\documentclass[12pt, onecolumn, titlepage]{article}
\usepackage{amsmath,amssymb}
\usepackage[dvips]{graphicx}
\usepackage{epsfig}
\usepackage{color}
\usepackage{refpkg,astrobib}
%\title{Multi-Order Slitless Imaging Spectroscopy}
\title{Simultaneous Imaging and Spectroscopy \\ of the EUV Sun:\\ 
       First Flight of \textit{MOSES}}
\author{Charles C. Kankelborg\\Department of Physics\\Montana State University}



% Detector specs
\newcommand{\CCDcols}{1024}
\newcommand{\CCDrows}{2048}
\newcommand{\micron}{$\mu$\negthickspace~m}
\newcommand{\pixsize}{13.5\,\micron} % pixel size
% \newcommand{\qe}{75~\negmedspace\%~@~300~\AA} % quantum efficiency (%)
\newcommand{\CCDtemp}{$< -30\, ^{\circ}$\!C}

% Optical specs
\newcommand{\arcmin}{\ensuremath{^{\prime}}}
\newcommand{\arcsec}{\ensuremath{^{\prime\prime}}}
\newcommand{\farcs}{.\negthickspace\arcsec}
\newcommand{\farcm}{.\negmedspace\arcmin}
\newcommand{\focal}{4.74~m} % focal length
\newcommand{\aperture}{8~cm square}
\newcommand{\fratio}{59}
\newcommand{\pixsubtent}{$0\farcs{59}$}
\newcommand{\gpitch}{950 lines/mm} % grating pitch
\newcommand{\lambdazero}{304\,\AA} % multilayer central wavelength
% \newcommand{\FWHM}{38\,\AA\ (1 bounce)} WRONG! % multilayer passband width
\newcommand{\dlambda}{29 m\AA} % milliangstroms per pixel
\newcommand{\dvelocity}{29.6\,km/s} % km/s per pixel
\newcommand{\fov}{20\arcmin\,$\times$\,10\arcmin} % field of view (arcmin)
%\newcommand{\psfzero}{$0\farcs{22}$ RMS} % spot size @ m = 0 (arcsec)
%\newcommand{\psfone}{$1\farcs{24}$ RMS} % spot size @ m = +-1 (arcsec)
%\newcommand{\instlinewidth}{52 m\AA} % instrument contribution to linewidth


\begin{document}

\maketitle

\clearpage
\tableofcontents

\clearpage
\section{Introduction}

We have demonstrated a new type of imaging spectrograph for
observing the EUV solar atmosphere.  The \textit{Multi-Order Solar EUV
Spectrograph} (\textit{MOSES}) sounding rocket payload was launched from White
Sands Missile Range February 8, 2006. High resolution images of the solar
atmosphere at $\lambda 304$\,\AA\ were obtained with line profile
information over a $20\times 10$ arc minute field of view (\S\,\ref{sec:flight}).

\textit{MOSES} is the first instrument to provide  \emph{simultaneous} imaging
and spectroscopy of dynamic events in EUV. A preliminary analysis of
the data is given in \S\,\ref{sec:prelimAnalysis}, showing two strong downflow
events in the transition region, and one explosive event. In two of the three
analyzed events, evidence is found for a two component line profile.

Much has been learned about the theory of this new instrument type
(\S\,\ref{sec:theory}). Based on these theoretical underpinnings, inversion
techniques have been developed and tested to support future in-depth analysis
of the rocket data (\S\,\ref{sec:inversion}).

In addition to the technical results which are the primary focus of this report,
the project has had considerable educational impact (\S\,\ref{sec:education}).

\section{Instrument Concept}

%The \textit{MOSES} rocket instrument represents a new kind of instrument for
%simultaneous imaging and spectroscopy over a narrow passband, with a wide field
%of view and high spatial resolution. 


The \textit{MOSES} concept, illustrated in figure \ref{fig:concept}, was first
described by \citeN{kankelborg00} and by \citeN{kankelborg01}. The instrument is
a slitless, objective grating spectrograph. By taking images at three spectral
orders $m = -1, 0, +1$, it is possible to acquire spatial and spectral
information simultaneously over a wide field of view (FOV). The passband is
limited to a few spectral lines by multilayer coatings so that the data inversion
for spectral information is tractable. In principle, detectors may be placed at
any number of spectral orders. If more orders are recorded, more spectral
information will be obtained.

\begin{figure}[htbp]
   \resizebox{\columnwidth}{!}{
   \rotatebox{0}{\input{instrument.pstex_t}}
   }
   \caption{Conceptual sketch of \textit{MOSES}.}
   \label{fig:concept}
\end{figure}


The theory of the instrument is worked out in the Appendix, \S\,\ref{sec:theory}.
A first analysis of the rocket data in \S\,\ref{sec:prelimAnalysis} validates the
concept of simultaneous imaging spectroscopy with \textit{MOSES}. What we have
learned so far about more advanced analysis techniques is presented in
\S\,\ref{sec:inversion}.



