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Aug 7, 2017, 9:32:48 AM (9 years ago)
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eugene
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incorporate comments from Ken and Chris

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  • trunk/doc/release.2015/systematics.20140411/systematics.tex

    r40105 r40108  
    9494$3\pi$ survey to characterize the behavior of the deep-depletion
    9595devices used in the Pan-STARRS\,1 Gigapixel Camera.  We have
    96 identified systematic variations in the photometric behavior and
    97 stellar profiles which are similar to the so-called tree rings
     96identified systematic spatial variations in the photometric behavior and
     97stellar profiles which are similar to the so-called Tree Rings
    9898identified in devices used by other wide-field cameras (DECam and
    99 Hypersuprime Camera).  The tree-ring features identified in these
     99Hypersuprime Camera).  The Tree-Ring features identified in these
    100100other cameras result from lateral electric fields which displace the
    101101electrons as they are transported in the silicon to the pixel
    102102location.  In contrast, we show that the photometric and morphological
    103103modifications observed in the GPC1 detectors are caused by variations
    104 in the vertical charge transportation range and resulting charge
     104in the vertical charge transportation rate and resulting charge
    105105diffusion variations.
    106106\end{abstract}
     
    110110
    111111\section{INTRODUCTION}\label{sec:intro}
     112
     113\note{KCC says: note what is unique to GPC1 vs other cameras}
    112114
    113115CCD detectors have evolved greatly since they were first introduced
     
    1481501990s \citep{Holland.1996}, CCDs made from thick, high-resistivity ($
    149151> 10 k\Omega$-cm) silicon were developed for astronomical instruments
    150 in the early 2000s\citep{Holland.2003}.  The high-resistivity of the
     152in the early 2000s \citep{Holland.2003}.  The high-resistivity of the
    151153silicon allows for depletion regions of hundreds of microns in depth,
    152154compared to \approx 10\micron\ for the low-resistivity silicon.  This
     
    158160to be absorbed, increasing quantum efficiency in the red.  Because
    159161these thick, deep-depletion devices have near-unity quantum efficiency
    160 across the whole a very wide spectral range, they have become the
    161 design of choice for many modern, large-scale CCD cameras (e.g.,
    162 Pan-STARRS GPC1, \citealt{2009amos.confE..40T}; Subaru Hypersuprime
    163 Camera, \citealt{2010SPIE.7735E..3FK}; Dark Energy Survey Camera,
     162across a very wide spectral range, they have become the design of
     163choice for many modern, large-scale CCD cameras (e.g., Pan-STARRS
     164GPC1, \citealt{2009amos.confE..40T}; Subaru Hypersuprime Camera,
     165\citealt{2010SPIE.7735E..3FK}; Dark Energy Survey Camera,
    164166\citealt{2015AJ....150..150F}).
    165167
     
    174176
    175177The effects of lateral electric fields are likewise identified as the
    176 cause of the so-called ``Tree-Rings'' observed in the flat-field,
     178cause of the so-called ``Tree Rings'' observed in the flat-field,
    177179astrometry, and photometry response of thick deep depletion detectors
    178 \citep{2014PASP..126..750P}.  These tree-ring patterns have been noted
     180\citep{2014PASP..126..750P}.  These Tree-Ring patterns have been noted
    179181in the flat-field response of deep depletion devices since their early
    180182testing \citep[see, e.g., Figure 2 in][]{2010SPIE.7735E..1RE} and were
     
    219221March 2014, PS1 was run under the aegis of the Pan-STARRS Science
    220222Consortium to perform a set of wide-field science surveys; since March
    221 2014, the telescope has been operated by the Pan-STARRS New Science
    222 Consortium (PSNSC).  Under the PS1SC, the largest survey, both in
    223 terms of area of the sky covered ($3\pi$ steradians) and fraction of
    224 observing time (56\%), was the \TPS\ in which the entire sky north of
    225 Declination $-30$\degrees\ was imaged up \approx 80 times over 4
    226 years.  These observations were distributed over five filters, \grizy,
    227 and have been astrometrically and photometrically calibrated to good
    228 precision \citep{magnier2017.calibration}.
     2232014, operations have been supported primarily by NASA's Near Earth
     224Object Observation program, see \cite{wainscoat.2015}.  Under the
     225PS1SC, the largest survey, both in terms of area of the sky covered
     226($3\pi$ steradians) and fraction of observing time (56\%), was the
     227\TPS\ in which the entire sky north of Declination $-30$\degrees\ was
     228imaged up \approx 80 times over 4 years.  These observations were
     229distributed over five filters, \grizy, and have been astrometrically
     230and photometrically calibrated to good precision
     231\citep{magnier2017.calibration}.
    229232
    230233% 2004SPIE.5489..667H == PS1.optics
     
