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\title{AIRS Module 12 Average SRF Fits For Data PHS386}

\author{Scott Hannon, L. Strow}\date{\today}

\begin{document}

\maketitle

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%\tableofcontents

%\listoffigures

%\begin{abstract}
%\end{abstract}

\section{Overview}
This document is really just an informal memo to George Aumann - it
won't make much sense to anybody else.

Here are some observations on retrieving the AIRS SRF from an average of
the M12 Bruker SRF "PHS386" data.  The average SRF used in the fits is
a simple average over the "good" channels without any attempt to shift
or stretch the individual channel measurements.  For M12, this makes
very little difference to the average SRF (the main effect of stretching
the SRFs to match the widths is to partially wash out the more distant
fringes in the wings).  The wavenumber scale is somewhat arbitrary; we
treat the average SRF as a channel at exactly 650 cm-1.

We are explicitely modeling the Bruker spectral response function by
integrating over the Bruker solid angle (including any offset due to
mis-alignment).  The resulting Bruker SRF is then convolved with the
trial parameterized AIRS SRF function George came up with (sum of a
pseudo- Lorentz and Gaussian functions with the same half widths, and
a total of 3 adjustable parameters).  The least-squares fit can
simultaneously vary the 3 AIRS SRF parameters, and any Bruker
mis-alignment angle.

\section{Figure 1}

Figure 1 shows the results of our attempt to retrieve the AIRS SRF.
The fit returned a Bruker off-axis angle of $\approx$0.5 degrees.
This is around 5 times larger than Lockheed-Martin's estimate of the
the maximum off-axis angle.

In the top plot, the Bruker response and retrieved AIRS SRF have been
shifted (0.104 and 0.029 cm-1 respectively) to force the maxima to
line up with the measured SRF.  This was done for sake of
presentation, since they are in fact shifted because of the Bruker
off-axis angle and FOV.  The middle plot is a blowup of the bottom few
percent of the convolved fit (AIRS and Bruker) SRF and data.  The
bottom plot shows the difference between the data and the computed
convolution of the best-fit AIRS SRF with the Bruker SRF.

\begin{center}
\begin{figure}
\includegraphics[width=0.9\textwidth]{t386srf_a.eps}
\caption{Average SRF Fit that converged to a 0.5 degree Bruker off-axis
angle. 
\tthonlyhtml{</A> <a href="t386srf_a.eps">Postscript version of figure.</a>}
}
\end{figure}
\end{center}

\section{Figure 2}

Figure 2 is similar to Figure 1, except the fit has been done with the
Bruker off-axis angle fixed at zero degrees.  In the top plot, the
AIRS SRF has been shifted by 0.104 cm-1 to align with the data and
Bruker SRF peaks.  Compare the middle and bottom plots to those in
Figure 1; the results in Figure 1 are noticeably better.  This seems
to suggest that there is either a significant off-axis angle of
$\approx$0.5 degrees, or, that the AIRS SRF is not perfectly
symmetric.

\begin{center}
\begin{figure}
\includegraphics[width=0.9\textwidth]{t386srf_d.eps}
\caption{Average SRF Fit with the Bruker off-axis angle set to zero.}
\tthonlyhtml{</A> <a href="t386srf_d.eps">Postscript version of figure.</a>}
\end{figure}
\end{center}

\section{Figure 3}

%LLS start
%Figure 3 shows the effect of uncertainty in the retrieved SRF's width on
%a simulated AIRS radiance.  The plot is for George's SRF with two
%resolutions, 1418 and 1449, corresponding to ignoring or taking into
%account the Bruker self apodization (for zero degrees off-axis).

Figure 3 shows the effect of uncertainty in the retrieved SRF
\emph{width} on a simulated AIRS radiance. The bottom plot shows the
difference in observed radiances for two different SRFs; (1) George's
SRF derived from a direct fit to the Bruker data (basically just
ignoring any broadening effects by the Bruker) and (2) the same SRF as
in (1) except its width was scaled down to account for the broadening
of the AIRS SRF by the Bruker.  This plot just illustrates what was
already known, namely that the Bruker at 3 cm OPD broadens the AIRS
SRF enough that it must be accounted for.

