OC-seislet: seislet transform construction with differential offset continuation

Acknowledgments

We thank BGP Americas for a partial financial support of this work. We thank Tamas Nemeth, Mauricio Sacchi, Sandra Tegtmeier-Last, and two anonymous reviewers for their constructive comments and suggestions. This publication was authorized by the Director, Bureau of Economic Geology, The University of Texas at Austin.

Appendix A

Review of differential offset continuation

In this appendix, we review the theory of differential offset continuation from Fomel (2003a,c). The partial differential equation for offset continuation (differential azimuth moveout) takes the form

$$\left( \mathbf{h}^T\,\mathbf{P}_{xx}\,\mathbf{h} - h^2\,{\partial... ...\right) \, = \, h\,t_n \, {\partial^2 P \over {\partial t_n \, \partial h}} \;,$$ (13)

where $P(\mathbf{x},\mathbf{h},t_n)$ is the seismic data in the midpoint-offset-time domain, $t_n$ is the time coordinate after the normal moveout (NMO) correction, $\mathbf{h}^T$ denotes the transpose of $\mathbf{h}$ , and $\mathbf{P}_{xx}$ is the tensor of the second-order midpoint derivatives.

A particularly efficient implementation of offset continuation results from a log-stretch transform of the time coordinate (Bolondi et al., 1982), followed by a Fourier transform of the stretched time axis. After these transforms, the offset-continuation equation takes the form

$$\left( \mathbf{h}^T\,\Tilde{\mathbf{P}}_{xx}\,\mathbf{h} - h^2\,\... ...al h^2} \right) \, = \, i\,\Omega\,h\,\frac{\partial \Tilde{P}}{\partial h} \;,$$ (14)

where $\Omega$ is the dimensionless frequency corresponding to the stretched time coordinate and $\Tilde{P} (\mathbf{x},\mathbf{h},\Omega)$ is the transformed data. As in other frequency-space methods, equation A-2 can be applied independently and in parallel on different frequency slices.

In the frequency-wavenumber domain, the extrapolation operator is defined by solving an initial-value problem for equation A-2. The analytical solution takes the form

$$\Tilde{\Tilde{P}}(\mathbf{k},\mathbf{h}_2,\Omega) = \Tilde{\Tilde... ...(\mathbf{k} \cdot \mathbf{h}_2)}{Z_{\lambda}(\mathbf{k} \cdot \mathbf{h}_1)}\;,$$ (15)

where $\Tilde{\Tilde{P}}(\mathbf{k},\mathbf{h},\Omega)$ is the double-Fourier-transformed data, $\lambda = (1 + i \Omega)/2$ , $Z_\lambda$ is the special function defined as

$$Z_{\lambda}(x) = \Gamma(1-\lambda)\,\left(x \over 2\right)^{\lambda}\, J_{-\lambda}(x)= {}_0F_1\left(;1-\lambda;-\frac{x^2}{4}\right)\;,$$ (16)

$\Gamma$ is the gamma function, $J_{-\lambda}$ is the Bessel function, and ${}_0F_1$ is the confluent hypergeometric limit function (Petkovsek et al., 1996). The wavenumber $\mathbf{k}$ in equation A-3 corresponds to the midpoint $\mathbf{x}$ in the original data domain. In high-frequency asymptotics, the offset-continuation operator takes the form

$$\Tilde{\Tilde{P}}(\mathbf{k},\mathbf{h}_2,\Omega) \approx \Tilde{... ...ga\,\psi\left(\frac{2 \mathbf{k} \cdot \mathbf{h}_1}{\Omega}\right)\right]}}\;,$$ (17)

where

$$F(\epsilon)=\sqrt{{1+\sqrt{1+\epsilon^2}} \over {2\,\sqrt{1+\epsilon^2}}}\, \exp\left({1-\sqrt{1+\epsilon^2}} \over 2\right)\;,$$ (18)

and

$$\psi(\epsilon)={1 \over 2}\,\left(1 - \sqrt{1+\epsilon^2} + \ln\left({1 + \sqrt{1+\epsilon^2}} \over 2\right)\right)\;.$$ (19)

The phase function $\psi$ defined in equation A-7 corresponds to the analogous term in the exact-log DMO and AMO (Liner, 1990; Zhou et al., 1996; Biondi and Vlad, 2002).

OC-seislet: seislet transform construction with differential offset continuation