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Rui Gong authored and Rui Gong committed May 30, 2024
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2 changes: 1 addition & 1 deletion book/tccs/pwstack/SConstruct
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from rsf.tex import *

End(options='reproduce',use='hyperref,listings,moreverb', color='ALL')
End(options='reproduce',use='hyperref,listings', color='ALL')
22 changes: 11 additions & 11 deletions book/tccs/pwstack/paper.tex
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Expand Up @@ -88,6 +88,8 @@ \section{Introduction}
with conventional NMO stack.

\section{Method}
\inputdir{.}

In geophysical estimation problems, regularization is
used to solve ill-posed problems by providing additional constraints on the estimated model.
Shaping regularization \cite[]{fomel,fomel2} implies a mapping of the input model $\mathbf{m}$ to the space of acceptable functions. The mapping is controlled by the shaping operator $\mathbf{S_m}$. In the linear case, the solution of the estimation problem using shaping regularization is defined as:
Expand Down Expand Up @@ -127,8 +129,6 @@ \section{Method}
trace inside a volume. In this application, we use the updated model to spread information
across the CMP gather using the estimated dip field.

\inputdir{.}

\plot{schem5}{width=0.7\textwidth}{Schematic of the PWC stacking algorithm. (a) Stack far offset trace T$_2$ with neighboring trace T$_1$,
(b) stack updated trace T'$_1$ with neighboring trace T$_0$, (c) final accumulated stack.}

Expand Down Expand Up @@ -158,13 +158,13 @@ \section{Method}
fashion while using shaping regularization (equation~\ref{eq:inv}) to yield a high resolution stack.

\section{Examples}
\inputdir{synseis}
In our first experiment, we generated a synthetic trace with a sampling interval of 1 ms and
used it as a reference trace. This trace was then inverse NMO corrected and subsampled to 4 ms
to produce a CMP gather (Figure~\ref{fig:cmp2}), which is the input data for PWC stack and conventional NMO and stack.
The result of applying a constant velocity NMO correction is displayed in Figure~\ref{fig:nmo0} and is used to compute
the dip field.

\inputdir{synseis}
\multiplot{2}{cmp2,nmo0}{width=0.45\textwidth}{(a) Synthetic CMP gather with 4-ms sampling interval and (b) constant velocity NMO-corrected gather to separate crossing events at far offsets.}
%\plot{nmodip}{width=0.45\textwidth}{(a) Synthetic CMP gather with 4 ms sampling interval and (b) constant velocity NMO corrected gather to separate crossing events at far offsets.}

Expand All @@ -190,6 +190,8 @@ \section{Examples}
\plot{niter}{width=0.7\textwidth}{Estimated PWC stack as function of iteration using shaping regularization
compared with conventional NMO stack and the reference trace.}

\multiplot{3}{spec5,spec6,speclow}{width=0.3\textwidth}{ Spectral comparison of the PWC stack (dashed magenta) with (a) conventional NMO and stack (solid blue) and (b) the reference trace (solid blue) with a 1-ms sampling interval. (c) PWC stack (dashed magenta) recovers lower frequencies in comparison with the conventional stack (dot dash black) and is consistent with the reference trace (solid blue).}

Low frequencies play an important role in seismic inversion for velocity and impedance models \cite[]{kroode}.
We next evaluate the algorithm's ability to recover low frequency information in the
estimated stack. We apply a low cut filter to the synthetic CMP gather to remove all of the useful low frequencies below 25 Hz.
Expand All @@ -207,7 +209,7 @@ \section{Examples}
The results indicate that the stretching effects that are prominent at far offsets and early times in the NMO-corrected gather
without a stretch mute are effectively reduced by implementing PWC stack.

\multiplot{3}{spec5,spec6,speclow}{width=0.3\textwidth}{ Spectral comparison of the PWC stack (dashed magenta) with (a) conventional NMO and stack (solid blue) and (b) the reference trace (solid blue) with a 1-ms sampling interval. (c) PWC stack (dashed magenta) recovers lower frequencies in comparison with the conventional stack (dot dash black) and is consistent with the reference trace (solid blue).}
\multiplot{3}{nmo3,nmo2,nsnmo}{width=0.3\textwidth}{(a) Conventional NMO correction, (b) NMO correction without stretch muting and (c) effective NMO using PWC stack.}

We next apply this method to a 2-D field dataset from the North Sea and compare the results to conventional NMO stack.
This dataset has 1,000 CMP locations and 800 samples per trace with a sampling interval of 4 ms.
Expand All @@ -220,6 +222,11 @@ \section{Examples}
in the PWC stacked section. Overall, events become more continuous and coherent throughout the section
using PWC stack, and resolution is noticeably improved.

\inputdir{elf}

\multiplot{2}{istack8,ungmres2}{width=0.8\textwidth}{NMO and stack using (a) conventional method and (b) shaping regularization using
plane-wave construction.}

\section{Conclusions}
Conventional NMO stack may result in lower resolution stacked sections due to distortions caused
by NMO correction and stretch muting. Treating the process of NMO and stack using regularized inversion
Expand All @@ -236,13 +243,6 @@ \section{Acknowledgments}
Seismology (TCCS) members for helpful discussions
and TCCS sponsors for their financial support.

\multiplot{3}{nmo3,nmo2,nsnmo}{width=0.3\textwidth}{(a) Conventional NMO correction, (b) NMO correction without stretch muting and (c) effective NMO using PWC stack.}

\inputdir{elf}

\multiplot{2}{istack8,ungmres2}{width=0.8\textwidth}{NMO and stack using (a) conventional method and (b) shaping regularization using
plane-wave construction.}

\onecolumn
\bibliographystyle{seg}
\bibliography{seg2016}

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