% Description of MOSES theory of operation 

Here we present the theory behind \textit{MOSES} measurements of spectral line
profiles. The equations are cast in a form that allows consideration of a system
with any number of spectral orders. The treatment is
readily generalizable to include dispersion in the vertical
as well has horizontal directions. An object $v(x,\lambda)$  forms images at
spectral orders $m$: 
\begin{equation}
I_m(x') = \int_B v(x'-m\lambda,\lambda)\,d\lambda,   \label{eq:mosis}
\end{equation}
where $x'$ is the detector coordinate and domain $B$ represents the passband of
the instrument. For the \textit{MOSES} rocket experiment,
$m=(-1,0,1)$.\footnote{We find this formulation more natural than the notation
developed previously in \citeN{kankelborg01}} For
convenience, position $x'$ on the detector, position $x$ on the Sun, and
wavelength $\lambda$ are all measured in pixels. The kernel is a projection
operator. For example, in the $m=1$ order, the above equation has $x = x' -
\lambda$. A pixel at position $x'$ receives light integrated along a slice
$x+\lambda = x'$ in the $x\lambda$-plane (compare figure \ref{fig:projection}).

It is also helpful to define an ``infinite'' spectral order as follows:
\begin{equation}
I_{\infty}(\lambda) = \int_F v(x,\lambda)\,dx,   \label{eq:Iinfinity}
\end{equation}
where $F$ is the instrument field of view. This projection is of considerable
interest, and will be taken up in \S\,\ref{sec:AvgSpectrum}. Once the
spectral orders are intercalibrated, then the projections of $v(x,\lambda)$ 
obey a normalization condition,
\begin{equation}
N = \int_F I_m\,dx 
   = \int_B I_{\infty}\,d\lambda = \int_B \int_F v\,dx\,d\lambda
   \label{eq:normalization}
\end{equation}
for all orders $m$. The normalization condition holds exactly if
the field of view is defined by a field stop prior to the dispersive element.
Otherwise, the projections of the three orders will cover slightly different
domains in $(x,\lambda)$ at the edges of the field of view.

\subsection{Inferring the integrated spectrum}
\label{sec:AvgSpectrum}
The average spectrum $I_{\infty}(\lambda)$ is not measured (though it could be,
in principle). We infer it by a combination of prior knowledge and reasonable
assumptions. The projection along the $x$-axis is special because it reveals the
sparseness of the spectrum; that is, $v(x,\lambda)$ is mostly zeroes
punctuated by a few spectral lines. For this reason, even a poor estimate of
$I_{\infty}$ greatly improves the inversion of equations \ref{eq:mosis} and
\ref{eq:Iinfinity} \cite{fox03}. 

The properties of the average spectrum, $I_{\infty}$, may be well known
due to prior observation; for example, previous observations indicate that  the
two spectral lines contained in the  \textit{MOSES} passband typically have a
Gaussian profile  (\citeNP{andretta00}). It is also possible to find the
position of a strong line and estimate its average line profile solely
from the data. If we assume (only as a first guess) that the image and
spectrum are separable,
\begin{equation}
v(x,\lambda) \sim \frac{1}{N} I_0(x)\,I_{\infty}(\lambda), \label{eq:assumption}
\end{equation} 
then the images at the dispersed spectral orders can be written in terms of the
image at $m=0$:
\begin{equation}
I_m(x) = \int_B \frac{1}{N}I_0(x-m\lambda)\,I_{\infty}(\lambda)\,d\lambda.
\end{equation}
Using the convolution theorem, this leads immediately to an independent
estimate of the fourier transform of the integrated spectrum for each 
$m \neq 0$. For example,
\begin{equation}
\tilde{I}_{\infty} = N \frac{\tilde{I}_{+1 }      }{\tilde{I}_0} 
                   = N \frac{\tilde{I}_{-1}^{\ast}}{\tilde{I}_0}.
\end{equation}
Equation \ref{eq:assumption} assumes that the spectrum varies only in
intensity. While this is not expected to be the case, we do expect these
solutions to approximate the average spectrum. This approach may be tested
quantitatively in at least two ways: (1) from simulation, using data from slit
spectrometers like \textit{SOHO}/SUMER; and (2) from real \textit{MOSES} data
by comparing the two independent solutions obtained from $I_{-1}$ and $I_{+1}$.
Our numerical experiments to date indicate that the profile of the strongest
line can be obtained in this way.

%What inherent ambiguities exist in the interpretation of \textit{MOSES} data?
%We may start by considering the null space of the integral equations, i.e. the
%set of perturbations to $v(x,\lambda)$ that do not affect the observed spectral
%orders. \cite{kankelborg01} have derived this null space, but the majority of
%null space elements also affect the positivity or smoothness of the solution,
%so that their contribution can be eliminated by regularization. The other
%obvious approach is to simply try inverting a wide variety of test data
%(contrived datasets for which the solution is known). The latter approach is
%less fundamental but perhaps more practical.

