Index: trunk/doc/release.2015/systematics.20140411/diffusion.tex
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+% \documentclass[iop,floatfix]{emulateapj}
+\documentclass[10pt,preprint]{aastex}
+% \pdfoutput=1
+
+% see latex.readme.txt for notes on using the PS1 template
+%\documentclass[12pt,preprint]{aastex}
+%\documentclass[manuscript]{aastex}
+%\documentclass[preprint2]{aastex}
+%\documentclass[preprint2,longabstract]{aastex}
+
+\RequirePackage{graphicx}
+\RequirePackage{color}
+\RequirePackage{code}
+\RequirePackage{pbox}
+\input{astro.sty}
+
+\usepackage[T1]{fontenc}% (2) specify encoding
+
+% online version may use color, but print version needs b/w
+\def\plotmode{col}
+%\def\plotmode{bw}
+
+%\def\plotext{pdf}
+\def\plotext{ps}
+
+%\def\picdir{/home/eugene/chipresid.20140404}
+%\def\picdir{/data/kukui.2/eugene/chipresid.20140404}
+\def\picdir{pics}
+
+% Pick a terse version of the title here;
+\shorttitle{Charge Diffusion Variations in PS1}
+\shortauthors{E.A. Magnier et al}
+\begin{document}
+\title{Charge Diffusion Variations in Pan-STARRS\,1 CCDs}
+
+% this is a crude trick to get the order of affiliations right.  These
+% names are used in the affiliations below.  The user needs to (1) set
+% the order and numbers to have the correct sequence in the author
+% list and (2) re-order the list at the bottom (and comment-out as needed)
+\def\IfA{1}
+\def\CfA{2}
+\def\MPIA{3}
+\def\Princeton{3}
+\def\USNO{4}
+\def\JHU{1}
+
+% This example has a first author from UH:
+\author{
+Eugene A. Magnier,\altaffilmark{\IfA}
+J.~L. Tonry, \altaffilmark{\IfA}
+D. Finkbeiner,\altaffilmark{\CfA}
+E. Schlafly,\altaffilmark{\MPIA}
+%PS Builder List
+W.~S. Burgett,\altaffilmark{\IfA}
+K.~C. Chambers,\altaffilmark{\IfA} 
+% L. Denneau,\altaffilmark{\IfA}
+% P. Draper,\altaffilmark{\DUR}
+H.~A. Flewelling,\altaffilmark{\IfA}
+% T. Grav,\altaffilmark{\IfA}
+% J. N. Heasley,\altaffilmark{\IfA}
+K. W. Hodapp,\altaffilmark{\IfA}
+% M. E. Huber,\altaffilmark{\IfA}
+% R. Jedicke,\altaffilmark{\IfA}
+N. Kaiser,\altaffilmark{\IfA}
+R.-P. Kudritzki,\altaffilmark{\IfA}
+% G. A. Luppino,\altaffilmark{\IfA}
+% R. H. Lupton,\altaffilmark{\Princeton}
+% E. A. Magnier,\altaffilmark{\IfA}
+N. Metcalfe,\altaffilmark{\DUH}
+% D. G. Monet,\altaffilmark{\USNO}
+% J.~S. Morgan,\altaffilmark{\IfA}
+% P. M. Onaka,\altaffilmark{\IfA}
+% P.~A. Price,\altaffilmark{\Princeton}
+% C.~W. Stubbs,\altaffilmark{\CfA}
+% W.~E. Sweeney,\altaffilmark{\IfA}
+% J.~L. Tonry, \altaffilmark{\IfA}
+R. J. Wainscoat,\altaffilmark{\IfA} and 
+C. Z. Waters,\altaffilmark{\IfA}
+PS Builders TBA
+} % this bracket terminates author list
+
+% The ordering here should be sequential, matching the sequence in the list of authors:
+\altaffiltext{\IfA}{Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu HI 96822}
+\altaffiltext{\CfA}{Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138}
+% \altaffiltext{\Princeton}{Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA}
+% \altaffiltext{\USNO}{US Naval Observatory, Flagstaff Station, Flagstaff, AZ 86001, USA}
+% \altaffiltext{\JHU}{Department of Physics and Astronomy, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA}
+\altaffiltext{\MPIA}{Max Planck Institute for Astronomy, K\"onigstuhl 17, D-69117 Heidelberg, Germany}
+\begin{abstract}
+
+Thick back-illuminated deep-depletion CCDs have superior quantum
+efficiency over previous generations of thinned and traditional thick
+CCDs.  As a result, they are being used for major wide-field imaging
+cameras in several projects.  We use observations from the Pan-STARRS
+$3\pi$ survey to characterize the behavior of the deep-depletion
+devices used in the Pan-STARRS\,1 Gigapixel Camera.  We have
+identified systematic spatial variations in the photometric behavior and
+stellar profiles which are similar to the so-called ``tree rings''
+identified in devices used by other wide-field cameras (DECam and
+Hypersuprime Camera).  The tree-ring features identified in these
+other cameras result from lateral electric fields which displace the
+electrons as they are transported in the silicon to the pixel
+location.  In contrast, we show that the photometric and morphological
+modifications observed in the GPC1 detectors are caused by variations
+in the vertical charge transportation rate and resulting charge
+diffusion variations.
+\end{abstract}
+
+% insert additional keywords as appropriate:
+\keywords{Surveys:\PSONE }
+
+\section{INTRODUCTION}\label{sec:intro}
+
+CCD detectors have evolved greatly since they were first introduced
+for astronomical imaging in the mid 1970s.  In addition to the
+well-known increases in the size of CCDs over the past 4 decades, CCD
+architecture has gone through three major evolutionary stages.  
+
+The first generation of CCDs used a silicon substrate a few hundred
+microns thick on top of which gate structures were deposited to define
+the pixels.  A positive voltage applied to the gate layers would
+create a shallow region (\approx 10 microns thick) in which the holes
+were depleted.  This ``depletion region'' acted as a potential well to
+trap electrons, specifically those generated by absorbed photons.  The
+thick silicon substrate required illumination from the ``front'' side
+with the thin gate structures to allow the photons to reach the
+depletion region and be detected.  These early CCDs had modest quantum
+efficiency as photons were easily absorbed by the several micron thick
+gate structures.  For an excellent review of the history of CCD
+development, see \cite{1992ASPC...23....1J}.
+
+Thinned, backside-illuminated CCDs such as the TI 3PCCD
+\citep{1981SPIE..290....6B} were developed to address the quantum
+efficiency limitations of the first generation thick CCDs.  The
+silicon substrate was removed using a chemical process, leaving a
+delicate device only \approx 10 - 20\micron\ thick, exposing the
+depletion region on the backside.  Photons entering the backside of
+the device are not blocked by the gate structures and thus more easily
+absorbed and detected.  Thinned backside-illuminated CCDs have high
+quantum efficiency to blue photons.  However, as the wavelength
+increases beyond \approx 800 nm, the silicon becomes more transparent
+to the photons, with a corresponding drop in quantum efficiency for
+red photons.  In addition, thin film interference between the entering
+photons and those reflecting off the front side of the CCD result in
+``fringe'' patterns for redder photons.
+
+Early generations of CCDs were made of low-resistivity (\approx 10 -
+50 $\Omega$-cm) silicon.  Following experiments beginning in the early
+1990s \citep{Holland.1996}, CCDs made from thick, high-resistivity ($
+> 10 k\Omega$-cm) silicon were developed for astronomical instruments
+in the early 2000s \citep{Holland.2003}.  The high-resistivity of the
+silicon allows for depletion regions of hundreds of microns in depth,
+compared to \approx 10\micron\ for the low-resistivity silicon.  This
+modification allows for a back-illuminated CCD with a relatively thick
+silicon subtrate of 75 - 300\micron.  Blue photons impinging on the
+back of the device are absorbed near the back surface of the device
+and are caried through the depletion region to the gates on the front
+side.  The thick silicon allows red photons to have a greater chance
+to be absorbed, increasing quantum efficiency in the red.  Because
+these thick, deep-depletion devices have near-unity quantum efficiency
+across a very wide spectral range, they have become the design of
+choice for many modern, large-scale CCD cameras (e.g., Pan-STARRS
+GPC1, \citealt{2009amos.confE..40T}; Subaru Hypersuprime Camera,
+\citealt{2010SPIE.7735E..3FK}; Dark Energy Survey Camera,
+\citealt{2015AJ....150..150F}).
+
+While these deep-depletion CCDs seem to be ideal, they do have
+features which can cause challenges for precise measurements.  As a
+result of the ``Brighter-Fatter Effect''
+\citep{2014JInst...9C3048A,2015JInst..10C5032G}, the profile of bright
+stars are measured to be wider than the profiles of faint stars.  The
+accepted interpretation is that the electric fields produced by the
+electrons accumulated from a star repel successive incoming electrons,
+with the repulsion increasing the more electrons have accumulated.
+
+The effects of lateral electric fields are likewise identified as the
+cause of the so-called ``tree rings'' observed in the flat-field,
+astrometry, and photometry response of thick deep depletion detectors
+\citep{2014PASP..126..750P}.  These tree-ring patterns have been noted
+in the flat-field response of deep depletion devices since their early
+testing \citep[see, e.g., Figure 2 in][]{2010SPIE.7735E..1RE} and were
+initially considered to be a sensitivity response which could be
+removed with a flat-field.  As discussed in detail by
+\cite{2014PASP..126..750P}, these tree rings are more correctly
+interpretted as variations in the effective pixel area due to
+migration of the electrons pushed by lateral electric fields induced
+by small changes in the doping used to set the resistivity of the
+silicon.  The changes in the effective area result in changes to the
+apparent flat-field response as well as the astrometric response of
+the detector.  More subtly, the flat-field response changes, since
+they do not reflect actual variations in sensitivity, can lead to
+systematic photometry errors for astronomical sources if the
+flat-field images are used in the standard fashion.
+
+In this paper, we examine the behavior of an apparently-similar kind
+of tree ring observed in the Pan-STARRS GPC1 CCDs.  Although we also
+observe the pixel effective area changes caused by lateral electric
+fields as described by \cite{2014PASP..126..750P}, we show below a
+second effect which is more important in driving systematic photometry
+errors.  We find that variations in charge diffusion, also resulting
+from changes in the silicon doping structures, affect both the
+observed stellar profiles as well as the photometry measured with
+profile fitting techniques.  In Section~\ref{sec:PS1}, we discuss the
+Pan-STARRS telescope, camera, and survey data used in this analysis.
+In Section~\ref{sec:tree.rings}, we present the tree-ring
+patterns as observed in several different types of measurements:
+flat-field response, systematic photometry residuals, systematic
+astrometric residuals, and stellar profile shape variations.  In
+Section~\ref{sec:discussion}, we discuss the interpretation of
+patterns we observe and present a simple model to explain the observed
+behavior.  We conclude with a discussion of the implications of this
+effect on astronomical measurements from deep depletion instruments
+
+\section{Pan-STARRS1}
+\label{sec:PS1}
+
+The 1.8m Pan-STARRS\,1 telescope (PS1), located on the summit of
+Haleakala on the Hawaiian island of Maui, has been surveying the sky
+regularly since May 2010 \citep{chambers2017}.  From May 2010 through
+March 2014, PS1 was run under the aegis of the Pan-STARRS Science
+Consortium to perform a set of wide-field science surveys; since March
+2014, operations have been supported primarily by NASA's Near Earth
+Object Observation program, see \cite{2015IAUGA..2251124W}.  Under the
+PS1SC, the largest survey, both in terms of area of the sky covered
+($3\pi$ steradians) and fraction of observing time (56\%), was the
+\TPS\ in which the entire sky north of Declination $-30$\degrees\ was
+imaged up \approx 80 times over 4 years.  These observations were
+distributed over five filters, \grizy, and have been astrometrically
+and photometrically calibrated to good precision
+\citep{magnier2017.calibration}.
+
+% 2004SPIE.5489..667H == PS1.optics
+% 2008SPIE.7014E..0DO == PS1.GPCB
+% 2009amos.confE..40T == PS1.GPCA
+% 2012ApJ...756..158S == ubercal
+The wide-field PS1 telescope optics \citep{2004SPIE.5489..667H} image
+a 3.3 degree field of view on a 1.4 gigapixel camera
+\citep[GPC1][]{2009amos.confE..40T}, with low distortion and generally
+good image quality.  The median seeing for the \TPS\ data vary
+somewhat by filter: (\grizy) = (1.31, 1.19, 1.11, 1.07, 1.02)
+arcseconds.  Routine observations are conducted remotely from the
+Advanced Technology Research Center in Kula, the main facility of the
+University of Hawaii's Institute for Astronomy operations on Maui.
+
+GPC1 \citep{2009amos.confE..40T}, currently the largest astronomical
+camera in terms of number of pixels, consists of a mosaic of 60
+edge-abutted $4800\times4800$ pixel detectors, with 10~$\mu$m pixels
+subtending 0.258~arcsec. These CCID58 detectors, manufactured by
+Lincoln Laboratory, are 75\micron-thick back-illuminated CCDs
+\citep{2006amos.confE..47T,2008SPIE.7021E..05T}.  Initial performance
+assessments are presented in \cite{2008SPIE.7014E..0DO}. The active,
+usable pixels cover \approx 80\% of the FOV.
+
+\subsection{Data Processing and Calibration}
+
+% PS1_IPP = \bibitem[Magnier(2006)]{PS1.IPP} Magnier, E.\ 2006,
+% Proceedings of The Advanced Maui Optical and Space Surveillance
+% Technologies Conference, Ed.: S. Ryan, The Maui Economic Development
+% Board, p.E5
+
+Images obtained by PS1 are processed by the Pan-STARRS Image
+Processing Pipeline (IPP;
+\citealp{2006amos.confE..50M,magnier2017.datasystem}).  All
+observations are processed nightly, with results sent to groups within
+the science consortium (i.e., PS1SC during the \TPS) performing
+short-term science projects (e.g., searching for transient and moving
+objects).  In addition, the \TPS\ dataset has been re-processed
+several times with improved calibration and analysis techniques.  To
+date (2017 July), 3 re-processings starting from raw pixel data have
+been performed.  The labels PV0, PV1, PV2, PV3 are used identify the
+nightly processing and successive re-processing versions.  PV3 has
+been used for the public release of the Pan-STARRS \TPS\ data via the
+{\it Barbara A. Mikulski Archive for Space Telescopes} (MAST) at the
+Space Telescope Science Institute.\footnote{http//panstarrs.stci.edu}
+
+The data processing and calibration operations are discussed in detail
+in elsewhere
+\citep{magnier2017.analysis,magnier2017.calibration,waters2017}.
+We re-visit here a number of points that are of significance to this
+study.  Images are processed following a fairly standard sequence of
+image detrending, source detection, and initial calibration
+(astrometric and photometric) of those detected sources.  Additional
+standard processing critical to PS1 science operations includes
+geometric transformation (`warping') and image combinations (summed
+stacks and differences).  For the purposes of this analysis, we are
+only considering the sources detected in the individual exposures from
+the initial analysis steps.
