Index: /trunk/doc/release.2015/ps1.analysis/analysis.tex =================================================================== --- /trunk/doc/release.2015/ps1.analysis/analysis.tex (revision 39813) +++ /trunk/doc/release.2015/ps1.analysis/analysis.tex (revision 39814) @@ -1,3 +1,3 @@ -\documentclass[iop,floatfix]{emulateapj} +\documentclass[iop,floatfix,onecolumn]{emulateapj} % \pdfoutput=1 @@ -93,14 +93,4 @@ \keywords{Surveys:\PSONE } -\section{OUTLINE} -\begin{verbatim} -Intro - Pan-STARRS background - Scope: Source Detection \& Characterization, Galaxy modeling - Requirements / Goals - Comparable programs - PSPhot -\end{verbatim} - \section{INTRODUCTION}\label{sec:intro} @@ -441,7 +431,4 @@ \subsubsection{Determination of the Peak Coordinates and Errors} -\note{this section is wrong: it is a poor estimator of the source - position errors. we gave up a reverted to using the FWHM / (S/N)} - We use the 9 pixels which include the source peak to fit for the position and position errors. We model the peak of the sources as a @@ -478,15 +465,4 @@ \end{verbatim} -which can be used to determine the errors on the coefficients: - -\begin{eqnarray} -\sigma^2_{00} & = & \sigma^2 (5/9) \\ -\sigma^2_{10} & = & \sigma^2 (1/6) \\ -\sigma^2_{01} & = & \sigma^2 (1/6) \\ -\sigma^2_{11} & = & \sigma^2 (1/6) \\ -\sigma^2_{20} & = & \sigma^2 (1/2) \\ -\sigma^2_{02} & = & \sigma^2 (1/2) \\ -\end{eqnarray} - The location of the peak is determined from the minimum of the bi-quadratic function above, and is given by: @@ -498,13 +474,5 @@ \end{eqnarray} -Applying error propagation to the above, we find: - -\begin{eqnarray} -\sigma_{Det}^2 & = & \sigma_{11}^2 (4 C_{11}^2) + \sigma_{20}^2 (16 C_{02}^2) + \sigma_{02}^2 (16 C_{20}^2) \\ -\sigma_{xn}^2 & = & \sigma_{11}^2 C_{01}^2 + \sigma_{01}^2 C_{11}^2 + \sigma_{02}^2 (4 C_{10}^2) + \sigma_{10}^2 (4 C_{02}^2) \\ -\sigma_{yn}^2 & = & \sigma_{11}^2 C_{10}^2 + \sigma_{10}^2 C_{11}^2 + \sigma_{20}^2 (4 C_{01}^2) + \sigma_{01}^2 (4 C_{20}^2) \\ -\sigma_{x}^2 & = & x^2 (\sigma_{xn}^2 / xn^2 + \sigma_{Det}^2 / Det^2) \\ -\sigma_{y}^2 & = & y^2 (\sigma_{yn}^2 / yn^2 + \sigma_{Det}^2 / Det^2) \\ -\end{eqnarray} +\note{error on the peak position} \subsection{PSF Determination} @@ -652,14 +620,14 @@ All candidate PSF objects are then fitted with the selected object model, allowing all of the parameters (PSF and independent) to vary in -the fit. PSPhot uses the Levenberg-Marqardt process for the -non-linear fitting. Non-linear fitting can be very computationally -intensive, particularly for if the starting parameters are far from -the minimization values. PSPhot uses the first and second moments to -make a good guess for the centroid and shape parameters for the PSF -models. \note{still true? In order to minimize the impact of close - neighbors, the variance values used in the fit are enhanced by a - fraction of the deviation of the particular pixel value from the - model guess.} Any objects which fail to converge in the fit are -flagged as invalid. +the fit. PSPhot uses the Levenberg-Marqardt method \note{REF, link to + psLibADD} for the non-linear fitting. Non-linear fitting can be +very computationally intensive, particularly for if the starting +parameters are far from the minimization values. PSPhot uses the +first and second moments to make a good guess for the centroid and +shape parameters for the PSF models. \note{still true? In order to + minimize the impact of close neighbors, the variance values used in + the fit are enhanced by a fraction of the deviation of the + particular pixel value from the model guess.} Any objects which +fail to converge in the fit are flagged as invalid. \note{does the variance enhancement introduce too much bias?} @@ -698,21 +666,13 @@ correction is judged to be the best model. -\subsubsection{Basic Deblending} - -The collection of identified peaks is examined to find peaks which are -'blended', that is, they are close enough together to make the -analysis of one of the sources difficult if performed in isolation. -Saturated stars also result in additional peaks which are likely to be -invalid; it is useful to restrict a saturated star to a single primary -position with associated neighboring peaks. - -The deblending process first searches for any peaks which are within -the image cell of another peak. All such groups are examined, -starting with the brightest source. An isophot is found about the -primary peak which is at least \code{DEBLEND\_SKY\_NSIGMA} times the sky -sigma above