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analysis.tex

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Dec 17, 2018, 10:41:29 AM (8 years ago)

Author:

eugene

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updates

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trunk/doc/release.2015/ps1.analysis/analysis.tex

-              r40559
+              r40584
 The photometric and astrometric precision goals for the Pan-STARRS\,1
 surveys were quite stringent: photmetric accuracy of 10
+surveys were quite stringent: photometric accuracy of 10
 millimagnitudes, relative astrometric accuracy of 10 milliarcseconds
 and absolute astrometric accuracy of 100 milliarcseconds with respect
 …
 In the process, the source analysis software was written using the
 data analysis C-code library written for the IPP, \code{psLib}
+\citep{psLib}.  Components of the photometry code were integrated into
+the IPP's mid-level astronomy data analysis toolkit called
+\code{psModules} \citep{psModules}.  The resulting software,
+`\code{psphot}', can be used either as a stand-alone C program, or as
+a set of library functions which may be integrated into other programs
+% \note{add refs to the psLib and psModules ADDs} : ref to online docs?
+\citep{magnier2017.datasystem}.  Components of the photometry code
+were integrated into the IPP's mid-level astronomy data analysis
+toolkit called \code{psModules} \citep{magnier2017.datasystem}.  The
+resulting software, `\code{psphot}', can be used either as a
+stand-alone C program, or as a set of library functions which may be
+integrated into other programs
 Several variants of \code{psphot} have been used in the PS1 PV3
 …
 \begin{figure}[htbp]
   \begin{center}
+  \includegraphics[width=\hsize]{{pics/FWHM.smooth.trend.ps1}.\plotext}
+  \caption{\label{fig:moments.window} Example of the biases encountered when measuring the second
+    moments.  A simulated image was generated using the PS1 PSF
+    profile.  Each panel corresponds to a different value of
+    $\sigma_w$, as marked.  The solid red line is the true FWHM of the
+    PSF used to generate the stars.  The blue solid line is the FWHM
+    of the window function ($2.35\sigma_w$).  The gray dots are the
+    FWHM derived from the measured second moments for stars in the
+    image.  The dotted blue line is the target (65\% of the window
+    function).  In this example, we would choose $\sigma_w$ between
+.5 and 0.8 arcseconds so the dotted blue line would match the
+    bright end of the gray dots.}
+% \includegraphics[width=0.6\hsize]{{pics/FWHM.smooth.trend.ps1}.\plotext}
+  \includegraphics[width=0.95\hsize]{{pics/FWHM.smooth.trend.ps1}.png}
+  \caption{\label{fig:moments.window} Example of the biases
+    encountered when measuring the second moments.  A simulated image
+    was generated using the PS1 PSF profile.  Each panel corresponds
+    to a different value of $\sigma_w$, corresponding to the window
+    FWHM values as marked.  The solid red line is the true FWHM of the
+    PSF used to generate the stars (1.4 arcsec in all cases).  The
+    blue solid line is the FWHM of the window function.  The gray dots
+    are the FWHM derived from the measured second moments for stars in
+    the image.  The median of this distribution (mag $< -10$) is
+    listed as ``obs''.  The ratio of the median FWHM to the FWHM of
+    the window function is listed as ``ratio'', while the ratio of the
+    median FWHM to the true stellar FWHM is listed as ``bias''.  The
+    dotted blue line is the target (65\% of the window function).  In
+    this example, we would choose $\sigma_w$ between 0.5 and 0.8
+    arcseconds so the dotted blue line would match the bright end of
+    the gray dots.   See discussion in the text for the choice of
+    target window.