    310313the relatively small number of images available at the time.  We have
    311314found that a single flat-field set can be used for all PS1
    312 observations to yield photometric consistency at the level of \approx
     315observations to yield photometric systematic errors at the level of \approx
    3133162\%.  PS1 benefits in this regard from the stability of having a
    314317single instrument which is rarely removed.
     
    319322aperture photometry performed using an aperture defined based on the
    320323image quality observed for a given chip.  The aperture diameter is set
    321 to be \approx 3.75 times the FWHM for the image.
     324to be 3.75 times the FWHM for the image.
    322325
    323326To improve the photometric systematic errors beyond the level achieved
     
    331334correction determined during the ubercal analysis
    332335\citep[see][]{2012ApJ...756..158S} consisted of an $8\times 8$ grid of
    333 corrections for each GPC1 chip and filter for each of 4 seasons.  The
    334 boundaries of those seasons are tentatively identified with
    335 modifications to the baffle structures or the system optics.  The
    336 critical point here is that the final effective flat-field image for
    337 the PV2 dataset is based on a dome-flat at the highest resolution,
    338 with very low resolution corrections based on photometry, resulting in
    339 photometric systmatic uncertainties in the range 7 - 12
    340 millimagnitudes, depending on the filter \citep{2013ApJS..205...20M}.
     336corrections for each GPC1 chip, corresponding to a correction for each
     337OTA ``cell'' and filter for each of 4 seasons.  The boundaries of
     338those seasons are tentatively identified with modifications to the
     339baffle structures or the system optics.  The critical point here is
     340that the final effective flat-field image for the PV2 dataset is based
     341on a dome-flat at the highest resolution, with very low resolution
     342corrections based on photometry, resulting in photometric systematic
     343uncertainties in the range 7 - 12 millimagnitudes, depending on the
     344filter \citep{2013ApJS..205...20M}.
    341345
    342346For all objects, positions are measured from the PSF model for the
     
    371375\end{table}
    372376
    373 For many of the GPC1 OTA CCDs, we observe a pattern in the photometric
    374 residuals which is similar in appearence to the Tree Rings described
    375 in the Dark Energy Camera (DECam) by \cite{2014PASP..126..750P}.  This
    376 pattern consists of systematic deviations which are consistent in a
    377 set of circular arcs centered on the corner of the CCD, as shown in
    378 Figure~\ref{fig:psfmags.by.filter}.  The details of the analysis used
    379 to generate Figure~\ref{fig:psfmags.by.filter} are given below.  For
    380 now, we note that the GPC1 CCDs are constructed by dividing the
    381 circular silicon wafer into 4 inscribed squares.  Thus the corners of
    382 the CCDs lie in the center of the silicon boule, just as the center of
    383 the circular Tree Rings described by \cite{2014PASP..126..750P} match
    384 the center of the boule from which they came.  This gives the
    385 impression that a similar mechanism is responsible for the pattern
    386 observed in the PS1 photometry and the DECam photometry, namely the
    387 diffusive effects of lateral electric field variations in the
    388 detectors.  In the next section, we will make the case that the
    389 patterns observed in the PS1 photometry residuals are {\em not} caused
    390 by this mechanism, but are instead caused by variations in the {\em
    391   vertical} electric field (the field direction perpendicular to the
    392 CCD surface).
     377For many of the GPC1 OTA CCDs, we observe a spatial pattern in the
     378photometric residuals for each device which is similar in appearence
     379to the Tree Rings described in the Dark Energy Camera (DECam) by
     380\cite{2014PASP..126..750P}.  This pattern consists of systematic
     381deviations which are consistent in a set of circular arcs centered on
     382the corner of the CCD, as shown in Figure~\ref{fig:psfmags.by.filter}.
     383The details of the analysis used to generate
     384Figure~\ref{fig:psfmags.by.filter} are given below.  For now, we note
     385that the GPC1 CCDs are constructed by dividing the circular silicon
     386wafer into 4 inscribed squares.  Thus the corners of the CCDs lie in
     387the center of the silicon boule, just as the center of the circular
     388Tree Rings described by \cite{2014PASP..126..750P} match the center of
     389the boule from which they came.  This gives the impression that a
     390similar mechanism is responsible for the pattern observed in the PS1
     391photometry and the DECam photometry, namely the diffusive effects of
     392lateral electric field variations in the detectors.  In the next
     393section, we will make the case that the patterns observed in the PS1
     394photometry residuals are {\em not} caused by this mechanism, but are
     395instead caused by variations in the {\em vertical} electric field (the
     396field direction perpendicular to the CCD surface).
    393397
    394398First, in this section, we will describe how we have measured the
    395 presence or absence of these tree-ring patterns in 5 types of data.
     399presence or absence of these Tree-Ring patterns in 5 types of data.
    396400For all of these examples, we use a single GPC1 CCD (XY40) to
    397401illustrate the effects in detail, but a similar set of effects are
    398402seen in many of the GPC1 detectors.  First, we show the residual PSF
    399 photometry.  Second, we show the residual Aperture photometry.  Third,
     403photometry.  Second, we show the residual aperture photometry.  Third,
    400404we show the astrometric residual patterns.  Fourth, we show the
    401405patterns observed in the flat-field images.  Finally, we show
     
