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\begin{titlepage}
\hrule
\noindent\textbf{EUV IMAGING SPECTROMETER}\\
\vspace{-0.7\baselineskip}\\
{\noindent\Huge\bf Hinode}
\vspace{2mm}
\hrule
\vspace{3mm}
\centerline{\bf EIS SOFTWARE NOTE No. 8}
\vspace{3mm}
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\noindent Version 1.0 \hfill 29 December 2016
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\begin{center}
{\Large\bf EIS Point Spread Function}\\
\mbox{}\\
\mbox{}\\
Ignacio Ugarte-Urra\\
Naval Research Laboratory\\
4555 Overlook Av. SW\\
Washington, D.C. 20375
U.S.A.\\
\mbox{}\\
ignacio.ugarte-urra@nrl.navy.mil\\
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\section{Overview}

The present EIS note describes the current understanding about spatial
resolution in the EIS instrument based on a preliminary analysis. The note is
simply informative and describes the work that has been done to characterize the
Point Spread Function. There is, therefore, no specific software in the EIS
SolarSoft distribution addressing this topic.

\section{Previous work}

\citet{korendyke2006} reports a pre-launch spatial resolution of $\sim$2\arcsec\
as measured from EUV emission from a Penning discharge lamp. They used lines
\ion{Mg}{3} and \ion{Ne}{3} lines in the 171--200 \AA\ range and \ion{He}{2}
and \ion{Ne}{3} in the 251--284 \AA. \citet{young2013} cites a 3\arcsec--4\arcsec\
estimated value from studies of transition region brightenings discussed on the EIS
Wiki\footnote{http://solarb.mssl.ucl.ac.uk:8080/eiswiki}. Here, we revisit this
topic and report our findings of small intensity features and a comparison of EIS
raster images to AIA/SDO and TRACE images.

\section{Transition region brightenings}
\label{sect:trb}
The response of the instrument to a point source results in the point spread
function (PSF). As there is no lamp on board EIS to perform this calibration on
orbit, to study the PSF we have to rely in a natural point source. We know from
higher resolution instruments that there are features on the Sun at smaller scales
than the pre-flight resolution estimate of 2\arcsec\ ($\sim$1450 km), so we can
work with the hypothesis that EIS observes small features that act as point sources
for the instrument. The dimensions of the resulting image will give us clues about
the optical performance.

\begin{figure}
\centerline{
\epsfxsize=8.5cm\epsfbox{eis_l1_20100112_134001.eps}
\epsfxsize=8.5cm\epsfbox{eis_l1_20071211_223042.eps}
}
\caption{Transition region brightenings observed with the 40\arcsec\ slit on Jan 12, 2010
(left set of panels) and December 11, 2007 (right panels). Shown here are images for two spectral
lines, a close-up, cross-sections along Solar X (in blue) and Solar Y (in green) and
Gaussian fits. The fits only include the points along the structure and the background
(filled circles). A 2D Gaussian fit for the \ion{Si}{7} line is shown in pink.}
\label{fig:trbright}
\end{figure}

\begin{figure}
\centerline{\epsfxsize=17cm\epsfbox{aia193_eis195_slit.eps}}
\caption{Comparison of an EIS raster ({\tt eis\_l0\_20100802\_105102.fits}) in 195.119 \AA\ to a composite AIA image
made out of a sequence of AIA 193 \AA\ images that mimics the EIS scanning. Several
degraded versions of the AIA composite are shown, each convolved by a Gaussian filter
with different FWHM as labeled.}
\label{fig:eisaiaslit}
\end{figure}