+
+% Magnier.belgium:
+% \bibitem[Magnier(2007)]{PS1.photometry} Magnier, E.\ 2007, The Future 
+% of Photometric, Spectrophotometric and Polarimetric Standardization, ASP Conference Series {\bf 364}, 153 
+
+%IPP astrometry (NOT USED)
+% \bibitem[Magnier {\it et al.}(2008)]{PS1.astrometry} Magnier, E.~A., Liu, 
+% M., Monet, D.~G., \& Chambers, K.~C.\ 2008, IAU Symposium, {\bf 248}, 553 
+
+As discussed in \cite{waters2017}, image detrending includes
+flat-field processing with a single epoch flat-field image for each
+filter.  The flat-field image used for this analysis has been
+generated by median-combining dome flat-field images (after
+pre-processing and pixel outlier rejections) and then multiplying by a
+photometric flat-field correction image generated by the analysis of a
+grid of images of a dense stellar field.  The purpose of this second
+step is to correct the basic flat-field image for errors arising from
+the non-uniformity of the illumination, from non-pixel uniformity due
+to the varying optical distorition across the field, and any other
+factors which may make the flat-field image inconsistent with stellar
+photometry, e.g., SED, filter band-pass variations, etc
+\citep[see][]{waters2017,2004PASP..116..449M,2007ASPC..364..153M}.
+This correction was made on a relatively coarse grid across the focal
+plane in order to accumulate sufficient statistics from the stars in
+the relatively small number of images available at the time.  We have
+found that a single flat-field set can be used for all PS1
+observations to yield photometric systematic errors at the level of \approx
+2\%.  PS1 benefits in this regard from the stability of having a
+single instrument which is rarely removed.
+
+Photometry of the PS1 images is performed using a
+point-spread-function (PSF) model as well as multiple kinds of
+apertures \citep{magnier2017.analysis}.  In this analysis, we refer to
+aperture photometry performed using an aperture defined based on the
+image quality observed for a given chip.  The aperture diameter is set
+to be 3.75 times the FWHM for the image.
+
+To improve the photometric systematic errors beyond the level achieved
+with a single (photometrically corrected) flat-field set, the PS1
+photometry is re-calibrated within the databasing system based on the
+properties of the measured photometry.  The calibration process is
+discussed by
+\cite{2012ApJ...756..158S,2013ApJS..205...20M,magnier2017.calibration}.
+As part of this process, several flat-field corrections have been
+determined.  For the PV2 analysis discussed here, a flat-field
+correction determined during the ubercal analysis
+\citep[see][]{2012ApJ...756..158S} consisted of an $8\times 8$ grid of
+corrections for each GPC1 chip, corresponding to a correction for each
+OTA ``cell'' and filter for each of 4 seasons.  The boundaries of
+those seasons are tentatively identified with modifications to the
+baffle structures or the system optics.  The critical point here is
+that the final effective flat-field image for the PV2 dataset is based
+on a dome-flat at the highest resolution, with very low resolution
+corrections based on photometry, resulting in photometric systematic
+uncertainties in the range 7 - 12 millimagnitudes, depending on the
+filter \citep{2013ApJS..205...20M}.
+
+For all objects, positions are measured from the PSF model for the
+brighter sources (using a non-linear fitting process) and from a
+simple centroid (1st moment) for the fainter source
+\citep{magnier2017.analysis}.  These position measurements are
+used in the astrometric analysis.  The astrometric calibration is
+discussed by \cite{magnier2017.calibration}; for the PV2
+dataset, the typical systematic floor is \approx 15 - 20
+milliarcsecond for individual measurements of brighter stars. 
+
+\section{Tree-Ring Patterns}
+\label{sec:tree.rings}
+
+\begin{table}
+% \tiny
+\begin{center}
+\caption{Systematic Trends : Standard deviation by filter\label{table:sigmas.by.filter}}
+\begin{tabular}{|l|rrrrr|}
+\hline
+{\bf Filter} & {\bf psf mags} & {\bf ap mags} & {\bf astrom} & {\bf smear} & {\bf flat} \\
+             & mmags         & mmags          & mas          & pixels$^2$  & mmags \\
+\hline
+\gps & 11.8 & 13 & 8.0  & 0.169 &  3.0 \\ 
+\rps & 10.9 & 12 & 6.7  & 0.133 &  2.2 \\
+\ips &  8.5 & 10 & 6.0  & 0.069 &  1.7 \\
+\zps &  8.7 & 12 & 5.5  & 0.052 &  3.2 \\
+\yps & 16.5 & 26 & 6.8  & 0.059 & 15.3 \\
+\hline
+\end{tabular}
+\end{center}
+\end{table}
+
+For many of the GPC1 OTA CCDs, we observe a spatial pattern in the
+photometric residuals for each device which is similar in appearence
+to the tree rings described in the Dark Energy Camera (DECam) by
+\cite{2014PASP..126..750P}.  This pattern consists of systematic
+deviations which are consistent in a set of circular arcs centered on
+the corner of the CCD, as shown in Figure~\ref{fig:psfmags.by.filter}.
+The details of the analysis used to generate
+Figure~\ref{fig:psfmags.by.filter} are given below.  For now, we note
+that the GPC1 CCDs are constructed by dividing the circular silicon
+wafer into 4 inscribed squares.  Thus the corners of the CCDs lie in
+the center of the silicon boule, just as the center of the circular
+tree rings described by \cite{2014PASP..126..750P} match the center of
+the boule from which they came.  This gives the impression that a
+similar mechanism is responsible for the pattern observed in the PS1
+photometry and the DECam photometry, namely the diffusive effects of
+lateral electric field variations in the detectors.  In the next
+section, we will make the case that the patterns observed in the PS1
+photometry residuals are {\em not} caused by this mechanism, but are
+instead caused by variations in the {\em vertical} electric field (the
+field direction perpendicular to the CCD surface).
+
+First, in this section, we will describe how we have measured the
+presence or absence of these tree-ring patterns in 5 types of data.
+For all of these examples, we use a single GPC1 CCD (XY40) to
+illustrate the effects in detail, but a similar set of effects are
+seen in many of the GPC1 detectors.  First, we show the residual PSF
+photometry.  Second, we show the residual aperture photometry.  Third,
+we show the astrometric residual patterns.  Fourth, we show the
+patterns observed in the flat-field images.  Finally, we show
+measurements derived from the second-moments of the stars.
+
+For all effects discussed below, we are measuring the mean value of
+the effect in 10x10 pixel superpixels across the detector.  The
+resulting images are all constructed so that a given superpixel
+represents the same range of true GPC1 XY40 pixels regardless of the
+type of measurement.  To generate the photometry, astrometry, or
+second-moment plots, measurements were extracted from the PV0 DVO
+database \citep{magnier2017.calibration} for observations covering
+the region ($\alpha$,$\delta$) = (90\degree\ -- 150\degree,
+-25\degree\ -- 10\degree).  This region of the sky provides a fairly
+high density of stars, but avoids the Galactic Plane where confusion
+may potentially contaminate the measurement.  We limit the analysis to
+good measurements (\ippmisc{PSF_QF} $>$ 0.85, see
+\citealt{magnier2017.analysis}) of likely stars ($|m_{psf} -
+m_{aper}| < 0.2$).  Only measurements with instrumental magnitude $<
+-8.0$ ($-2.5\log \mbox{cts sec}^{-1} < -8.0$) are included to ensure
+reasonable signal-to-noise per measurement.  We require at least 2
+measurements in a given filter and at least 5 measurements total for
+any star included in the analysis.
+
+\subsection{Photometric Residuals}
+
+% PSF Magnitudes
+\def\figwidth{5.2in}
+\def\jumpleft{-2.6in}
+\def\capwidth{2.4in}
+\begin{figure*}[htbp]
+\begin{center}
+\parbox[b]{\figwidth}{\includegraphics[width=\figwidth]{\picdir/dmag.\plotext}}
+\hspace{\jumpleft}
+\parbox[b]{\capwidth}{
+\caption{PSF Magnitude residuals by filter (\grizy).  White boxes are
+  GPC1 cells which have been masked due to poor response.  Superpixels
+  representing regions of $10\times10$ pixels are used to determine
+  the median deviation for measurements at the given chip pixel
+  location compared with the average photometry for the given
+  object.} \label{fig:psfmags.by.filter}}
+\end{center}
+\end{figure*}
+
+% Aperture Magnitudes
+\begin{figure*}[htbp]
+\begin{center}
+\parbox[b]{\figwidth}{\includegraphics[width=\figwidth]{\picdir/dapmag.\plotext}}
+\hspace{\jumpleft}
+\parbox[b]{\capwidth}{
+\caption{Aperture Magnitude residuals by filter (\grizy).  White boxes
+  are GPC1 cells which have been masked due to poor response.
+  Superpixels representing regions of $10\times10$ pixels are used to
+  determine the median deviation for measurements at the given chip
+  pixel location compared with the average photometry for the given
+  object.  } \label{fig:apmags.by.filter}}
+\end{center}
+\end{figure*}
+
+Figure~\ref{fig:psfmags.by.filter} shows the 2D patterns of PSF
+photometry residuals.  In this case, we select PSF magnitude
+measurements for detections of stars which fall in the given
+superpixel.  We subtract each measurement from the average magnitude
+for the object in the selected filter ($\delta m_{psf} =
+\overline{m}_{psf} - m_{psf}$) to determine the residual magnitude,
+excluding as an outlier any measurement with $|\delta m_{psf}| > 0.5$.
+For a given superpixel, we measure the median of the $\delta m_{psf}$
+distribution.  The figure shows $\delta m_{psf}$ for each filter
+(\grizy).  The dynamic range of the color scale is from -20 to +20
+millimagnitudes for all 5 plots.
+
+The tree-ring pattern is clearly visible for the four blue filters,
+but finging dominates the pattern for \yps.  Small offsets of
+individual cells are also apparent for \zps.  While the patterns are
+clear across the image, the signal-to-noise of the structures per
+pixel is not very strong in these images.  The per-pixel standard
+deviations of these plots are listed in
+Table~\ref{table:sigmas.by.filter}.  The per-pixel standard deviation
+is comparable to the amplitude of the correlated structures, so we
+need to integrate along the radial structures to make stronger
+statements about these patterns.
+
+Figure~\ref{fig:apmags.by.filter} shows the equivalent measurement for
+aperture photometry instead of PSF photometry.  The finging
+pattern again dominates the plot for \yps, but the tree rings are not
+seen in any of the filters.  A diagonal pattern is visible in \gps
+which is not observed in the PSF magnitudes.  While the per-pixel
+scatter is somewhat (10\% to 20\%) higher for these aperture
+magnitudes than for the PSF magnitudes
+(Table~\ref{table:sigmas.by.filter}), a structure with the amplitude
+of the PSF magnitude tree-rings would certainly have been obvious.
+
+\subsection{Astrometric Residuals}
+
+% astrometry radial term
+\begin{figure*}[htbp]
+\begin{center}
+\parbox[b]{\figwidth}{\includegraphics[width=\figwidth]{\picdir/drad.\plotext}}
+\hspace{\jumpleft}
+\parbox[b]{\capwidth}{
+\caption{Astrometric residuals of the displacement in the radial
+  direction, relative to the chip coordinate -5,4960 (upper left
+  corner), by filter (\grizy).  White boxes are GPC1 cells which have
+  been masked due to poor response.  Superpixels representing regions
+  of $10\times10$ pixels are used to determine the median deviation
+  for measurements at the given chip pixel location compared with the
+  average astrometry for the given
+  object. } \label{fig:astrom.by.filter}}
+\end{center}
+\end{figure*}
+
+Figure~\ref{fig:astrom.by.filter} shows a similar type of measurement
+for astrometric residuals.  To generate this plot, we use the same
+selection of measurements for astrometry as for photometry.  In this
+case, we extract the residual in both the RA and DEC directions
+($\delta RA = \overline{RA} - RA_i$, $\delta DEC = \overline{DEC} -
+DEC_i$) and rotate these values to the chip coordinate system ($\delta
+X,\delta Y$) using our knowledge of the chip orientation on the sky.
+We again exclude as bad any measurement with $|\delta X|$ or $|\delta
+Y| > 0.5$ arcsec before measuring the median values for each
+superpixel.  We have determined the approximate center of the circular
+tree-ring pattern as (-5,4960) for this particular chip based on the
+pattern of the X astrometry displacements.  Using this coordinate as the center
+of the pattern, we have converted the $\delta X,\delta Y$ offsets into
+$\delta R,\delta \theta$ measurements ($\delta R$ : radial component
+away from the center, $\delta \theta$ : tangential component).
+
+Figure~\ref{fig:astrom.by.filter} shows the 2D patterns of $\delta R$
+for each filter (\grizy).  The dynamic range of the color scale is
+from -20 to +20 milliarcseconds for all 5 plots.  A tree-ring
+pattern is visible for all five filters, with systematic structures
+following a circular pattern centered on the chip corner; the finging
+pattern is not apparent in the \yps\ astrometry.  The per-pixel
+standard deviations of these plots area listed in
+Table~\ref{table:sigmas.by.filter}.  The signal-to-noise of these
+structures is again somewhat weak, but the pattern is clearly visible
+in these figures.
+
+\subsection{Flat-field Structures}
+
+% flat-field residual
+\begin{figure*}[htbp]
+\begin{center}
+\parbox[b]{\figwidth}{\includegraphics[width=\figwidth]{\picdir/dflat.\plotext}}
+\hspace{\jumpleft}
+\parbox[b]{\capwidth}{
+\caption{Flat-field high-frequency structues, by filter (\grizy).
+  White boxes are GPC1 cells which have been masked due to poor
+  response.  Flat-field images generated using a tunable laser have
+  been combined (see text); a smoothed version has been subtracted to
+  high-pass the response.  Flat-field pixels are averaged for
+  $10\times10$ superpixels. } \label{fig:flats.by.filter}}
+\end{center}
+\end{figure*}
+
+% 2012ApJ...750...99T = Tonry et al PS1 phot system
+Figure~\ref{fig:flats.by.filter} shows the high-spatial-frequency
+structures in the flat-field images.  For this measurement, we have
+used a set of monochromatic flat-field images obtained with a tunable
+laser.  The laser is used to illuminate our flat-field screen which is
+then observed by the PS1 telescope.  These flat-field images were
+obtained 2011 Feb 09 as part of a campaign to study the PS1 system
+response \citep{2012ApJ...750...99T}.  Flats were obtain in a set of
+4nm steps sampling the spectral response curve of each filter.  To
+enhance the signal-to-noise, we have median-combined a set of 6 flats
+at the wavelength center of the corresponding filter.