\section{Payload Design}

\subsection{Optics}

\begin{figure}[htbp]
   \resizebox{\columnwidth}{!}{
   \rotatebox{0}{\input{moses_layout_sketch.pstex_t}}
   }
   \caption{Sketch of the \textit{MOSES} optical layout. Incident
   radiation from the right is imaged by an off-axis concave grating G1 onto 
   detectors at three spectral orders $m=-1,0,+1$. The optical path is folded
   by a flat mirror.}
   \label{fig:layout}
\end{figure}

The optical configuration of \textit{MOSES} is illustrated schematically in
figure \ref{fig:layout}. Instrument optical design parameters are summarized in
table \ref{table:optics}. The design was ray traced and optimized by Roger Thomas
at Goddard Space Flight Center (GSFC) \cite{thomas01}. The major optical elements
are shown in figure \ref{fig:instrumentAssy}. Thin film aluminum filters (not
shown) cover the open end of each detector housing to make a light tight box.

\begin{figure*}[htb]
\begin{center}
\resizebox{\textwidth}{!}{\input{instrumentAssy.pstex_t}}
\caption{Layout of the \textit{MOSES} instrument. Not shown are the
   baffles, front aperture plate, shutter and LN2 plumbing. Most
   of the payload electronics and the H-alpha video imager are on
   the opposite side of the optical table.}
\label{fig:instrumentAssy}
\end{center}
\end{figure*}

\begin{table}[!htb]
         \caption{Optical Characteristics}
         \label{table:optics}
         \begin{center}
                  \begin{tabular}{|rl|} \hline
                                Focal length    & \focal                        \\
                                Aperture        & \aperture\, (\textit{f}/\fratio) \\
                                Pixel subtent   & \pixsubtent ,
                                                  \dlambda , 29\,km/s \\
                                FOV             & \fov \\
                                Grating         & 9.48\,m sphere,
                                                  \gpitch \\
                                Coatings        & B$_4$C/Mg$_2$Si multilayer,\\
                                                & Reflectivity $\sim 0.4$ ,\\
                                                & $\lambda$\lambdazero \\
                                Filters         & 1500\,\AA\ Al on Ni mesh, \\
                                                & 38\,mm diameter \\
                                                & 1 each at $m=\pm1$, 2 at $m=0$.\\
                                Lines           & He\,\textsc{ii} 303.8\,\AA, 
                                                  Si\,\textsc{xi} 303.3\,\AA\\
                                %    & \FWHM\, FWHM \\
                                \hline
                  \end{tabular}
         \end{center}
\end{table}

The grating flown on \textit{MOSES} was a Zerodur substrate with spherical figure
and laminar rulings fabricated by Carl Zeiss Laser Optics. Precise control of the
groove depth allows an optimal the distribution of light in the three central
spectral orders, where our detectors are placed. The fold flat is ULE with a
protected aluminum coating on the reverse side for instrument alignment.
Multilayer coatings were applied to the gratings and fold flat at GSFC.


Robust kinematic mounts for the primary (grating) and secondary (fold flat)
mirror mounts were patterned after mounts manufactured for the LMSAL
\textit{Solar Plasma Diagnostics Experiment} (\textit{SPDE}). Precision
hemispherical pads locate the front surface (3) and sides (2+1) of each mirror,
with spring loaded set screws opposing each hemisphere along a straight load path
through the mirror. The set screws are inserted almost to maximum spring
deflection, leaving approximately 0.001 in.\ of travel on the spring. If the mirror
position or orientation is shifted during vibration, the springs return it
immediately to alignment. The mount design makes it possible to remove the
grating and replace it with a grating ruled for work in visible light, without
changing the alignment. This  arrangement was very helpful for optical
integration and testing.

\subsection{Detectors} The detectors are three $2048 \times 1024$,
rear-illuminated E2V CCD42-20 devices at spectral orders $m=-1,0,+1$. The readout
electronics is a three-channel variant of the \textit{Solar-B}/EIS system,
contributed by Mullard Space Science Laboratory (MSSL). The power supply is also
a copy of the corresponding EIS system. Thermoelectric coolers (figure
\ref{fig:blockdiagram}) regulate CCD temperature. An aluminum block, kept cold
with liquid nitrogen (LN2) until launch, is the heat dump for the thermal control
system.

\begin{figure}[!p]
   \begin{center}
   \resizebox{0.5\textheight}{!}{  % Something is wrong with the fig size!
   \rotatebox{90}{\input{blockdiagram.pstex_t}}
   }
   \end{center}
   \caption{\textit{MOSES} electronics system.}
   \label{fig:blockdiagram}
\end{figure}