    408412type of measurement.  To generate the photometry, astrometry, or
    409413second-moment plots, measurements were extracted from the PV0 DVO
    410 database for observations covering the region ($\alpha$,$\delta$) =
    411 (90\degree\ -- 150\degree, -25\degree\ -- 10\degree).  This region of
    412 the sky provides a fairly high density of stars, but avoids the
    413 Galactic Plane where confusion may potentially contaminate the
    414 measurement.  We limit the analysis to good measurements
    415 (\ippmisc{PSF_QF} $>$ 0.85) of likely stars ($|m_{psf} - m_{aper}| <
    416 0.2$).  Only measurements with instrumental magnitude $< -8.0$
    417 ($-2.5\log \mbox{cts sec}^{-1} < -8.0$) are included to ensure
     414database \citep{magnier.2017.calibration} for observations covering
     415the region ($\alpha$,$\delta$) = (90\degree\ -- 150\degree,
     416-25\degree\ -- 10\degree).  This region of the sky provides a fairly
     417high density of stars, but avoids the Galactic Plane where confusion
     418may potentially contaminate the measurement.  We limit the analysis to
     419good measurements (\ippmisc{PSF_QF} $>$ 0.85, see
     420\citealt{magnier.2017.analysis}) of likely stars ($|m_{psf} -
     421m_{aper}| < 0.2$).  Only measurements with instrumental magnitude $<
     422-8.0$ ($-2.5\log \mbox{cts sec}^{-1} < -8.0$) are included to ensure
    418423reasonable signal-to-noise per measurement.  We require at least 2
    419 measurements in a given filter and 5 measurements total for any star
    420 included in the analysis.
     424measurements in a given filter and at least 5 measurements total for
     425any star included in the analysis.
    421426
    422427\subsection{Photometric Residuals}
     
    428433\parbox{\figwidth}{\includegraphics[width=\figwidth]{\picdir/dmag.g.\plotext}}
    429434\parbox{\figwidth}{
    430 \caption{PSF Magnitude residuals by Filter
    431  } \label{fig:psfmags.by.filter}}
     435\caption{PSF Magnitude residuals by Filter.  \note{expand colorscale
     436    bars, make clearer labels} } \label{fig:psfmags.by.filter}}
    432437
    433438\includegraphics[width=\figwidth]{\picdir/dmag.r.\plotext}
     
    468473millimagnitudes for all 5 plots.
    469474
    470 The tree-ring pattern is clearly visible for the four blue filters,
     475The Tree-Ring pattern is clearly visible for the four blue filters,
    471476but finging dominates the pattern for \yps.  Small offsets of
    472477individual cells are also apparent for \zps.  While the patterns are
     
