Index: trunk/doc/release.2015/ps1.analysis/analysis.tex =================================================================== --- trunk/doc/release.2015/ps1.analysis/analysis.tex (revision 40700) +++ trunk/doc/release.2015/ps1.analysis/analysis.tex (revision 40705) @@ -417,10 +417,10 @@ \end{itemize} -\section{\nocode{psphot} Analysis Process} +\section{Basic Analysis} \subsection{Overview} -The \ippprog{psphot} analysis is divided into several major stages, as -listed below. +The basic \ippprog{psphot} analysis is divided into several major +stages, as listed below. \begin{enumerate} @@ -441,7 +441,4 @@ properties (aperture or PSF) -\item {\bf Extended Source Analysis} Detailed measurements relevant to - galaxies and/or other extended (non-PSF) sources. - \item {\bf Aperture corrections} Measure the curve-of-growth, spatial aperture variations, and background-error corrections. @@ -450,4 +447,12 @@ difference image, variance image, etc, as selected. \end{enumerate} + +In addition to this basic sequence, additional analysis steps may be +performed. An ``extended source'' analysis mode is available to +measure photometry and morphology of galaxies and other resolved +sources. Forced photometry may be performed for both point-like and +extended sources. A special mode is available for the photometry of +sources detected in difference images. These different modes are +discussed in their own sections below. Table~\ref{tab:measurements} lists the types of @@ -1938,5 +1943,222 @@ \code{PM_SOURCE_MODE2_MATCHED} set. -\subsection{Extended Source Analysis} +\subsection{Aperture Correction and Total Aperture Fluxes} +\label{sec:aperture.correction} + +A PSF model will always fail to describe the flux of the stellar +sources at some level. For high-precision photometry, we need to be +able to correct for the difference between the PSF model fluxes and +the total flux of the sources. In the end, all astronomical +photometry is in some sense a relative measurement between two images. +Whether the goal is calibration of a science image taken at one +location to a standard star image at another location, or the goal is +simply the repetitive photometry of the same star at the same location +in the image, it is always necessary to compare the photometry between +two images. If this measurement is to be consistent, then the +measurement must represent the flux of the stars in the same way +regardless of the conditions under which the images were taken, at +least within some range of normal image conditions. So, for example, +two images with different image quality, or with different tracking +and focus errors, will have different PSF models. To the extent the +PSF model is inaccurate, the measured flux of the same source in the +two images will be different (even assuming all other atmospheric and +instrumental effects have been corrected). The amplitude of the error +will by determined by how inconsistently the models represent the +actual source flux. + +Aperture photometry attempts to avoid these problems, but introduces +other difficulties. In aperture photometry, if a large enough +aperture is chosen, the amount of flux which is lost will be a small +fraction of the total source flux. Even more importantly, as the +image conditions change, the amount lost will change by an even +smaller fraction, at least for a large aperture. This can be seen by +the fact that the dominant variations in the image quality are in the +focus, tracking and seeing. All of these errors initially affect the +cores of the stellar images, rather than the wide wings. The wide +wings are largely dominated by scattering in the optics and scattering +in the atmosphere. The amplitude and distribution of these two +scattering functions do not change significantly or quickly for a +single telescope and site. Aperture photometry can then be used to +correct the PSF photometry. + +The difficulty for aperture photometry is the need to make an accurate +measurement of the local background for each source. As the aperture +grows, errors in the measurement of the sky flux start to become +dominant. If the aperture is too small, then variations in the image +quality are dominant. The brighter is the source, the smaller is the +error introduced by the large size of the aperture. However, the +number of very bright stars is limited in any image, and of course the +brighter stars are more likely to suffer from non-linearity or +saturation. + +\begin{figure*}[htbp] + \begin{center} + \includegraphics[width=\hsize,clip]{pics/{mag.resid.psf}.png} + \caption{\label{fig:mag.resid.psf} PSF Photometry demonstration. + The bottom panel shows the difference of the measured PSF + photometry for stars in the first image of the STS sequence + compared to the next 17 images, after correction for a relative + zero point. Black dots are from stars for which both measurements + have {\tt PSF\_QF} $> 0.95$, while grey dots have lower {\tt + PSF\_QF} values. The top three panels show histograms in three + instrumental magnitude ranges for the magnitude difference divided + by the reported measurement error: $N\sigma = (m_0 - m_1) / + \sqrt{\sigma_0^2 + \sigma_1^2}$. The red curves are Gaussian fits + to these histograms, with the measured standard deviations in the + upper-right corners of the plots. The instrumental magnitude + ranges are listed in the upper-left corners of the three plots and + the boundaries are marked as vertical red lines in the lower plot. + } + \end{center} +\end{figure*} + +\begin{figure*}[htbp] + \begin{center} + \includegraphics[width=\hsize,clip]{pics/{mag.resid.aper}.png} + \caption{\label{fig:mag.resid.aper} Aperture Photometry + demonstration. The plots show identical measurements to those in + Figure~\ref{fig:mag.resid.psf}, but for aperture photometry, as discussed in + Section~\ref{sec:aperture.correction}, rather than PSF photometry.} + \end{center} +\end{figure*} + +In order to thread the needle between these effects, \ippprog{psphot} +measures the aperture photometry on a modest-sized aperture, and then +uses the PSF model to extrapolate to a large aperture. When the PSF +fluxes are calculated, the aperture flux for the modest-sized aperture +is also determined. The aperture is a circular aperture with radius +set to a fixed multiple (\code{PSF_APERTURE_SCALE}) of $\sigma_w$, the +width of the Gaussian window function determined based on the analysis +of the second moments (see Section~\ref{sec:moments}). For the PV3 +$3\pi$ analysis, the aperture window radius is $4.5 \times \sigma_w$, +while the large reference aperture radius is set to 25 pixels +(\code{PSF_REF_RADIUS} = 6\farcs4). These corrected aperture +magnitudes are saved in the output catalogs as \code{AP_MAG}, the +uncorrected aperture magnitudes are saved as \code{AP_MAG_RAW}, and +the radius used to measure the raw aperture flux is saved as +\code{AP_MAG_RADIUS}. The corresponding flux and the flux error are +saved as \code{AP_FLUX} and \code{AP_FLUX_SIG}. + +With these aperture magnitudes in hand, it is now possible to make an +average correction to the PSF magnitudes to bring the PSF and aperture +magnitudes to the same system. This correction is measured using the +same stars from which the PSF model is measured, as long as the PSF +magnitude error for the star is less than 0.03 mag. The correction is +calculated using the weighted average of the values $m_{\rm AP} - +m_{\rm PSF}$. Since the PSF may vary across the image, the correction +is determined as a function of position in the image. Like the PSF +model, the 2D variations of the aperture correction may be modeled as +a polynomial or via interpolation in a grid. For the $3\pi$ PV3 +analysis, a grid with a maximum of $6\times 6$ samples per GPC1 chip +image was used. The reported PSF magnitudes for all objects have this +aperture correction applied. + +% growth curve analysis in psphot: +% in psphotChoosePSF : call psphotMakeGrowthCurve +% in psphotMakeGrowthCurve : boolean GROWTH_FROM_SOURCES, use +%% pmGrowthCurveGenerateFromSources or +%% pmGrowthCurveGenerate (uses PSF model only) +%% GROWTH_FROM_SOURCES is set to TRUE for default recipe + +%% ApTrend: +%% in psphotApResid, called by psphotReadout near the end of the +%% analysis +%% ApTrend = f(x,y) for (apMag - psfMag) for psfMagErr <= 0.03 +%% apMag is growth curve corrected +%% psfMag is raw + +%% raw psfMag and raw apMag are measured +%% apMag = apMagRaw + growth curve correction (from apRadius to 25 pix +%% = PSF_REF_RADIUS) +%% psfMag = psfMagRaw + aperture trend ( + growth curve) + +% How important is this effect? Consider a typical bright source with a +% flux of (say) 40,000 counts in an image of background 1000 counts per +% pixel, with FWHM of 4 pixels. In principle, the flux of this source +% should be measurable with an accuracy of roughly 0.57\% +% ($\frac{\sqrt{40000 + 1000 \times 12}}{40000}$). However, the +% measurement of the sky is limited at some finite level by Poisson +% statistics. If we are required to use an aperture of (say) 25 pixels +% in radius (eg, 5 arcseconds for an 0.2 arcsec / pixel detector), and +% we have an annulus of twice this radius to measure the local sky, then +% we will have an error of XXX. +% +% \note{outline the variation of {\em ApResid} as a function of +% magnitude}. + +%%% \ippprog{psphot} measures the aperture correction ({\em ApResid}) for every PSF +%%% candidate source, then calculates the trend of this correction as a +%%% function of the magnitude. This trend is fitted with a line. The +%%% resulting function can be used to determine the effective aperture +%%% correction for an infinite flux source and the average bias inherent +%%% in the sky measurement for the image. The scatter of the +%%% PSF-candidate source measurements about this trend is a measure of how +%%% well we can measure photometry from the image by applying the specific +%%% PSF model. The slope of this trend is a measure of the bias in the +%%% local sky measurment for each source. In principal, the measured sky +%%% levels could be modified by this bias. More generally, the measured +%%% bias in a collection of images could