\subsection{Electronics}

A simplified block diagram of the \textit{MOSES} electronics is shown in figure
\ref{fig:blockdiagram}. Data handling and experiment control is through a single
experiment computer, a Diamond Systems Hercules EBX CPU board running a minimal
Linux distribution. A PC/104+ stack on the CPU board contains COTS cards for
serial communications, camera data interface, and high speed telemetry. Heat from
the motherboard and other stack elements are removed via heat pipes to a manifold
that is linked to the heat dump LN2 line. Figure \ref{fig:PC104stack} shows 
the PC/104+ stack.
.


\begin{figure*}[!htbp]
\begin{center}
\includegraphics[width=\textwidth]{PC104stack.eps}
\caption{The \textit{MOSES} PC/104+ stack. Heat pipes (thick copper tubes) run
from the components with greatest power dissipation to the manifold at right.}
\label{fig:PC104stack}
\end{center}
\end{figure*}



All science data is stored onboard in a 1\,GB flash disk. As much science data as
possible is sent to high speed telemetry during flight. Serial uplink and
downlink channels allow two-way communication with the experiment computer,
including a stream of detailed housekeeping data that is displayed on the
\textit{MOSES} EGSE. Standard ASCL uplinks are available to the computer through
the DIO interface that is integral to the CPU board. The DIO interface also
controls subsystem power and the shutter.

Custom circuits were designed, built and tested at MSU for power control,
randomization and filtering of the high speed telemetry stream, CCD temperature
control, pressure monitoring, timer/uplink interface, conditioning of analog
data, and driving the shutter mechanism.



\section{Calibration}

A thorough EUV calibration of \textit{MOSES} was undertaken in three stages:
\begin{enumerate}
   \item CCD flat field measurements obtained at Goddard Space Flight Center.
   \item BNL and NIST measurements of the MOSES gratings and multilayers.
   \item End-to-end calibration of the instrument at Rutherford Appleton
   Laboratory.
\end{enumerate}
The following sections expand on the specific motivation and results of each
calibration measurement.

\subsection{Flat Fields}

The spectroscopic measurement to be made by \textit{MOSES} is sensitive to
differences between, and small-scale variations within, the three images obtained
simultaneously at $m=-1, 0, 1$. Consequently, flat fielding of the three spectral
orders is especially important.

\textit{MOSES} has an unvignetted optical design. The range of field angles
subtended by the detectors is too small to show any response variation due to the
multilayer rocking curve. Consequently, the only
response variation over the field of view is due to the CCD detectors. 

A flat fielding experiment was undertaken by Lewis Fox, Roger Thomas and Marvin
Schwartz at GSFC in April, 2004.

The EUV hollow cathode source does not provide uniform illumination: it contains
large-scale variations due to the geometry of the hollow cathode and the
aperture, as well as small-scale variations due to the filter mesh. Our best
measurements of the beam profile are the flat field images themselves, taken at a
series of overlapping positions within the beam.  The flat field and beam
profile are derived by an iterative fitting procedure. Sample results are shown in
figures \ref{fig:flat}, \ref{fig:flatHistequal} and  \ref{fig:beam}.

\begin{figure*}[!htbp]
\begin{center}
\includegraphics[width=\textwidth]{det1ff_042605-2ndFlat.eps}
\caption{Flat field image for detector 1 ($m=-1$).}
\label{fig:flat}
\end{center}
\end{figure*}

\begin{figure*}[!htbp]
\begin{center}
\includegraphics[width=\textwidth]{det1ff_042605-2ndFlatHistequal.eps}
\caption{Flat field image for detector 1 ($m=-1$). The image has been
histogram equalized to bring out subtle details.}
\label{fig:flatHistequal}
\end{center}
\end{figure*}

\begin{figure*}[!htbp]
\begin{center}
\includegraphics[width=\textwidth]{det1ff_042605-2ndBeam2.eps}
\caption{Beam image derived from the detector 1 flat field data. Since images were
taken at several positions, the recovered beam image is wider than the detector.}
\label{fig:beam}
\end{center}
\end{figure*}
% Note: had to downsample the Beam image to 2048 x whatever to keep LaTeX from
% cropping the right 30% or so wantonly!!!

Analysis of flat field data for the detectors used at orders $m=-1$ and $m=0$
shows a very flat response (figure \ref{fig:flat}). The flat fields correspond
well with the lists of dark pixels provided by MSSL when the CCDs were procured.
They also show subtle details associated with the CCD manufacturing process
(figure \ref{fig:flatHistequal}).  The corresponding EUV beam estimates
demonstrate recovery of the faint shadow of the filter mesh (figure
\ref{fig:beam}).  The flat field software needs  revision to improve its handling
of signal statistics, and in its present state does not work for the $m=1$
detector. We will return to the analysis this fall, and  final flat fields for
all three detectors are expected to be available soon.