%LLS stop

\begin{center}
\begin{figure}
\includegraphics[width=0.9\textwidth]{t386srf_b.eps}
\caption{Effect of ignoring broadening by the Bruker in the AIRS SRF measurements.}
\tthonlyhtml{</A> <a href="t386srf_b.eps">Postscript version of figure.</a>}
\end{figure}
\end{center}

\section{Figure 4}

Figure 4 shows the effect of uncertainty in the retrieved SRF's
\emph{shape and width} on a simulated AIRS radiance.

The middle plot shows differences in radiances computed using; (1) a
SRF fit directly to the Bruker data (George's SRF) subsequently
reduced in width to account for the broadening by the Bruker, compared
to using; (2) our (UMBC) fitted SRF which takes the Bruker broadening
(sinc function) into account during the fit.  George's SRF width was
reduced so it had the same width as our fitted SRF.  The Bruker
off-axis angle was set to zero for the UMBC fit.

The bottom plot shows how two different UMBC fitted SRFs affect
radiances.  One SRF was the same as in the middle plot, namely with
the Bruker off-axis angle set to zero degrees.  The other SRF came
from a fit where the Bruker off-axis angle was varied, giving a best
off-axis angle of 0.5 degrees.  These two fits result in slightly
different AIRS SRF widths, but identical convolved widths (AIRS
convolved with the Bruker SRF).

\begin{center}
\begin{figure}
\includegraphics[width=0.9\textwidth]{t386srf_c.eps}
\caption{Sensitivity of radiances to various SRF fitting methods.  See 
  text for details.}
\tthonlyhtml{</A> <a href="t386srf_c.eps">Postscript version of figure.</a>}
\end{figure}
\end{center}


\section{Conclusions}

%It does not seem possible to reliably retrieve the AIRS SRF from the
%data.

%The AIRS SRFs appear to be significantly higher quality than those
%used in early modeling of the retrieval of the SRF from the Bruker
%data.  In fact, it may be that the 15 \um\ AIRS SRFs have ``frequency''
%components that cannot be captured with only a 3 cm OPD.  

At this point, we cannot really say if the AIRS SRF is slightly
asymmetric, or if the Bruker off-axis angle is significantly larger
than expected.  The results presented here do suggest that further
testing (longer OPD for the Bruker?, smaller FOV?) may be needed to
adequately characterize the long-wave AIRS arrays.  These small,
somewhat subtle effects do produce differences in the observed
radiances larger than requirements, some are almost \emph{10x} larger
than the single-spot noise.

Examination of short-wave SRF data might tell us more about any
asymmetry in the AIRS SRF since broadening by the Bruker will be less
important for these arrays.  However, it is not clear that we should
expect similar asymmetries in the AIRS SRF in different arrays. If
these asymmetries are due to optics they could vary (smoothly?) across
the focal plane?

Simultaneous gas cell, SRF measurements could also tell us more
definitive information about any Bruker off-axis mis-alignments.  More
work is needed to determine exact procedures for these test,
especially at 15 \um.

We also need to see if different AIRS SRF parameterizations fit the
data the same, but produce significantly different radiances.  We
might try using a voigt function (convolution of a Gaussian and
Lorentz) instead of George's sum of a Lorentz and Gaussian as an
example.  It may be that the shorter-wave SRF data will help us choose
the best parameterization for the AIRS SRF shape if we can show that
the SRF shape does not vary significantly over the focal plane.

In addition, 15 \um\ gas cell data should give us some good information 
on the AIRS SRF shape, at least down to the 1\%? level or so.  

%There are just too many uncertainties and insufficient resolution.  A
%large part of the difficulty is due to the Bruker self apodization.  The
%field of view angle (2.86 degrees) is a bit large, and the 2.88 cm optical
%path difference is a bit small.  George's equation for the SRF shape has
%3 adjustable parameters, and alllows for a wide variation in SRF shape.
%It may be possible to put some constraints on the SRF shape and width if
%we had good gas cell or air gap data to look at.  We'll begin looking
%into this with the data we already have.


%\begin{displaymath}
%R_{gas} = R(T_{BB}) \tau_{gas} \tau_{optics} + R(T_{room}) (1-\tau_{gas} \tau_{optics})
%\end{displaymath}

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%\section{Graphs:  Blackbody Temperature of 360K}

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