be used to improve the model +%%% fitting or sky fitting portion of the software the remove the bias +%%% term. + +\ippprog{psphot} allows a collection of PSF model functions to be tried on all +PSF candidate sources. For each model test, the above corrected +ApResid scatter is measured. The PSF model function with the smallest +value for the ApResid scatter is then used by \ippprog{psphot} as the best PSF +model for this image. The number of models to be tested is specified +by the configuration keyword \code{PSF_MODEL_N}. The configuration +variables \code{PSF_MODEL_0}, \code{PSF_MODEL_1}, through +\code{PSF_MODEL_N - 1} specify the names of the models which should be +tested. + +\subsection{Stellar Photometry Example} + +To illustrate the quality of the stellar photometry as measured with +PSF and aperture magnitudes, we show the results of an analysis of a +set of 18 images obtained by PS1 19 February 2010. These images were +obtained for the stellar transit survey ``Pan-Planets'' +\citep{2016A&A...587A..49O} and thus target a relatively dense +Galactic plane field. The observations were obtained with +approximately consistent pointing, reducing our sensitivity to +small-scale variations in the flat-field structures. + +Figures~\ref{fig:mag.resid.psf} and ~\ref{fig:mag.resid.aper} show +comparisons of the PSF and aperture photometry measured for these 18 +images. In these figures, the photometry has been measured using the +configuration for \ippprog{psphot} as used for the full PV3 +\ippstage{chip} analysis. The first image of the sequence is compared +to the remaining 17 images. A relative zero point correction is +applied, measured as the median of the photometry difference for stars +with signal-to-noise greater than 50. The combined error is reported +and used to generate the histograms shows in the figures. From these +two figures, one can observe the trade-off between PSF and aperture +photometry. For the brightest instrumental magnitudes, corresponding +to signal-to-noise ratios greater than roughly 300, aperture +magnitudes provide a more consistent measurement of the stellar +fluxes, while the PSF magnitudes are more reliable for fainter +sources. Catastophic failures or extreme outliers are also reduced +for the PSF photometry. + +We largely attribute the behavior on the bright end to systematic +errors in the photometry due to our inability to perfectly represent +the shape of the PSF. The PSF of stars at the bright end will depend +on the brightness because of the ``Brighter-Fatter'' effect +\citep[][]{2014JInst...9C3048A,2015JInst..10C5032G}, in which the +charge already present in the pixels will force the newly arriving +photoeletrons to be systematically pushed away from the accumulating +stellar image, but we do not include a brightness term in our PSF +model. Detector or electronic non-linearity may also affect the PSF +shape and thus the PSF photometry, though non-linearity will affect +the reported photometry for both PSF and aperture magnitudes. + +We believe the observed behavior at the faint end is primarily a +result of the increased crowding. Aperture photometry is more +adversely affected by close neighbors than PSF photometry. Compared +to the formal errors, the faint PSF photometry is the most reliable, +with the aperture photometry degrading rapidly as the flux of the star +decreases. + +\section{Extended Source Analysis} +\label{sec:extended.source} After the initial, fast analysis of the image relying primarily on the @@ -2013,5 +2235,5 @@ % if |b| > 20.0 + 15.0 exp(-long^2 / (2 * 50^2)) -\subsubsection{Radial Profiles} +\subsection{Radial Profiles} \label{sec:radial.profile.v2} @@ -2082,5 +2304,5 @@ % \note{these profiles are not saved in PSPS} -\subsubsection{Petrosian Radii and Magnitudes} +\subsection{Petrosian Radii and Magnitudes} \label{sec:petrosian} @@ -2141,5 +2363,5 @@ -\subsubsection{Convolved Galaxy Model Fits} +\subsection{Convolved Galaxy Model Fits} \label{sec:galaxy.conv.fit} @@ -2347,5 +2569,5 @@ any of the parameters. -\subsubsection{Fixed Aperture Photometry} +\subsection{Fixed Aperture Photometry} \label{sec:fixed.aperture.photom} @@ -2397,141 +2619,85 @@ % last bin is first with inner radius >= skyRadius -\subsection{Aperture Correction and Total Aperture Fluxes} -\label{sec:aperture.correction} - -A PSF model will always fail to describe the flux of the stellar -sources at some level. For high-precision photometry, we need to be -able to correct for the difference between the PSF model fluxes and -the total flux of the sources. In the end, all astronomical -photometry is in some sense a relative measurement between two images. -Whether the goal is calibration of a science image taken at one -location to a standard star image at another location, or the goal is -simply the repetitive photometry of the same star at the same location -in the image, it is always necessary to compare the photometry between -two images. If this measurement is to be consistent, then the -measurement must represent the flux of the stars in the same way -regardless