\subsection{Component Measurements}

Reflectivity measurements for the \textit{MOSES} fold flat and gratings were
made at NIST's SURF III and Brookhaven's X24C beamlines, respectively
\cite{owens05}. These measurements demonstrate the high throughput (nearly 40\%)
and strong out-of-band suppression of the B$_4$C/Mg$_2$Si multilayer coatings.
The coatings also passed standard pull tests and showed little change after
annealing for 6 hours at $100^{\circ}$\,C. To our knowledge, \textit{MOSES} is
the first space mission to use this new multilayer, which shows great promise
for future applications.

\citeN{owens05} found that, allowing for multilayer efficiency, the grating
groove efficiency is very close to the optimum specification of 50\% at $m=0$ and
25\% at $m =\pm 1$. Diffraction into higher orders is strongly suppressed.


\subsection{Measurements at RAL}
\label{sec:RAL}

A secondary goal of the rocket mission is calibration of \textit{SOHO}/CDS. The
\textit{MOSES} instrument, as configured for the 2006 flight, was  calibrated
end-to-end at Rutherford Appleton Laboratory (RAL) in Spring 2005. The photometric
accuracy of this calibration relies on the well-characterized PTB hollow
cathode source, which has been calibrated by direct comparison the BESSY
synchrotron. The effort of RAL was supported by the CDS investigation. 

A first analysis of the RAL calibration data was completed by \citeN{thomas06}.
The response to He\,\textsc{ii}\ $\lambda303.8$\,\AA\ radiation is 0.0084 DN per
incident photon in $m=\pm 1$, 0.0042 in $m=0$.


\section{Optical Testing}

The \textit{MOSES} gratings and flat mirrors were tested individually by the
manufacturers to meet specification, and subsequently at GSFC by Maria Nowak to
verify compliance. The measurements made included wavefront error, radius of
curvature, groove spacing (via placement of the spectral orders), and AFM
measurements of groove depth and profile. All delivered components met or
exceeded (and often far exceeded) their design specifications. 


\subsection{Instrument Focus}

The primary mirror (grating) has an $f$-ratio of about 60 and a radius greater
than 9\,m. Measuring this large radius proved challenging. For grating G1 which
was flown, five trials at GSFC \cite{nowak04} gave an average radius of
9482.5\,mm, with a standard deviation of 1.2\,mm and a spread of 3.2\,mm among
the measurements. The EUV depth of field for the instrument is 0.8\,mm. Similar
difficulty was encountered at MSU in locating the best instrument focus.  Our
first efforts used a Foucault test. More consistent results were obtained by
autocollimation. 

In the most successful version of the focus experiment, a large aperture, high
quality Newtonian telescope was autocollimated using a retroreflector and a
green He/Ne laser. The laser illuminated a 25\,$\mu$m spatial filter in a $\sim
5:1$ eyepiece projection. The Newtonian precision focuser was equipped with a
micrometer dial gauge for repeatable adjustments. Retroreflected light was
returned through a beam splitter behind the spatial filter to a photodiode. Best
focus (maximum return signal) was determined by curve fitting. Then the
collimated beam was used to form an image in the \textit{MOSES} central order.
This was done for a series of positions of the Newtonian precision focuser, the
best focus being found by fitting the curve of spot size vs.\ focal position.
The focus correction for \textit{MOSES} was then determined by a straightforward
optical calculation.  The outboard orders cannot be tested independently for
focus because the grating dispersion is set for EUV; for the outboard orders, we
rely on the symmetry of the grating and measure the detectors into place with
reference to the central order.

Because of the slow beam of the \textit{MOSES} system, the focus procedure
depends on fitting curves for very shallow extrema. The improvement of our
techniques for focus determination will be a priority for future flights. We
also intend to set up the grating in a Wadsworth mount to diagnose any curvature
of the ruling pattern that might lead to asymmetric focal lengths in the
outboard orders.



\section{Flight 36.193}
\label{sec:flight}

\begin{figure*}[!tp]
   \begin{center}
   \resizebox{!}{0.9\textheight}{
   \rotatebox{90}{\input{moses3sec.pstex_t}}
   }
   \end{center}
   \caption{A sample 3\,s exposure, showing all three spectral orders,  from the
   2006-Feb-09 \textit{MOSES} flight. the grayscale has been altered to compress
   the dynamic range; it is otherwise unprocessed.}
   \label{fig:moses3sec}
\end{figure*}


\textit{MOSES} was launched successfully on a Terrier Black Brant sounding
rocket from White Sands Missile Range on 2006 Feb 8 at 18:44\,UT. Live video
from the onboard H$\alpha$ imager confirmed nominal pointing. Apogee was close
to the nominal 260\,km. All science data was recovered from the intact payload.
Backup science data was received via high speed telemetry.

The exposure sequence for the February 8 flight was well chosen thanks to the
end-to-end calibration at Rutherford Appleton Laboratory (\S\,\ref{sec:RAL}).
Twenty-seven science exposures, ranging from 0.25-24\,s, were obtained above
our nominal observing threshold of 160\,km altitude. Because of the variation
in atmospheric opacity, the range of \emph{effective} exposure times is
approximately 0.13-24\,s.