+
+In order to mask pixels which do not flatten well, we generate a copy
+of the image smoothed with a Gaussian kernel with $\sigma = 1.5$
+pixels.  Any pixels in the smoothed image which deviate from the
+median value in the image by more than 4 standard deviations are
+masked.  We generate the superpixel image by averaging the unmasked
+pixels associated with each superpixel.  
+
+Figure~\ref{fig:flats.by.filter} shows the superpixel images for the
+flat-fields in the five filters.  These flat-field images are
+displayed as fractional deviations relative to the median flat-field
+image and can thus be compared to the magnitude residuals.  When
+flattening an image, these flat-fields would be divided into the flux
+of the raw image.  The residuals are thus defined in the sense that a
+positive feature in these flats which does {\em not} represent a real
+quantum efficiency deviation would induce a {\em reduction} in the
+measured flux in those pixels, and thus a {\em negative} deviation in
+$\delta m_{psf}$ as defined above.  The dynamic range of the color
+scale in these plots is -0.01 to +0.01.  The tree-ring pattern is
+strong in the (\gps,\rps,\ips) images, but nearly swamped by fringing
+in \zps, and completely lost to finging in \yps.  A diagonal banding
+pattern is seen in \gps: this features is thought to be due to the
+lithography process used to generate the CCD.  A blob can also been
+seen covering 4 cells near the center of this chip; this is apparently
+a deposit of some kind on the detector.  Both of the latter two
+effects behave like quantum efficiency variations and are removed well
+by standard flat-field techniques.  Note that a small amount of the
+diagonal banding pattern remains in the aperture magnitude residuals
+for \gps.  For the rest of this article, we ignore these features and
+concentrate on the tree ring features.
+
+In order to suppress the large-scale structures for a quantitative
+analysis of the tree rings, we high-pass filter the superpixel image
+by subtracting a copy smoothed with a Gaussian of $\sigma = 3.0$
+superpixels.
+
+\subsection{Second Moments}
+
+% Smear Images
+\begin{figure*}[htbp]
+\begin{center}
+\parbox[b]{\figwidth}{\includegraphics[width=\figwidth]{\picdir/smear.\plotext}}
+\hspace{\jumpleft}
+\parbox[b]{\capwidth}{
+\caption{Average residual smear variations, by filter (\grizy).  White
+  boxes are GPC1 cells which have been masked due to poor response.
+  The residual smear ($\sigma^2_{\mbox{major}} + \sigma^2_{\mbox{minor}}$) has been
+  determined after the after PSF second moments have been subtracted
+  for each image; these values are averaged for each $10\times10$
+  superpixels.  } \label{fig:smear.by.filter}}
+\end{center}
+\end{figure*}
+
+% Shear Images
+\begin{figure*}[htbp]
+\begin{center}
+\parbox[b]{\figwidth}{\includegraphics[width=\figwidth]{\picdir/shear.\plotext}}
+\hspace{\jumpleft}
+\parbox[b]{\capwidth}{
+\caption{Average residual shear variations, by filter (\grizy).  White
+  boxes are GPC1 cells which have been masked due to poor response.
+  The residual shear ($\sigma^2_{\mbox{major}} - \sigma^2_{\mbox{minor}}$) has been
+  determined after the after PSF second moments have been subtracted
+  for each image; these values are averaged for each $10\times10$
+  superpixels.  } \label{fig:shear.by.filter}}
+\end{center}
+\end{figure*}
+
+During the image analysis, the second moments are measured for all
+stars.  The values can be used to assess changes in the shape of stars
+on the image.  To measure changes in the shapes, we have extracted the
+second moments for all stellar detections, subject to the same
+selections as for the photometry and astrometry residuals (good stars,
+multiple detections).  The second moments are measured with a Gaussian
+weighting function, with the $\sigma_{w}$ scaled by the PSF size so
+that the $\sigma$ measured for PSF stars is \approx 65\% of
+$\sigma_{w}$.  (Note that, since the measured $\sigma$ of stellar
+objects is biased down by the weighting function, this is not quite
+the same as having $\sigma_{w} = 1.6$ times the true PSF $\sigma$, see
+discussion in \citealt{magnier2017.analysis}).  For each stellar
+detection, we extract the values $M_{xx,xy,yy} = \sum F_i w_i (x^2, x
+y, y^2) / \sum F_i w_i$.  For each exposure, we find the median second
+moments for PSF objects on this chip (XY40) and subtract those median
+values from the instantaneous measurements of $M_{xx,xy,yy}$.  We then
+determine the median of the residual second moments for each
+superpixel, resulting in 3 images ($\delta M_{xx,xy,yy}$) for each
+filter.
+
+Using the second moment images, we can construct certain interesting
+combinations, inspired by discussions of lensing measurements \citep{1995ApJ...449..460K}:
+\begin{eqnarray}
+e_0 & = & \delta M_{xx} + \delta M_{yy}  \\ 
+e_1 & = & \delta M_{xx} - \delta M_{yy}  \\
+e_2 & = & \sqrt{e_1^2 + 4 \delta M_{xy}}
+\end{eqnarray}
+For a 2D Gaussian profile with an elliptical contour, these values are
+related to the shape of the elliptical contour as follows:
+\begin{eqnarray}
+e_0 & = & \sigma^2_{\mbox{major}}  + \sigma^2_{\mbox{minor}} \\
+e_1 & = & (\sigma^2_{\mbox{major}}  - \sigma^2_{\mbox{minor}}) \cos (2 \theta) \\
+e_2 & = & \sigma^2_{\mbox{major}}  - \sigma^2_{\mbox{minor}}
+\end{eqnarray}
+Where $\sigma_{\mbox{major}}$ and $\sigma_{\mbox{minor}}$ are the
+major and minor axis dimensions of the ellipse and $\theta$ is the
+position angle.  Thus, $e_0$ is a measurement of the change in the
+size of the stellar PSFs as a function of position in the detector
+(``smear''), $e_2$ is a measurement of the change in ellipticity of
+the stellar PSFs (``shear''), and we can determine the angle of the
+PSF ellipticity from the $e_1$ term.
+
+Figure~\ref{fig:smear.by.filter} shows the spatial trend of $e_0$, the {\em
+  smear}.  This value corresponds to the increase or decrease in
+the circularly-symmetric component of the image size.  The dynamic
+range of these images is -0.3 to +0.3 pixel$^2$. A tree-ring
+pattern is visible for all 5 filters, though \yps is dominated by the
+fringing pattern.  Structures with relatively low spatial frequencies
+can also be seen.
+
+Figure~\ref{fig:shear.by.filter} shows the spatial trend of $e_2$, the
+{\em shear}.  This value is positive definite and is plotted with a
+color scale ranging from -0.02 to 0.22 pixel$^2$.  We can also
+determine the orientation of the corresponding ellipse.  Overlayed on
+Figure~\ref{fig:shear.by.filter} is a set of vectors representing the
+ellipse orientation as a function of postion.  The length of the
+vectors corresponds to the value of $\sigma^2_{major} -
+\sigma^2_{minor}$.  The tree-ring structure is {\em not} apparent
+in this figure for any filter.  The spatial variations are
+low-frequency and unrelated to the radial trend from the upper-left
+corner.
+
+\subsection{Correlations Between Tree-Ring Patterns}
+
+% All Effects in r-band
+\begin{figure*}[htbp]
+\begin{center}
+\parbox[b]{\figwidth}{\includegraphics[width=5.0in]{\picdir/all.effects.r.\plotext}}
+\caption{All 6 measured effects for \rps.  This figure illustrates the
+  different spatial structure observed for each of the 6 patterns
+  measured in this work.  The PSF magnitude (upper-left) and smear
+  residuals (lower-left) have a very clear common tree-ring structure,
+  while the astrometric residual (middle-left) and flat-field
+  residuals (middle-right) have their own common tree-ring pattern with
+  much higher frequencies than the previous two effects.  Aperture
+  magnitude (upper-right) and shear residuals (lower-right) do not
+  show a strong signal consistent with either of the two patterns.} \label{fig:all.effects.rband}
+\end{center}
+\end{figure*}
+
+\begin{table}
+% \tiny
+\begin{center}
+\caption{Systematic Trends : Correlations by filter\label{table:correlation.by.filter}}
+\begin{tabular}{|l|rrrr|}
+\hline
+{\bf Filter} & {\bf smear} & {\bf psf mags} & {\bf astrom} & {\bf flat} \\
+\hline
+\gps & 1.00 & 1.00 &  1.00 & 1.00 \\ 
+\rps & 0.78 & 0.84 &  0.84 & 0.76 \\
+\ips & 0.40 & 0.50 &  0.66 & 0.64 \\
+\zps & 0.16 & 0.26 &  0.37 & 0.33 \\
+\yps & 0.10 & 0.10 &  0.25 & 0.30 \\
+\hline
+\end{tabular}
+\end{center}
+\end{table}
+
+Tree-ring patterns are clearly seen in 4 of the measurement types
+above: the PSF photometry, the astrometry, the flat-field, and the
+smear terms.  As discussed above, the signal-to-noise per pixel in the
+plots of the systematic trends is relatively low (\approx 1.0).  While
+the tree-ring patterns are apparent in many of these figures,
+there are also some other systematic structures which may degrade the
+signal further.
+
+To quantatatively compare the tree-ring trends between
+filters and between the types of measurements, we need to measure the
+tree-ring structure explicitly and filter out the other effects if
+possible.  To do this, we have applied a high-pass filter to all of
+the relevant images (PSF photometry residuals, astrometric residuals
+in the radial direction, flat-field residuals, and second moment smear
+terms) to remove unrelated spatial structures.  We have then measured
+the median of the signal in radial bins centered on (-5,4960) across
+an arc from $\phi$ = -20\degrees\ to -50\degrees (as measured relative
+to the top row of the images.  We have selected a small fraction of
+the arc to minimize the error associated with the choice of the
+pattern center and to avoid several bad cells near the bottom of the
+chip.
+
+% \note{include the arc on one of the figures?}
+
+% \note{do plots of all filter pairs in a triangle?  is that interesting?}
+
+For a given type of measurement, the systematic effect is strongly
+correlated between filters.  The strongest correlation is the smear
+term: Figure~\ref{fig:smear.trends} shows the correlation of the smear
+pattern between \gps\ and the other four filters. Even \yps\ is
+strongly correlated with \gps\ despite the presence of the fringe
+pattern.  PSF photometric residuals are also correlated between
+filters, as shown in Figure~\ref{fig:psfmag.trends}.  Here, the
+\yps\ correlation with \gps\ is quite weak: the fringing pattern
+dominates the tree rings for PSF photometry.  The radial component of
+the astrometric residual is also well correlated between filters, with
+no loss of correlation due to fringing in \yps. Finally, the
+flat-field residuals are generally correlated between filters, but
+both \zps\ and \yps\ are affected by fringing.  For \yps, the
+correlation is completely washed out by the very strong fringing
+pattern.
+
+For all four types of measurements, the slope of the fitted lines are
+listed in Table~\ref{table:correlation.by.filter}.  There is a
+consistency in the trend from \gps, with the strongest systematic
+tree-ring effects to \yps, with the weakest effects.  Note that the
+second moment smear and astrometry terms have different relative
+strength in \yps\ compared with \gps.
+
+% smear trends by filter
+\def\figwidth{6.5in}
+\begin{figure*}[htbp]
+\begin{center}
+\includegraphics[width=\figwidth]{\picdir/smear.trends.\plotext}
+\caption{Correlation of the smear ($\sigma^2_{\mbox{major}} +
+  \sigma^2_{\mbox{minor}}$) signal in \gps\ with the other 4 bands:
+  \rps\ (upper-left),  \ips\ (upper-right), \zps\ (lower-left), \yps\ (lower-right).
+} \label{fig:smear.trends}
+\end{center}
+\end{figure*}
+
+% psfmag trends by filter
+\def\figwidth{6.5in}
+\begin{figure*}[htbp]
+\begin{center}
+\includegraphics[width=\figwidth]{\picdir/psfmag.trends.\plotext}
+\caption{Correlation of the PSF magnitude residuals ($\delta m_{psf}$)
+  in \gps\ with the other 4 bands: \rps\ (upper-left), \ips\
+  (upper-right), \zps\ (lower-left), \yps\ (lower-right).
+} \label{fig:psfmag.trends}
+\end{center}
+\end{figure*}
+
+% astrom trends by filter
+\def\figwidth{6.5in}
+\begin{figure*}[htbp]
+\begin{center}
+\includegraphics[width=\figwidth]{\picdir/astrom.trends.\plotext}
+\caption{Correlation of the radial astrometric residual displacement ($\delta R$)
+  in \gps\ with the other 4 bands: \rps\ (upper-left), \ips\
+  (upper-right), \zps\ (lower-left), \yps\ (lower-right).
+} \label{fig:astrom.trends}
+\end{center}
+\end{figure*}
+
+% flat trends by filter
+\def\figwidth{6.5in}
+\begin{figure*}[htbp]
+\begin{center}
+\includegraphics[width=\figwidth]{\picdir/flat.trends.\plotext}
+\caption{Correlation of the flat-field tree-ring structures in \gps\
+  with the other 4 bands: \rps\ (upper-left), \ips\ (upper-right), \zps\
+  (lower-left), \yps\ (lower-right).  } \label{fig:flat.trends}
+\end{center}
+\end{figure*}
+
+An important question is the relationship of the tree-ring
+pattern between the different types of measurements.  Different models
+for the tree-ring structures make different predictions about the
+correlations between different effects.  Note the very different
+spatial structure between the different measurements in a given
+filter: the radial variations do not all follow the same patterns.
+Instead, we find the following relationships hold:
+
+First, the PSF magnitude residuals and the second-moment smear trends
+are strongly anti-correlated: regions which have larger PSFs than the
+mean tend to have smaller measured PSF fluxes than the mean (note that
+$\delta m_{psf}$ is defined so that positive values correspond to
+larger fluxes).  These trends are shown in
+Figure~\ref{fig:smear.vs.psfmag}.  
+
+Second, the radial derivative of the smear is anti-correlated with the
+radial component of the astrometric residuals: $\frac{\partial
+  (\sigma^2_{major} + \sigma^2_{minor})}{\partial radius} \sim \delta
+R$ (see Figure~\ref{fig:dsmear.vs.astrom}).
+
+Finally, the radial derivative of the radial component of the
+astrometric residual is anti-correlated with the flat-field residual
+errors: $\frac{\partial \delta R}{\partial radius} \sim \delta flat$
+(see Figure~\ref{fig:dastrom.vs.flat}.  This last relationship is
+somewhat weakly measured.  Because of the periodic nature of the Tree
+Rings, it is also difficult to be completely certain that the
+flat-field is proportional to the derivative of the astrometry
+residual, rather than the astrometry residual being proportional to
+the derivative of the flat-field.  The correlation is somewhat weaker
+for derivative of the flat-field vs astrometry residual.  The
+correlation is very weak between the flat-field and the astrometry
+residual values without a derivative.  We are convinced that we have
+the sense of the derivative correct by examination of specific
+features in each imaage.