of the conditions under which the images were taken, at -least within some range of normal image conditions. So, for example, -two images with different image quality, or with different tracking -and focus errors, will have different PSF models. To the extent the -PSF model is inaccurate, the measured flux of the same source in the -two images will be different (even assuming all other atmospheric and -instrumental effects have been corrected). The amplitude of the error -will by determined by how inconsistently the models represent the -actual source flux. - -Aperture photometry attempts to avoid these problems, but introduces -other difficulties. In aperture photometry, if a large enough -aperture is chosen, the amount of flux which is lost will be a small -fraction of the total source flux. Even more importantly, as the -image conditions change, the amount lost will change by an even -smaller fraction, at least for a large aperture. This can be seen by -the fact that the dominant variations in the image quality are in the -focus, tracking and seeing. All of these errors initially affect the -cores of the stellar images, rather than the wide wings. The wide -wings are largely dominated by scattering in the optics and scattering -in the atmosphere. The amplitude and distribution of these two -scattering functions do not change significantly or quickly for a -single telescope and site. Aperture photometry can then be used to -correct the PSF photometry. - -The difficulty for aperture photometry is the need to make an accurate -measurement of the local background for each source. As the aperture -grows, errors in the measurement of the sky flux start to become -dominant. If the aperture is too small, then variations in the image -quality are dominant. The brighter is the source, the smaller is the -error introduced by the large size of the aperture. However, the -number of very bright stars is limited in any image, and of course the -brighter stars are more likely to suffer from non-linearity or -saturation. - -In order to thread the needle between these effects, \ippprog{psphot} -measures the aperture photometry on a modest-sized aperture, and then -uses the PSF model to extrapolate to a large aperture. When the PSF -fluxes are calculated, the aperture flux for the modest-sized aperture -is also determined. The aperture is a circular aperture with radius -set to a fixed multiple (\code{PSF_APERTURE_SCALE}) of $\sigma_w$, the -width of the Gaussian window function determined based on the analysis -of the second moments (see Section~\ref{sec:moments}). For the PV3 -$3\pi$ analysis, the aperture window radius is $4.5 \times \sigma_w$, -while the large reference aperture radius is set to 25 pixels -(\code{PSF_REF_RADIUS} = 6\farcs4). These corrected aperture -magnitudes are saved in the output catalogs as \code{AP_MAG}, the -uncorrected aperture magnitudes are saved as \code{AP_MAG_RAW}, and -the radius used to measure the raw aperture flux is saved as -\code{AP_MAG_RADIUS}. The corresponding flux and the flux error are -saved as \code{AP_FLUX} and \code{AP_FLUX_SIG}. - -With these aperture magnitudes in hand, it is now possible to make an -average correction to the PSF magnitudes to bring the PSF and aperture -magnitudes to the same system. This correction is measured using the -same stars from which the PSF model is measured, as long as the PSF -magnitude error for the star is less than 0.03 mag. The correction is -calculated using the weighted average of the values $m_{\rm AP} - -m_{\rm PSF}$. Since the PSF may vary across the image, the correction -is determined as a function of position in the image. Like the PSF -model, the 2D variations of the aperture correction may be modeled as -a polynomial or via interpolation in a grid. For the $3\pi$ PV3 -analysis, a grid with a maximum of $6\times 6$ samples per GPC1 chip -image was used. The reported PSF magnitudes for all objects have this -aperture correction applied. - -% growth curve analysis in psphot: -% in psphotChoosePSF : call psphotMakeGrowthCurve -% in psphotMakeGrowthCurve : boolean GROWTH_FROM_SOURCES, use -%% pmGrowthCurveGenerateFromSources or -%% pmGrowthCurveGenerate (uses PSF model only) -%% GROWTH_FROM_SOURCES is set to TRUE for default recipe - -%% ApTrend: -%% in psphotApResid, called by psphotReadout near the end of the -%% analysis -%% ApTrend = f(x,y) for (apMag - psfMag) for psfMagErr <= 0.03 -%% apMag is growth curve corrected -%% psfMag is raw - -%% raw psfMag and raw apMag are measured -%% apMag = apMagRaw + growth curve correction (from apRadius to 25 pix -%% = PSF_REF_RADIUS) -%% psfMag = psfMagRaw + aperture trend ( + growth curve) - -% How important is this effect? Consider a typical bright source with a -% flux of (say) 40,000 counts in an image of background 1000 counts per -% pixel, with FWHM of 4 pixels. In principle, the flux of this source -% should be measurable with an accuracy of roughly 0.57\% -% ($\frac{\sqrt{40000 + 1000 \times 12}}{40000}$). However, the -% measurement of the sky is limited at some finite level