Most of the 5 minute sequence consists of a well-exposed, rapid cadence movie
of 3 second exposures. An example 3\,s exposure is displayed in figure
\ref{fig:moses3sec}, which shows the central order image with insets of the
outboard orders. Image resolution and dynamic range are outstanding: the
J-shaped filament near the center of the image shows multiple resolved dark
strands, yet the brightest element of the emerging active region (near the
bottom of the figure) has barely saturated. Dark images at operating
temperature were obtained before flight and during ascent.

Coordinated observations were carried out by \textit{TRACE}, \textit{SoHO} CDS,
EIT, and several ground-based instruments. The \textit{MOSES} launch window was
timed to coincide with best \textit{TRACE} observing during eclipse season.



\section{Analysis}

The following outline summarizes the \textit{MOSES} data analysis plan.
\begin{enumerate}
   \item Reduction (\S\,\ref{sec:prelimAnalysis})
   \begin{enumerate}
      \item Dark subtraction, flat fields
      \item \emph{Coregistration} of $m=-1,0,+1$ to $0.2$ pixel (6\,km/s) 
      \item Removal of drift
   \end{enumerate}
   \item Scientific analysis
   \begin{enumerate}
      \item Preliminary analysis (\S\,\ref{sec:prelimAnalysis})
      \begin{enumerate}
         \item Spectral signatures in \emph{difference} between orders
         \item Centroid features; ``parallax'' 
            $\Rightarrow$ doppler shift
      \end{enumerate}
      \item Inversions (\S\,\ref{sec:inversion})
      \begin{enumerate}
         \item Compensate for \emph{PSF}, different in each order
         \item Inversion algorithms---apply globally or locally
      \end{enumerate}
   \end{enumerate}
\end{enumerate}
The next two sections present a preliminary analysis and describe the
capabilities we have developed for inversion of the data.


\subsection{Data reduction and preliminary analysis}
\label{sec:prelimAnalysis}

Figure \ref{fig:moses3sec} shows a minimally processed \textit{MOSES} image.
After dark subtraction and removal of flat fields, the images from the three
spectral orders must be aligned as accurately as possible. All spectral
measurements depend on this step. We use a low-order polynomial coordinate
transformation for coregistration, allowing for the slight distortion
that is present in the outboard ($m=\pm 1$) images:
\begin{align}
   x' &= \sum_{i=0}^{1} \sum_{j=0}^{1} a_{ij} x^i y^j, \\
   y' &= \sum_{i=0}^{1} \sum_{j=0}^{1} b_{ij} x^i y^j.
\end{align}
The images were deemed aligned when the sum of the square of the residuals,
normalized by intensity (in the manner of $\chi_R^2$) was minimized as a
function of the polynomial warping coefficients $a_{ij}$ and $b_{ij}$.\footnote{In this application, the most consistent results are found when
$\chi_R^2$ is normalized by the number of \emph{overlapping} pixels, which is
itself a function of the alignment parameters.} By aligning the three orders, we
effectively assume a zero mean doppler shift over the FOV; the measurement of
nonzero mean doppler shift would require calibration of the instrument with an
extremely well-collimated EUV line source, which was not available to us.  The
alignments of the three spectral orders were repeatable from exposure to
exposure, to better than 0.2 pixel RMS. No measurable relative movement of the
detectors should occur in flight, so the median alignment across all of the 3\,s
exposures was taken as the alignment for the entire sequence.

In a difference image, say $I_+ - I_0$, we expect doppler shifts to show up as
black-and-white bipolar features because of dispersion. We will now briefly
summarize the properties of three observed events that show strong spectral
signatures in $ I_+ - I_0$. These features are identified in figure
\ref{fig:features}.

\begin{figure*}[!tp]
   \begin{center}
   \resizebox{!}{0.9\textheight}{
   \rotatebox{90}{\input{features.pstex_t}}}
   \end{center}
   \caption{Location of features analyzed in this section. Exposure 19 of 27,
   $m=0$, 3\,s. The image is histogram equalized to bring out faint details.}
   \label{fig:features}
\end{figure*}

The three panels of figure \ref{fig:feature1} show Feature 1 in $I_+$, $I_0$,
and $I_+ - I_0$. The bipole in the third panel indicates a strong redshift.
Over time, an extended bipolar lobe of the same sense grows out of the bottom
of this feature. By comparing the centroids of the feature in $I_+$ and $I_0$ (figure
\ref{fig:doppler1}), it is possible to derive an average doppler shift of
36\,km/s. The dispersed order shows a bimodal shape. Evidently, there are two
distinct components: a strong redshift and a weaker blueshift.