+
+\begin{table}
+% \tiny
+\begin{center}
+\caption{Systematic Trends : Correlations between trends\label{table:correlation.by.trend}}
+\begin{tabular}{|l|rrr|}
+\hline
+{\bf Filter} & {\bf psf mags} & {\bf $\grad$ smear} & {\bf $\grad$ astrom} \\
+             & {\bf vs smear} & {\bf vs astrom}     & {\bf vs flat}        \\
+\hline
+\gps & -0.056 & -0.060 & -0.47  \\ 
+\rps & -0.071 & -0.073 & -0.45  \\
+\ips & -0.077 & -0.095 & -0.45  \\
+\zps & -0.082 & -0.078 & -0.17  \\
+\hline
+\end{tabular}
+\end{center}
+\end{table}
+
+% smear vs psfmag
+\def\figwidth{6.5in}
+\begin{figure*}[htbp]
+\begin{center}
+\includegraphics[width=\figwidth]{\picdir/smear.vs.psfmag.\plotext}
+\caption{Correlation of the PSF magnitude residuals ($\delta m_{PSF}$)
+  with the smear ($\sigma^2_{\mbox{major}} + \sigma^2_{\mbox{minor}}$)
+  signal for \gps\ (upper-left), \rps\ (upper-right), \ips\ (lower-left),
+  \zps\ (lower-right).
+} \label{fig:smear.vs.psfmag}
+\end{center}
+\end{figure*}
+
+% dsmear vs astrom
+\def\figwidth{6.5in}
+\begin{figure*}[htbp]
+\begin{center}
+\includegraphics[width=\figwidth]{\picdir/dsmear.vs.astrom.\plotext}
+\caption{
+Correlation of the radial astrometric residual displacement ($\delta
+R$) with the derivative of the smear ($\partial
+\sigma^2_{\mbox{major}} + \sigma^2_{\mbox{minor}}$) signal with
+respect to the radial postion for \gps\ (upper-left), \rps\
+(upper-right), \ips\ (lower-left), \zps\ (lower-right).
+} \label{fig:dsmear.vs.astrom}
+\end{center}
+\end{figure*}
+
+% dastrom vs flat
+\def\figwidth{6.5in}
+\begin{figure*}[htbp]
+\begin{center}
+\includegraphics[width=\figwidth]{\picdir/dastrom.vs.flat.\plotext}
+\caption{
+Correlation of the derivative of the radial astrometric residual
+displacement ($\delta R$) with respect to the radial position with the
+flat-field tree-ring signal for \gps\ (upper-left), \rps\ (upper-right),
+\ips\ (lower-left), \zps\ (lower-right).
+} \label{fig:dastrom.vs.flat}
+\end{center}
+\end{figure*}
+
+\section{Discussion}
+\label{sec:discussion}
+
+These trends measured above (Section~\ref{sec:tree.rings}) help to
+illuminate the underlying causes of these different effects.
+
+First, if we consider the smear pattern
+(Figure~\ref{fig:smear.by.filter}), the measurement shows that the
+intrinsic sizes of the stellar images are varying in a radial sense
+between the different tree-ring regions.  Although images experience
+an average image quality (due to seeing and focus) across the chip
+which may vary substantially from exposure to exposure, stars landing
+in the different tree-ring regions are consistently somewhat
+larger or somewhat smaller than that average.
+
+Next, we can explain the correlation between the PSF photometry
+residuals and the observed smear (Figure~\ref{fig:smear.vs.psfmag}).
+In the photometry analysis, we model the PSF allowing for some spatial
+variation in the shape.  However, we have a limited number of stars to
+measure any spatial variation.  Thus the 2D variations are sampled on
+a very coarse (e.g., $3 \times 3$) grid for each chip: the PSF
+parameters may vary smoothly across the chip following the bilinear
+interpolation between the $3 \times 3$ grid points.  Thus, the spatial
+scale on which we model PSF variations is much larger than the spatial
+scale on which PSF variations are actually occuring, as illustrated
+by the changes in the smear plot (Figure~\ref{fig:smear.by.filter}).
+When the true PSF is larger than the model PSF, our model fits
+systematically underestimate the amount of flux in a given object.
+Conversely, when the true PSF is smaller, we overestimate the flux -- this
+type of offset is a typical effect when mis-estimating the PSF size.
+The slope of the trend depends on the mean typical seeing for the
+given filter.  For example, the \gps\ seeing is typically 1.3\arcsec,
+corresponding to a Gaussian $\sigma$ of 2.15 pixels.  A smearing of
+$\sigma^2_{major} + \sigma^2_{minor} = 0.1$ pixels$^2$ would increase
+the size by about 0.02 pixels, or 1\%, roughly consistent with the
+observed photometric deviation of about 5 to 10 millimags for this
+amount of smearing.
+
+The correlation between the flat-field structures and the radial
+derivative of the astrometric residual displacements in the radial
+direction (Figure~\ref{fig:dastrom.vs.flat}) is consistent with radial
+variations in the plate-scale.  The tree-rings observed by DES are
+completely attributed to effective plate scale changes.  Effective
+plate scale changes result in flat-field deviations because the
+flat-field illumination is a source of constant surface brightness.
+Pixels see a varying amount of flux depending on their effective area.
+This changing plate scale also affects the astrometry since these
+variations occur on spatial scales much smaller than the astrometric
+model.  In this description of the tree rings, the flat-field
+deviations are $-1 \times \frac{\partial \delta R}{\partial r}$.  The
+best-fit slopes of our correlations are \approx 0.5, but the
+signal-to-noise is rather low.  A slope of -1 appears to be consistent
+with our measurements.
+
+The fact that the PSF ellipticity changes are {\em not} correlated
+with the tree-ring structure (Figure~\ref{fig:shear.by.filter}) tells us
+that, unlike the case for DES, the effective plate-scale changes seen
+in the flat-field and astrometry signals are not the dominant cause of
+the PSF photometry errors.  Also, the fact that we do not measure
+significant aperture photometry errors correlated with the tree rings
+confirms this point.  The amplitude of the flat-field errors are 1-2
+millimagnitudes, much smaller than the PSF photometry errors, and far
+below the pixel-to-pixel noise in the aperture magnitude residuals.
+It is likely in our opinion that the plate-scale changes causing the
+flat-field and astrometry effects is affecting both the ellipticity
+and the aperture magnitudes, but the level of the effect is too small
+to see given the other systematic structures (in the shear plot) and
+the noise level (in the aperture magnitudes).
+
+Finally, the correlation between the smear structures and the
+astrometry residuals shows that these two effects are connected.
+Although the correlation is weak in Figure~\ref{fig:dsmear.vs.astrom},
+careful inspection of the location of the these two tree ring patterns
+shows that the locations of the rings in the radial astrometric
+residual images occurs at the boundaries between regions with
+substantially different values of the smear signal.
+
+We suggest that the underlying connection between all of these
+tree-ring effects is the pattern of the doping variations in the
+silicon.  As discussed by \cite{2014PASP..126..750P}, the tree-ring
+patterns seen by the DES team are caused by lateral electic fields in
+the detector silicon (in the plane of the CCD wafer) generated by
+variations in the space charges embedded in the silicon, in turn
+coming from low-level changes in the doping as the silicon boule is
+grown.  We conclude that the astrometric and flat-field variations
+seen in our detectors are caused by these same types of doping
+variations.  The changes in the smear (and thus the PSF magnitudes)
+are apparently also related to the doping variations.  The lateral
+electric fields which introduce the astrometry and flat-field
+variations occur at the boundary between regions with higher and lower
+space charges from the dopant.  Regions with high (or low) space
+charge density thus correspond to regions with relatively high (or
+low) amounts of smear; the astrometric deviations follow the gradient
+between these regions.
+
+We interpret the changes in the {\em smear} term as changes in the
+amount of charge diffusion as the photoelectrons travel to the bottom
+of the pixel well.  The blue filters exhibit the strongest changes in
+the amount of smear.  These are also the filters for which the
+detected electrons have travelled the longest distance in the silicon,
+and are thus most affected by diffusion effects.  Charge diffusion (as
+opposed to the charge drift caused by the lateral electric fields)
+results in a Gaussian smearing of the stellar profile: as the
+photoelectrons migrate from the site where they were generated by the
+incoming photon to the bottom of the pixel well, they follow a random
+walk in the plane of the detector.  The longer the electrons take to
+make the journey down to the bottom of the pixel, the further they are
+able to wander from their creation coordinate in the detector.
+Following the discussion in \cite{Holland.2003}, the amount of charge
+diffusion is thus related to the velocity of the electrons in the
+direction of the optical axis: $\sigma \sim \sqrt{2Dt}$ where $\sigma$
+is the size of the smearing kernel, $t$ is the time required for the
+electrons to traverse the thickness of the silicon wafer, and $D$ is
+the diffusion coefficient.  The velocity of the photoelectron, and
+thus the time to traverse the silicon, is related to the vertical
+electric fields in the silicon, which are caused by a combination of
+the applied voltages and the distribution of the space charges from
+the dopant.  As shown by \cite{Holland.2003}, the charge diffusion is
+related to the space charge density by $\sigma \sim
+\rho^{-\frac{1}{2}}$ (their equation 6).  Regions with high space
+charge densities increase the migration speed of the photoelectrons
+and reduce the amount of charge diffusion smearing; and vice versa for
+regions of low space-charge densities. 
+
+In summary, the variations in the space-charge density caused by
+variations in the dopant result in regions of higher and lower charge
+diffusion, and in turn regions with PSF photometry systematic
+residuals.  The lateral gradients in the space-charge density induce
+lateral electric fields which in turn cause lateral motions of the
+photoelectrons, resulting in astrometric and flat-field deviations.
+
+The DES team did not detect these charge diffusion variations.  In
+that case, the amplitude of the photometric effects due to the lateral
+field are dominant; these include both the modification of the
+flat-field as well as PSF fitting errors due to the changing PSF sizes
+introduced by the varying effective pixels sizes.  If the smearing
+effect reported here were as large for DES compared with the lateral
+PSF size changes as they are for GPC1, then the reported PSF
+photometry residuals for would have had very different
+characteristics.  We conclude that, for DES, the lateral effects are
+much larger than the diffusion variations, compared with GPC1.  The
+relative amplitude of these two effects depends on the details of the
+applied voltages, the amplitude of the space-charge density variations
+compared with the typical space-charge density, and the detector
+thicknesses.  It is beyond the scope of this article to model these
+effects in detail.
+
+% http://adsabs.harvard.edu/abs/2006NIMPA.568...41K
+
+\section{Conclusion}
+
+The tree rings observed in the Pan-STARRS GPC1 data show (at least)
+two effects, though they are related.  First, the images are
+experiencing circularly-symmetric changes in the PSF size correlated
+with the tree-ring pattern.  These PSF size changes drive errors in
+the PSF photometry on the scale of a few millimagnitudes, are also
+correlated with the tree-ring pattern.  These PSF size changes are
+consistent with changes in the charge diffusion, which also introduces
+a circularly symmetric smearing.
+
+In addition, there are radial plate-scale changes correlated with the
+tree rings.  These plate-scale changes introduce a flat-field errors
+on the scale of \approx 1 millimagnitude and astrometric errors in the
+scale of 2-3 milliarcseconds.  The observed relationship between the
+flat-field deviations and the radial derivative of the astrometric
+deviations confirms this interpretation \citep[see also discussion
+  in][]{2014PASP..126..750P}.
+
+The spatial correlation of the gradient in the smear variations and
+the astrometric variations imply that both of these two types of tree
+ring effects are related, even though they manifest through different
+mechanisms.  We conclude that the variations in both the vertical charge
+diffusion and the lateral charge migration are driven by changes
+in the electric field structures in the silicon due to the same
+variations in the doping structures in the silicon.
+
+% The small-scale variations in the charge diffusion observed in these
+% devices has not been reported for DECam, Hypersuprime Cam, or
+% prototype LSST sensors.  
+
+The small-scale variations in the charge diffusion observed in the
+Pan-STARRS detectors represents a new type of systematic effect in
+deep depletion devices.  This feature, if present in other detectors,
+could manifest in systematic errors in several ways.  Like in the
+Pan-STARRS analysis example, the charge diffusion variations result in
+fine-structure in the observed stellar point-spread functions.  For
+very precise photometry or morphological analysis, it will be
+necessary for the PSF models to account for the extra charge
+diffusion.  Unlike the non-uniform pixel-size effects, correction of
+the PSF photometry cannot simply be performed as an average flat-field
+correction on the measurements after they have been processed.  
+The additional smearing acts as a convolution with a Gaussian kernel
+of fixed size for a given filter.  The photometry bias is a function
+of the fractional change of the PSF size.  Thus, the introduced error
+depends on the average PSF for the image in question: an image with
+good image quality will suffer larger PSF model errors than an image
+with poor image quality.  To account for this effect in a rigorous
+way, the analysis should use the measured diffusion variations to
+modify the model PSFs as a function of position before they are used
+for the image analysis.
+
+The charge diffusion variations may also have an impact on
+spectroscopic measurements.  Modern, precise spectroscopic
+measurements rely on precise measurements of the stellar line
+profiles.  If such an analysis ignores variations in the charge
+diffusion, the measured line widths may be systematically biased.
+
+This analysis points to the importance of careful instrumental
+characterization, especially for those instruments which are used for
+large-scale surveys with largely automatic data analysis systems and
+stringent precision goals.
+
+\acknowledgments
+
+The Pan-STARRS1 Surveys (PS1) have been made possible through
+contributions of the Institute for Astronomy, the University of
+Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its
+participating institutes, the Max Planck Institute for Astronomy,
+Heidelberg and the Max Planck Institute for Extraterrestrial Physics,
+Garching, The Johns Hopkins University, Durham University, the
+University of Edinburgh, Queen's University Belfast, the
+Harvard-Smithsonian Center for Astrophysics, the Las Cumbres
+Observatory Global Telescope Network Incorporated, the National
+Central University of Taiwan, the Space Telescope Science Institute,
+the National Aeronautics and Space Administration under Grant
+No. NNX08AR22G issued through the Planetary Science Division of the
+NASA Science Mission Directorate, the National Science Foundation
+under Grant No. AST-1238877, the University of Maryland, and Eotvos
+Lorand University (ELTE) and the Los Alamos National Laboratory.