+}
   \end{center}
 \end{figure}
 …
 appropriate aperture in which the moments are measured.  We also apply
 a ``window function'', down-weighting the pixels by a Gaussian,
 centered on the object, with size $\sigma_W$ chosen to be large
+centered on the object, with size $\sigma_w$ chosen to be large
 compared to the PSF size, $\sigma_{\rm PSF}$.  This window function
 reduces the noise of the measurement of the moments by suppressing the
 noisy pixels at high radial distance as well as by reducing the
 contaminating effects of neighboring stars.  The choice of $\sigma_W$
+contaminating effects of neighboring stars.  The choice of $\sigma_w$
 and the aperture is an iterative process: for a given value of
 $\sigma_W$, the PSF stars will have a measured value of the PSF size,
+$\sigma_w$, the PSF stars will have a measured value of the PSF size,
 $\sigma^{\prime}_{\rm PSF}$ which different from the true value due to
 the effect of the window function.  The measured value of the PSF size
 …
 radial profile of the PS1 PSF model with $\sigma_{\rm PSF}$
 corresponding to a FWHM of 1.4 arcseconds.  As the window function
 $\sigma_W$ is increased, the measured FWHM for the bright simulated
 stars rises to meet the truth value.  For small values of $\sigma_W$,
+$\sigma_w$ is increased, the measured FWHM for the bright simulated
+stars rises to meet the truth value.  For small values of $\sigma_w$,
 fainter stars are biased to low measured values of the FWHM.  For
+large values of $\sigma_W$, the faint stars are biased to higher
+values and the scatter increases.
+large values of $\sigma_w$, the faint stars are biased to higher
+values and the scatter increases.  We attempt to minimize the scatter
+and trends with magnitude at the cost of overall bias.
 In a real image, we do not know the true value of the PSF size.  If we
 …
 brightness.
 To choose the value of $\sigma_W$, we try a sequence of values
+To choose the value of $\sigma_w$, we try a sequence of values
 spanning a range guaranateed to contain any reasonable seeing values.
 The values are specified in the \code{psphot} recipe as
 \code{PSF.SIGMA.VALUES} and have the following values for PS1 PV3: (1,
 , 3, 4.5, 6, 9, 12, 18) pixels $\approx$ (0.26, 0.51, 0.77, 1.15,
 .54, 2.3, 3.1, 4.6) arcseconds.  For each of these $\sigma_W$ values,
+.54, 2.3, 3.1, 4.6) arcseconds.  For each of these $\sigma_w$ values,
 we then select candidate PSF stars based on the distribution of the
 measured $\sigma^{\prime}_{\rm PSF}$ in the two principal directions:
 …
 \frac{\sigma_{x} + \sigma{y}}{2 \sigma_w}$, i.e., the ratio of the
 window size to the observed PSF size.  We interpolate to find a value
 of $\sigma_W$ for which $\rho_\sigma$ is expected to be 0.65.  We use
+of $\sigma_w$ for which $\rho_\sigma$ is expected to be 0.65.  We use
 an aperture with a radius of 4$\sigma_w$ to select the pixels for the
 measurement of the moments.
 …
 \subsubsection{PSF Model vs Source Model}
+\label{sec:Source.Model}
 The point-spread-function (PSF) of an image describes the shape of all
 …
 selected by \code{psphot}, though sources which have more than a
 certain number of saturated pixels are excluded at this stage.  The
 program then examines the 2-D plane of $\sigma_x, \sigma_y$ in search
+program then examines the 2-D plane of $M_{x,x}, M_{y,y}$ in search
 of a concentrated clump of sources (see
 Figure~\ref{fig:moment.class}).  To do this, it constructs an
 artificial image with pixels representing the value of $\sigma_x,
 \sigma_y$, using $0.1 \sigma_w$ as the size of a pixel in this
 artificial image.  The binned $\sigma_x, \sigma_y$ plane is then
+artificial image with pixels representing the value of $M_{x,x},
+M_{y,y}$, using $0.1 \sigma^2_w$ as the size of a pixel in this
+artificial image.  The binned $M_{x,x}, M_{y,y}$ plane is then
 examined to find a significant peak.  Unless the image is extremely
 sparse, such a peak will be well-defined and should represent the
 …
 % (\sigma_x^2) not \sigma_x,\sigma_y)
-\note{re-work wording above reflecting comment above}
 Once a peak has been detected in this plane, the centroid and second
 moments of this peak are measured.  All sources which land within 2
 …
   \begin{center}
   \includegraphics[width=\hsize]{{pics/moment.class}.\plotext}
   \caption{\label{fig:moment.class} Illustration of PSF star selection using the FWHM derived
     from the second moments in $X_{\rm ccd}$ and $Y_{\rm ccd}$
+  \caption{\label{fig:moment.class} Illustration of PSF star selection
+    using the second moments in $X_{\rm ccd}$ and $Y_{\rm ccd}$
     directions.  The dominant clump is located in this diagram.