by Poisson -% statistics. If we are required to use an aperture of (say) 25 pixels -% in radius (eg, 5 arcseconds for an 0.2 arcsec / pixel detector), and -% we have an annulus of twice this radius to measure the local sky, then -% we will have an error of XXX. -% -% \note{outline the variation of {\em ApResid} as a function of -% magnitude}. - -%%% \ippprog{psphot} measures the aperture correction ({\em ApResid}) for every PSF -%%% candidate source, then calculates the trend of this correction as a -%%% function of the magnitude. This trend is fitted with a line. The -%%% resulting function can be used to determine the effective aperture -%%% correction for an infinite flux source and the average bias inherent -%%% in the sky measurement for the image. The scatter of the -%%% PSF-candidate source measurements about this trend is a measure of how -%%% well we can measure photometry from the image by applying the specific -%%% PSF model. The slope of this trend is a measure of the bias in the -%%% local sky measurment for each source. In principal, the measured sky -%%% levels could be modified by this bias. More generally, the measured -%%% bias in a collection of images could be used to improve the model -%%% fitting or sky fitting portion of the software the remove the bias -%%% term. - -\ippprog{psphot} allows a collection of PSF model functions to be tried on all -PSF candidate sources. For each model test, the above corrected -ApResid scatter is measured. The PSF model function with the smallest -value for the ApResid scatter is then used by \ippprog{psphot} as the best PSF -model for this image. The number of models to be tested is specified -by the configuration keyword \code{PSF_MODEL_N}. The configuration -variables \code{PSF_MODEL_0}, \code{PSF_MODEL_1}, through -\code{PSF_MODEL_N - 1} specify the names of the models which should be -tested. +\subsection{Galaxy Model Simulations} + +To test the galaxy model analysis, we have generated a series of +simulated images containing both stars and galaxies on which we have +run the \ippprog{psphot} PSF-convolved galaxy model fitting analysis. +The images generated for this analysis have dimensions of $4000 \times +4000$ pixels with a spatial scale of 0.25 arcseconds per pixel. The +images are generated using an effective exposure time of 30 seconds, +with zero points matching the PS1 \rps-filter, and a realistic sky +brightness of 20.86 magnitudes per square arcsecond. The stars are +injected into these images with fluxes drawn from a realistic stellar +luminosity function and random spatial locations. For each image, the +same underlying simulated stellar population was used. Galaxies are +injected into the image with positions on a regularly spaced grid with +separation of 120 pixels. The galaxies are injected using Exponential +and DeVaucouleur profiles in separate simulation runs. The major axis +values are randomly distributed between 1 and 10 pixels (0.25 - 2.5 +arcseconds) while the aspect ratios are randomly chosen in a range +from 0.25 to 1.0. The position angles are set by the sequence in the +image and allowed to vary from 0 to 180 degrees. The images are then +convolved with a PSF model using the \code{PS1_V1} profile ($\kappa = +0.2$) and noise is added using Poisson statistics for the detected photons. + +For the figures below, we present results as a function of the (input) +instrumental magnitude of the galaxy minus the instrumental magnitude +corresponding to the stellar $5 \sigma$ detection limit. We make the +simplifying assumption that the stellar detection threshold +encapsulates enough information about the sensitivity of the images +that this magnitude difference may be used to compare the results +shown here to images with other depths. Thus this and subsequent +figures may be compared with the reported detection limits from the +PS1 $3\pi$ survey. Note for reference that the typical stellar +detection limits in the PS1 $3\pi$ stack images are (\grizy) = (23.3, +23.2, 23.1, 22.3, 21.4). The minimum Kron magnitudes for which galaxy +model fits were performed for the PV3 analysis +(Section~\ref{sec:extended.source}) thus correspond to -1.6 to -1.8 in +these plots. + +Figure~\ref{fig:galaxy.complete} shows completeness for the detection +of the Exponential and DeVaucouleur model galaxies. This analysis +does not indicate if the galaxy was detected {\em as a galaxy} (\ie, +was the extended nature of the source sufficiently clear), only if +the source was detected by the peak-finding algorithm. As expected, +the more compact galaxies are more likely to be detected; Exponential +profile galaxies, with a broader light distribution for the same +effective radius, are less likely to be detected for the same +magnitude than DeVaucouleur profile galaxies. + +\begin{figure}[htbp] + \begin{center} + \includegraphics[width=\hsize,clip]{pics/{galaxy.exp.complete}.png} + \includegraphics[width=\hsize,clip]{pics/{galaxy.dev.complete}.png} + \caption{\label{fig:galaxy.complete} Top: Completeness curves for + simulated galaxies with Exponential