+
+\note{Ken: please add NASA ops grants}
+
+\bibliographystyle{apj}
+\bibliography{lib}{}
+%\input{analysis.bbl}
+
+\end{document}
+
+%% Some refs to be added as appropriate:
+% Bernstein DEC astrometry : arxiv 1703.01679
+% Baumer et al arxiv 1706.07400 (Flat-fielding)
Index: trunk/doc/release.2015/systematics.20140411/systematics.tex
===================================================================
--- trunk/doc/release.2015/systematics.20140411/systematics.tex	(revision 40132)
+++ 	(revision )
@@ -1,1165 +1,0 @@
-% \documentclass[iop,floatfix]{emulateapj}
-\documentclass[10pt,preprint]{aastex}
-% \pdfoutput=1
-
-% see latex.readme.txt for notes on using the PS1 template
-%\documentclass[12pt,preprint]{aastex}
-%\documentclass[manuscript]{aastex}
-%\documentclass[preprint2]{aastex}
-%\documentclass[preprint2,longabstract]{aastex}
-
-\RequirePackage{graphicx}
-\RequirePackage{color}
-\RequirePackage{code}
-\RequirePackage{pbox}
-\input{astro.sty}
-
-\usepackage[T1]{fontenc}% (2) specify encoding
-
-% online version may use color, but print version needs b/w
-\def\plotmode{col}
-%\def\plotmode{bw}
-
-%\def\plotext{pdf}
-\def\plotext{ps}
-
-%\def\picdir{/home/eugene/chipresid.20140404}
-%\def\picdir{/data/kukui.2/eugene/chipresid.20140404}
-\def\picdir{pics}
-
-% Pick a terse version of the title here;
-\shorttitle{Charge Diffusion Variations in PS1}
-\shortauthors{E.A. Magnier et al}
-\begin{document}
-\title{Charge Diffusion Variations in Pan-STARRS\,1 CCDs}
-
-% this is a crude trick to get the order of affiliations right.  These
-% names are used in the affiliations below.  The user needs to (1) set
-% the order and numbers to have the correct sequence in the author
-% list and (2) re-order the list at the bottom (and comment-out as needed)
-\def\IfA{1}
-\def\CfA{2}
-\def\MPIA{3}
-\def\Princeton{3}
-\def\USNO{4}
-\def\JHU{1}
-
-% This example has a first author from UH:
-\author{
-Eugene A. Magnier,\altaffilmark{\IfA}
-J.~L. Tonry, \altaffilmark{\IfA}
-D. Finkbeiner,\altaffilmark{\CfA}
-E. Schlafly,\altaffilmark{\MPIA}
-%PS Builder List
-W.~S. Burgett,\altaffilmark{\IfA}
-K.~C. Chambers,\altaffilmark{\IfA} 
-% L. Denneau,\altaffilmark{\IfA}
-% P. Draper,\altaffilmark{\DUR}
-H.~A. Flewelling,\altaffilmark{\IfA}
-% T. Grav,\altaffilmark{\IfA}
-% J. N. Heasley,\altaffilmark{\IfA}
-K. W. Hodapp,\altaffilmark{\IfA}
-% M. E. Huber,\altaffilmark{\IfA}
-% R. Jedicke,\altaffilmark{\IfA}
-N. Kaiser,\altaffilmark{\IfA}
-R.-P. Kudritzki,\altaffilmark{\IfA}
-% G. A. Luppino,\altaffilmark{\IfA}
-% R. H. Lupton,\altaffilmark{\Princeton}
-% E. A. Magnier,\altaffilmark{\IfA}
-N. Metcalfe,\altaffilmark{\DUH}
-% D. G. Monet,\altaffilmark{\USNO}
-% J.~S. Morgan,\altaffilmark{\IfA}
-% P. M. Onaka,\altaffilmark{\IfA}
-% P.~A. Price,\altaffilmark{\Princeton}
-% C.~W. Stubbs,\altaffilmark{\CfA}
-% W.~E. Sweeney,\altaffilmark{\IfA}
-% J.~L. Tonry, \altaffilmark{\IfA}
-R. J. Wainscoat,\altaffilmark{\IfA} and 
-C. Z. Waters,\altaffilmark{\IfA}
-PS Builders TBA
-} % this bracket terminates author list
-
-% The ordering here should be sequential, matching the sequence in the list of authors:
-\altaffiltext{\IfA}{Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu HI 96822}
-\altaffiltext{\CfA}{Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138}
-% \altaffiltext{\Princeton}{Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA}
-% \altaffiltext{\USNO}{US Naval Observatory, Flagstaff Station, Flagstaff, AZ 86001, USA}
-% \altaffiltext{\JHU}{Department of Physics and Astronomy, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA}
-\altaffiltext{\MPIA}{Max Planck Institute for Astronomy, K\"onigstuhl 17, D-69117 Heidelberg, Germany}
-\begin{abstract}
-
-Thick back-illuminated deep-depletion CCDs have superior quantum
-efficiency over previous generations of thinned and traditional thick
-CCDs.  As a result, they are being used for major wide-field imaging
-cameras in several projects.  We use observations from the Pan-STARRS
-$3\pi$ survey to characterize the behavior of the deep-depletion
-devices used in the Pan-STARRS\,1 Gigapixel Camera.  We have
-identified systematic spatial variations in the photometric behavior and
-stellar profiles which are similar to the so-called ``tree rings''
-identified in devices used by other wide-field cameras (DECam and
-Hypersuprime Camera).  The tree-ring features identified in these
-other cameras result from lateral electric fields which displace the
-electrons as they are transported in the silicon to the pixel
-location.  In contrast, we show that the photometric and morphological
-modifications observed in the GPC1 detectors are caused by variations
-in the vertical charge transportation rate and resulting charge
-diffusion variations.
-\end{abstract}
-
-% insert additional keywords as appropriate:
-\keywords{Surveys:\PSONE }
-
-\section{INTRODUCTION}\label{sec:intro}
-
-CCD detectors have evolved greatly since they were first introduced
-for astronomical imaging in the mid 1970s.  In addition to the
-well-known increases in the size of CCDs over the past 4 decades, CCD
-architecture has gone through three major evolutionary stages.  
-
-The first generation of CCDs used a silicon substrate a few hundred
-microns thick on top of which gate structures were deposited to define
-the pixels.  A positive voltage applied to the gate layers would
-create a shallow region (\approx 10 microns thick) in which the holes
-were depleted.  This ``depletion region'' acted as a potential well to
-trap electrons, specifically those generated by absorbed photons.  The
-thick silicon substrate required illumination from the ``front'' side
-with the thin gate structures to allow the photons to reach the
-depletion region and be detected.  These early CCDs had modest quantum
-efficiency as photons were easily absorbed by the several micron thick
-gate structures.  For an excellent review of the history of CCD
-development, see \cite{1992ASPC...23....1J}.
-
-Thinned, backside-illuminated CCDs such as the TI 3PCCD
-\citep{1981SPIE..290....6B} were developed to address the quantum
-efficiency limitations of the first generation thick CCDs.  The
-silicon substrate was removed using a chemical process, leaving a
-delicate device only \approx 10 - 20\micron\ thick, exposing the
-depletion region on the backside.  Photons entering the backside of
-the device are not blocked by the gate structures and thus more easily
-absorbed and detected.  Thinned backside-illuminated CCDs have high
-quantum efficiency to blue photons.  However, as the wavelength
-increases beyond \approx 800 nm, the silicon becomes more transparent
-to the photons, with a corresponding drop in quantum efficiency for
-red photons.  In addition, thin film interference between the entering
-photons and those reflecting off the front side of the CCD result in
-``fringe'' patterns for redder photons.
-
-Early generations of CCDs were made of low-resistivity (\approx 10 -
-50 $\Omega$-cm) silicon.  Following experiments beginning in the early
-1990s \citep{Holland.1996}, CCDs made from thick, high-resistivity ($
-> 10 k\Omega$-cm) silicon were developed for astronomical instruments
-in the early 2000s \citep{Holland.2003}.  The high-resistivity of the
-silicon allows for depletion regions of hundreds of microns in depth,
-compared to \approx 10\micron\ for the low-resistivity silicon.  This
-modification allows for a back-illuminated CCD with a relatively thick
-silicon subtrate of 75 - 300\micron.  Blue photons impinging on the
-back of the device are absorbed near the back surface of the device
-and are caried through the depletion region to the gates on the front
-side.  The thick silicon allows red photons to have a greater chance
-to be absorbed, increasing quantum efficiency in the red.  Because
-these thick, deep-depletion devices have near-unity quantum efficiency
-across a very wide spectral range, they have become the design of
-choice for many modern, large-scale CCD cameras (e.g., Pan-STARRS
-GPC1, \citealt{2009amos.confE..40T}; Subaru Hypersuprime Camera,
-\citealt{2010SPIE.7735E..3FK}; Dark Energy Survey Camera,
-\citealt{2015AJ....150..150F}).
-
-While these deep-depletion CCDs seem to be ideal, they do have
-features which can cause challenges for precise measurements.  As a
-result of the ``Brighter-Fatter Effect''
-\citep{2014JInst...9C3048A,2015JInst..10C5032G}, the profile of bright
-stars are measured to be wider than the profiles of faint stars.  The
-accepted interpretation is that the electric fields produced by the
-electrons accumulated from a star repel successive incoming electrons,
-with the repulsion increasing the more electrons have accumulated.
-
-The effects of lateral electric fields are likewise identified as the
-cause of the so-called ``tree rings'' observed in the flat-field,
-astrometry, and photometry response of thick deep depletion detectors
-\citep{2014PASP..126..750P}.  These tree-ring patterns have been noted
-in the flat-field response of deep depletion devices since their early
-testing \citep[see, e.g., Figure 2 in][]{2010SPIE.7735E..1RE} and were
-initially considered to be a sensitivity response which could be
-removed with a flat-field.  As discussed in detail by
-\cite{2014PASP..126..750P}, these tree rings are more correctly
-interpretted as variations in the effective pixel area due to
-migration of the electrons pushed by lateral electric fields induced
-by small changes in the doping used to set the resistivity of the
-silicon.  The changes in the effective area result in changes to the
-apparent flat-field response as well as the astrometric response of
-the detector.  More subtly, the flat-field response changes, since
-they do not reflect actual variations in sensitivity, can lead to
-systematic photometry errors for astronomical sources if the
-flat-field images are used in the standard fashion.
-
-In this paper, we examine the behavior of an apparently-similar kind
-of tree ring observed in the Pan-STARRS GPC1 CCDs.  Although we also
-observe the pixel effective area changes caused by lateral electric
-fields as described by \cite{2014PASP..126..750P}, we show below a
-second effect which is more important in driving systematic photometry
-errors.  We find that variations in charge diffusion, also resulting
-from changes in the silicon doping structures, affect both the
-observed stellar profiles as well as the photometry measured with
-profile fitting techniques.  In Section~\ref{sec:PS1}, we discuss the
-Pan-STARRS telescope, camera, and survey data used in this analysis.
-In Section~\ref{sec:tree.rings}, we present the tree-ring
-patterns as observed in several different types of measurements:
-flat-field response, systematic photometry residuals, systematic
-astrometric residuals, and stellar profile shape variations.  In
-Section~\ref{sec:discussion}, we discuss the interpretation of
-patterns we observe and present a simple model to explain the observed
-behavior.  We conclude with a discussion of the implications of this
-effect on astronomical measurements from deep depletion instruments
-
-\section{Pan-STARRS1}
-\label{sec:PS1}
-
-The 1.8m Pan-STARRS\,1 telescope (PS1), located on the summit of
-Haleakala on the Hawaiian island of Maui, has been surveying the sky
-regularly since May 2010 \citep{chambers2017}.  From May 2010 through
-March 2014, PS1 was run under the aegis of the Pan-STARRS Science
-Consortium to perform a set of wide-field science surveys; since March
-2014, operations have been supported primarily by NASA's Near Earth
-Object Observation program, see \cite{2015IAUGA..2251124W}.  Under the
-PS1SC, the largest survey, both in terms of area of the sky covered
-($3\pi$ steradians) and fraction of observing time (56\%), was the
-\TPS\ in which the entire sky north of Declination $-30$\degrees\ was
-imaged up \approx 80 times over 4 years.  These observations were
-distributed over five filters, \grizy, and have been astrometrically
-and photometrically calibrated to good precision
-\citep{magnier2017.calibration}.
-
-% 2004SPIE.5489..667H == PS1.optics
-% 2008SPIE.7014E..0DO == PS1.GPCB
-% 2009amos.confE..40T == PS1.GPCA
-% 2012ApJ...756..158S == ubercal
-The wide-field PS1 telescope optics \citep{2004SPIE.5489..667H} image
-a 3.3 degree field of view on a 1.4 gigapixel camera
-\citep[GPC1][]{2009amos.confE..40T}, with low distortion and generally
-good image quality.  The median seeing for the \TPS\ data vary
-somewhat by filter: (\grizy) = (1.31, 1.19, 1.11, 1.07, 1.02)
-arcseconds.  Routine observations are conducted remotely from the
-Advanced Technology Research Center in Kula, the main facility of the
-University of Hawaii's Institute for Astronomy operations on Maui.
-
-GPC1 \citep{2009amos.confE..40T}, currently the largest astronomical
-camera in terms of number of pixels, consists of a mosaic of 60
-edge-abutted $4800\times4800$ pixel detectors, with 10~$\mu$m pixels
-subtending 0.258~arcsec. These CCID58 detectors, manufactured by
-Lincoln Laboratory, are 75\micron-thick back-illuminated CCDs
-\citep{2006amos.confE..47T,2008SPIE.7021E..05T}.  Initial performance
-assessments are presented in \cite{2008SPIE.7014E..0DO}. The active,
-usable pixels cover \approx 80\% of the FOV.
-
-\subsection{Data Processing and Calibration}
-
-% PS1_IPP = \bibitem[Magnier(2006)]{PS1.IPP} Magnier, E.\ 2006,
-% Proceedings of The Advanced Maui Optical and Space Surveillance
-% Technologies Conference, Ed.: S. Ryan, The Maui Economic Development
-% Board, p.E5
-
-Images obtained by PS1 are processed by the Pan-STARRS Image
-Processing Pipeline (IPP;
-\citealp{2006amos.confE..50M,magnier2017.datasystem}).  All
-observations are processed nightly, with results sent to groups within
-the science consortium (i.e., PS1SC during the \TPS) performing
-short-term science projects (e.g., searching for transient and moving
-objects).  In addition, the \TPS\ dataset has been re-processed
-several times with improved calibration and analysis techniques.  To
-date (2017 July), 3 re-processings starting from raw pixel data have
-been performed.  The labels PV0, PV1, PV2, PV3 are used identify the
-nightly processing and successive re-processing versions.  PV3 has
-been used for the public release of the Pan-STARRS \TPS\ data via the
-{\it Barbara A. Mikulski Archive for Space Telescopes} (MAST) at the
-Space Telescope Science Institute.\footnote{http//panstarrs.stci.edu}
-
-The data processing and calibration operations are discussed in detail
-in elsewhere
-\citep{magnier2017.analysis,magnier2017.calibration,waters2017}.
