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trunk/doc/release.2015/ps1.analysis/analysis.tex
r40589 r40590 121 121 \note{the beginning section needs to be updated to mention the DR1 and 122 122 DR2 releases, the PV0-PV3 analysis versions, and the basic idea of 123 the IPP stages ).123 the IPP stages}. 124 124 125 125 This is the fourth in a series of seven papers describing the … … 1734 1734 \item Exponential profile : $f = I_0 e^{-\rho}$ 1735 1735 \item DeVaucouleur profile \citep{1948AnAp...11..247D}: $f = I_0 e^{-\rho^{1/4}}$ 1736 \item S ersic \citep{1963BAAA....6...41S} : $f = I_0 e^{-\rho^{1/n}}$1736 \item S\'ersic \citep{1963BAAA....6...41S} : $f = I_0 e^{-\rho^{1/n}}$ 1737 1737 \end{itemize} 1738 1738 where $\rho$ is a normalized radial term: $\rho = … … 1743 1743 x_0, Y_{\rm chip} - y_0$). Including the normalization ($I_0$) and a 1744 1744 local sky value, the Exponential and DeVaucouleur profiles have 7 free 1745 parameters and the S ersic profile has the additional free parameter of1746 the S ersic index $n$. In this stage, the galaxy model is convolved1745 parameters and the S\'ersic profile has the additional free parameter of 1746 the S\'ersic index $n$. In this stage, the galaxy model is convolved 1747 1747 with an approximation to our best guess for the PSF model at the 1748 1748 location of the galaxy. … … 1772 1772 quantive the relationships between the first radial moment used to 1773 1773 calculated a Kron Magnitude and the effective radius for different 1774 S ersic index values, $n$. Since the Exponential and DeVaucouleur1775 models are equivalent to S ersic models with $n$ = 1 and 4,1774 S\'ersic index values, $n$. Since the Exponential and DeVaucouleur 1775 models are equivalent to S\'ersic models with $n$ = 1 and 4, 1776 1776 respectively, this work can be used to generate the initial effective 1777 1777 radius values for all 3 model types. Once the effective radius is … … 1780 1780 generate a guess for the normalization, applying an appropriate scale 1781 1781 factor based on the ($R_{xx}$, $R_{yy}$ , $R_{xy}$) values, generated 1782 by integrating normalized S ersic models and determining the1782 by integrating normalized S\'ersic models and determining the 1783 1783 relationship between the central intensity and the integrated flux as 1784 a function of the Sersic index. 1785 1786 \note{special handling for central pixel} 1784 a function of the S\'ersic index. 1787 1785 1788 1786 The PSF-convolved galaxy model fitting analysis uses the … … 1859 1857 1860 1858 For the Exponential and DeVaucouleur fits, all parameters are fitted 1861 in the non-linear minimization stage. For the S ersic model, we do not1859 in the non-linear minimization stage. For the S\'ersic model, we do not 1862 1860 fit the index within the Levenberg-Marquardt analysis. Instead, we 1863 1861 start with a coarse grid search over a range of possible index values, 1864 1862 ($n = 0.5, 1.0, 1.5, 2.0, 3.0, 4.0, 5.0, 6.0$) and a range of possible 1865 1863 values for $R_{\rm eff}$ based on the value of $R_1$, the first radial 1866 moment. For a given value of the S ersic index, the $R_{\rm eff}$ is1864 moment. For a given value of the S\'ersic index, the $R_{\rm eff}$ is 1867 1865 related to the 1st radial moment by the scale factor specificy by 1868 1866 Graham \& Driver. We use the observed value of the 1st radial moment … … 1874 1872 We next perform 3 Levenberg-Marquardt minimization fits allowing the 1875 1873 shape parameters ($R_{xx}$, $R_{yy}$ , $R_{xy}$) and the normalization 1876 to be fitted, holding the centroid ($x_0, y_0$), S ersic index $n$, and1874 to be fitted, holding the centroid ($x_0, y_0$), S\'ersic index $n$, and 1877 1875 sky constant. In these fits, the index $n$ is set to the minimum 1878 1876 value previously calculated as well as values halfway to the next, and … … 1887 1885 % Graham & Driver : Graham A. W., Driver S. P. 2005, PASA 22, 118 1888 1886 % DOI: https://doi.org/10.1071/AS05001 1887 1888 The central pixel of the S\'ersic, DeVaucouleur, and Exponential 1889 models require special handling. When comparing an analytical model 1890 to the pixelized image recorded by a CCD, one normally treats the 1891 value in a pixel as equivalent to the value of the model at the center 1892 of the pixel. However, in reality, the number of counts observed in a 1893 pixel represents the integral of the surface brightness across the 1894 area of the pixel. This average will be equal to the central surface 1895 brightness times the area of a pixel as long as the second and higher 1896 derivatives of the analytical model are zero. However, if the first 1897 and second derivatives are non-zero, the curvature of the function 1898 within the pixel will make the integral differ from the central 1899 surface brightness times a fixed pixel area. If the curvature of the 1900 model function is sufficiently large, this difference will have a 1901 significant impact on the evaluation of the model. This situation is 1902 particularly true for the central portion of the S\'ersic-like model 1903 functions. 