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Stanford Exploration Project\\
Geophysics Department\\
Mitchell Building\\
Stanford, CA 94305 \\
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January 11th, 2007\\
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To the Associate Editor, Geophysics:\\
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Re: Paper GEO-2006-0303
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This letter is in response to the review received December 7$^{th}$,
2006 of paper GEO-2006-0303 entitled  {\it Riemannian wavefield
  extrapolation: Non-orthogonal coordinate systems}.  I thank the
Associate Editor and three reviewers for their comments, and for
catching a number of mathematical and language typos in the
manuscript.  I respond to the reviewer's comments in two sections
below.  In 'General Comments', I address issues that were common to
most or all reviewers.  This is followed by point-by-point responses
of more specific concerns.

\subsection{General Comments}

1) Efficiency and Computational Costs - Most of the reviewers
   stated that the manuscript should include a commentary on the
   efficiency of the RWE algorithm relative to Cartesian-based
   extrapolation.  I have added an additional section after the
   extrapolation examples entitled ``Implementation Costs''. 
   The accompanying table presents the results of a benchmark test
   that conveys the additional overhead costs of the RWE approach.  I
   also discuss the extra memory requirements required to hold
   the additional coefficients in memory.

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2) Additional Migration example - I agree with the reviewers that
   another RWE migration example would be informative.  (A migration
   example was included in the Expanded Abstract volumes of the 2005
   and 2006 SEG Annual conventions.)  However, I am not entirely sure
   that a conventional migration example best demonstrates the
   approach.  RWE is most effective when it is used to model
   wavefields that have a particular directionality (e.g., Green's 
   functions or overturning waves).  Recently, I submitted a 'Letter
   to Geophysics' that uses one-way RWE operators as the forward
   modeling and migration component of waveform inversion.  In
   particular, I show that RWE on a coordinate system oriented in the
   direction of the overturning wavefield accurately models the early
   arrivals important for waveform inversion.  I would argue that this
   'migration' example better demonstrates the utility of the RWE
   approach.   (The document is attached in support.)
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   If the AE feels that a conventional migration results is a
   requirement for publication, though, then I am willing to generate
   a Sigsbee migration image as suggested.  Because this task is not
   just a 'minor revision', I would ask that AE for an additional 3
   weeks to complete what I feel is a 'moderate revision'. 

\subsection{Response to the Associate Editor}

1) The $a_i$ in equation 12 is supposed to represent the elements of
the vector $\mathbf{a}$ directly below equation 12.  I agree that this
is a confusing notation.  This is modified to read ``where
$\mathbf{a}$ is a vector of non-stationary coefficients''. 
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2) The editorial comments have been addressed accordingly.

\subsection{Reviewer 1}

1) I have included a third panel in figure 6 that compares the RWE-
   and Cartesian-based extrapolation result to a two-way
   finite-difference propagation result.  
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2) The extra 't' has been added to 'he'.

\subsection{Reviewer 2}

1) The reviewer is correct that the step from equation 6 to 7 is strictly
   valid only for a medium of constant slowness.  In practice, this
   problem is addressed using approximation techniques that develop
   the standard extrapolation operators (e.g. PSPI, SSF, etc).   I 
   have added the following sentence to reflect this: 
\begin{quote}
   Note that the use
   of these dual operators is strictly accurate only for constant
   slowness functions.  Situations where $s$ spatially varies leads to
   a non-physical and simultaneous space-wavenumber dependence.
   However, this is routinely handled through various approximations
   that are discussed below.''
\end{quote}
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2) I agree with the reviewer that the word ``extended split-step
Fourier'' is confusing.  What I am trying to convey is that one can
use extend the single split-step Fourier approach to `multi-coefficient
split-step Fourier''.  This is required because we are now dealing
with up to 10 mixed-domain coefficients.  I have changed the word
``extended'' throughout the manuscript such that it reads
``multi-coefficient''.  
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3) Equations C-3 and C-9 have been changed to be in accordance
with the correct expression in equation 11.

\subsection{Reviewer 3} 

1) I have added the following sentence at the bottom of page 8 stating
   explicitly which extrapolation operator is used to generate the
   examples: ``All that results in
   the following sections were generated with the combined PSPI plus
   SSF correction extrapolation operator.''
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2) This point is covered in the last sentence of the paragraph two on
   page 11: 
\begin{quote}
``Note that the propagation creates explainable boundary
artifacts: reflections on the left are due to a truncated
coordinate system and hyperbolic diffractions on the right are
caused truncated plane-waves.'' 
\end{quote}
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3) I have included a third panel in figure 6 that compares the RWE-
and Cartesian-based extrapolation result to a two-way
finite-difference propagation result.  This panel demonstrates that
the RWE-generated Green's function is quite close (at least
kinematically) to the expected result.
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4) The figure captions for both figures 6 and 8 have been expanded in
detail. 
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I look forward to hearing the response from Geophysics regarding the
manuscript corrections. 
\\ 
\noindent Sincerely,

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\noindent Jeff Shragge\\
Stanford Exploration Project \\
Geophysics Department\\
Stanford University

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