\begin{figure*}[!p]
\begin{center}
\includegraphics[width=\textwidth]{feature-1690-0800-09.eps}
\caption{Detail  of Feature 1. The three panels are $I_+$, $I_0$,
and $I_+ - I_0$. The third panel has a 20\arcsec grid. From exposure 9 of 27,
$\sim$18:45:41\,UT, 3\,s duration.}
\label{fig:feature1}
\end{center}
\end{figure*}

\begin{figure*}[!p]
\begin{center}
\includegraphics[width=\textwidth]{feature-1690-0800-doppler1.eps}
\caption{Intensity profiles along an east-west line through the core
of Feature 1. From exposure 9, $\sim$18:45:41\,UT, 3\,s.}
\label{fig:doppler1}
\end{center}
\end{figure*}


Feature 2 (figure \ref{fig:feature2}) shows the strongest doppler shift found so
far in the data set. The feature bifurcates into two bright kernels, both
redshifted. The stronger redshift is 89\,km/s, clearly supersonic for transition
region temperatures (figure \ref{fig:doppler2}).

\begin{figure*}[!p]
\begin{center}
\includegraphics[width=\textwidth]{feature-1880-0950-26.eps}
\caption{Detail of Feature 2. The three panels
are $I_+$, $I_0$, and $I_+ - I_0$. From exposure 26, $\sim$18:49:07\,UT, 0.75\,s. The band at top is
off the edge of the detector.}
\label{fig:feature2}
\end{center}
\end{figure*}

\begin{figure*}[!p]
\begin{center}
\includegraphics[width=\textwidth]{feature-1880-0950-doppler2.eps}
\caption{Intensity profiles along an east-west line through the strongest
velocity signal in Feature 2. From exposure 26, $\sim$18:49:07\,UT, 0.75\,s.}
\label{fig:doppler2}
\end{center}
\end{figure*}

Feature 3  begins with a line broadening (figure \ref{fig:feature3})
characteristic of a transition region explosive event. The northern end of a
filament, seen in absorption, lies just to the west. Approximately one minute
after the peak of the line broadening event, jets are seen shooting north and
south from the bright kernel, parallel to the filament (figure
\ref{fig:jet3}). The bidirectional jets are offset, not from a common
origin. The velocity estimations (figures \ref{fig:core3} and \ref{fig:jet3})
are both taken from the same exposure as figure \ref{fig:feature3jet}. In figure
\ref{fig:core3}, the broad line profile of the bright core is evident. The
average velocity is essentially zero, but there appear to be two components: a
bright peak with a weak blue shift and a dim peak with a strong red shift.
Figure \ref{fig:jet3} shows the velocity estimate, $-21$\,km/s, for the blue
shifted (northern) jet.

\begin{figure*}[!p]
\begin{center}
\includegraphics[width=\textwidth]{feature-1530-0780-20.eps}
\caption{Detail of Feature 3. The three panels
are $I_+$, $I_0$, and $I_+ - I_0$. From exposure 20, $\sim$18:47:42\,UT, 3\,s.}
\label{fig:feature3}
\end{center}
\end{figure*}


\begin{figure*}[!p]
\begin{center}
\includegraphics[width=\textwidth]{feature-1530-0780-24.eps}
\caption{Detail of Feature 3. The three panels
are $I_+$, $I_0$, and $I_+ - I_0$. From exposure 24, $\sim$18:48:52\,UT, 6\,s.}
\label{fig:feature3jet}
\end{center}
\end{figure*}


\begin{figure*}[!p]
\begin{center}
\includegraphics[width=\textwidth]{feature-1530-0780-core3.eps}
\caption{Intensity profiles along an east-west line through the bright core of
Feature 3. From exposure 24, $\sim$18:48:52\,UT, 6\,s.}
\label{fig:core3}
\end{center}
\end{figure*}


\begin{figure*}[!p]
\begin{center}
\includegraphics[width=\textwidth]{feature-1530-0780-jet3.eps}
\caption{Intensity profiles along an east-west line through the upper
(northern) jet of Feature 3, showing a blueshift. From exposure 24,
$\sim$18:48:52\,UT, 6\,s.}
\label{fig:jet3}
\end{center}
\end{figure*}

\clearpage

\subsection{Inversion techniques}
\label{sec:inversion}

Much can be gained from \textit{MOSES} data using simple analysis techniques like those
demonstrated in section \ref{sec:prelimAnalysis}. A step beyond this is inversion to
obtain spectral information in every pixel.