     Galaxies tend to have a range of sizes and thus spread out above
 …
 \begin{table}
 \caption{\label{tab:psf.order.nstars} Minimum number of stars required
   for a given order of the PSF 2D variations.}\vspace{-0.5cm}
+  for a given order of the PSF 2D variations.} % \vspace{-0.5cm}
 \begin{center}
 \begin{tabular}{llll}
 …
 PSF.
+\subsubsection{Full PSF Model Fitting}
+% \note{review the discussion below}
+Once a PSF model has been selected for an image, \code{psphot} attempts to
+fit all of the detected sources, above a user-defined signal-to-noise
+ratio with the PSF model.  For these fits, the dependent parameters
+are fixed by the PSF model and only the 4 independent source model
+parameters are allowed to vary in the fit.  \code{psphot} again uses
+Levenberg-Marquardt minimization for the non-linear fitting.  The
+sources are fitted in their S/N order, starting with the brightest and
+working down to the user-specified limit, with the other sources
+subtracted as discussed above.
+\note{code review for the next bit}
+Once a solution has been achieved for a source, \code{psphot} attempts to
+judge the quality of the PSF model as a representation of the source
+shape.  To do this, it calculates the next step of the minimization
+{\em allowing the shape parameters to vary}.  This step, essentially
+the Gauss-Newton minimization distance from the current local minimum,
+should be very small if the source is well represented by the PSF, but
+large if the PSF is not a good representation of the source flux.  The
+model quality is judged by the change in the two shape parameters
+which represent the 2D size of the source.  For the case of the
+elliptical Gaussian, these two parameters are $\sigma_x$ and
+$\sigma_y$.  For a generic model, the shape parameters may be defined
+differently, but there should always be two parameters which scale the
+source size in two dimensions.  Currently, \code{psphot} requires the two
+relevant shape parameters to be the first two dependent parameters in
+the list of model parameters (ie, parameters 4 \& 5).
+The expected distribution of the variation of the two shape parameters
+will be a function of the signal-to-noise of the source in question
+and the value of the shape parameter itself.  The expected standard
+deviation on the shape parameter is, eg, $\sigma_x / {\rm S/N}$.  If
+the source is well-represented by the PSF, then the shape parameter
+values should be close to their minimization value.  We can thus ask,
+for each source, given the measured amplitude of the Gauss-Newton
+step, how many standard deviations from the expected value (of 0.0) is
+this particular value?  Sources for which the variation in the shape
+parameters is a large positive number of standard deviations are
+likely to be better represented by a larger flux distribution than the
+PSF (eg, a Galaxy or Comet, etc).  Sources for which the variation in
+the shape parameters is a large negative number of standard deviations
+are likely to be better represented by a smaller flux distribution
+than the PSF (ie, a cosmic ray or other defect).  A user-defined
+number of standard deviations is used to select these two cases, and
+to flag the source as a likely galaxy (really meaning 'extended') or
+as a likely defect.
+At this stage of the analysis, \code{psphot} uses two additional indicators
+to identify good and poor PSF fits.  The first of these is the
+signal-to-noise ratio.  It is possible for the peak finding algorithm
+to identify peaks in locations which are not actually a normal peak.
+Some of these cases are in the edges of saturated, bleeding columns
+from bright stars, in the nearly flat halos of very bright stars, and
+so on.  In these cases, a local peak exists, with a lower nearby sky
+region.  However, the fitted PSF model cannot converge on the peak
+because it is very poorly defined (perhaps only existing in the
+smoothed image).  The fit can either fail to converge or it can
+converge on a fit with very low or negative peak flux / flux
+normalization.  \code{psphot} will flag any non-convergent PSF fit and any
+source with PSF S/N ratio lower than a user-defined cutoff.  It is
+also useful to identify very poor fits by setting a maximum Chi-Square
+cutoff for sources.
+As the sources are fitted to the PSF model, those which survive the
+exclusion stage are subtracted from the image.  The subtraction
+process modifies the image pixels (removing the fitted flux, though
+not the locally fitted background) but does not modify the mask or the
+variance images.  The signal-to-noise ratio in the image after
+subtraction represents the significance of the remaining flux.  If the
+subtractions are sufficiently accurate models of the PSF flux
+distribution, the remaining flux should be below 1 $\sigma$
+significance.  In practice the cores of bright stars are poorly
+represented and may have larger residual significance.