profiles. Bottom: + Completeness curves for simulated galaxies with DeVaucouleur + profiles. The curves are shown as a function of the difference + between the injected instrumental magnitude of the galaxy and the + magnitude corresponding to the $5\sigma$ detection threshold for a + PSF-like source. The black curves shows the compleness for all + galaxies. The three colored curves show the completeness for + three major axis ranges. Compact galaxies are more likely to be + detected since peaks are detected after convolution with the + PSF. } + \end{center} +\end{figure} + +\begin{figure*}[htbp] + \begin{center} + \includegraphics[width=\hsize,clip]{pics/{galaxy.exp.params}.png} + \caption{\label{fig:exp.complete} Parameter recovery for simulated + galaxies with Exponential profiles. } + \end{center} +\end{figure*} + +\begin{figure*}[htbp] + \begin{center} + \includegraphics[width=\hsize,clip]{pics/{galaxy.dev.params}.png} + \caption{\label{fig:dev.complete} Parameter recovery for simulated + galaxies with DeVaucouleur profiles. } + \end{center} +\end{figure*} \section{Forced Photometry Modes} @@ -2679,13 +2845,4 @@ \label{sec:lensing.params} -\begin{verbatim} -* background : KSB, related (mention Deacon et al here or at the end?) -* second moments are discussed above (same values, including window function as given) -* write out the KSB formalism -* stellar parameters using PSF stars -* output parameters -* this is only done on warp -- move to ForceWarp section? -\end {verbatim} - Weak-lensing studies frequently use non-parametric measurements of the ellipticities of galaxies to quantify the strength of gravitational @@ -2706,23 +2863,28 @@ applied the techinique to PTF data to search for binary stars and \cite{2017MNRAS.468.3499D} used the same technique to search for -binary companions to known ultracool dwarfs using Pan-STARRS $\3pi$ +binary companions to known ultracool dwarfs using Pan-STARRS $3\pi$ data. The work by \cite{2017MNRAS.468.3499D} used images and their own analysis of the pixels with the program Sextractor \citep{Bertin.ref}. -For the Pan-STARRS $\3pi$ PV3 analysis, we have measured the full set -of KSB lensing parameters for \note{which subset?} of the data to -enable both lensing studies and binary / multiple star searches. Here -we describe the measurements as performed within \ippprog{psphot}, -reviewing the mathematical framework as described by -\cite{1995ApJ...449..460K} and \cite{1998ApJ...504..636H}. +For the Pan-STARRS $3\pi$ PV3 analysis, we have measured the full set +of KSB lensing parameters for all objects with measured second moments +(i.e.,, excluding saturated stars, suspected cosmic rays, and other +likely defects) of the data to enable both lensing studies and binary +/ multiple star searches. Here we describe the measurements as +performed within \ippprog{psphot}, reviewing the mathematical +framework as described by \cite{1995ApJ...449..460K} and +\cite{1998ApJ...504..636H}. The goal of the KSB technique is to measure the intrinsic ellipticity of objects (i.e., galaxies, in the case of weak lensing studies) as -would be observed sky on the without instrumental effects. The -analysis starts with the observed ellipticity of the object as -represented by the two polarization components derived from the second -moments (see Section~\ref{sec:moments}): +would be observed sky on the without instrumental effects and to +determine the impact weak graviational lensing would have on the +observed shapes, after correction for the instrumental effects. The +analysis starts with the observed ellipticity of objects as represented +by the two polarization components derived from the second moments +(see Section~\ref{sec:moments}): \begin{eqnarray} +\label{eqn:polarization} e_1 = \frac{M_{xx} - M_{yy}}{M_{xx} + M_{yy}} \\ e_2 = \frac{2 M_{xy}}{M_{xx} + M_{yy}}. \\ @@ -2738,124 +2900,129 @@ $e_2$ and low absoluate values of $e_1$. -\note{need for the window function}. - -The observed ellipticity of an object observed in a real instrument +Note that in our analysis of the second moments, we are applying a +Gaussian window function to down-weight the noise contributions from +pixels at high radii and low flux (see Section~\ref{sec:moments}). +This type of window function is also assumed in the KSB formalism, and +is represented in the equations below as $W$. + +The measured ellipticity of an object observed in a real instrument will be affected by the point spread function of the instrument. To first order, the effect on the polarization components can be -described as a combination of ``smear'', in which the observed shape -is more circularized (driving $e_1,e_2$ to low absolute values) and -``shear'', in which the observed shape is stretched in one direction +described as a combination of the circularly symmetric seeing disc, +which smears the observed shapes (driving $e_1,e_2$ to low absolute +values) and the shearing effect of the anisotropic component of the +PSF, in which the observed shape is stretched in one direction relative to the others (driving $e_1,e_2$ to