In Figure~\ref{fig:trbright} we show slot images for the smallest features that
we could find. On the left, there is an active region point-like brightening
particularly prominent in transition region lines such as \ion{Mg}{6} 269.0 \AA\
and \ion{Si}{7} 275.3 \AA. The feature also has a coronal counterpart in lines
such as \ion{Fe}{11} 180.4 \AA\ (not shown here). To characterize its dimensions
we fit a Gaussian function to the well resolved cross-section along the Solar X
direction. The result is a full-width-half (FWHM) of 3 pixels in both spectral
lines, that is 3\arcsec. This sets an upper limit to the spatial resolution of
the instrument in that direction expressed in terms of the PSF. Along the Solar Y
direction the structure is larger, around 6 pixels. An elliptical PSF is a possibility
and we have, for that purpose, fitted a 2D Gaussian for the stronger \ion{Si}{7} case.
This possibility is discussed further in Section~\ref{sect:asym}. In this case we
can not rule out that the source is extended along that direction and the asymmetry
is just a consequence of the asymmetric extension. For that reason, we show in
the same figure a different transition brightening, a weaker one, that suggests
that indeed EIS can image smaller features, in this particular case a 3.5 pixels
FWHM. We should point out that the uncertainties in those fitted widths can be of
the order of a pixel.

\begin{figure}
\centerline{\epsfxsize=17cm\epsfbox{aia193_eis195_slot.eps}}
\caption{Comparison of an EIS slot raster ({\tt eis\_l0\_20100618\_111034}) in
195.119 \AA\ to an AIA 193 \AA\ image convolved with different Gaussian PSF values.}
\label{fig:eisaiaslot}
\end{figure}


\begin{figure}
\centerline{\epsfxsize=16cm\epsfbox{hpw_run_eis2imager_trace.b.eps}}
\caption{Comparison of an EIS \ion{Fe}{10} 184.5 \AA\ raster
({\tt eis\_l0\_20071211\_102542})to a TRACE
171 \AA\ image convolved with different PSF values.}
\label{fig:eistrace}
\end{figure}

\section{AIA comparisons}
In this section we proceed with an alternative approach to understand the EIS performance
by comparing it to other instruments that observe the same source at the same wavelength,
but higher resolution. That is the case for AIA/SDO with a plate scale of 0.6\arcsec\ per
pixel and a FWHM PSF of $\sim$1.2\arcsec\ \citep{grigis2012}.

In Figure~\ref{fig:eisaiaslit} we compare an EIS \ion{Fe}{12} 195.119 \AA\ raster to an
AIA composite image made of portions of several 193\arcsec\ images at different times
to mimic the EIS scanning along the Solar X direction. Taking the AIA composite image as
the `true' scale for the natural source, we convolve the image with several Gaussians of
different FWHM in AIA pixels to infer the effect of the EIS PSF. We find that a PSF with
a FWHM of 5--6 pixels ($\sim$3--3.6\arcsec) returns the closest match to
the EIS raster, in line with the results obtained in the transition region brightenings
analysis. The comparison between an EIS 40\arcsec\ slot image (Figure~\ref{fig:eisaiaslot})
and a single AIA image is consistent with this value.

Finally, we also compare an EIS raster to an image from the TRACE instrument with a pixel
scale of 0.5\arcsec\ per pixel. Figure~\ref{fig:eistrace} shows a comparison between EIS
\ion{Fe}{10} 184.5 \AA\ and TRACE 171 \AA. The best match in this particular case is around
4 FWHM pixels, that is $\sim$2\arcsec. We have looked at the Solar Y dimensions of some of
the  small features in the raster and found a range of sizes of 2.5\arcsec -- 4 \arcsec,
with the smaller fit 2.5$\pm$1.42 pixels (H.P. Warren, private communication).

\begin{figure}[htbp!]
% \begin{wrapfigure}{r}{0.6\textwidth}
\centerline{\epsfxsize=11cm\epsfbox{psf_fullSun.eps}}
  % \includegraphics[width=0.58\textwidth]{psf_fullSun.eps}
\caption{Simulation of the effect of an elliptical PSF to the Doppler shift
measurement in the 195.119 \AA\ line. From left to right: spectral profiles
for the two different source regions, intensity map at peak spectral intensity,
PSF used in the convolution and Doppler map after single Gaussian fit to the
line profiles.}
\label{fig:limb}
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\end{figure}