The inversion of \textit{MOSES} data is equivalent to tomography because each
spectral order is a projection through position-and-wavelengh space (figure
\ref{fig:projection}). The data are sparse, with only a few projections. Yet,
the object $v(x,\lambda)$ itself is sparse because it contains mainly zeroes and
a few spectral lines. In rough terms, the inversion works if there are as many
projections as there are free parameters in the spectrum.

\begin{figure*}[htb]
\begin{center}
\resizebox{\textwidth}{!}{\input{projection2.pstex_t}}
\caption{A visualization of the equivalence of \textit{MOSES} data inversion to
tomography. Three spectral orders are shown. There are only a small number of
projections, so the data set is sparse.}
\label{fig:projection}
\end{center}
\end{figure*}

A variety of inversion algorithms have been explored using test data. The key
results are that (1) all the inversion techniques systematically underestimate
the doppler shifts, but (2) even with no compensation for systematic error, the
reconstructions allow doppler shifts to be determined with sub-pixel (0.2-0.5
pixel RMS) accuracy \cite{fox03}.  Test data have included generated test
patterns, SERTS spectrograms, and SUMER data cubes. The test algorithms include
Fourier backprojection (iterated for positivity), a Pixon algorithm, and Smooth
Multiplicative Algebraic Reconstruction Technique (SMART). The first two
algorithms are reported in Fox \& Kankelborg. The rest of the section will
discuss the Fourier technique, its relation to singular value decomposition
(SVD), and the SMART algorithm.

\subsubsection{Fourier inversion and SVD}
\label{sec:fourier}
The Fourier Slice theorem (e.g.\ \citeNP{kak99}) gives a direct pseudoinverse
for equations \ref{eq:mosis} and \ref{eq:Iinfinity}. The Fourier transform
of each  of our projections corresponds with a slice through the origin of the
Fourier transform of the object $v(x,\lambda)$:
\begin{equation}
\begin{split}
\label{eq:fourierSlice}
\tilde{I}_m(k_x) &= \tilde{v}(u, mu), \\
\tilde{I}_{\infty}(k_{\lambda}) &= \tilde{v}(0, k_x).
\end{split}
\end{equation}
The wedges in between these slices of 2D Fourier space are undetermined, and
therefore constitute the null space---the set of objects in $x\lambda$-space
that produce a zero signature in all projections (\S\,\ref{sec:nullspace}). The
pseoudoinverse is implemented by zeroing the null space elements and using the
Fourier-transformed data directly to fill the slices in $\tilde{v}$. It is no
coincidence that this sounds much like SVD. The slices overlap slightly near the
origin (or only at the origin, if there are just 3 orders like \textit{MOSES}).
If this overlap region is treated so that the result fits the data in a
least-squares sense, then the Fourier pseudoinverse is identical to the SVD
pseudoinverse. 

The above reasoning (as well as the logic of projection) implies that null space
elements have no DC component. They are oscillatory in nature, with a zero
average along each of the measured (or inferred) projections
(\S\,\ref{sec:nullspace}). \citeN{kankelborg01} derive the same properties of the
null space by a different approach, expressing it interms of a compact,
oscillating kernel function $\kappa(x,\lambda)$. Even crude knowledge of
$I_{\infty}$ constrains $v(x,\lambda)$ to be zero or very small over most of its
domain; this combined with the positivity requirement leaves little room for the
null space to operate, because the negative parts of any null space element must
fall within the spectral line profile and be small enough in magnitude to avoid
negatives. Since we expect line profiles to be smooth on the scale of the thermal
width, a local smoothness constraint in the $\lambda$-direction can further
reduce the ambiguity.

By itself, the pseudoinverse leads to unphysical solutions: they show a
cross-hatched ringing pattern with many negative intensities. In the Fourier
inversion method developed by \citeN{kankelborgFourier02} (also
\citeNP{kankelborgFourier03}; \citeNP{fox03}), a trial solution is iteratively
made nonnegative in real space, while the relevant slices in Fourier space are
corrected to fit the data using equations \ref{eq:fourierSlice}. The null space
wedges become filled in such a way as to make the solution positive. Local
smoothness is intrinsic to this procedure.

\subsubsection{SMART inversion}
\label{sec:smart}

SMART (Smooth Multiplicative Algebraic Reconstruction Technique,
\citeNP{kankelborgSMART}) is related to long-established algebraic
reconstruction techniques (ART) developed for computed tomography
\cite{kak99}. SMART differs from ART in that it begins with a positive
initial guess and refines the fit to all spectral orders iteratively by
multiplicative intensity adjustments. Positivity is thus inherent in the
algorithm. Variable smoothing is  introduced at each iteration, with the goal
of convergence at $\chi^2_R=1$. SMART is much faster than any of the techniques
we have described previously, though the results from all these
algorithms are similar. Results with a SUMER data cube have shown good
separation of a weak coronal spectral line (near the noise floor, in fact) from
a strong transition region spectral line (figure \ref{fig:sumer2}). The same
reconstruction shows 0.2 pixel RMS accuracy in doppler shift reconstruction for
the strong TR line (figure \ref{fig:sumer1}). In this example, velocities are
computed only where there are 150 or more counts per pixel. The largest doppler
shifts are about 1.5 pixels.