+\subsubsection{Blended Sources}
+Sources which are blended with other sources are fitted together as a set of
+PSFs.  A single multi-source fit is performed on all blended peaks.
+The resulting fits are evaluated independently and any which are
+determined to be PSFs are subtracted from the image.
+\subsubsection{Double Sources}
+Sources which are judged to be non-PSF-like are confronted with two
+possible alternative choices.  First, the source is fitted with a
+double-source model.  In this pass, the assumption is made that there
+are two neighboring sources, but the peaks are blended together, or
+otherwise not distinguished.  The initial guess for the two peaks is
+made by splitting the flux of the single source in half and locating
+the two starting peaks at +/- 2 pixels from the original peak along
+the direction of the semi-major axis of the sources, as measured from
+the second moments.  In order for the two-source model to be accepted,
+both sources must be judged as a valid PSF source.  Otherwise, the
+double-PSF model is rejected and the source is fitted with the
+available non-PSF model or models.
+\subsubsection{Radial Profile Wings}
+We attempt to measure the radial profile of sources in order to find
+the radius at which the flux of the source is matches the sky.  In
+this analysis, a series of up to 25 radial bins with power-law spacing
+are defined and the flux of the source in each annulus is measured.
+The ``sky radius'' is defined to be the radius at which the (robust
+median) flux in the annulus is within 1 $\sigma$ of the local sky
+level.  If this limit is not reached before the slope of the flux from
+one annulus to the next is less than a user-defined limit, then the
+annulus at which the slope reaches this limit is used to define the
+sky radius.  These values are saved in the output smf / cmf files, but
+not sent to the PSPS.  The sky radius value is used below in the
+calculation of the Kron magnitude.
+\subsubsection{Kron Magnitudes}
+\label{sec:kron.mags}
+Preliminary Kron radius and flux values \citep{1980ApJS...43..305K}
+are calculated soon after sources are detected
+(Section~\ref{sec:moments}).  However, these preliminary values are
+not accurate due to the window-functions applied.  After sources have
+been characterized and the PSF model is well-determined, the Kron
+parameters are re-calculated more carefully.  In this version of the
+calculation, following the algorithm described by \cite{sextractor},
+the image is first smoothed by Gaussian kernel with $\sigma = 1.7$
+pixels, corresponding to a FWHM of 1.0\arcsec\ in the PS1 stack
+images.  Next, the Kron radius is determined in an iterative process:
+the first radial moment is measured using the pixels in an aperture
+$\times$ the first radial moment from the previous iteration.  On the
+first iteration, the sky radius is used in place of the first radial
+moment.  By default, 2 iterations are performed.  The Kron radius is
+defined to be 2.5$\times$ the first radial moment.  The Kron flux is
+the sum of pixel fluxes within the Kron radius.  We also calculate the
+flux in two related annular apertures: the Kron inner flux is the sum
+of pixel values for the annulus $R_1 < r < 2.5 R_1$, while the Kron
+outer flux is the sum of pixel values for $2.5 R_1 < r < 4 R_1$.
+Two details in the calculation above should be noted.  First, for
+faint sources, noise in the measurement of the 1st radial moment may
+result in an excessively small aperture for the successive
+calculations.  The window used for the calculations is constrained to
+be at least the size of the aperture based on the PSF stars
+(Section~\ref{sec:moments}).  At the other extreme, noise may make the radius
+grow excessively, resulting in an unrealistically low effective
+surface brightness.  The aperture is constrained to be less than a
+maximum value defined such that the minimum surface brightness is
+/2$times$ the effective surface brightness of a point source detected at the
+$5\sigma$ limit.
+Second, the measurement of the 1st radial moment includes a filter to
+reduce contamination from outlier pixels.  Pairs of pixels on
+opposites sides of the central pixel are considered together.  The
+geometric mean of the two fluxes is used to replace the flux values.
+If the source has 180\degree\ symmetry, this operation has no impact.
+However, if one of the two pixels is unusually high, the value will be
+surpressed by the matched pixel on the other side.  This trick has the
+effect of reducing the impact of pixels which include flux from near
+neighbors.