\section{Asymmetry}
\label{sect:asym}
As discussed in Section~\ref{sect:trb}, it is possible that
the EIS PSF is asymmetric. Figure~\ref{fig:trbright} shows that an asymmetric
2D Gaussian can fit the spatial distribution of the intensity of a point source
candidate. In that particular example, we find some degree of inclination. This is
relevant because an inclination in the elliptical spot image formed on the detector
can lead to systematic Doppler shift signatures. The slot data, nevertheless, needs
to be treated with care because the image on the detector mixes spatial and spectral
information. Any flows on the source can broaden the image in the spectral direction.
A bi-directional flow could produce an inclination on the detector if the red and
blue components of the flow are at different Y-positions.

There are, however, other indications for an elliptical PSF from narrow slit scans. Appendix B
in \citet{young2012} discusses ``Offsets between intensity and velocity structures in
EIS data'' and argues that an elliptical inclined PSF would explain characteristic
Doppler signatures in certain EIS datasets. In particular, regions with strong gradients
such as the solar limb or coronal bright points within a coronal hole. The effect of an
elliptical and inclined PSF is simulated in Figure~\ref{fig:limb}. It shows the effect
for a disk surrounded by brighter emission, analogue to the observation of a coronal
hole against the limb brightening in a coronal line. The source consists of two
regions (1 and 2) with two characteristic spectral profiles (left panel in the figure).
In the experiment, we convolve their signal with a symmetric (2\arcsec$\times$2\arcsec)
and an asymmetric inclined (2\arcsec$\times$5\arcsec) Gaussian profile. We also include
a case that is very similar to the PSF numbers found earlier in the fit to a transition
region brightening. After the convolution, we fit the spectra of every pixel with a single
Gaussian profile and display the Doppler shift displacements. The figure demonstrates
that an inclined PSF can introduce systematic features at reasonable velocity amplitudes
where there are gradients. The redshifted rim at the North pole limb has been reported by
\citet{tian2010} from an EIS raster. We have found this feature in other datasets, but
we should note that it is not present in every EIS raster of the poles.

\section{Conclusion}
The current preliminary analysis of EIS data is consistent with a PSF with a FWHM of $\sim$3
pixels (3\arcsec). There are indications that the PSF may be elliptical in shape
and inclined with respect to the slit. Users are encouraged to look for the effect
in their data and report their findings to the EIS team.

\begin{thebibliography}{}

\bibitem[Brooks et al.(2012)]{brooks2012} Brooks, D.~H., Warren, H.~P., \& Ugarte-Urra, I.\ 2012, ApJL, 755, L33

\bibitem[Grigis et al.(2012)]{grigis2012} Grigis, P., Su, Y. \& Weber, M. \ 2012, AIA Team Online Document: http://hesperia.gsfc.nasa.gov/ssw/sdo/aia/idl/psf/DOC/psfreport.pdf

\bibitem[Korendyke et al.(2006)]{korendyke2006} Korendyke, C.~M., Brown, C.~M., Thomas, R.~J., et al.\ 2006, Appl. Optics, 45, 8674

\bibitem[Lang et al.(2006)]{lang06}
  Lang, J., Kent, B. J., Paustian, W., et al., 2006,
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\bibitem[Tian et al.(2010)]{tian2010} Tian, H., Tu, C., Marsch, E., He, J., \& Kamio, S.\ 2010, ApJL, 709, L88

\bibitem[Young et al.(2012)]{young2012} Young, P.~R., O'Dwyer, B., \& Mason, H.~E.\ 2012, ApJ, 744, 14

\bibitem[Young et al.(2013)]{young2013} Young, P.~R., Doschek, G.~A., Warren, H.~P., \& Hara, H.\ 2013, ApJ, 766, 127


\end{thebibliography}


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