    481486Figure~\ref{fig:apmags.by.filter} shows the equivalent measurement for
    482487aperture photometry instead of PSF photometry.  The finging
    483 pattern again dominates the plot for \yps, but the tree-rings are not
     488pattern again dominates the plot for \yps, but the Tree Rings are not
    484489seen in any of the filters.  A diagonal pattern is visible in \gps
    485490which is not observed in the PSF magnitudes.  While the per-pixel
     
    518523Y| > 0.5$ arcsec before measuring the median values for each
    519524superpixel.  We have determined the approximate center of the circular
    520 tree-ring pattern as (-5,4960) for this particular chip.  Using this
    521 coordinate as the center of the pattern, we have converted the $\delta
    522 X,\delta Y$ offsets into $\delta R,\delta \theta$ measurements
    523 ($\delta R$ : radial component away from the center, $\delta \theta$ :
    524 tangential component).
     525Tree-Ring pattern as (-5,4960) for this particular chip based on the
     526pattern of the X astrometry displacements.  Using this coordinate as the center
     527of the pattern, we have converted the $\delta X,\delta Y$ offsets into
     528$\delta R,\delta \theta$ measurements ($\delta R$ : radial component
     529away from the center, $\delta \theta$ : tangential component).
    525530
    526531Figure~\ref{fig:astrom.by.filter} shows the 2D patterns of $\delta R$
    527532for each filter (\grizy).  The dynamic range of the color scale is
    528 from -20 to +20 milliarcseconds for all 5 plots.  A tree-ring-like
     533from -20 to +20 milliarcseconds for all 5 plots.  A Tree-Ring-like
    529534pattern is visible for all five filters, with systematic structures
    530 following a circular pattern centered on the chip corner.; the finging
     535following a circular pattern centered on the chip corner; the finging
    531536pattern is not apparent in the \yps\ astrometry.  The per-pixel
    532537standard deviations of these plots is listed in
     
    562567obtained 2011 Feb 09 as part of a campaign to study the PS1 system
    563568response \citep{2012ApJ...750...99T}.  Flats were obtain in a set of
    564 4nm steps.  To enhance the signal-to-noise, we have median-combined a
    565 set of 6 flats at the center of the corresponding filter.
    566 
    567 In order to mask pixels which do not flatten well, we generate a
    568 a copy of the image smoothed with a Gaussian kernel with
    569 $\sigma = 1.5 pixels$.  Any pixels in the smoothed image which deviate
    570 from the median value in the image by more than 4 standard deviations
    571 is masked.  We generate the superpixel image by averaging the unmasked
    572 pixels associated with each superpixel.  We then high-pass filter the
    573 superpixel image by subtracting a copy smoothed with a Gaussian of
    574 $\sigma = 3.0$. 
     5694nm steps sampling the spectral response curve of each filter.  To
     570enhance the signal-to-noise, we have median-combined a set of 6 flats
     571at the wavelength center of the corresponding filter.
     572
     573In order to mask pixels which do not flatten well, we generate a copy
     574of the image smoothed with a Gaussian kernel with $\sigma = 1.5$
     575pixels.  Any pixels in the smoothed image which deviate from the
     576median value in the image by more than 4 standard deviations are
     577masked.  We generate the superpixel image by averaging the unmasked
     578pixels associated with each superpixel.  In order to suppress
     579large-scale gradients in the flat-field response, we high-pass filter
     580the superpixel image by subtracting a copy smoothed with a Gaussian of
     581$\sigma = 3.0$.
    575582
    576583Figure~\ref{fig:flats.by.filter} shows the remaining high-frequency
     
    584591measured flux in those pixels, and thus a {\em negative} deviation in
    585592$\delta m_{psf}$ as defined above.  The dynamic range of the color
    586 scale in these plots is -0.01 to +0.01.  The tree-ring-like pattern is
     593scale in these plots is -0.01 to +0.01.  The Tree-Ring-like pattern is
    587594strong in the (\gps,\rps,\ips) images, but nearly swamped by fringing
    588595in \zps, and completely lost to finging in \yps.  A diagonal banding
    589596similar to the aperture residuals is seen in \gps.
     597
     598\note{CZW asks about the blob in the flat-field response.  KCC asks
     599  about the brick-wall pattern.  discuss these and fringing so we can
     600  move on to the tree rings}
    590601
    591602\subsection{Second Moments}
     
