{P}_0 = {\partial \over {\partial t_n}}\, \int_{\left(h_1/h\right)\,t_n}^... ...r t_n^2}-{h_1^2 \over t_1^2}\right)\, \left(t_n^2-t_1^2\right)}\right)\,dt_1\;,$$ (78)

$$\widetilde{P}_1 = \int_{\left(h_1/h\right)\,t_n}^{t_n} h_1\,\widetilde{P}^{(1)}_1\l... ...2 \over t_1^2}\right)\, \left(t_n^2-t_1^2\right)}\right)\,{dt_1 \over t_1^2}\;,$$ (79)

$$\widetilde{P}^{(j)}_1(t_1,k) = \int\,P\,^{(j)}_1(t_1,y_1)\exp (-iky_1)\,dy_1\;(j=0,1)\;,$$ (80)

$$\widetilde{P}(t_n,h,k) = \int\,P(t_n,h,y)\exp (-iky)\,dy\;(j=0,1)\;.$$ (81)

Setting the output offset to zero, we obtain the following DMO-like integral operators in the $t$-$k$ domain:

$$\widetilde{P}(t_0,0,k)= H(t_0)\,\left(\widetilde{P}_0(t_0,k) + t_0\,\widetilde{P}_1(t_0,k)\right)\;,$$ (82)

where

$$\widetilde{P}_0(t_0,k) = - {\partial \over {\partial t_0}}\... ...\left({{k\,h_1}\over t_1}\, \sqrt{t_1^2-t_0^2}\right)\,dt_1\;,$$ (83)

$$\widetilde{P}_1(t_0,k) = - \int_{t_0}^{\infty} h_1\,\wideti... ...}\over t_1}\, \sqrt{t_1^2-t_0^2}\right)\,{dt_1 \over t_1^2}\;,$$ (84)

the wavenumber $k$ corresponds to the midpoint axis $y$, and $J_0$ is the zeroth-order Bessel function. The Fourier transform of (83) and (84) with respect to the time variable $t_0$ reduces to known integrals (Gradshtein and Ryzhik, 1994) and creates explicit DMO-type operators in the frequency-wavenumber domain, as follows:

$$\widetilde{\widetilde{P}}_0(\omega_0,k) = i\, \int_{-\infty... ...n{\left(\omega_0\,\vert t_1\vert\,A\right)} \over A} \,dt_1\;,$$ (85)

$$\widetilde{\widetilde{P}}_1(\omega_0,k) = i\, \int_{-\infty... ...ga_0\,\vert t_1\vert\,A\right)} \over A} {dt_1 \over t_1^2}\;,$$ (86)

where

$$A=\sqrt{1+{(k\,h_1)^2 \over (\omega_0\,t_1)^2}}\;,$$ (87)

$$\widetilde{\widetilde{P}}_j(\omega_0,k)= \int\,\widetilde{P}_j(t_0,k)\,\exp (i\omega_0 t_0)\,dt_0\;.$$ (88)

It is interesting to note that the first term of the continuation to zero offset (85) coincides exactly with the imaginary part of Hale's DMO operator (Hale, 1984). However, unlike Hale's, operator (82) is causal, which means that its impulse response does not continue to negative times. The non-causality of Hale's DMO and related issues are discussed in more detail by Stovas and Fomel (1996).

Though Hale's DMO is known to provide correct reconstruction of the geometry of zero-offset reflections, it does not account properly for the amplitude changes (Black et al., 1993). The preceding section of this paper shows that the additional contribution to the amplitude is contained in the second term of the OC operator (64), which transforms to the second term in the DMO operator (82). Note that this term vanishes at the input offset equal to zero, which represents the case of the inverse DMO operator.

Considering the inverse DMO operator as the continuation from zero offset to a non-zero offset, we can obtain its representation in the $t$-$k$ domain from equations (77-79) as

$$\widetilde{P}(t_n,h,k) = H(t_n) {\partial \over {\partial t... ..._0\left({{k\,h}\over t_n}\, \sqrt{t_n^2-t_0^2}\right)\,dt_0\;,$$ (89)

Fourier transforming equation (89) with respect to the time variable $t_0$ according to equation (88), we get the Fourier-domain version of the ``amplitude-preserving'' inverse DMO:

$$\widetilde{P}(t_n,h,k) = {H(t_n) \over {2\,\pi}}\,{\partial... ...\vert t_n\vert\,A\right)} \over {\omega_0\,A}}} \,d\omega_0\;,$$ (90)

$$A=\sqrt{1+{(k\,h)^2 \over (\omega_0\,t_n)^2}}\;.$$ (91)