larger absolute values). -With sufficient understanding of the image PSF, both shear and smear -terms can be corrected. - -The change in the observed polarization of an object due to the - + +KSB and HFK quantify the change in the observed polarization due to +the smearing effect of the PSF with +\begin{equation} + \delta e^{\rm sm}_\alpha = P^{\rm sm}_{\alpha, \beta} p_{\beta} +\end{equation} +$p_\beta$ is a measurement of the +anisotropy of the PSF (see below), and $P^{\rm sm}_{\alpha,\beta}$ is +the ``Smear Polarizability'' of the object, defined as \begin{eqnarray} -X^{sh}_{1,1} = T^{-1} \sum f \left[ 2W(x^2 + y^2) + 2W^\prime (x^2 - y^2)^2 \\ -X^{sh}_{1,2} = T^{-1} \sum f \left[ 4W^\prime(x^2 - y^2) x y \\ -X^{sh}_{2,2} = T^{-1} \sum f \left[ 2W(x^2 + y^2) + 8W^\prime x^2 y^2 \\ + P^{\rm sm}_{\alpha \beta} = X^{\rm sm}_{\alpha \beta} - e_\alpha e^{\rm sm}_\beta +\end{eqnarray} +where +\begin{eqnarray} +X^{\rm sm}_{1,1} &=& \frac{1}{T} \sum f \left[ W + 2W^\prime r^2 + W^{\prime \prime} (x^2 - y^2)^2 \right] \\ +X^{\rm sm}_{1,2} &=& \frac{1}{T} \sum f \left[ 2W^{\prime\prime} (x^2 - y^2) x y \right] \\ +X^{\rm sm}_{2,2} &=& \frac{1}{T} \sum f \left[ W + 2W^\prime r^2 + 4W^{\prime \prime} x^2 y^2 \right] \end{eqnarray} - +and \begin{eqnarray} -e^{sh}_1 = 2 e_1 + 2 T^{-1} \sum f W^\prime (x^2 + y^2) (x^2 - y^2) \\ -e^{sh}_2 = 2 e_2 + 2 T^{-1} \sum f W^\prime (x^2 + y^2) 2 x y \\ +e^{\rm sm}_1 &=& \frac{1}{T} \sum f \left[ 2W^\prime + W^{\prime \prime} (x^2 + y^2) \right] (x^2 - y^2) \\ +e^{\rm sm}_2 &=& \frac{1}{T} \sum f \left[ 2W^\prime + W^{\prime \prime} (x^2 + y^2) \right] 2 x y. \end{eqnarray} - +In these equations, $T = M_{xx} + M_{yy}$ and $W$ is the window +function applied when measuring the second moments. The terms +$W^\prime$ and $W^{\prime \prime}$ are the derivatives of the window +function with respect to $r^2 = x^2 + y^2$. Since the window function +is a circularly-symmetric Gaussian with width $\sigma_w$, the +derivatives are simply $W^\prime = -\frac{1}{2\sigma^2_w} W$ and +$W^{\prime \prime} = \frac{1}{4\sigma^4_w} W$. + +The elements of the equations above can be written in terms of the second and higher-order +moments calculated in Section~\ref{sec:moments}: \begin{eqnarray} -X^{sm}_{1,1} = T^{-1} \sum f \left[ W + 2W^\prime (x^2 + y^2) + W^{\prime \prime} (x^2 - y^2)^2 \\ -X^{sm}_{1,2} = T^{-1} \sum f \left[ 2W^{\prime\prime} (x^2 - y^2) x y \\ -X^{sm}_{2,2} = T^{-1} \sum f \left[ W + 2W^\prime (x^2 + y^2) + 4W^{\prime \prime} x^2 y^2 \\ +X^{\rm sm}_{1,1} &=& \frac{1}{T} \left[ 1 - \frac{R_2}{\sigma^{2}} + \frac{(M_{xxxx} - 2 M_{xxyy} + M_{yyyy})}{4 \sigma^{4}} \right] \\[0.1in] +X^{\rm sm}_{1,2} &=& \frac{1}{T} \left[ \frac{(M_{xyyy} - M_{xxxy})}{2 \sigma^{4}} \right] \\[0.1in] +X^{\rm sm}_{2,2} &=& \frac{1}{T} \left[ 1 - \frac{R_2}{\sigma^{2}} + \frac{ M_{xxyy}}{\sigma^{4}} \right] \end{eqnarray} - +and \begin{eqnarray} -e^{sm}_1 = T^{-1} \sum f \left[ 2W^\prime + W^{\prime \prime} (x^2 + y^2) \right] (x^2 - y^2) \\ -e^{sm}_2 = T^{-1} \sum f \left[ 2W^\prime + W^{\prime \prime} (x^2 + y^2) \right] 2 x y \\ +e^{\rm sm}_1 &=& \frac{1}{T} \left[ \frac{M_{xx} - M_{yy}}{\sigma^{2}} + \frac{M_{xxxx} - M_{yyyy}}{4 \sigma^{4}} \right] \\[0.1in] +e^{\rm sm}_2 &=& \frac{1}{T} \left[ \frac{(M_{xxxy} + M_{xyyy})}{2\sigma^{4}} - \frac{2 M_{xy}}{\sigma^{2}} \right] \end{eqnarray} - - -@ARTICLE{2017MNRAS.468.3499D, - author = {{Deacon}, N.~R. and {Magnier}, E.~A. and {Best}, W.~M.~J. and - {Liu}, M.~C. and {Dupuy}, T.~J. and {Chambers}, K.~C. and {Draper}, P.~W. and - {Flewelling}, H. and {Metcalfe}, N. and {Tonry}, J.~L. and {Wainscoat}, R.~J. and - {Waters}, C.}, - title = "{Identification of partially resolved binaries in Pan-STARRS 1 data}", - journal = {\mnras}, -archivePrefix = "arXiv", - eprint = {1702.05491}, - primaryClass = "astro-ph.SR", - keywords = {binaries: visual, brown dwarfs}, - year = 2017, - month = jul, - volume = 468, - pages = {3499-3515}, - doi = {10.1093/mnras/stx440}, - adsurl = {http://adsabs.harvard.edu/abs/2017MNRAS.468.3499D}, - adsnote = {Provided by the SAO/NASA Astrophysics Data System} -} - -@ARTICLE{1995ApJ...449..460K, - author = {{Kaiser}, N. and {Squires}, G. and {Broadhurst}, T.}, - title = "{A Method for Weak Lensing Observations}", - journal = {\apj}, - eprint = {astro-ph/9411005}, - keywords = {COSMOLOGY: OBSERVATIONS, COSMOLOGY: DARK MATTER, GALAXIES: FORMATION, COSMOLOGY: GRAVITATIONAL LENSING, COSMOLOGY: LARGE-SCALE STRUCTURE OF UNIVERSE}, - year = 1995, - month = aug, - volume = 449, - pages = {460}, - doi = {10.1086/176071}, - adsurl = {http://adsabs.harvard.edu/abs/1995ApJ...449..460K}, - adsnote = {Provided by the SAO/NASA Astrophysics Data System} -} -@ARTICLE{1998ApJ...504..636H, - author = {{Hoekstra}, H. and {Franx}, M. and {Kuijken}, K. and {Squires}, G. - }, - title = "{Weak Lensing Analysis of CL 1358+62 Using Hubble Space Telescope Observations}", - journal = {\apj}, - keywords = {GALAXIES: CLUSTERS: INDIVIDUAL ALPHANUMERIC: CL 1358+62, GALAXIES: FUNDAMENTAL PARAMETERS, COSMOLOGY: GRAVITATIONAL LENSING, galaxies: clusters: individual (Cl 1358 + 62), Galaxies: Fundamental Parameters, Cosmology: Gravitational Lensing}, - year = 1998, - month = sep, - volume = 504, - pages = {636-660}, - doi = {10.1086/306102}, - adsurl = {http://adsabs.harvard.edu/abs/1998ApJ...504..636H}, - adsnote = {Provided by the SAO/NASA Astrophysics Data System} -} -@ARTICLE{2005ApJ...626.1070H, - author = {{Hoekstra}, H. and {Wu}, Y. and {Udalski}, A.