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Appendix A

Local plane-waves operators

Local plane-wave operators model seismic data (Fomel, 2002) . The mathematical basis is the local plane differential equation

$\displaystyle \frac{\partial P}{\partial x} + \sigma \frac{\partial P}{\partial t} = 0\;,$

(29)

where

is the wave field and $\sigma$ the local slope field (Claerbout, 1992). In the case of a constant slope, the solution of equation A-1 is a simple plane wave $P(t,x) = f(t - \sigma x)$ where

is an arbitrary waveform. Assuming that the slope $\sigma=\sigma(t,x)$ varies in time and space, one can design a local operator to propagate each trace to its neighbors. Let $\mathbf{s}$ represent a seismic section as a collection of traces: $\mathbf{s} = \left[\mathbf{s}_1 \; \mathbf{s}_2 \; \ldots \; \mathbf{s}_N\right]^T$ , where $\mathbf{s}_k$ corresponds to

for $k=1,2,\ldots$ A plane-wave destruction operator (PWD) effectively predicts each trace from its neighbor and subtracts the prediction from the original trace (Fomel, 2002). In linear operator notation, the plane-wave destruction operation can be defined as

$\displaystyle \mathbf{r} = \mathbf{D(\sigma) s}\;,$

(30)

where $\mathbf{r}$ is the destruction residual, and $\mathbf{D}$ is the destruction operator defined as

$\displaystyle \mathbf{D}(\sigma) = \left[\begin{array}{ccccc} \mathbf{I} & 0 & ... ...0 & \cdots & - \mathbf{P}_{N-1,N}(\sigma) & \mathbf{I} \end{array}\right]\;,$

(31)

where $\mathbf{I}$ stands for the identity operator, and $\mathbf{P}_{i,j}(\sigma)$ describes prediction of trace

from trace

. Prediction of a trace consists of shifting the original trace along dominant event slopes $\sigma$ . The dominant slopes are estimated by minimizing the prediction residual $\mathbf{r}$ in a least-squares sense. Since the prediction operators A-3 depends on the slopes themselves, the inverse problem is nonlinear and must be solved in a iterative fashion by subsequent linearizations. We employ shaping regularization (Fomel, 2007a) for controlling the smoothness of the estimated slope field.

Once the local slope field $\sigma$ has been computed, prediction of a trace from a distant neighbor can be accomplished by simple recursion. Predicting trace from trace is

$\displaystyle \mathbf{P}_{1,k} = \mathbf{P}_{k-1,k} \cdots \mathbf{P}_{2,3} \mathbf{P}_{1,2}\;.$

(32)

If $\mathbf{s}_r$ is a reference trace, then the prediction of trace $\mathbf{s}_k$ is $\mathbf{P}_{r,k} \mathbf{s}_r$ . Fomel (2010) called the recursive operator $\mathbf{P}_{r,k}$ predictive painting. The elementary prediction operators in equation A-3 spread information from a given trace to its neighbors recursively by following the local structure of seismic events. Figure 7 in the main text illustrates the painting concept.

Appendix B

Mathematical derivation of slope-based Dix inversion

The Dix formula (Dix, 1955) can be written in the differential form

$\displaystyle \hat{\mu}=\frac{d}{{d\tau _{0}}}\left[ {\tau _{0}\mu (\tau _{0})}\right] ,$

(33)

where $\hat{\mu}$ is the interval parameter corresponding to zero-slope time ${\tau _{0}}$ and ${\mu (\tau _{0})}$ is the vertically-variable general effective parameter. Using the chain rule, we rewrite the Dix's formula B-1 as follows:

$\displaystyle \hat{\mu}=\frac{d}{{d\tau }}\left[ {\tau _{0}(\tau )\mu (\tau )}\... ...] <tex2html_comment_mark>476 \left[ \frac{d{\tau }_{0}}{{d\tau }}\right] ^{-1}.$

(34)