Comparing operator (90) with Ronen's version of inverse DMO (Ronen, 1987), one can see that if Hale's DMO is denoted by ${\bf D}_{t_0}\,{\bf H}$, then Ronen's inverse DMO is ${\bf H^{T}\,D}_{-t_0}$, while the amplitude-preserving inverse (90) is ${\bf D}_{t_n}\,{\bf H^T}$. Here ${\bf D}_t$ is the derivative operator $\left( \partial \over \partial t\right)$, and ${\bf H^T}$ stands for the adjoint operator defined by the dot-product test

$${\bf (Hm,d)=(m,H^{T}d)},$$ (92)

where the parentheses denote the dot product:

$${\bf (m_1,m_2)}=\int\!\int\,m_1(t_n,y)\,m_2(t_n,y)\,dt_n\,dy\;. \nonumber$$

In high-frequency asymptotics, the difference between the amplitudes of the two inverses is simply the Jacobian term ${d\,t_0 \over d\,t_n}$, asymptotically equal to ${t_0 \over t_n}$. This difference corresponds exactly to the difference between Black's definition of amplitude preservation (Black et al., 1993) and the definition used in Born DMO (Bleistein, 1990; Liner, 1991), as discussed above. While operator (90) preserves amplitudes in the Born DMO sense, Ronen's inverse satisfies Black's amplitude preservation criteria. This means Ronen's operator implies that the ``geometric spreading'' correction (multiplication by time) has been performed on the data prior to DMO.

To construct a one-term DMO operator, thus avoiding the estimation of the offset derivative in (72), let us consider the problem of inverting the inverse DMO operator (90). One of the possible approaches to this problem is the least-squares iterative inversion, as proposed by Ronen (1987). This requires constructing the adjoint operator, which is Hale's DMO (or its analog) in the case of Ronen's method. The iterative least-squares approach can account for irregularities in the data geometry (Ronen, 1994; Ronen et al., 1991) and boundary effects, but it is computationally expensive because of the multiple application of the operators. An alternative approach is the asymptotic inversion, which can be viewed as a special case of preconditioning the adjoint operator (Chemingui and Biondi, 1996; Liner and Cohen, 1988). The goal of the asymptotic inverse is to reconstruct the geometry and the amplitudes of the reflection events in the high-frequency asymptotic limit.

According to Beylkin's theory of asymptotic inversion, also known as the generalized Radon transform (Beylkin, 1985), two operators of the form

$$D(\omega)=\int X(t,\omega)\,M(t)\, \exp\left[i\omega \phi (t,\omega)\right]\,dt$$ (93)

and

$$\widetilde{M}(t)=\int Y(t,\omega)\,D(\omega)\, \exp\left[-i\omega \phi (t,\omega)\right]\,d\omega$$ (94)

constitute a pair of asymptotically inverse operators ( $\widetilde{M}(t)$ matching $M(t)$ in the high-frequency asymptotics) if

$$X(t,\omega)\,Y(t,\omega)={Z(t,\omega) \over {2\,\pi}}\;,$$ (95)

where $Z$ is the ``Beylkin determinant''

$$Z(t,\omega)=\left\vert\partial \omega \over \partial \hat{\o... ...{\omega}=\omega\,{\partial \phi(t,\omega) \over \partial t}\;.$$ (96)

With respect to the high-frequency asymptotic representation, we can recast (90) in the equivalent form by moving the time derivative under the integral sign:

$$\widetilde{P}(t_n,k) \approx {H(t_n) \over {2\,\pi}}\,\mbox... ...\left(-i \omega_0\,\vert t_n\vert\,A\right) \,d\omega_0\right]$$ (97)

Now the asymptotic inverse of (98) is evaluated by means of Beylkin's method (94)-(95), which leads to an amplitude-preserving one-term DMO operator of the form

$$\widetilde{\widetilde{P}}_0(\omega_0,k) = \mbox{Im}\left[ \... ...exp\left(i \omega_0\,\vert t_1\vert\,A\right) \,dt_1\right]\;,$$ (98)

where

$$B = A^2 {\partial \over \partial \omega_0}\left(\omega_0\, {... ...t_n\,A)} \over \partial t_n}\right) = A^{-1}\,(2\,A^2 - 1)\;.$$ (99)

The amplitude factor (100) corresponds exactly to that of Born DMO (Bleistein, 1990) in full accordance with the conclusions of the asymptotic analysis of the offset-continuation amplitudes. An analogous result can be obtained with the different definition of amplitude preservation proposed by Black et al. (1993). In the time-and-space domain, the operator asymptotically analogous to (99) is found by applying either the stationary phase technique (Liner, 1990; Black et al., 1993) or Goldin's method of discontinuities (Goldin, 1990,1988), which is the time-and-space analog of Beylkin's asymptotic inverse theory (Stovas and Fomel, 1996). The time-and-space asymptotic DMO operator takes the form

$$P_0(t_0,y) = {\bf D}^{1/2}_{-t_0}\,\int w_0(\xi;h_1,t_0)\, P^{(0)}_1(\theta^{(-)}(\xi;h_1,0,t_0),y_1-\xi)\,d\xi\;,$$ (100)

where the weighting function $w_0$ is defined as

$$w_0(\xi;h_1,t_0)=\sqrt{t_0 \over {2\,\pi}}\, {{h_1\,(h_1^2+\xi^2)} \over (h_1^2-\xi^2)^2}\;.$$ (101)

Theory of differential offset continuation