-We re-visit here a number of points that are of significance to this
-study.  Images are processed following a fairly standard sequence of
-image detrending, source detection, and initial calibration
-(astrometric and photometric) of those detected sources.  Additional
-standard processing critical to PS1 science operations includes
-geometric transformation (`warping') and image combinations (summed
-stacks and differences).  For the purposes of this analysis, we are
-only considering the sources detected in the individual exposures from
-the initial analysis steps.
-
-% Magnier.belgium:
-% \bibitem[Magnier(2007)]{PS1.photometry} Magnier, E.\ 2007, The Future 
-% of Photometric, Spectrophotometric and Polarimetric Standardization, ASP Conference Series {\bf 364}, 153 
-
-%IPP astrometry (NOT USED)
-% \bibitem[Magnier {\it et al.}(2008)]{PS1.astrometry} Magnier, E.~A., Liu, 
-% M., Monet, D.~G., \& Chambers, K.~C.\ 2008, IAU Symposium, {\bf 248}, 553 
-
-As discussed in \cite{waters2017}, image detrending includes
-flat-field processing with a single epoch flat-field image for each
-filter.  The flat-field image used for this analysis has been
-generated by median-combining dome flat-field images (after
-pre-processing and pixel outlier rejections) and then multiplying by a
-photometric flat-field correction image generated by the analysis of a
-grid of images of a dense stellar field.  The purpose of this second
-step is to correct the basic flat-field image for errors arising from
-the non-uniformity of the illumination, from non-pixel uniformity due
-to the varying optical distorition across the field, and any other
-factors which may make the flat-field image inconsistent with stellar
-photometry, e.g., SED, filter band-pass variations, etc
-\citep[see][]{waters2017,2004PASP..116..449M,2007ASPC..364..153M}.
-This correction was made on a relatively coarse grid across the focal
-plane in order to accumulate sufficient statistics from the stars in
-the relatively small number of images available at the time.  We have
-found that a single flat-field set can be used for all PS1
-observations to yield photometric systematic errors at the level of \approx
-2\%.  PS1 benefits in this regard from the stability of having a
-single instrument which is rarely removed.
-
-Photometry of the PS1 images is performed using a
-point-spread-function (PSF) model as well as multiple kinds of
-apertures \citep{magnier2017.analysis}.  In this analysis, we refer to
-aperture photometry performed using an aperture defined based on the
-image quality observed for a given chip.  The aperture diameter is set
-to be 3.75 times the FWHM for the image.
-
-To improve the photometric systematic errors beyond the level achieved
-with a single (photometrically corrected) flat-field set, the PS1
-photometry is re-calibrated within the databasing system based on the
-properties of the measured photometry.  The calibration process is
-discussed by
-\cite{2012ApJ...756..158S,2013ApJS..205...20M,magnier2017.calibration}.
-As part of this process, several flat-field corrections have been
-determined.  For the PV2 analysis discussed here, a flat-field
-correction determined during the ubercal analysis
-\citep[see][]{2012ApJ...756..158S} consisted of an $8\times 8$ grid of
-corrections for each GPC1 chip, corresponding to a correction for each
-OTA ``cell'' and filter for each of 4 seasons.  The boundaries of
-those seasons are tentatively identified with modifications to the
-baffle structures or the system optics.  The critical point here is
-that the final effective flat-field image for the PV2 dataset is based
-on a dome-flat at the highest resolution, with very low resolution
-corrections based on photometry, resulting in photometric systematic
-uncertainties in the range 7 - 12 millimagnitudes, depending on the
-filter \citep{2013ApJS..205...20M}.
-
-For all objects, positions are measured from the PSF model for the
-brighter sources (using a non-linear fitting process) and from a
-simple centroid (1st moment) for the fainter source
-\citep{magnier2017.analysis}.  These position measurements are
-used in the astrometric analysis.  The astrometric calibration is
-discussed by \cite{magnier2017.calibration}; for the PV2
-dataset, the typical systematic floor is \approx 15 - 20
-milliarcsecond for individual measurements of brighter stars. 
-
-\section{Tree-Ring Patterns}
-\label{sec:tree.rings}
-
-\begin{table}
-% \tiny
-\begin{center}
-\caption{Systematic Trends : Standard deviation by filter\label{table:sigmas.by.filter}}
-\begin{tabular}{|l|rrrrr|}
-\hline
-{\bf Filter} & {\bf psf mags} & {\bf ap mags} & {\bf astrom} & {\bf smear} & {\bf flat} \\
-             & mmags         & mmags          & mas          & pixels$^2$  & mmags \\
-\hline
-\gps & 11.8 & 13 & 8.0  & 0.169 &  3.0 \\ 
-\rps & 10.9 & 12 & 6.7  & 0.133 &  2.2 \\
-\ips &  8.5 & 10 & 6.0  & 0.069 &  1.7 \\
-\zps &  8.7 & 12 & 5.5  & 0.052 &  3.2 \\
-\yps & 16.5 & 26 & 6.8  & 0.059 & 15.3 \\
-\hline
-\end{tabular}
-\end{center}
-\end{table}
-
-For many of the GPC1 OTA CCDs, we observe a spatial pattern in the
-photometric residuals for each device which is similar in appearence
-to the tree rings described in the Dark Energy Camera (DECam) by
-\cite{2014PASP..126..750P}.  This pattern consists of systematic
-deviations which are consistent in a set of circular arcs centered on
-the corner of the CCD, as shown in Figure~\ref{fig:psfmags.by.filter}.
-The details of the analysis used to generate
-Figure~\ref{fig:psfmags.by.filter} are given below.  For now, we note
-that the GPC1 CCDs are constructed by dividing the circular silicon
-wafer into 4 inscribed squares.  Thus the corners of the CCDs lie in
-the center of the silicon boule, just as the center of the circular
-tree rings described by \cite{2014PASP..126..750P} match the center of
-the boule from which they came.  This gives the impression that a
-similar mechanism is responsible for the pattern observed in the PS1
-photometry and the DECam photometry, namely the diffusive effects of
-lateral electric field variations in the detectors.  In the next
-section, we will make the case that the patterns observed in the PS1
-photometry residuals are {\em not} caused by this mechanism, but are
-instead caused by variations in the {\em vertical} electric field (the
-field direction perpendicular to the CCD surface).
-
-First, in this section, we will describe how we have measured the
-presence or absence of these tree-ring patterns in 5 types of data.
-For all of these examples, we use a single GPC1 CCD (XY40) to
-illustrate the effects in detail, but a similar set of effects are
-seen in many of the GPC1 detectors.  First, we show the residual PSF
-photometry.  Second, we show the residual aperture photometry.  Third,
-we show the astrometric residual patterns.  Fourth, we show the
-patterns observed in the flat-field images.  Finally, we show
-measurements derived from the second-moments of the stars.
-
-For all effects discussed below, we are measuring the mean value of
-the effect in 10x10 pixel superpixels across the detector.  The
-resulting images are all constructed so that a given superpixel
-represents the same range of true GPC1 XY40 pixels regardless of the
-type of measurement.  To generate the photometry, astrometry, or
-second-moment plots, measurements were extracted from the PV0 DVO
-database \citep{magnier2017.calibration} for observations covering
-the region ($\alpha$,$\delta$) = (90\degree\ -- 150\degree,
--25\degree\ -- 10\degree).  This region of the sky provides a fairly
-high density of stars, but avoids the Galactic Plane where confusion
-may potentially contaminate the measurement.  We limit the analysis to
-good measurements (\ippmisc{PSF_QF} $>$ 0.85, see
-\citealt{magnier2017.analysis}) of likely stars ($|m_{psf} -
-m_{aper}| < 0.2$).  Only measurements with instrumental magnitude $<
--8.0$ ($-2.5\log \mbox{cts sec}^{-1} < -8.0$) are included to ensure
-reasonable signal-to-noise per measurement.  We require at least 2
-measurements in a given filter and at least 5 measurements total for
-any star included in the analysis.
-
-\subsection{Photometric Residuals}
-
-% PSF Magnitudes
-\def\figwidth{5.2in}
-\def\jumpleft{-2.6in}
-\def\capwidth{2.4in}
-\begin{figure*}[htbp]
-\begin{center}
-\parbox[b]{\figwidth}{\includegraphics[width=\figwidth]{\picdir/dmag.\plotext}}
-\hspace{\jumpleft}
-\parbox[b]{\capwidth}{
-\caption{PSF Magnitude residuals by filter (\grizy).  White boxes are
-  GPC1 cells which have been masked due to poor response.  Superpixels
-  representing regions of $10\times10$ pixels are used to determine
-  the median deviation for measurements at the given chip pixel
-  location compared with the average photometry for the given
-  object.} \label{fig:psfmags.by.filter}}
-\end{center}
-\end{figure*}
-
-% Aperture Magnitudes
-\begin{figure*}[htbp]
-\begin{center}
-\parbox[b]{\figwidth}{\includegraphics[width=\figwidth]{\picdir/dapmag.\plotext}}
-\hspace{\jumpleft}
-\parbox[b]{\capwidth}{
-\caption{Aperture Magnitude residuals by filter (\grizy).  White boxes
-  are GPC1 cells which have been masked due to poor response.
-  Superpixels representing regions of $10\times10$ pixels are used to
-  determine the median deviation for measurements at the given chip
-  pixel location compared with the average photometry for the given
-  object.  } \label{fig:apmags.by.filter}}
-\end{center}
-\end{figure*}
-
-Figure~\ref{fig:psfmags.by.filter} shows the 2D patterns of PSF
-photometry residuals.  In this case, we select PSF magnitude
-measurements for detections of stars which fall in the given
-superpixel.  We subtract each measurement from the average magnitude
-for the object in the selected filter ($\delta m_{psf} =
-\overline{m}_{psf} - m_{psf}$) to determine the residual magnitude,
-excluding as an outlier any measurement with $|\delta m_{psf}| > 0.5$.
-For a given superpixel, we measure the median of the $\delta m_{psf}$
-distribution.  The figure shows $\delta m_{psf}$ for each filter
-(\grizy).  The dynamic range of the color scale is from -20 to +20
-millimagnitudes for all 5 plots.
-
-The tree-ring pattern is clearly visible for the four blue filters,
-but finging dominates the pattern for \yps.  Small offsets of
-individual cells are also apparent for \zps.  While the patterns are
-clear across the image, the signal-to-noise of the structures per
-pixel is not very strong in these images.  The per-pixel standard
-deviations of these plots are listed in
-Table~\ref{table:sigmas.by.filter}.  The per-pixel standard deviation
-is comparable to the amplitude of the correlated structures, so we
-need to integrate along the radial structures to make stronger
-statements about these patterns.
-
-Figure~\ref{fig:apmags.by.filter} shows the equivalent measurement for
-aperture photometry instead of PSF photometry.  The finging
-pattern again dominates the plot for \yps, but the tree rings are not
-seen in any of the filters.  A diagonal pattern is visible in \gps
-which is not observed in the PSF magnitudes.  While the per-pixel
-scatter is somewhat (10\% to 20\%) higher for these aperture
-magnitudes than for the PSF magnitudes
-(Table~\ref{table:sigmas.by.filter}), a structure with the amplitude
-of the PSF magnitude tree-rings would certainly have been obvious.
-
-\subsection{Astrometric Residuals}
-
-% astrometry radial term
-\begin{figure*}[htbp]
-\begin{center}
-\parbox[b]{\figwidth}{\includegraphics[width=\figwidth]{\picdir/drad.\plotext}}
-\hspace{\jumpleft}
-\parbox[b]{\capwidth}{
-\caption{Astrometric residuals of the displacement in the radial
-  direction, relative to the chip coordinate -5,4960 (upper left
-  corner), by filter (\grizy).  White boxes are GPC1 cells which have
-  been masked due to poor response.  Superpixels representing regions
-  of $10\times10$ pixels are used to determine the median deviation
-  for measurements at the given chip pixel location compared with the
-  average astrometry for the given
-  object. } \label{fig:astrom.by.filter}}
-\end{center}
-\end{figure*}
-
-Figure~\ref{fig:astrom.by.filter} shows a similar type of measurement
-for astrometric residuals.  To generate this plot, we use the same
-selection of measurements for astrometry as for photometry.  In this
-case, we extract the residual in both the RA and DEC directions
-($\delta RA = \overline{RA} - RA_i$, $\delta DEC = \overline{DEC} -
-DEC_i$) and rotate these values to the chip coordinate system ($\delta
-X,\delta Y$) using our knowledge of the chip orientation on the sky.
-We again exclude as bad any measurement with $|\delta X|$ or $|\delta
-Y| > 0.5$ arcsec before measuring the median values for each
-superpixel.  We have determined the approximate center of the circular
-tree-ring pattern as (-5,4960) for this particular chip based on the
-pattern of the X astrometry displacements.  Using this coordinate as the center
-of the pattern, we have converted the $\delta X,\delta Y$ offsets into
-$\delta R,\delta \theta$ measurements ($\delta R$ : radial component
-away from the center, $\delta \theta$ : tangential component).
-
-Figure~\ref{fig:astrom.by.filter} shows the 2D patterns of $\delta R$
-for each filter (\grizy).  The dynamic range of the color scale is
-from -20 to +20 milliarcseconds for all 5 plots.  A tree-ring
-pattern is visible for all five filters, with systematic structures
-following a circular pattern centered on the chip corner; the finging
-pattern is not apparent in the \yps\ astrometry.  The per-pixel
-standard deviations of these plots area listed in
-Table~\ref{table:sigmas.by.filter}.  The signal-to-noise of these
-structures is again somewhat weak, but the pattern is clearly visible
-in these figures.
-
-\subsection{Flat-field Structures}
-
-% flat-field residual
-\begin{figure*}[htbp]
-\begin{center}
-\parbox[b]{\figwidth}{\includegraphics[width=\figwidth]{\picdir/dflat.\plotext}}
-\hspace{\jumpleft}
-\parbox[b]{\capwidth}{
-\caption{Flat-field high-frequency structues, by filter (\grizy).
-  White boxes are GPC1 cells which have been masked due to poor
-  response.  Flat-field images generated using a tunable laser have
-  been combined (see text); a smoothed version has been subtracted to
-  high-pass the response.  Flat-field pixels are averaged for
-  $10\times10$ superpixels. } \label{fig:flats.by.filter}}
-\end{center}
-\end{figure*}
-
-% 2012ApJ...750...99T = Tonry et al PS1 phot system
-Figure~\ref{fig:flats.by.filter} shows the high-spatial-frequency
-structures in the flat-field images.  For this measurement, we have
-used a set of monochromatic flat-field images obtained with a tunable
-laser.  The laser is used to illuminate our flat-field screen which is
-then observed by the PS1 telescope.  These flat-field images were
-obtained 2011 Feb 09 as part of a campaign to study the PS1 system
-response \citep{2012ApJ...750...99T}.  Flats were obtain in a set of
-4nm steps sampling the spectral response curve of each filter.  To
-enhance the signal-to-noise, we have median-combined a set of 6 flats
-at the wavelength center of the corresponding filter.