}, - title = "{An Algorithm to Detect Blends with Eclipsing Binaries in Planet Transit Searches}", - journal = {\apj}, - eprint = {astro-ph/0501353}, - keywords = {Stars: Binaries: Eclipsing, Stars: Planetary Systems}, - year = 2005, - month = jun, - volume = 626, - pages = {1070-1078}, - doi = {10.1086/430299}, - adsurl = {http://adsabs.harvard.edu/abs/2005ApJ...626.1070H}, - adsnote = {Provided by the SAO/NASA Astrophysics Data System} -} -@ARTICLE{2013ApJS..206...18T, - author = {{Terziev}, E. and {Law}, N.~M. and {Arcavi}, I. and {Baranec}, C. and - {Bloom}, J.~S. and {Bui}, K. and {Burse}, M.~P. and {Chorida}, P. and - {Das}, H.~K. and {Dekany}, R.~G. and {Kraus}, A.~L. and {Kulkarni}, S.~R. and - {Nugent}, P. and {Ofek}, E.~O. and {Punnadi}, S. and {Ramaprakash}, A.~N. and - {Riddle}, R. and {Sullivan}, M. and {Tendulkar}, S.~P.}, - title = "{Millions of Multiples: Detecting and Characterizing Close-separation Binary Systems in Synoptic Sky Surveys}", - journal = {\apjs}, -archivePrefix = "arXiv", - eprint = {1210.4550}, - primaryClass = "astro-ph.SR", - keywords = {binaries: close, methods: data analysis, stars: statistics, surveys, techniques: image processing }, - year = 2013, - month = jun, - volume = 206, - eid = {18}, - pages = {18}, - doi = {10.1088/0067-0049/206/2/18}, - adsurl = {http://adsabs.harvard.edu/abs/2013ApJS..206...18T}, - adsnote = {Provided by the SAO/NASA Astrophysics Data System} -} +where $R_2 = M_{xx} + M_{yy}$. + +KSB and HFK use the observed ellipticities of stars and the smear +polarizability of the stars to estimate the anisotropy due to the PSF: +\begin{eqnarray} +p_\alpha = \frac{e^*_{\alpha}}{P^{{\rm sm},*}_{\alpha \alpha}} +\end{eqnarray} +where the terms with the $*$ represent parameters measured on stars. + +%% \begin{eqnarray} +%% p_1 &=& M_{xx} - M_{yy} \\ +%% p_2 &=& 2 M_{xy} +%% \end{eqnarray} + +Similarly, the impact of shear can be quantified by the ``Shear +Polarizabilty'' in a similar fashion: +\begin{equation} + \delta e^{\rm sh}_\alpha = P^{\rm sh}_{\alpha, \beta} p_{\beta} +\end{equation} +where now the shear polarizability $P^{\rm sh}_{\alpha \beta}$ is +defined as +\begin{eqnarray} + P^{\rm sh}_{\alpha \beta} = X^{\rm sh}_{\alpha \beta} - e_\alpha e^{\rm sh}_\beta +\end{eqnarray} +where +\begin{eqnarray} +X^{\rm sh}_{1,1} &=& \frac{1}{T} \sum f \left[ 2W(x^2 + y^2) + 2W^\prime (x^2 - y^2)^2 \right] \\ +X^{\rm sh}_{1,2} &=& \frac{1}{T} \sum f \left[ 4W^\prime(x^2 - y^2) x y \right] \\ +X^{\rm sh}_{2,2} &=& \frac{1}{T} \sum f \left[ 2W(x^2 + y^2) + 8W^\prime x^2 y^2 \right] +\end{eqnarray} +and +\begin{eqnarray} +e^{\rm sh}_1 &=& 2 e_1 + \frac{2}{T} \sum f W^\prime (x^2 + y^2) (x^2 - y^2) \\ +e^{\rm sh}_2 &=& 2 e_2 + \frac{2}{T} \sum f W^\prime (x^2 + y^2) 2 x y. +\end{eqnarray} + +Re-writing in terms of the second and higher-order moments calculated +in Section~\ref{sec:moments}, we find: +\begin{eqnarray} +X^{\rm sh}_{1,1} &=& \frac{1}{T} \left[ 2 R_2 - \frac{(M_{xxxx} - 2 M_{xxyy} + M_{yyyy})}{\sigma^{2}} \right] \\ +X^{\rm sh}_{1,2} &=& \frac{1}{T} \left[ \frac{2(M_{xyyy} - M_{xxxy})}{\sigma^{2}} \right] \\ +X^{\rm sh}_{2,2} &=& \frac{1}{T} \left[ 2 R_2 - \frac{4 M_{xxyy}}{\sigma^{2}} \right] +\end{eqnarray} +and +\begin{eqnarray} +e^{\rm sh}_1 &=& \frac{1}{T} \left[ 2 (M_{xx} - M_{yy}) + \frac{( M_{yyyy} - M_{xxxx})}{\sigma^{2}} \right] \\ +e^{\rm sh}_2 &=& \frac{1}{T} \left[ 4 M_{xy} - \frac{2 (M_{xxxy} + M_{xyyy})}{\sigma^{2}} \right] +\end{eqnarray} + +In the Pan-STARRS PV3 analysis, we have measured the elements of the +smear polarizability ($X^{\rm sm}_{\alpha \beta}$, $e^{\rm + sm}_\alpha$) and the shear polarizability ($X^{\rm sh}_{\alpha + \beta}$, $e^{\rm sh}_\alpha$) for all objects on each of the warp +images. We have also selected only the PSF stars from the images and +interpolated a smoothed version of these parameters to the location of +the objects, using the grid described above to interpolate the PSF +parameters. We also determine the interpolated PSF ellipticities +($e^*_1, e^*_2$) from the equivalent smooth grid. Thus, for every +object in the $3\pi$ survey, we are able to report the PSF and object +elements of the KSB analysis. These lensing parameters are measured +for each of the warps, and then averaged over all warps for each of +the filters. The average values are calculated by including only +measurements from the same warp detection used in the average +photometry (nominally, the primary skycell; see Paper V, Section +5.4.4) and excluding any measurements for which the \code{PSF_QF} or +\code{PSF_QF_PERFECT} is less than 0.85. + +\note{example of using the lensing elements for binaries?} \section{Difference Image Photometry}