+\note{give a test example}
 \subsubsection{Source Size Assessment}
 \label{sec:source.size}
-\note{is this in the right place?}
 After the PSF model has been fitted to all sources, and the Kron flux
 …
 \note{how are / were these parameters set?}
+\subsubsection{Full PSF Model Fitting}
+% \note{review the discussion below}
+% gaussSigma = MOMENTS_GAUSS_SIGMA from recipe (initially)
+% gaussSigma = Sigma used for window function, $\sigma_w$
+% fitRadius, apRadius = (fitScale, apScale) * gaussSigma
+% fitScale = 7.0
+% apScale = 4.5
+Once a PSF model has been selected for an image, \code{psphot}
+attempts to fit all of the detected sources, with signal-to-noise
+ratio greater than a user-defined limit, with the PSF model.  In the
+PV3 analysis of the $3\pi$ survey data, this limit was set to a
+signal-to-noise ratio of 20.0 for all analysis stages.  In these fits,
+the dependent parameters are fixed by the PSF model and only the 4
+independent source model parameters are allowed to vary in the fit.
+\code{psphot} again uses Levenberg-Marquardt minimization for the
+non-linear fitting.  The sources are fitted in their S/N order,
+starting with the brightest and working down to the user-specified
+limit, with the other sources subtracted as discussed above.
+For the PSF model fitting, only pixels within a circular aperture
+scaled based on the seeing are used.  The radius of the circular
+aperture is set to be a fixed multiple 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 PSF fit window radius is $7 \times \sigma_w$.
+Sources which are blended with other sources are fitted together as a
+set of PSFs.  Blended objects are identified by first searching for
+objects for which the PSF fit windows overlap.  For a group of such
+neighboring objects, a contour is determined in the flux image at
+$25\%$ of the peak of the brightest source in the group.  All objects
+lying within this contour are treated as blends of this brightest
+source.  If other objects in this group exist, the brightest object
+not already assigned to a blend is used to define the contour for
+blends of this next object.  All objects in the image are tested as
+possible blends.  A single multi-source fit is performed on each group
+of blended peaks.
+%% Once a solution has been achieved for a source, \code{psphot} attempts to
+%% judge the quality of the PSF model as a representation of the source
+%% shape.  To do this, it calculates the next step of the minimization
+%% {\em allowing the shape parameters to vary}.  This step, essentially
+%% the Gauss-Newton minimization distance from the current local minimum,
+%% should be very small if the source is well represented by the PSF, but
+%% large if the PSF is not a good representation of the source flux.  The
+%% model quality is judged by the change in the two shape parameters
+%% which represent the 2D size of the source.  For the case of the
+%% elliptical Gaussian, these two parameters are $\sigma_x$ and
+%% $\sigma_y$.  For a generic model, the shape parameters may be defined
+%% differently, but there should always be two parameters which scale the
+%% source size in two dimensions.  Currently, \code{psphot} requires the two
+%% relevant shape parameters to be the first two dependent parameters in
+%% the list of model parameters (ie, parameters 4 \& 5).
+%%
+%% The expected distribution of the variation of the two shape parameters
+%% will be a function of the signal-to-noise of the source in question
+%% and the value of the shape parameter itself.  The expected standard
+%% deviation on the shape parameter is, eg, $\sigma_x / {\rm S/N}$.  If
+%% the source is well-represented by the PSF, then the shape parameter
+%% values should be close to their minimization value.  We can thus ask,
+%% for each source, given the measured amplitude of the Gauss-Newton
+%% step, how many standard deviations from the expected value (of 0.0) is
+%% this particular value?  Sources for which the variation in the shape
+%% parameters is a large positive number of standard deviations are
+%% likely to be better represented by a larger flux distribution than the
+%% PSF (eg, a Galaxy or Comet, etc).  Sources for which the variation in
+%% the shape parameters is a large negative number of standard deviations
+%% are likely to be better represented by a smaller flux distribution
+%% than the PSF (ie, a cosmic ray or other defect).  A user-defined
+%% number of standard deviations is used to select these two cases, and
+%% to flag the source as a likely galaxy (really meaning 'extended') or
+%% as a likely defect.