At first, let us consider the VTI NMO velocity as an effective parameter, hence ${\mu =}V_{N}^{2}$ . Using the expression for ${\tau _{0}(\tau )}$ (equation 15) and $V_{N}^{2}{(\tau )}$ (equation 16), we obtain, after some algebra,

$\displaystyle \frac{{d\tau _{0}}}{{d\tau }}=\frac{1}{{2\tau _{0}}}\frac{{\tau (2ND+\tau (N_{\tau }D-D_{\tau }N))}}{{D^{2}}},$

(35)

$\displaystyle \frac{{d\left[ {\tau _{0}(\tau )V_{N}^{2}(\tau )}\right] }}{{d\ta... ...{\tau }-\tau DRN_{\tau }-3\tau NRD_{\tau }+4DNR}\right] }}{{pD^{3}N\tau _{0}}},$

(36)

where ${R_{\tau }=}\partial {R/}\partial \tau$ , ${Q_{\tau }=}\partial {Q/}\partial \tau$ and

$\displaystyle N_{\tau }=\partial N/\partial \tau =\left( {3\tau -6pR}\right) R_{\tau }+p\tau Q_{\tau }+3R+pQ,$

(37)

$\displaystyle D_{\tau }=\partial D/\partial \tau =\left( {3\tau +2pR}\right) R_{\tau }+p\tau Q_{\tau }+3R+pQ.$

(38)

Inserting equation B-3 and B-4 in B-2 leads to

$\displaystyle \hat{V}_{N}=-\frac{{16\tau R^{2}}}{{pDN}}\frac{{\left[ {6\tau DNR... ...{\tau }+4DNR}\right] }}{{\left[ {2ND+\tau N_{\tau }D-\tau D_{\tau }N}\right] }}$

(39)

To compute the interval $\hat{V}_H$ or $\hat{\eta}$ , we employ as effective value ${\mu =}S$ $V_{N}^{4}$ as described in equation 10. In this case, the modified Dix formula (equation B-2) can be rewritten as follows

$\displaystyle \hat{S}=\frac{1}{{\hat{V}_{N}^{4}(\tau )}}\frac{d}{{d\tau }}\left... ...au )V_{N}^{4}(\tau )}\right] \left[ \frac{d{\tau }_{0}}{{d\tau }}\right] ^{-1},$

(40)

which, after substituting the chain relation for the interval ${\hat{V}_{N}^{2}(\tau )=}\frac{{d\left[ {\tau _{0}(\tau )V_{N}^{2}(\tau )}\right] }}{{d\tau }}\left[ \frac{d{\tau }_{0}}{{d\tau }}\right] ^{-1}$ and some simplifications, leads to the relation

$\displaystyle \hat{S}={S(\tau )}\frac{{V_{N}^{2}(\tau )}}{{\hat{V}_{N}^{2}(\tau... ...] \left[ \frac{d{\tau }_{0}}{{<tex2html_comment_mark>480 d\tau }}\right] ^{-1},$

(41)

which involves only the mapping relations for the zero-slope time $\tau_{0}(\tau)$ (formula 15) effective NMO velocity $V_{N}^{2}(\tau )$ (formula 16), and the anellipticity parameter obtained by equation 18 as $S(\tau )=1+8\eta (\tau )$ . From the interval parameter $\hat{S}$ , we can go back to interval $\hat{V}_{H}^{2}=\hat{V}_{N}^{2} (\hat{S}+3)/4$ and $\hat{\eta}=(\hat{S}-1)/8$ .