    641652detection, we extract the values $M_{xx,xy,yy} = \sum F_i w_i (x^2, x
    642653y, y^2) / \sum F_i w_i$.  For each exposure, we find the median second
    643 moments for PSF objects on this chip (XY40) and subtract that median
    644 value from the instantaneous measurements of $M_{xx,xy,yy}$.  We then
     654moments for PSF objects on this chip (XY40) and subtract those median
     655values from the instantaneous measurements of $M_{xx,xy,yy}$.  We then
    645656determine the median of the residual second moments for each
    646657superpixel, resulting in 3 images ($\delta M_{xx,xy,yy}$) for each
     
    661672e_2 & = & \sigma^2_{\mbox{major}}  - \sigma^2_{\mbox{minor}}
    662673\end{eqnarray}
    663 Where $\sigma_{\mbox{major}}$ and $\sigma_{\mbox{minor}}$ are the major and minor axis
    664 dimensions of the ellipse and $\theta$ is the position angle. 
    665 Thus, $e_0$ is a measurement of the change in the size of the stellar
    666 PSFs as a function of position in the detector (``smear''), $e_2$ is a measurement
    667 of the change in ellipticity of the stellar PSFs (``shear''), and we
    668 can determine the angle of the PSF ellipticity from the $e_1$ term.
     674Where $\sigma_{\mbox{major}}$ and $\sigma_{\mbox{minor}}$ are the
     675major and minor axis dimensions of the ellipse and $\theta$ is the
     676position angle.  Thus, $e_0$ is a measurement of the change in the
     677size of the stellar PSFs as a function of position in the detector
     678(``smear''), $e_2$ is a measurement of the change in ellipticity of
     679the stellar PSFs (``shear''), and we can determine the angle of the
     680PSF ellipticity from the $e_1$ term.
    669681
    670682Figure~\ref{fig:smear.by.filter} shows the spatial trend of $e_0$, the {\em
    671683  smear}.  This value corresponds to the increase or decrease in
    672684the circularly-symmetric component of the image size.  The dynamic
    673 range of these images is -0.3 to +0.3 pixel$^2$. A tree-ring-like
     685range of these images is -0.3 to +0.3 pixel$^2$. A Tree-Ring-like
    674686pattern is visible for all 5 filters, though \yps is dominated by the
    675687fringing pattern.  Structures with relatively low spatial frequencies
     
    683695ellipse orientation as a function of postion.  The length of the
    684696vectors corresponds to the value of $\sigma^2_{major} -
    685 \sigma^2_{minor}$.  The tree-ring-like structure is {\em not} apparent
     697\sigma^2_{minor}$.  The Tree-Ring-like structure is {\em not} apparent
    686698in this figure for any filter.  The spatial variations are
    687699low-frequency and unrelated to the radial trend from the upper-left
     
    692704\begin{table}
    693705\caption{Systematic Trends : Correlations by filter\label{table:correlation.by.filter}}
     706\note{reconsider the column order}
    694707% \tiny
    695708\begin{center}
     
    708721\end{table}
    709722
    710 Tree-ring-like patterns are clearly seen in 4 of the measurement types
     723Tree-Ring-like patterns are clearly seen in 4 of the measurement types
    711724above: the PSF photometry, the astrometry, the flat-field, and the
    712725smear terms.  As discussed above, the signal-to-noise per pixel in the
    713726plots of the systematic trends is relatively low (\approx 1.0).  While
    714 the tree-ring-like patterns are apparent in many of these figures,
     727the Tree-Ring-like patterns are apparent in many of these figures,
    715728there are also some other systematic structures which may degrade the
    716729signal further.
    717730
    718 To quantatatively compare the tree-ring-like trends between
     731To quantatatively compare the Tree-Ring-like trends between
    719732filters and between the types of measurements, we need to measure the
    720 tree-ring structure explicitly and filter out the other effects if
     733Tree-Ring structure explicitly and filter out the other effects if
    721734possible.  To do this, we have applied a high-pass filter to all of
    722 the relevant images (PSF Photometry residuals, Astrometric residuals
     735the relevant images (PSF photometry residuals, astrometric residuals
    723736in the radial direction, flat-field residuals, and second moment smear
    724737terms) to remove unrelated spatial structures.  We have then measured
     