+After the PSF model is fitted to each object, \code{psphot} makes an
+assessment of the quality of the PSF fits.  First, it checks that the
+non-linear fitting process has converged with a valid fit.  The fit
+for an object can fail if there are too few valid pixels, due to
+masking or proximity to an edge, or if the parameters are driven to
+extreme values which are not permitted.  In addition, it is possible
+for the peak finding algorithm to identify peaks in locations which
+are not actually a normal peak.  Some of these cases are in the edges
+of saturated, bleeding columns from bright stars, in the nearly flat
+halos of very bright stars, and so on.  In these cases, a local peak
+exists, with a lower nearby sky region.  However, the fitted PSF model
+cannot converge on the peak because it is very poorly defined (perhaps
+only existing in the smoothed image).  In these cases, \code{psphot}
+flags the object with the bad bit \code{PM_SOURCE_MODE_FAIL}.  It is
+also possible in this type of case for the fit to result in a very low
+or negative value for the flux normalization parameter.  Source for
+which the peak is less than 0.02 are also marked as failing the
+non-linear PSF fit (\code{PM_SOURCE_MODE_FAIL}).
+Poor fits are also identified by the signal-to-noise and the $\chi^2$
+value of the resulting fit.  If a source has a PSF S/N ratio lower
+than a user-defined cutoff (set to 2.0 for the PV3 analysis of the
+$3\pi$ survey), the non-linear PSF fit will be rejected.  If the
+Chi-Square per degree of freedom is greater than a user-defined limit
+(set to 50.0 for the PV3 analysis of the $3\pi$ survey), the
+non-linear PSF fit will be rejected.  These sources are marked with
+the flag bit (\code{PM_SOURCE_MODE_POOR}).
+Sources which are pass the above tests are marked as having a valid
+non-linear PSF model fit (\code{PM_SOURCE_MODE_SATSTAR}).  Among these
+sources, those for which the peak flux is greater than the saturation
+limit are marked as saturated stars (\code{PM_SOURCE_MODE_SATSTAR}).
+These model fits should be consisdered with caution, but the fluxes
+and positions may have some validity (see Section~\ref{Saturation}).
+As the sources are fitted to the PSF model, those which survive the
+exclusion stage are subtracted from the image.  The subtraction
+process modifies the image pixels (removing the fitted flux, though
+not the locally fitted background) but does not modify the mask or the
+variance images.  The signal-to-noise ratio in the image after
+subtraction represents the significance of the remaining flux.  If the
+subtractions are sufficiently accurate models of the PSF flux
+distribution, the remaining flux should be below 1 $\sigma$
+significance.  In practice the cores of bright stars are poorly
+represented and may have larger residual significance.
+For sources in groups of blended stars, the resulting fits are
+evaluated independently.  Any which are determined to be valid PSF
+fits are subtracted from the image and kept for future analysis.
+\subsubsection{Double and Extended Sources}
+Sources which are judged to be non-PSF-like are confronted with two
+possible alternative choices.  First, the source is fitted with a
+double-source model.  In this pass, the assumption is made that there
+are two neighboring sources, but the peaks are not resolved.  The
+initial guess for the two peaks is made by splitting the flux of the
+single source in half and locating the two starting peaks at +/- 2
+pixels from the original peak along the direction of the semi-major
+axis of the sources, as measured from the second moments.  In order
+for the two-source model to be accepted, both sources must be judged
+as a valid PSF source.  Otherwise, the double-PSF model is rejected
+and the source is fitted with the available non-PSF model or models.
 \subsubsection{Non-PSF Sources}
 Once every source (above the S/N cutoff) has been confronted with the
+PSF model, the sources which are thought to be galaxies (extended) can
+now be fit with appropriate models for the galaxies (or other likely
+extended shapes).  Again, the fitting stage starts with the brightest
+sources (as judged by the rough S/N measured from the moments
+aperture) and working to a user defined S/N limit.
+\code{psphot} will use the user-selected galaxy model to attempt the galaxy
+model fits.  In the configuration system, the keyword \code{GAL_MODEL}
+is set to the model of interest.  All suspected extended sources are
+fitted with the model, allowing all of the parameters to float.  The
+initial parameter guesses are critical here to achieving convergence
+on the model fits in a reasonable time.  The moments and the pixel
+flux distribution are used to make the initial parameter guess.  Many
+of the source parameters can be accurately guessed from the first and
+second moments.  The power-law slope can be guessed by measuring the
+isophotal level at two elliptical radii and comparing the ratio to
+that expected.
+For each of the galaxy models (in fact for all source models), a
+function is defined which examines the fit results and determines if
+the fit can be consider as a success or a failure.  The exact criteria
+for this decision will depend on the details of the model, and so this
+level of abstraction is needed.  For example, in some case, the range
+of valid values for each of the parameters must be considered in the
+fit assessment.  In other cases, we may choose to use only the
+parameter errors and the fit Chi-Square value.