\begin{figure*}[!bp]
\begin{center}
\includegraphics[width=\textwidth]{sumer2.eps}
\caption{Reconstruction of integrated line intensities from a SUMER data cube by
the SMART algorithm.}
\label{fig:sumer2}
\end{center}
\end{figure*}

\begin{figure}[!tp]
\begin{center}
\resizebox{0.5\textwidth}{!}{\input{sumer1.pstex_t}}
\caption{Estimation of doppler shifts from a SUMER data cube by the SMART
algorithm. The middle panel shows the result of doppler shift estimation using
slit spectrometer data with the same signal-to-noise ratio.}
\label{fig:sumer1}
\end{center}
\end{figure}

 
\section{Conclusions} 

A new concept in high resolution imaging spectroscopy has been explored, and its
usefulness has been demonstrated for observing the solar atmosphere in EUV.
\textit{MOSES} obtains narrowband images at rapid cadence like \textit{TRACE}
\cite{schrijverNewView99}, but provides enough spectral information to measure
doppler  shifts anywhere within the field of view. Data from the first rocket
flight have already yielded uniquely comprehensive data on transition
region downflows and an explosive event.

Much more will be learned from the \textit{MOSES} data when we invert to obtain
spectra. We have elaborated a theory of the instrument, developed several
inversion algorithms and completed successful inversions of simulated data.
Before inversion algorithms can be applied to the rocket data, we must deduce the
exact nature of the differences in PSF among the three spectral orders, and
develop analysis techniques to compensate for these differences in the
inversion. Solving this problem is the next goal of our analysis plan.

Though not discussed in the body of this report, we see evidence of the
Si\,\textsc{xi} $\lambda 303.3$\,\AA\ line when we compare the  \textit{MOSES}
difference images with EIT 171\,\AA\ data. This evidence, and our prior simulation
experience, suggest that spectral inversions will yield a usable Si\,\textsc{xi}
coronal image. 
% Doppler shift measurement is unlikely since the coronal line is
% typically an order of magnitude weaker than the He\,\textsc{ii} line.

The \textit{MOSES} rocket payload demonstrates the feasibility of a class of
instruments with  new observational capabilities. It is hoped that this
development will find fruitful application in future space science missions
(\citeNP{2001orbiter}; \citeNP{2001orbiter2}; \citeNP{harra04};
\citeNP{deforest04}). 

\section*{Acknowledgments}

The success of \textit{MOSES} owes much to collaborators around the world. Roger
Thomas and collaborators at GSFC provided optical design, optical  testing,
multilayer coatings, and extensive support of calibration efforts. Mullard Space
Science Laboratory (MSSL), with funding from PPARC, contributed a 3-CCD version
of the EIS camera. The MSSL team provided invaluable assistance at solving
interface issues and supporting our use of the camera. Marilyn Bruner influenced
the optomechanical design and provided many words of wisdom. Her optical table
system, developed for \textit{SPDE}, flew on \textit{MOSES} thanks to the cooperation of
Lockheed-Martin Solar and Astrophysics Laboratory (LMSAL). The CDS program,
Richard Harrison, Barry Kent and colleagues at Rutherford Appleton Laboratory
invested many euros and hours in supporting our end-to-end calibration. Mission
Manager Bill Payne, NSROC electrical engineers Jim Diehl and John Aasen, and the
SPARCS team at WSMR provided unstinting, highly professional support of the
mission from concept through launch. 

The development and launch of \textit{MOSES} was supported by the NASA Solar \&
Heliospheric Low Cost Access to Space program, grant NAG5-10997. Additional
funding to support MSU students involved in \textit{MOSES} was provided by the
MSU Undergraduate Scholars Program, which is in turn supported by the Montana
Space Grant Consortium. Lewis Fox has been supported for three years through a
NASA GSRP award.

\appendix
\clearpage
\begin{center}
{\large APPENDIX}
\end{center}
\addcontentsline{toc}{section}{APPENDIX}


\section{Theory}
\label{sec:theory}
\input{moses_theory.tex}


\section{Education}
\label{sec:education}

The \textit{MOSES} mission has had far reaching educational impacts. Most of the
payload, including optomechanics, custom circuits and software, was designed and
built by undergraduate students under the mentorship of the PI and his graduate
student.  More than 30 students contributed meaningfully to this work. Many of
these students have graduated and taken high tech jobs, including one working at
LMSAL as a mechanical engineer on \textit{STEREO}/SECCHI.

Graduate student J.\ Lewis Fox will base his dissertation on  \textit{MOSES}.
Mr.\ Fox joined the project shortly after its inception in 2001. He led the team
that developed the instrument electronics, did most of the system engineering,
wrote the interface specifications for the payload, performed EUV calibration
experiments at GSFC and RAL, and did most of the instrument operations during
flight. He has developed and tested algorithms for data inversion
(\citeNP{fox02}; \citeNP{fox03}), and started modeling transition region
turbulence \cite{2003VelocityRedistribution} that might explain observed
linewidths and intensities.

%The \textit{MOSES} instrument is the first sounding rocket experiment developed
%at Montana State University. 





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