the local background and which is otherwise -\code{DEBLEND\_PEAK\_FRACTION} of the primary peak central pixel flux. -Any secondary sources which are contained within this isophot are -considered to be blended peaks associated with the primary peak. +\subsection{Very Bright Stars} +\note{flesh out} + +The PSF modeling code fails to fit the wings of highly saturated stars +if the core of the star is too contaminated by saturated pixels. For +stars with estimated instrumental magnitudes brighter than XXX, we fit +and subtract a radial profile modeled with a spline (?). + +\subsection{PSF vs CR vs Extended} \subsection{Bright Source Analysis} @@ -1071,5 +1031,164 @@ tested. -\subsubsection{Types of Object / PSF models currently available} +\subsection{Radial Profiles} + +Galaxies with regular profiles, such as elliptical galaxies and +regular spiral galaxies, may be described as primarily a radial +surface brightness profile, with additional structure acting as a +perturbation on that profile. For many galaxies, the azimuthal shape +at a given isophotal level may be described as an elliptical contour. +To first order, a galaxy may be well decribed with a single elliptical +contour and radial profile. + +In order to facilitate the Petrosian photometry analysis below, PSPhot +generates a radial profile for each suspected galaxy. This analysis +starts by generating a radial profile in 24 azimuthal segments. Near +the center of the galaxy, the profile is defined for radial +coordinates in steps of 1 pixel, with the closest pixel values +interpolated to that radial position. Further from the center, +profile is defined using the median of the pixels landing in an +annular segment of size $\delta R = r \sin \theta$, rounded up to the +nearest integer pixel value. The median of all pixels within a +rectangular approximation to the radial wedge is used. + +The resulting 24 radial profiles are subject to contamination from +neighboring sources or to NAN values from masked pixels. To clean the +profiles, pairs of radial profiles from opposite sides of the source +are compared. Any masked values are replaced by the corresponding +value in the other profile. The minimum of both profiles is the kept +for both profiles. The result of this analysis is a set of profiles +of the form $f_i(r_i)$. In this case, $f_i$ is effectively the +surface brightness for each radius in instrumental counts per pixel. + +The surface brightness profiles are then used to define the radial +contour at a specific isophotal level. This contour will be used to +rescale the radial profiles into a single set of profiles normalized +by the elliptical contour. This contour is defined by determining the +median radius for profile bins with surface brightness in the range +$F_{\rm min} + 0.1 F_{\rm range}$ to $F_{\rm min} + 0.5 F_{\rm + range}$. The result of this analysis is a value for the radius as a +function of the angle for a well-defined surface brightness regime. +We then determine the elliptical shape parameters for this elliptical +contour: $R_{\rm major}, R_{\rm minor}, \theta$. This ellipse is then +used to redefine a single radial profile normalized by the elliptical +contour: +\[ +\rho = \sqrt{\frac{x^2}{S^2_{xx}} + \frac{y^2}{S^2_{yy}} + x y S_{xy} \\ +\] + +The surface brightness values are sampled at a number of radial +annuli, with the radii defined in the configuration ({\tt + RADIAL.ANNULAR.BINS.LOWER \& RADIAL.ANNULAR.BINS.UPPER}). For each +source, the resulting surface brightness profile is saved in the +output cmf-file as an N-element value in the FITS table ({\tt + PROF\_SB}). The flux at each radial position and the fill-factor +(fraction of pixels used to the total possible) as also saved as +equal-length vectors in the FITS table ({\tt PROF\_FLUX and + PROF\_FILL}). The values of the radial bins are saved in the cmf +header ({\tt RMIN\_NN, RMAX\_NN}). + +\note{these profiles are not saved in PSPS} + +\subsection{Petrosian Radii and Magnitudes} + +Petrosian (REF) defined an adaptive aperture based on a ratio of +surface brightnesses. The motivation is to define an aperture which +can be determined for galaxies without significant biases as a +function of distance. Since surface brightness in a resolved object +is conserved, using a ratio of surface brightness to define a spatial +scale results in a spatial scale which is constant regardless of +galaxy distance. + +In the classic definition, a reference radius, R90 +is specified as the radius at which