+All galaxy model fits which are successful are then subtracted from
+the image as is done for the successful PSF model fits.  Of course,
+the background flux is retained, with the result that only the source
+is subtracted from the image.  Again, the variance image is (currently)
+not modified.
+PSF model, the sources which are thought to be extended (resolved) can
+now be fit with an appropriate model (e.g., galaxy profile or other
+likely extended shapes).  Again, the fitting stage starts with the
+brightest sources (as judged by the rough S/N measured from the
+moments aperture) and working to a user defined S/N limit.
+\code{psphot} will use the user-selected extended source model to
+attempt these fits.  In the configuration system, the keyword
+\code{EXT_MODEL} is set to the model of interest.  All suspected
+extended sources are fitted with the model, allowing all of the
+parameters to float.  The initial parameter guesses are critical here
+to achieving convergence on the model fits in a reasonable time.  The
+moments and the pixel flux distribution are used to make the initial
+parameter guess.  Many of the source parameters can be accurately
+guessed from the first and second moments.  The power-law slope can be
+guessed by measuring the isophotal level at two elliptical radii and
+comparing the ratio to that expected.
+For each type of extended source model (in fact for all source
+models), a function is defined which examines the fit results and
+determines if the fit can be consider as a success or a failure.  The
+exact criteria for this decision depends on the details of the model,
+and so this level of abstraction is needed.  For example, in some
+case, the range of valid values for each of the parameters must be
+considered in the fit assessment.  In other cases, we may choose to
+use only the parameter errors and the fit Chi-Square value.
+All extended source model fits which are successful are then
+subtracted from the image as is done for the successful PSF model
+fits.  The background flux is retained, with the result that only the
+source is subtracted from the image.  At this stage, the variance
+image is not modified.
+For the single exposure (\ippstage{camera}) and \ippstage{stack} image
+analysis, these galaxy model fits are only used internally to generate
+a clean object-subtracted residual image.  For the PV3 analysis of the
+$3\pi$ survey, these model fits were saved in the output catalog
+files, but not loaded to the public database.  The \code{QGAUSS}
+extended source model was used for the PV3 analysis (see
+Section~\ref{sec:Source.Model}).  The convolved galaxy model fits (see
+Section~\ref{sec:galaxy.conv.fit}) and the forced galaxy model fits
+(see Section~\ref{sec:galaxy.forced.fit}) provide more reliable and
+physically-motivated galaxy models.
+For the difference image analysis, a trailed object model is used for
+the extended sources; these model fit parameters are passed to the
+Moving Object Processing System \citep[MOPS][]{2013PASP..125..357D}.
 \subsection{Faint Source Analysis}
 …
 Petrosian flux is contained.
-\subsubsection{Radial Profile Wings}
-We attempt to measure the radial profile of sources in order to find
-the radius at which the flux of the source is matches the sky.  In
-this analysis, a series of up to 25 radial bins with power-law spacing
-are defined and the flux of the source in each annulus is measured.
-The ``sky radius'' is defined to be the radius at which the (robust
-median) flux in the annulus is within 1 $\sigma$ of the local sky
-level.  If this limit is not reached before the slope of the flux from
-one annulus to the next is less than a user-defined limit, then the
-annulus at which the slope reaches this limit is used to define the
-sky radius.  These values are saved in the output smf / cmf files, but
-not sent to the PSPS.  The sky radius value is used below in the
-calculation of the kron magnitude.
-\subsubsection{Kron Magnitudes}
-\label{sec:kron.mags}
-Preliminary Kron radius and flux values \citep{1980ApJS...43..305K}
-are calculated soon after sources are detected
-(Section~\ref{sec:moments}).  However, these preliminary values are
-not accurate due to the window-functions applied.  After sources have
-been characterized and the PSF model is well-determined, the Kron
-parameters are re-calculated more carefully.  In this version of the
-calculation, following the algorithm described by \cite{sextractor},
-the image is first smoothed by Gaussian kernel with $\sigma = 1.7$
-pixels, corresponding to a FWHM of 1.0\arcsec\ in the PS1 stack
-images.  Next, the Kron radius is determined in an iterative process:
-the first radial moment is measured using the pixels in an aperture
-$\times$ the first radial moment from the previous iteration.  On the
-first iteration, the sky radius is used in place of the first radial
-moment.  By default, 2 iterations are performed.  The Kron radius is
-defined the be 2.5$\times$ the first radial moment.  The Kron flux is
-the sum of pixel fluxes within the Kron radius.  We also calculate the
-flux in two related annular apertures: the Kron inner flux is the sum
-of pixel values for the annulus $R_1 < r < 2.5 R_1$, while the Kron
-outer flux is the sum of pixel values for $2.5 R_1 < r < 4 R_1$.