In the case of isotropy or elliptical anisotropy ( $\hat{V}_{N}=\hat{V}_{H}$ ), equations B-3 and B-4 simplify to

$\displaystyle \frac{{d\tau _{0}}}{{d\tau }}=\frac{{2\tau -p(R+\tau R_{\tau })}}{{2\tau _{0}<tex2html_comment_mark>481 }},$

(42)

$\displaystyle \frac{{d\left[ {\tau _{0}(\tau )V_{N}^{2}(\tau )}\right] }}{{d\ta... ...}{2}\frac{{pR(R+\tau R_{\tau })-2R_{\tau }\tau ^{2}}}{{\tau _{0}p(pR-\tau )}}%$

(43)

Inserting equations B-10 and B-11 in formula B-2, we get

$\displaystyle \hat{V}_{N}=\frac{1}{{p(pR-\tau )}}\frac{{pR(R+\tau R_{\tau })-2R_{\tau }\tau ^{2}}}{{{2\tau -p(R+\tau R_{\tau })}}}.$

(44)

This equation is the analog of equation 15 in (Fomel, 2008).

Appendix C

Mathematical derivation of stripping equations

Starting from the integral representation of $\tau$ - signature in equation 7, we arrive at the expression of the slope and the curvature fields as the following integrals:

$\displaystyle R (\tau_{0},p)$	$\displaystyle =$	$\displaystyle \int\limits_{0}^{\tau _{0}}\mathcal{F'}(p,\xi)d\xi ,$	(45)
$\displaystyle Q (\tau_{0},p)$	$\displaystyle =$	$\displaystyle \int\limits_{0}^{\tau _{0}}\mathcal{F''}(p,\xi)d\xi ,$	(46)

where

$\displaystyle \mathcal{F}(\tau_{0},p)=\sqrt{\frac{1-\hat{V}_{H}^{2}(\tau_{0})p^{2}}{1-[\hat{V}_{H}^{2}(\tau_{0} )-\hat{V}_{N}^{2}(\tau_{0} )]p^{2}}}\;,$

$\mathcal{F'}=\frac{d\mathcal{F}}{dp}$ and $\mathcal{F''}=\frac{d^2\mathcal{F}}{dp^2}$ . According to equation 7, it descends that

$\displaystyle \tau_{0,\tau} (\tau ,p) =\left[ \dfrac{\partial \tau_{0}}{\partial \tau}\right] ^{-1} =\dfrac{1}{\mathcal{F}(p,\tau)}.$

(47)

Therefore, applying the chain rule

$\displaystyle \dfrac{\partial }{\partial \tau} = \left[ \dfrac{\partial \tau}{\... ...l \tau_{0}}=\dfrac{1}{\mathcal{F}(\tau,p)}\dfrac{\partial}{ \partial \tau_{0}},$

(48)

we obtain

$\displaystyle R_{\tau}(\tau ,p)$	$\displaystyle =$	$\displaystyle \dfrac{1}{\mathcal{F}(\tau,p)} \dfrac{\partial R}{\partial \tau_0} =\dfrac{\mathcal{F'}(\tau ,p)}{\mathcal{F}(\tau ,p)}\;,$	(49)
$\displaystyle Q_{\tau}(\tau ,p)$	$\displaystyle =$	$\displaystyle \dfrac{1}{\mathcal{F}(\tau,p)}\dfrac{\partial Q}{\partial \tau_0} =\dfrac{\mathcal{F''}(\tau, p)}{\mathcal{F}(\tau ,p)}\;.$	(50)

Solving equations C-5 and C-6 for $\hat{V}_N$ and $\hat{V}_H$ leads to the stripping equations 24 and 25 in the main text. Alternatively, if we replace the $\tau$ derivative of the curvature (equation C-4 ) with the squared derivative of the zero-slope traveltime $\tau^2_{0,\tau}$ (equation C-2 ) and solve again for $\hat{V}_N$ and $\hat{V}_H$ , we obtain Fowler's equations 27 and 28 in the main text.

Note that no approximations were made here, other than Alkhalifah's acoustic approximation in equation 7. In the case of isotropic or elliptical anisotropy ( $\hat{V}_{N}=\hat{V}_{H}$ ), one can just solve equation C-5 for $\hat{V}_{N}$ obtaining equation 23 in the main text.

Next: About this document ... Up: Casasanta & Fomel: Velocity-independent Previous: Acknowledgments

2011-06-25