    730743chip.
    731744
     745\note{include the arc on one of the figures?}
     746
     747\note{do plots of all filter pairs in a triangle?  is that interesting?}
     748
    732749For a given type of measurement, the systematic effect is strongly
    733750correlated between filters.  The strongest correlation is the smear
     
    738755filters, as shown in Figure~\ref{fig:psfmag.trends}.  Here, the
    739756\yps\ correlation with \gps\ is quite weak: the fringing pattern
    740 dominates the tree-rings for PSF photometry.  The radial component of
     757dominates the Tree Rings for PSF photometry.  The radial component of
    741758the astrometric residual is also well correlated between filters, with
    742759no loss of correlation due to fringing in \yps. Finally, the
     
    749766listed in Table~\ref{table:correlation.by.filter}.  There is a
    750767consistency in the trend from \gps, with the strongest systematic
    751 tree-ring effects to \yps, with the weakest effects.  Note that the
     768Tree-Ring effects to \yps, with the weakest effects.  Note that the
    752769second moment smear and astrometry terms have different relative
    753770strength in \yps\ compared with \gps.
     
    758775\begin{center}
    759776\includegraphics[width=\figwidth]{\picdir/smear.trends.\plotext}
    760 \caption{Smear : correlation between filters
     777\caption{Smear : correlation between filters \note{include trend slopes in plots?}
    761778} \label{fig:smear.trends}
    762779\end{center}
     
    793810\end{figure*}
    794811
    795 An important question is the relationship of the tree-ring-like
     812An important question is the relationship of the Tree-Ring-like
    796813pattern between the different types of measurements.  Different models
    797 for the tree-ring structures make different predictions about the
     814for the Tree-Ring structures make different predictions about the
    798815correlations between different effects.  Note the very different
    799816spatial structure between the different measurements in a given
     
    817834errors: $\frac{\partial \delta R}{\partial radius} \sim \delta flat$
    818835(see Figure~\ref{fig:dastrom.vs.flat}.  This last relationship is
    819 somewhat weakly measured.  Because of the periodic nature of the tree
    820 rings, it is also difficult to be completely certain that the
     836somewhat weakly measured.  Because of the periodic nature of the Tree
     837Rings, it is also difficult to be completely certain that the
    821838flat-field is proportional to the derivative of the astrometry
    822839residual, rather than the astrometry residual being proportional to
     
    885902(Figure~\ref{fig:smear.by.filter}), the measurement shows that the
    886903intrinsic size of the stellar images is varying in a radial sense
    887 between the different tree-ring regions.  Although images experience
     904between the different Tree-Ring regions.  Although images experience
    888905an average image quality (due to seeing and focus) across the chip
    889906which may vary substantially from exposure to exposure, stars landing
    890 in the different tree-ring-like regions are consistently somewhat
     907in the different Tree-Ring-like regions are consistently somewhat
    891908larger or somewhat smaller than that average.
    892909
     