-Two details in the calculation above should be noted.  First, for
-faint sources, noise in the measurement of the 1st radial moment may
-result in an excessively small aperture for the successive
-calculations.  The window used for the calculations is constrained to
-be at least the size of the aperture based on the PSF stars
-(Section~\ref{sec:moments}).  At the other extreme, noise may make the radius
-grow excessively, resulting in an unrealistically low effective
-surface brightness.  The aperture is constrained to be less than a
-maximum value defined such that the minimum surface brightness is
-/2$times$ the effective surface brightness of a source detected at the
-$5\sigma$ limit.
-Second, the measurement of the 1st radial moment includes a filter to
-reduce contamination from outlier pixels.  Pairs of pixels on
-opposites sides of the central pixel are considered together.  The
-geometric mean of the two fluxes is used to replace the flux values.
-If the source has 180\degree\ symmetry, this operation has no impact.
-However, if one of the two pixels is unusually high, the value will be
-surpressed by the matched pixel on the other side.  This trick has the
-effect of reducing the impact of pixels which include flux from near
-neighbors.
-\note{give a test example}
 \subsubsection{Convolved Galaxy Model Fits}
+\label{sec:galaxy.conv.fit}
 In the galaxy model fittting stage, sources which meet certain
 …
 tested.
+\begin{table*}
+\begin{center}
+\caption{\label{tab:measurements} \nocode{psphot} measurements performed} % \vspace{-0.5cm}
+\begin{tabular}{lcccc}
+\hline
+\hline
+{\bf Measurement} & {\bf Camera} & {\bf Stack} & {\bf Forced Warp} & {\bf Diff} \\
+\hline
+  Background                 & Y & Y & Y & N$^1$ \\
+  Peaks                      & Y & Y & N & Y \\
+  Footprints                 & Y & Y & N & Y \\
+  Moments                    & Y & Y & Y & Y \\
+  PSF Model                  & Y & Y & Y & N$^2$ \\
+  Bright Star Profile        & Y & Y & N & Y \\
+  Non-Linear PSF Fits        & Y & Y & N & N \\
+  Source-Size Tests          & Y & Y & N & Y \\
+  Unconvolved Galaxy Model   & Y & Y & N & N \\
+  Unconvolved Streak Model   & N & N & N & Y \\
+  Linear PSF Fits            & Y & Y & Y & Y \\
+  Radial Profiles            & Y & Y & N & Y \\
+  Petrosian Fluxes           & N & Y & Y & N \\
+  Kron Fluxes                & Y & Y & Y & Y \\
+  Convolved Galaxy Models    & N & Y & N & N \\
+  Fixed Aperture Photometry  & N & Y & Y & N \\
+  Convolved, Fixed Apertures & N & Y & N & N \\
+  Aperture Corrections       & Y & Y & Y & N \\
+  Forced PSF Fluxes          & N & N & Y & N \\
+  Forced Galaxy Models       & N & N & Y & N \\
+  Lensing Parameters         & N & Y & Y & N \\
+\hline
+\hline
+\multicolumn{5}{l}{$^1$ Background subtraction is performed by {\tt ppSub} before calling {\tt psphot}} \\
+\multicolumn{5}{l}{$^2$ PSF modeling is perfom by {\tt ppSub} on the input warps before calling {\tt psphot}} \\
+\end{tabular}
+\end{center}
+\end{table*}
 \subsection{Output Formats}
 …
 is marked as completed.
+\subsection{Forced Photometry : PSFs}
+\subsection{Forced Photometry : galaxies}
+\subsection{Forced Galaxy Models}
+\label{sec:galaxy.forced.fit}
 The convolved galaxy models are also re-measured on the
 …
 * background model description (see waters)

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