\subsection{Null space} 
\label{sec:nullspace}
The inversion of the integral equations \ref{eq:mosis} and \ref{eq:Iinfinity}
for $v(x,\lambda)$ is mathematically ill-posed. In principle, the three
projections obtained by \textit{MOSES} (four including $I_{\infty}$) may be
expected to contain less information than the original object (four functions
of one variable compared to a function of two variables). However, prior
knowledge about the spectrum, including its sparseness, smoothness and
positivity, makes the problem tractable in practice. An understanding of the
null space of the equations will clarify this point.

The null space is the set of possible variations of the object $v(x,\lambda)$
that lead to identical measurements. \S\,\ref{sec:fourier} points out that
the null space corresponds to a set of definite locations in the Fourier
transform plane $k_x, k_{\lambda}$. \citeNP{kankelborg01} have derived the null
space in the coordinate basis, and the following presentation follows their
derivation. Treating $I_{\infty}$ as just one of the projections $I_m$, we
express our integral equations in terms of a linear operator $T_m$:
\begin{equation}
   I_m = T_m \, v(x,\lambda).
\end{equation}
If some other interpretation $v'$ leads to the same data $I_m$, then
\begin{equation}
   I_m = T_m\, v'(x,\lambda) \quad \Rightarrow \quad T_m\, (v' - v) = 0.
\end{equation}
The null space is the set of functions $\mathcal{N} (x,\lambda)$ satisfying
\begin{equation}
   T_m \, \mathcal{N} = 0 \quad \,\forall m.
\end{equation}
Any such null space element can be written 
\begin{equation}
   \label{eq:convolvekg}
   \mathcal{N} (x,\lambda) = \int_F dx' \int_B d\lambda'\, 
      \kappa (x-x',\lambda - \lambda') \, g(x,\lambda).
\end{equation}
The function $g$ must be zero at the boundaries of $B$. If a field stop is used,
then $g$ must also be zero at the boundaries of the FOV, $F$. The kernel of
equation \ref{eq:convolvekg} is a differential operator,
\begin{equation}
   \kappa (x,\lambda) =
   \left[ \prod_m \hat{e}_m \cdot \nabla \right] \delta (x) \, \delta (\lambda),
\end{equation}
\begin{equation}
   \hat{e}_m = \frac{m \hat{x}}{\sqrt{1+m^2}} 
             - \frac{\hat{\lambda}}{\sqrt{1+m^2}}.
\end{equation}
The operator $\hat{e}_m \cdot \nabla$ is a partial derivative along the
projection axis for order $m$. For example,
\[
   \hat{e}_{0} \cdot \nabla =  -\frac{\partial}{\partial \lambda}, \quad
   \hat{e}_{\pm 1} \cdot \nabla = \frac{1}{\sqrt{2}} \left( 
      \pm \frac{\partial}{\partial x}
      -   \frac{\partial}{\partial \lambda}
      \right) , \quad
   \hat{e}_{\infty} \cdot \nabla =  \frac{\partial}{\partial x}.
\]
Since the detectors are pixellated, it is appropriate to construct $\kappa$ in
discrete space. For the \textit{MOSES} rocket instrument, it takes the form:
\begin{equation}
   \kappa = 
   \begin{pmatrix}
      0  & +1  & -1  &  0 \\
     -1  &  0  &  0  & +1 \\
     +1  &  0  &  0  & -1 \\
      0  & -1  & +1  &  0 \\
   \end{pmatrix}.
\end{equation}
This is the smallest array of pixels having the property that the sum along any
row, column or diagonal is zero; hence, the projection is zero for all $3+1$
spectral orders. Any null in a pixellated $x\lambda$-space must be a superposition
of such patterns, as expressed by equation \ref{eq:convolvekg}.

The above considerations are pivotal to solving the inverse problem. Successful
inversion schemes use an estimate for $I_{\infty}$ because it enforces the
sparseness that is inherent to real, physical line spectra. Since the null
functions have equal positive and negative flux, the positiviy requirement is
very powerful. Null functions that yield positive solutions consist of 
oscillations which, because of $I_{\infty}$, are confined within the envelope
of the line profile(s) and thus have a small period in the $\lambda$ direction.
These remaining nulls are suppressed in our inversion schemes by assuming local
smoothness of the solution. This may seem arbitrary, but it is an application
of Occam's razor: the simplest solution is \emph{a priori} most likely.
Moreover, oscillations in $\lambda$ smaller than the thermal width of the
spectral line are not physically plausible.


Noise has not been considered in this discussion. Of course noise is inherent
to the measurement process. Many inverse problems (such as image deblurring and
differential emission measure) have kernels with smoothing properties that lead
to high noise susceptibility. The kernel of equations \ref{eq:mosis} and
\ref{eq:Iinfinity} is a projection operator and therefore does not have this
undesirable property. Our trial inversions have confirmed that even direct
(Fourier) inversions of \textit{MOSES} data do not amplify the noise in the
data.