-
-In order to mask pixels which do not flatten well, we generate a copy
-of the image smoothed with a Gaussian kernel with $\sigma = 1.5$
-pixels.  Any pixels in the smoothed image which deviate from the
-median value in the image by more than 4 standard deviations are
-masked.  We generate the superpixel image by averaging the unmasked
-pixels associated with each superpixel.  
-
-Figure~\ref{fig:flats.by.filter} shows the superpixel images for the
-flat-fields in the five filters.  These flat-field images are
-displayed as fractional deviations relative to the median flat-field
-image and can thus be compared to the magnitude residuals.  When
-flattening an image, these flat-fields would be divided into the flux
-of the raw image.  The residuals are thus defined in the sense that a
-positive feature in these flats which does {\em not} represent a real
-quantum efficiency deviation would induce a {\em reduction} in the
-measured flux in those pixels, and thus a {\em negative} deviation in
-$\delta m_{psf}$ as defined above.  The dynamic range of the color
-scale in these plots is -0.01 to +0.01.  The tree-ring pattern is
-strong in the (\gps,\rps,\ips) images, but nearly swamped by fringing
-in \zps, and completely lost to finging in \yps.  A diagonal banding
-pattern is seen in \gps: this features is thought to be due to the
-lithography process used to generate the CCD.  A blob can also been
-seen covering 4 cells near the center of this chip; this is apparently
-a deposit of some kind on the detector.  Both of the latter two
-effects behave like quantum efficiency variations and are removed well
-by standard flat-field techniques.  Note that a small amount of the
-diagonal banding pattern remains in the aperture magnitude residuals
-for \gps.  For the rest of this article, we ignore these features and
-concentrate on the tree ring features.
-
-In order to suppress the large-scale structures for a quantitative
-analysis of the tree rings, we high-pass filter the superpixel image
-by subtracting a copy smoothed with a Gaussian of $\sigma = 3.0$
-superpixels.
-
-\subsection{Second Moments}
-
-% Smear Images
-\begin{figure*}[htbp]
-\begin{center}
-\parbox[b]{\figwidth}{\includegraphics[width=\figwidth]{\picdir/smear.\plotext}}
-\hspace{\jumpleft}
-\parbox[b]{\capwidth}{
-\caption{Average residual smear variations, by filter (\grizy).  White
-  boxes are GPC1 cells which have been masked due to poor response.
-  The residual smear ($\sigma^2_{\mbox{major}} + \sigma^2_{\mbox{minor}}$) has been
-  determined after the after PSF second moments have been subtracted
-  for each image; these values are averaged for each $10\times10$
-  superpixels.  } \label{fig:smear.by.filter}}
-\end{center}
-\end{figure*}
-
-% Shear Images
-\begin{figure*}[htbp]
-\begin{center}
-\parbox[b]{\figwidth}{\includegraphics[width=\figwidth]{\picdir/shear.\plotext}}
-\hspace{\jumpleft}
-\parbox[b]{\capwidth}{
-\caption{Average residual shear variations, by filter (\grizy).  White
-  boxes are GPC1 cells which have been masked due to poor response.
-  The residual shear ($\sigma^2_{\mbox{major}} - \sigma^2_{\mbox{minor}}$) has been
-  determined after the after PSF second moments have been subtracted
-  for each image; these values are averaged for each $10\times10$
-  superpixels.  } \label{fig:shear.by.filter}}
-\end{center}
-\end{figure*}
-
-During the image analysis, the second moments are measured for all
-stars.  The values can be used to assess changes in the shape of stars
-on the image.  To measure changes in the shapes, we have extracted the
-second moments for all stellar detections, subject to the same
-selections as for the photometry and astrometry residuals (good stars,
-multiple detections).  The second moments are measured with a Gaussian
-weighting function, with the $\sigma_{w}$ scaled by the PSF size so
-that the $\sigma$ measured for PSF stars is \approx 65\% of
-$\sigma_{w}$.  (Note that, since the measured $\sigma$ of stellar
-objects is biased down by the weighting function, this is not quite
-the same as having $\sigma_{w} = 1.6$ times the true PSF $\sigma$, see
-discussion in \citealt{magnier2017.analysis}).  For each stellar
-detection, we extract the values $M_{xx,xy,yy} = \sum F_i w_i (x^2, x
-y, y^2) / \sum F_i w_i$.  For each exposure, we find the median second
-moments for PSF objects on this chip (XY40) and subtract those median
-values from the instantaneous measurements of $M_{xx,xy,yy}$.  We then
-determine the median of the residual second moments for each
-superpixel, resulting in 3 images ($\delta M_{xx,xy,yy}$) for each
-filter.
-
-Using the second moment images, we can construct certain interesting
-combinations, inspired by discussions of lensing measurements \citep{1995ApJ...449..460K}:
-\begin{eqnarray}
-e_0 & = & \delta M_{xx} + \delta M_{yy}  \\ 
-e_1 & = & \delta M_{xx} - \delta M_{yy}  \\
-e_2 & = & \sqrt{e_1^2 + 4 \delta M_{xy}}
-\end{eqnarray}
-For a 2D Gaussian profile with an elliptical contour, these values are
-related to the shape of the elliptical contour as follows:
-\begin{eqnarray}
-e_0 & = & \sigma^2_{\mbox{major}}  + \sigma^2_{\mbox{minor}} \\
-e_1 & = & (\sigma^2_{\mbox{major}}  - \sigma^2_{\mbox{minor}}) \cos (2 \theta) \\
-e_2 & = & \sigma^2_{\mbox{major}}  - \sigma^2_{\mbox{minor}}
-\end{eqnarray}
-Where $\sigma_{\mbox{major}}$ and $\sigma_{\mbox{minor}}$ are the
-major and minor axis dimensions of the ellipse and $\theta$ is the
-position angle.  Thus, $e_0$ is a measurement of the change in the
-size of the stellar PSFs as a function of position in the detector
-(``smear''), $e_2$ is a measurement of the change in ellipticity of
-the stellar PSFs (``shear''), and we can determine the angle of the
-PSF ellipticity from the $e_1$ term.
-
-Figure~\ref{fig:smear.by.filter} shows the spatial trend of $e_0$, the {\em
-  smear}.  This value corresponds to the increase or decrease in
-the circularly-symmetric component of the image size.  The dynamic
-range of these images is -0.3 to +0.3 pixel$^2$. A tree-ring
-pattern is visible for all 5 filters, though \yps is dominated by the
-fringing pattern.  Structures with relatively low spatial frequencies
-can also be seen.
-
-Figure~\ref{fig:shear.by.filter} shows the spatial trend of $e_2$, the
-{\em shear}.  This value is positive definite and is plotted with a
-color scale ranging from -0.02 to 0.22 pixel$^2$.  We can also
-determine the orientation of the corresponding ellipse.  Overlayed on
-Figure~\ref{fig:shear.by.filter} is a set of vectors representing the
-ellipse orientation as a function of postion.  The length of the
-vectors corresponds to the value of $\sigma^2_{major} -
-\sigma^2_{minor}$.  The tree-ring structure is {\em not} apparent
-in this figure for any filter.  The spatial variations are
-low-frequency and unrelated to the radial trend from the upper-left
-corner.
-
-\subsection{Correlations Between Tree-Ring Patterns}
-
-% All Effects in r-band
-\begin{figure*}[htbp]
-\begin{center}
-\parbox[b]{\figwidth}{\includegraphics[width=5.0in]{\picdir/all.effects.r.\plotext}}
-\caption{All 6 measured effects for \rps.  This figure illustrates the
-  different spatial structure observed for each of the 6 patterns
-  measured in this work.  The PSF magnitude (upper-left) and smear
-  residuals (lower-left) have a very clear common tree-ring structure,
-  while the astrometric residual (middle-left) and flat-field
-  residuals (middle-right) have their own common tree-ring pattern with
-  much higher frequencies than the previous two effects.  Aperture
-  magnitude (upper-right) and shear residuals (lower-right) do not
-  show a strong signal consistent with either of the two patterns.} \label{fig:all.effects.rband}
-\end{center}
-\end{figure*}
-
-\begin{table}
-% \tiny
-\begin{center}
-\caption{Systematic Trends : Correlations by filter\label{table:correlation.by.filter}}
-\begin{tabular}{|l|rrrr|}
-\hline
-{\bf Filter} & {\bf smear} & {\bf psf mags} & {\bf astrom} & {\bf flat} \\
-\hline
-\gps & 1.00 & 1.00 &  1.00 & 1.00 \\ 
-\rps & 0.78 & 0.84 &  0.84 & 0.76 \\
-\ips & 0.40 & 0.50 &  0.66 & 0.64 \\
-\zps & 0.16 & 0.26 &  0.37 & 0.33 \\
-\yps & 0.10 & 0.10 &  0.25 & 0.30 \\
-\hline
-\end{tabular}
-\end{center}
-\end{table}
-
-Tree-ring patterns are clearly seen in 4 of the measurement types
-above: the PSF photometry, the astrometry, the flat-field, and the
-smear terms.  As discussed above, the signal-to-noise per pixel in the
-plots of the systematic trends is relatively low (\approx 1.0).  While
-the tree-ring patterns are apparent in many of these figures,
-there are also some other systematic structures which may degrade the
-signal further.
-
-To quantatatively compare the tree-ring trends between
-filters and between the types of measurements, we need to measure the
-tree-ring structure explicitly and filter out the other effects if
-possible.  To do this, we have applied a high-pass filter to all of
-the relevant images (PSF photometry residuals, astrometric residuals
-in the radial direction, flat-field residuals, and second moment smear
-terms) to remove unrelated spatial structures.  We have then measured
-the median of the signal in radial bins centered on (-5,4960) across
-an arc from $\phi$ = -20\degrees\ to -50\degrees (as measured relative
-to the top row of the images.  We have selected a small fraction of
-the arc to minimize the error associated with the choice of the
-pattern center and to avoid several bad cells near the bottom of the
-chip.
-
-% \note{include the arc on one of the figures?}
-
-% \note{do plots of all filter pairs in a triangle?  is that interesting?}
-
-For a given type of measurement, the systematic effect is strongly
-correlated between filters.  The strongest correlation is the smear
-term: Figure~\ref{fig:smear.trends} shows the correlation of the smear
-pattern between \gps\ and the other four filters. Even \yps\ is
-strongly correlated with \gps\ despite the presence of the fringe
-pattern.  PSF photometric residuals are also correlated between
-filters, as shown in Figure~\ref{fig:psfmag.trends}.  Here, the
-\yps\ correlation with \gps\ is quite weak: the fringing pattern
-dominates the tree rings for PSF photometry.  The radial component of
-the astrometric residual is also well correlated between filters, with
-no loss of correlation due to fringing in \yps. Finally, the
-flat-field residuals are generally correlated between filters, but
-both \zps\ and \yps\ are affected by fringing.  For \yps, the
-correlation is completely washed out by the very strong fringing
-pattern.
-
-For all four types of measurements, the slope of the fitted lines are
-listed in Table~\ref{table:correlation.by.filter}.  There is a
-consistency in the trend from \gps, with the strongest systematic
-tree-ring effects to \yps, with the weakest effects.  Note that the
-second moment smear and astrometry terms have different relative
-strength in \yps\ compared with \gps.
-
-% smear trends by filter
-\def\figwidth{6.5in}
-\begin{figure*}[htbp]
-\begin{center}
-\includegraphics[width=\figwidth]{\picdir/smear.trends.\plotext}
-\caption{Correlation of the smear ($\sigma^2_{\mbox{major}} +
-  \sigma^2_{\mbox{minor}}$) signal in \gps\ with the other 4 bands:
-  \rps\ (upper-left),  \ips\ (upper-right), \zps\ (lower-left), \yps\ (lower-right).
-} \label{fig:smear.trends}
-\end{center}
-\end{figure*}
-
-% psfmag trends by filter
-\def\figwidth{6.5in}
-\begin{figure*}[htbp]
-\begin{center}
-\includegraphics[width=\figwidth]{\picdir/psfmag.trends.\plotext}
-\caption{Correlation of the PSF magnitude residuals ($\delta m_{psf}$)
-  in \gps\ with the other 4 bands: \rps\ (upper-left), \ips\
-  (upper-right), \zps\ (lower-left), \yps\ (lower-right).
-} \label{fig:psfmag.trends}
-\end{center}
-\end{figure*}
-
-% astrom trends by filter
-\def\figwidth{6.5in}
-\begin{figure*}[htbp]
-\begin{center}
-\includegraphics[width=\figwidth]{\picdir/astrom.trends.\plotext}
-\caption{Correlation of the radial astrometric residual displacement ($\delta R$)
-  in \gps\ with the other 4 bands: \rps\ (upper-left), \ips\
-  (upper-right), \zps\ (lower-left), \yps\ (lower-right).
-} \label{fig:astrom.trends}
-\end{center}
-\end{figure*}
-
-% flat trends by filter
-\def\figwidth{6.5in}
-\begin{figure*}[htbp]
-\begin{center}
-\includegraphics[width=\figwidth]{\picdir/flat.trends.\plotext}
-\caption{Correlation of the flat-field tree-ring structures in \gps\
-  with the other 4 bands: \rps\ (upper-left), \ips\ (upper-right), \zps\
-  (lower-left), \yps\ (lower-right).  } \label{fig:flat.trends}
-\end{center}
-\end{figure*}
-
-An important question is the relationship of the tree-ring
-pattern between the different types of measurements.  Different models
-for the tree-ring structures make different predictions about the
-correlations between different effects.  Note the very different
-spatial structure between the different measurements in a given
-filter: the radial variations do not all follow the same patterns.
-Instead, we find the following relationships hold:
-
-First, the PSF magnitude residuals and the second-moment smear trends
-are strongly anti-correlated: regions which have larger PSFs than the
-mean tend to have smaller measured PSF fluxes than the mean (note that
-$\delta m_{psf}$ is defined so that positive values correspond to
-larger fluxes).  These trends are shown in
-Figure~\ref{fig:smear.vs.psfmag}.  
-
-Second, the radial derivative of the smear is anti-correlated with the
-radial component of the astrometric residuals: $\frac{\partial
-  (\sigma^2_{major} + \sigma^2_{minor})}{\partial radius} \sim \delta
-R$ (see Figure~\ref{fig:dsmear.vs.astrom}).
-
-Finally, the radial derivative of the radial component of the
-astrometric residual is anti-correlated with the flat-field residual
-errors: $\frac{\partial \delta R}{\partial radius} \sim \delta flat$
-(see Figure~\ref{fig:dastrom.vs.flat}.  This last relationship is
-somewhat weakly measured.  Because of the periodic nature of the Tree
-Rings, it is also difficult to be completely certain that the
-flat-field is proportional to the derivative of the astrometry
-residual, rather than the astrometry residual being proportional to
-the derivative of the flat-field.  The correlation is somewhat weaker
-for derivative of the flat-field vs astrometry residual.  The
-correlation is very weak between the flat-field and the astrometry
-residual values without a derivative.  We are convinced that we have
-the sense of the derivative correct by examination of specific
-features in each imaage.
