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\begin{flushright}
Stanford Exploration Project\\
Geophysics Department\\
Mitchell Building\\
Stanford, CA 94305 \\
\end{flushright}
\noindent
\today \\
To the Associate Editor, Geophysics:\\
\\
Re: Paper GEO-2007-0117\\
\\
This letter is in response to the review received Oct 5$^{th}$, 2007
of paper GEO-2007-0117.R1 entitled  {\it Riemannian wavefield
  extrapolation: Non-orthogonal coordinate systems}.  I thank the four
reviewers for their comments, and for catching a number of
mathematical and language typos in the manuscript.  I respond to the
individual reviewer's comments below.   
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\par
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I look forward to hearing the response from Geophysics regarding the
manuscript corrections. \\
\\ 
\noindent Sincerely,

\vspace{.15in}
\noindent Jeff Shragge\\
Stanford Exploration Project \\
Geophysics Department\\
Stanford University

\newpage

\subsection{Response to Reviewer 1}
\noindent
1) x has been changed to ${\bf x}$.
\par
\noindent
2) I have changed $\mathcal{U}$ and $\mathcal{I}$ to $U$ and $I$.
\par
\noindent
3) Actually, I only apply a single SSF correction step; however, I
   apply a mask that applies the correct local shift.  This is the
   reason why there are fewer calls to the SSF operator than the PSPI
   in Table 1.
\par
\noindent
4) A smoother version of the model has been used, and the ray-field is
   smoother.

\subsection{Response to Reviewer 2}
\noindent
Abstract: I have made all the edits recommended by the reviewer.
\par \noindent
Introduction: I have made the edit recommended by the
reviewer.
\par \noindent
Acoustic ...: I have made all the edits as recommended by the
reviewer.
\par \noindent

Numerical modeling examples: 1) These techniques are fairly easy to
implement in the code (as they are independent of the chosen
coordinate system).  However, I am choosing to keep the artifacts in
the images as they communicate common phenomena to the readers.  The
text now gives the well-known specific example of cosine boundary
tapers.

\par\noindent
Numerical modeling examples: 2) The main reason why this figure is
included is because it demonstrates that this meshcan be
used for plane-wave migration in a dynamic coordinate system.  The
plane waves are therein defined between 1000m and 9000m;
however, previously, this was not explicitly stated in the text.
Because the waves are of this breadth, they do spread to both sides of
the model as they overturn.   

\par\noindent

Generating triplication-free coordinate meshes: I agree with the
reviewer in all of the raised points.  Actually, these comments were
exactly what I was trying to convey to the reader.  That they were
taken differently by the reviewer indicates to me that my explanation
was not sufficiently clear.  I have rewritten this paragraph to state
more explicitly what I believe the reviewer was commenting on:
\begin{quote}
Differences in the modeled amplitudes at and above the salt interface
in the upper two panels are attributed to differences between
finite difference and one-way wavefield extrapolation implementations.
Finite differences better models amplitudes in the presence of
velocity gradients in the propagation direction.  Thus, 
incident energy is more accurately partitioned at the top sediment-salt 
interface leading to the lower (and more correct) amplitudes of
subsalt multi-pathed arrivals.  The RWE wavefield underestimates the
reflection contribution and allows significantly more energy to be
transmitted into and through the salt body.  This modeling inaccuracy
leads to the more pronounced multipathing below the salt body and the 
more complicated wavefield behavior relative to the Cartesian
wavefield example.  Incorporating higher-order terms
(Zhang, 2002) into the RWE formalism, though, would likely
diminish the differences between the finite difference and RWE results.
\end{quote}

\subsection{Response to Reviewer 3}
\noindent
I have gone through the text and have changed all of the ${\rm i}=\sqrt(-1)$
to be consitently ${\rm i}$ (rather than $i$).  I chose to use the Roman font
to prevent confusion, since $i$ and $j$ are often used for tensor
indicies  (e.g. $g^{ij}$).

\subsection{Response to Associate Editor}
\noindent
The prevailing issue discussed by this reviewer was that I did not
distinguish explicitly between my non-orthogonal development and that
of Sava and Fomel (2005).  I have rewritten the paragraph in question
according to the reviewer's suggestions and now moreee explicitly
state the differences between my metric tensor from that of Sava and
Fomel (2005). 
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\begin{quote}
The derivation of the 3D RWE acoustic wave equation deviates here from
that found in Sava and Fomel (2005), who represent the metric tensor
with only four independent coefficients.  The remaining two
independent coefficients are explicitly set to zero in order to
satisfy the semi-orthogonal geometry restriction.  In this
development, I follow a more general approach that represents the metric
tensor with the full six independent coefficients.  This is the key
extension developed in this paper that leads to the greater
flexibility in coordinate system design discussed below.
\end{quote}
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\par
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A second issue was that figure panels should be labeled 'a','b','c'
rather than 'upper left' etc.  I have rewritten the text and redone
the figures accordingly. 
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\par
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The reviewer also suggested a number of grammatical edits and wording
changes.  I have incorporated almost all into the updated version of
the text. 




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