the flux + +To measure the Petrosian radius and flux, we start by defining a +series of radial apertures with power-law spacing: $r_{i + 1} = 1.25 +r_i$. We calculate the surface brightness for the annulus from $r_i - +r_{i+1}$ by calculating the median of the values in the range $r_i / +\sqrt{1.25}$ to $r_{i+1} \sqrt{1.25}$ and dividing the the effective +area of the annulus corresponding to $r_i - r_{i+1}$. + +For any annulus $i$ spanning the radii $r_{\rm min}$ to $r_{\rm max} = +\Beta r_{\rm min}$, the +Petrosian Ratio for that annulus is defined as the ratio of the +surface brightness in the annulus to the average surface brigthness +within $r_{\rm max}$. The Petrosian Radius is defined to be $r_{\rm + max}$ for the annulus for which the Petrosian Ratio = 0.2, i.e., the +point on the galaxy radial profile at which the surface brightness is +20\% of the average surface brightness at that point. + +We determine the Petrosian Radius for the galaxy by quadratic +interpolation between the last two of the fixed annuli with Petrosian +Ratio $> 0.2$ and the first annulus with Petrosian Ratio $< 0.2$. In +general, the Petrosian Ratio for a galaxy with a regular morphology +(spiral or elliptical) is falling monotonically, so this interpolation +is unambiguous. However, irregular galaxy morphologies, noise, and/or +significant masking can cause the Petrosian Ratio to have rises as +well as drops. We track the Petrosian Ratio until the value is no +longer significant ($\sigma_{\rm Ratio} < 2 {\rm Ratio}$). If the +Petrosian Ratio drops below 0.2 for more than one radius, we choose +the largest such radius. + +Once the Petrosian Radius has been determined, we can now measure the +Petrosian Flux : this is defined to be the total flux within an +aperture corresponding to 2 $\times$ the Petrosian Radius. Using the +Petrosian Flux, we can calculate two other interesting radii: $R_{50}$ +and $R_{90}$, the radii inside which 50\% and 90\% of the total +Petrosian flux is contained. + +\subsection{Kron Magnitudes} + + + +\subsection{Convolved Galaxy Model Fits} + +In the galaxy model fittting stage, sources which meet certain +criteria are fitted with analytical models for galaxies. The +available models for the PV3 analysis were: +\begin{itemize} +\item Exponential profile : $f = I_0 e^{\frac{-r}{r_0}}$ +\item DeVaucouleur profile : $f = I_0 e^{\frac{-r^{1/4}}{r_0}}$ +\item Sersic : $f = I_0 e^{\frac{-r^{1/n}}{r_0}}$ +\end{itemize} + +In this stage, the galaxy model is convolved with our best guess for +the PSF model at the location of the galaxy. For the PV3 analysis, +all sources detected in the 'bright source' analysis step (S/N > 20 ?) +were fitted with all three galaxy models, unless (a) the morphological +test identified the source as a likely cosmic ray (\ref{CR}) +or (b) the peak of the PSF profile was above the saturation limit +\note{for the chip? cell?}. Sources in the denser portions of the +Galactic plane and bulge were not included in the analysis. This +restriction limited the total time spent on the galaxy modeling +analysis at the expense of galaxy photometry in the plane (though Kron +photometry is available for those objects). + +The Galactic Plane region was defined by $|b| > b_{\rm min}$ where +$b_{\rm min} = b_0 + r_b e^{\frac{-l^2}{2 \sigma_b^2}}$. For the PV3 +analysis, $b_0 = XX$, $r_b = XX$, $\sigma_b = XX$. + +The galaxy models are fitted using the same Levenberg-Marquart +minimization code use for the other non-linear fitting stages. In the +convolved galaxy fit, the galaxy model image and the model derivative +images are convolved with the psf at each iteration. WRITE out the +chi-square and show how this is separated out as a set of images. For +the Exponential and DeVaucouleur fits, all parameters are fitted in +the non-linear minimization stage. For the Sersic model fits, there +is too much degeneracy (yes?) between ???. We determine the Sersic +index using a grid search, using the non-linear minimization for the +remaining parameters on each grid search step. The index is fitted in +the following values (XXXXX). + +With XXXM galaxies to It is important to make an initial guess for the model parameters +which is reasonably close to the best fit value, + +\subsection{Convolved Radial Aperture Photometry} + +\subsection{Forced Photometry : PSFs} + +\subsection{Forced Photometry : galaxies} + +\subsection{Types of Object / PSF models currently available} \note{the discussion of the model types needs to be extended}