-
-\begin{table}
-% \tiny
-\begin{center}
-\caption{Systematic Trends : Correlations between trends\label{table:correlation.by.trend}}
-\begin{tabular}{|l|rrr|}
-\hline
-{\bf Filter} & {\bf psf mags} & {\bf $\grad$ smear} & {\bf $\grad$ astrom} \\
-             & {\bf vs smear} & {\bf vs astrom}     & {\bf vs flat}        \\
-\hline
-\gps & -0.056 & -0.060 & -0.47  \\ 
-\rps & -0.071 & -0.073 & -0.45  \\
-\ips & -0.077 & -0.095 & -0.45  \\
-\zps & -0.082 & -0.078 & -0.17  \\
-\hline
-\end{tabular}
-\end{center}
-\end{table}
-
-% smear vs psfmag
-\def\figwidth{6.5in}
-\begin{figure*}[htbp]
-\begin{center}
-\includegraphics[width=\figwidth]{\picdir/smear.vs.psfmag.\plotext}
-\caption{Correlation of the PSF magnitude residuals ($\delta m_{PSF}$)
-  with the smear ($\sigma^2_{\mbox{major}} + \sigma^2_{\mbox{minor}}$)
-  signal for \gps\ (upper-left), \rps\ (upper-right), \ips\ (lower-left),
-  \zps\ (lower-right).
-} \label{fig:smear.vs.psfmag}
-\end{center}
-\end{figure*}
-
-% dsmear vs astrom
-\def\figwidth{6.5in}
-\begin{figure*}[htbp]
-\begin{center}
-\includegraphics[width=\figwidth]{\picdir/dsmear.vs.astrom.\plotext}
-\caption{
-Correlation of the radial astrometric residual displacement ($\delta
-R$) with the derivative of the smear ($\partial
-\sigma^2_{\mbox{major}} + \sigma^2_{\mbox{minor}}$) signal with
-respect to the radial postion for \gps\ (upper-left), \rps\
-(upper-right), \ips\ (lower-left), \zps\ (lower-right).
-} \label{fig:dsmear.vs.astrom}
-\end{center}
-\end{figure*}
-
-% dastrom vs flat
-\def\figwidth{6.5in}
-\begin{figure*}[htbp]
-\begin{center}
-\includegraphics[width=\figwidth]{\picdir/dastrom.vs.flat.\plotext}
-\caption{
-Correlation of the derivative of the radial astrometric residual
-displacement ($\delta R$) with respect to the radial position with the
-flat-field tree-ring signal for \gps\ (upper-left), \rps\ (upper-right),
-\ips\ (lower-left), \zps\ (lower-right).
-} \label{fig:dastrom.vs.flat}
-\end{center}
-\end{figure*}
-
-\section{Discussion}
-\label{sec:discussion}
-
-These trends measured above (Section~\ref{sec:tree.rings}) help to
-illuminate the underlying causes of these different effects.
-
-First, if we consider the smear pattern
-(Figure~\ref{fig:smear.by.filter}), the measurement shows that the
-intrinsic sizes of the stellar images are varying in a radial sense
-between the different tree-ring regions.  Although images experience
-an average image quality (due to seeing and focus) across the chip
-which may vary substantially from exposure to exposure, stars landing
-in the different tree-ring regions are consistently somewhat
-larger or somewhat smaller than that average.
-
-Next, we can explain the correlation between the PSF photometry
-residuals and the observed smear (Figure~\ref{fig:smear.vs.psfmag}).
-In the photometry analysis, we model the PSF allowing for some spatial
-variation in the shape.  However, we have a limited number of stars to
-measure any spatial variation.  Thus the 2D variations are sampled on
-a very coarse (e.g., $3 \times 3$) grid for each chip: the PSF
-parameters may vary smoothly across the chip following the bilinear
-interpolation between the $3 \times 3$ grid points.  Thus, the spatial
-scale on which we model PSF variations is much larger than the spatial
-scale on which PSF variations are actually occuring, as illustrated
-by the changes in the smear plot (Figure~\ref{fig:smear.by.filter}).
-When the true PSF is larger than the model PSF, our model fits
-systematically underestimate the amount of flux in a given object.
-Conversely, when the true PSF is smaller, we overestimate the flux -- this
-type of offset is a typical effect when mis-estimating the PSF size.
-The slope of the trend depends on the mean typical seeing for the
-given filter.  For example, the \gps\ seeing is typically 1.3\arcsec,
-corresponding to a Gaussian $\sigma$ of 2.15 pixels.  A smearing of
-$\sigma^2_{major} + \sigma^2_{minor} = 0.1$ pixels$^2$ would increase
-the size by about 0.02 pixels, or 1\%, roughly consistent with the
-observed photometric deviation of about 5 to 10 millimags for this
-amount of smearing.
-
-The correlation between the flat-field structures and the radial
-derivative of the astrometric residual displacements in the radial
-direction (Figure~\ref{fig:dastrom.vs.flat}) is consistent with radial
-variations in the plate-scale.  The tree-rings observed by DES are
-completely attributed to effective plate scale changes.  Effective
-plate scale changes result in flat-field deviations because the
-flat-field illumination is a source of constant surface brightness.
-Pixels see a varying amount of flux depending on their effective area.
-This changing plate scale also affects the astrometry since these
-variations occur on spatial scales much smaller than the astrometric
-model.  In this description of the tree rings, the flat-field
-deviations are $-1 \times \frac{\partial \delta R}{\partial r}$.  The
-best-fit slopes of our correlations are \approx 0.5, but the
-signal-to-noise is rather low.  A slope of -1 appears to be consistent
-with our measurements.
-
-The fact that the PSF ellipticity changes are {\em not} correlated
-with the tree-ring structure (Figure~\ref{fig:shear.by.filter}) tells us
-that, unlike the case for DES, the effective plate-scale changes seen
-in the flat-field and astrometry signals are not the dominant cause of
-the PSF photometry errors.  Also, the fact that we do not measure
-significant aperture photometry errors correlated with the tree rings
-confirms this point.  The amplitude of the flat-field errors are 1-2
-millimagnitudes, much smaller than the PSF photometry errors, and far
-below the pixel-to-pixel noise in the aperture magnitude residuals.
-It is likely in our opinion that the plate-scale changes causing the
-flat-field and astrometry effects is affecting both the ellipticity
-and the aperture magnitudes, but the level of the effect is too small
-to see given the other systematic structures (in the shear plot) and
-the noise level (in the aperture magnitudes).
-
-Finally, the correlation between the smear structures and the
-astrometry residuals shows that these two effects are connected.
-Although the correlation is weak in Figure~\ref{fig:dsmear.vs.astrom},
-careful inspection of the location of the these two tree ring patterns
-shows that the locations of the rings in the radial astrometric
-residual images occurs at the boundaries between regions with
-substantially different values of the smear signal.
-
-We suggest that the underlying connection between all of these
-tree-ring effects is the pattern of the doping variations in the
-silicon.  As discussed by \cite{2014PASP..126..750P}, the tree-ring
-patterns seen by the DES team are caused by lateral electic fields in
-the detector silicon (in the plane of the CCD wafer) generated by
-variations in the space charges embedded in the silicon, in turn
-coming from low-level changes in the doping as the silicon boule is
-grown.  We conclude that the astrometric and flat-field variations
-seen in our detectors are caused by these same types of doping
-variations.  The changes in the smear (and thus the PSF magnitudes)
-are apparently also related to the doping variations.  The lateral
-electric fields which introduce the astrometry and flat-field
-variations occur at the boundary between regions with higher and lower
-space charges from the dopant.  Regions with high (or low) space
-charge density thus correspond to regions with relatively high (or
-low) amounts of smear; the astrometric deviations follow the gradient
-between these regions.
-
-We interpret the changes in the {\em smear} term as changes in the
-amount of charge diffusion as the photoelectrons travel to the bottom
-of the pixel well.  The blue filters exhibit the strongest changes in
-the amount of smear.  These are also the filters for which the
-detected electrons have travelled the longest distance in the silicon,
-and are thus most affected by diffusion effects.  Charge diffusion (as
-opposed to the charge drift caused by the lateral electric fields)
-results in a Gaussian smearing of the stellar profile: as the
-photoelectrons migrate from the site where they were generated by the
-incoming photon to the bottom of the pixel well, they follow a random
-walk in the plane of the detector.  The longer the electrons take to
-make the journey down to the bottom of the pixel, the further they are
-able to wander from their creation coordinate in the detector.
-Following the discussion in \cite{Holland.2003}, the amount of charge
-diffusion is thus related to the velocity of the electrons in the
-direction of the optical axis: $\sigma \sim \sqrt{2Dt}$ where $\sigma$
-is the size of the smearing kernel, $t$ is the time required for the
-electrons to traverse the thickness of the silicon wafer, and $D$ is
-the diffusion coefficient.  The velocity of the photoelectron, and
-thus the time to traverse the silicon, is related to the vertical
-electric fields in the silicon, which are caused by a combination of
-the applied voltages and the distribution of the space charges from
-the dopant.  As shown by \cite{Holland.2003}, the charge diffusion is
-related to the space charge density by $\sigma \sim
-\rho^{-\frac{1}{2}}$ (their equation 6).  Regions with high space
-charge densities increase the migration speed of the photoelectrons
-and reduce the amount of charge diffusion smearing; and vice versa for
-regions of low space-charge densities. 
-
-In summary, the variations in the space-charge density caused by
-variations in the dopant result in regions of higher and lower charge
-diffusion, and in turn regions with PSF photometry systematic
-residuals.  The lateral gradients in the space-charge density induce
-lateral electric fields which in turn cause lateral motions of the
-photoelectrons, resulting in astrometric and flat-field deviations.
-
-The DES team did not detect these charge diffusion variations.  In
-that case, the amplitude of the photometric effects due to the lateral
-field are dominant; these include both the modification of the
-flat-field as well as PSF fitting errors due to the changing PSF sizes
-introduced by the varying effective pixels sizes.  If the smearing
-effect reported here were as large for DES compared with the lateral
-PSF size changes as they are for GPC1, then the reported PSF
-photometry residuals for would have had very different
-characteristics.  We conclude that, for DES, the lateral effects are
-much larger than the diffusion variations, compared with GPC1.  The
-relative amplitude of these two effects depends on the details of the
-applied voltages, the amplitude of the space-charge density variations
-compared with the typical space-charge density, and the detector
-thicknesses.  It is beyond the scope of this article to model these
-effects in detail.
-
-% http://adsabs.harvard.edu/abs/2006NIMPA.568...41K
-
-\section{Conclusion}
-
-The tree rings observed in the Pan-STARRS GPC1 data show (at least)
-two effects, though they are related.  First, the images are
-experiencing circularly-symmetric changes in the PSF size correlated
-with the tree-ring pattern.  These PSF size changes drive errors in
-the PSF photometry on the scale of a few millimagnitudes, are also
-correlated with the tree-ring pattern.  These PSF size changes are
-consistent with changes in the charge diffusion, which also introduces
-a circularly symmetric smearing.
-
-In addition, there are radial plate-scale changes correlated with the
-tree rings.  These plate-scale changes introduce a flat-field errors
-on the scale of \approx 1 millimagnitude and astrometric errors in the
-scale of 2-3 milliarcseconds.  The observed relationship between the
-flat-field deviations and the radial derivative of the astrometric
-deviations confirms this interpretation \citep[see also discussion
-  in][]{2014PASP..126..750P}.
-
-The spatial correlation of the gradient in the smear variations and
-the astrometric variations imply that both of these two types of tree
-ring effects are related, even though they manifest through different
-mechanisms.  We conclude that the variations in both the vertical charge
-diffusion and the lateral charge migration are driven by changes
-in the electric field structures in the silicon due to the same
-variations in the doping structures in the silicon.
-
-% The small-scale variations in the charge diffusion observed in these
-% devices has not been reported for DECam, Hypersuprime Cam, or
-% prototype LSST sensors.  
-
-The small-scale variations in the charge diffusion observed in the
-Pan-STARRS detectors represents a new type of systematic effect in
-deep depletion devices.  This feature, if present in other detectors,
-could manifest in systematic errors in several ways.  Like in the
-Pan-STARRS analysis example, the charge diffusion variations result in
-fine-structure in the observed stellar point-spread functions.  For
-very precise photometry or morphological analysis, it will be
-necessary for the PSF models to account for the extra charge
-diffusion.  Unlike the non-uniform pixel-size effects, correction of
-the PSF photometry cannot simply be performed as an average flat-field
-correction on the measurements after they have been processed.  
-The additional smearing acts as a convolution with a Gaussian kernel
-of fixed size for a given filter.  The photometry bias is a function
-of the fractional change of the PSF size.  Thus, the introduced error
-depends on the average PSF for the image in question: an image with
-good image quality will suffer larger PSF model errors than an image
-with poor image quality.  To account for this effect in a rigorous
-way, the analysis should use the measured diffusion variations to
-modify the model PSFs as a function of position before they are used
-for the image analysis.
-
-The charge diffusion variations may also have an impact on
-spectroscopic measurements.  Modern, precise spectroscopic
-measurements rely on precise measurements of the stellar line
-profiles.  If such an analysis ignores variations in the charge
-diffusion, the measured line widths may be systematically biased.
-
-This analysis points to the importance of careful instrumental
-characterization, especially for those instruments which are used for
-large-scale surveys with largely automatic data analysis systems and
-stringent precision goals.
-
-\acknowledgments
-
-The Pan-STARRS1 Surveys (PS1) have been made possible through
-contributions of the Institute for Astronomy, the University of
-Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its
-participating institutes, the Max Planck Institute for Astronomy,
-Heidelberg and the Max Planck Institute for Extraterrestrial Physics,
-Garching, The Johns Hopkins University, Durham University, the
-University of Edinburgh, Queen's University Belfast, the
-Harvard-Smithsonian Center for Astrophysics, the Las Cumbres
-Observatory Global Telescope Network Incorporated, the National
-Central University of Taiwan, the Space Telescope Science Institute,
-the National Aeronautics and Space Administration under Grant
-No. NNX08AR22G issued through the Planetary Science Division of the
-NASA Science Mission Directorate, the National Science Foundation
-under Grant No. AST-1238877, the University of Maryland, and Eotvos
-Lorand University (ELTE) and the Los Alamos National Laboratory.
-
-\note{Ken: please add NASA ops grants}
-
-\bibliographystyle{apj}
-\bibliography{lib}{}
-%\input{analysis.bbl}
-
-\end{document}
-
-%% Some refs to be added as appropriate:
-% Bernstein DEC astrometry : arxiv 1703.01679
-% Baumer et al arxiv 1706.07400 (Flat-fielding)