    895912the PSF, allowing for some spatial variation in the shape.  However,
    896913we have a limited number of stars to measure any spatial variation.
    897 Thus the 2D variation are sampled on a very coarse (e.g., 3x3) grid
    898 for each chip: the PSF parameters may vary smoothly across the chip
    899 following the bilinear interpolation between the 3x3 grid points.
    900 Thus, the spatial scale on which we model PSF variations is much
    901 larger than the spatial scale on which PSF variations are apparently
    902 occuring, as illustrated by the changes in the smear plot.  When the
    903 true PSF is larger than the model PSF, our model fits systematically
    904 underestimate the amount of flux in a given object.  Conversely, when
    905 the PSF is smaller, we overestimate the flux -- this type of offset is
    906 a typical effect when mis-estimating the PSF size.  The slope of the
    907 trend depends on the mean typical seeing for the given filter.  For
    908 example, the \gps\ seeing is typically 1.3\arcsec, corresponding to a
    909 Gaussian $\sigma$ of 2.15 pixels.  A smearing of $\sigma^2_{major} +
    910 \sigma^2_{minor} = 0.1$ pixels$^2$ would increase the size by about
    911 0.02 pixels, or 1\%, roughly consistent with the observed photometric
    912 deviation of about 5 to 10 millimags for this amount of smearing.
     914Thus the 2D variation are sampled on a very coarse (e.g., $3 \times
     9153$) grid for each chip: the PSF parameters may vary smoothly across
     916the chip following the bilinear interpolation between the $3 \times 3$
     917grid points.  Thus, the spatial scale on which we model PSF variations
     918is much larger than the spatial scale on which PSF variations are
     919apparently occuring, as illustrated by the changes in the smear plot.
     920When the true PSF is larger than the model PSF, our model fits
     921systematically underestimate the amount of flux in a given object.
     922Conversely, when the PSF is smaller, we overestimate the flux -- this
     923type of offset is a typical effect when mis-estimating the PSF size.
     924The slope of the trend depends on the mean typical seeing for the
     925given filter.  For example, the \gps\ seeing is typically 1.3\arcsec,
     926corresponding to a Gaussian $\sigma$ of 2.15 pixels.  A smearing of
     927$\sigma^2_{major} + \sigma^2_{minor} = 0.1$ pixels$^2$ would increase
     928the size by about 0.02 pixels, or 1\%, roughly consistent with the
     929observed photometric deviation of about 5 to 10 millimags for this
     930amount of smearing.
    913931
    914932The relationship between the flat-field residual and the astrometric
    915933gradient is consistent with radial variations in the plate-scale.  The
    916 tree-rings observed by DES are completely attributed to effective
     934Tree-Rings observed by DES are completely attributed to effective
    917935plate scale changes.  Effective plate scale changes would result in
    918936flat-field deviations since the flat-field illumination is a source of
     
    927945
    928946The fact that the PSF ellipticity changes are {\em not} correlated
    929 with the tree ring structure tells us that the effective plate-scale
    930 changes seen in the flat-field and astrometry signals are not also the
     947with the Tree-Ring structure tells us that the effective plate-scale
     948changes seen in the flat-field and astrometry signals are not the
    931949dominant cause of the PSF photometry errors.  Also, the fact that we
    932950do not measure significant aperture photometry errors correlated with
    933 the tree rings confirms this point.  The amplitude of the flat-field
     951the Tree Rings confirms this point.  The amplitude of the flat-field
    934952errors are 1-2 millimagnitudes, much smaller than the PSF photometry
    935953errors, and far below the pixel-to-pixel noise in the aperture
     
    953971\section{Conclusion}
    954972
    955 The tree rings observed in the Pan-STARRS GPC1 data show (at least)
     973The Tree Rings observed in the Pan-STARRS GPC1 data show (at least)
    956974two effects, though they are related.  First, the images are
    957975experiencing circularly-symmetric changes in the PSF size correlated
    958 with the tree-ring pattern.  These PSF size changes drive errors in
    959 the PSF photometry which the are also correlated with the tree ring
     976with the Tree-Ring pattern.  These PSF size changes drive errors in
     977the PSF photometry which the are also correlated with the Tree-Ring
    960978pattern on the scale of a few millimagnitudes.  These PSF size changes
    961979are consistent with changes in the charge diffusion, which also
     
    963981
    964982In addition, there are radial plate-scale changes correlated with the
    965 tree rings.  These plate-scale changes introduce a flat-field errors
     983Tree Rings.  These plate-scale changes introduce a flat-field errors
    966984on the scale of \approx 1 millimagnitude and astrometric errors in the
    967985scale of 2-3 milliarcseconds.  The observed relationship between the
     
    970988  in][]{2014PASP..126..750P}.
    971989
    972 The vertical diffusion variations and the lateral charge migration are
    973 both driven by the same variations in the doping structures.  This
    974 point is clear from the spatial correlation of the gradient in the
    975 smear variations and the astrometric variations.
     990The spatial correlation of the gradient in the smear variations and
     991the astrometric variations imply that both of these two types of tree
     992ring effects are related, even though they manifest through different
     993mechanisms.  We suspect that the variations in both the vertical charge
     994diffusion and the lateral charge migration are driven by changes
     995in the electric field structures in the silicon due to the same
     996variations in the doping structures in the silicon.
    976997
    977998% The small-scale variations in the charge diffusion observed in these
     
    10291050Lorand University (ELTE) and the Los Alamos National Laboratory.
    10301051
     1052\note{add NASA ops grant(s)}
     1053
    10311054\bibliographystyle{apj}
    10321055\bibliography{lib}{}
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