1904 1905 %% this can be seen by writing the taylor expansion of the function 1906 %% about the center of the pixel. do this? 1907 1908 In order to accurately compare the observed galaxy flux distribution 1909 to a model, it is necessary to integrate the pixel flux for a given 1910 set of model parameter values. This could be done in a numerical 1911 fashion, but in practice brute-force evaluation of the numerical 1912 integral is computationally very expensive, requiring many evaluations 1913 of the model function. Within \ippprog{psphot}, we bypass this 1914 problem by defining a set of pre-calculated images for the central 9 1915 pixels (the $3 \times 3$ grid of pixels centered on the peak). These 1916 pixel images are defined at higher resolution, with 11 subpixels per 1917 real CCD pixel. The pre-calculated images are generated for a series 1918 of values for the following parameters: S\'ersic index, effective 1919 radius, axial ratio. We then select the closest image to our specific 1920 case, and integrate over the true sub-pixels relevant for our position 1921 and model. We have thus turned the problem from thousands of 1922 evaluations of the full galaxy model to \approx 100 straight 1923 additions, or up to $6 \times$ that number if we interpolate between 1924 any of the parameters. 1889 1925 1890 1926 \subsubsection{Convolved Radial Aperture Photometry} … … 2109 2145 \section{Forced Photometry Modes} 2110 2146 2111 \note{edit this section to remove references to the IPP stages; just refer to the psphot concepts}2112 2113 2147 Traditionally, projects which use multiple exposures to increase the 2114 2148 depth and sensitivity of the observations have generated something 2115 equivalent to the \ippstage{stack}images produced by the IPP analysis2149 equivalent to the stack images produced by the IPP analysis 2116 2150 (c.f, CFHT Legacy survey, COSMOS, etc). In theory, the photometry of 2117 the \ippstage{stack}images produces the ``best'' photometry catalog,2151 the stack images produces the ``best'' photometry catalog, 2118 2152 with best sensitivity and the best data quality at all magnitudes. In 2119 2153 practice, these images have some significant limitations due to the … … 2130 2164 that point. Because of the high mask fraction, the exposures which 2131 2165 contributed to pixels at one location may be somewhat different just a 2132 few tens of pixels away. In the end, the \ippstage{stack}images have2166 few tens of pixels away. In the end, the stack images have 2133 2167 a effective point spread function which is not just variable, but 2134 2168 changing significantly on small scales in a highly textured fashion. 2135 2169 2136 2170 Any measurement which relies on a good knowledge of the PSF at the 2137 location of an object either needs to determine the PSF variations2138 present in the \ippstage{stack} image or the measurement will be2139 somewhat degraded. The highly textured PSF variations make this a 2140 very challenging problem: not only would such a PSF model require an 2141 unusually fine-grained PSF model, there would likely not be enough PSF 2142 stars in a given \ippstage{stack} image to determine the model at the 2143 resolution required. The IPP photometry analysis code uses a PSF 2144 model with 2D variations using a grid of at most $6\times 6$ samples 2145 per skycell, a number reasonably well-matched to the density of stars 2146 at most moderate Galactic latitudes. This scale is far too large to 2147 track the fine-grainedchanges apparent in the stack images.2148 2149 Thus PSF photometry as well as convolved galaxy models in the stack 2150 are degraded by the PSF variations. Aperture-like measurements are in 2151 general not as affected by the PSF variations, as long as the aperture2152 in question is large compared to the FWHM of the PSF.2171 location of an object needs to determine the PSF variations present in 2172 the stack image or the measurement will be somewhat degraded. The 2173 highly textured PSF variations make this a very challenging problem: 2174 not only would such a PSF model require an unusually fine-grained PSF 2175 model, there would likely not be enough PSF stars in a given stack 2176 image to determine the model at the resolution required. The IPP 2177 photometry analysis code uses a PSF model with 2D variations using a 2178 grid of at most $6\times 6$ samples per skycell, a number reasonably 2179 well-matched to the density of stars at most moderate Galactic 2180 latitudes. This scale is far too large to track the fine-grained 2181 changes apparent in the stack images. 2182 2183 As a result, PSF photometry as well as convolved galaxy models in the 2184 stack are degraded by the PSF variations. Aperture-like measurements 2185 are in general not as affected by the PSF variations, as long as the 2186 aperture in question is large compared to the FWHM of the PSF. 2153 2187 2154 2188 %% The IPP team initially explored the option of convolving each input … … 2158 2192 The IPP analysis solves this problem by starting with the sources 2159 2193 detected in the stack images and performing forced photometry on the 2160 individual warp images used to generate the stack. This 2161 forced-photometry analysis is performed using the 2194 individual warp images used to generate the stack, and then combining 2195 the resulting measurements to determine a high-quality average value. 2196 This forced-photometry analysis is performed using the 2162 2197 \ippprog{psphotFullForce} variant of \ippprog{psphot}. 2163 2198 … … 2176 2211 image; the measured flux may even be negative due to statistical 2177 2212 fluctuations. When combined together, these low-significance 2178 measurements will result in a signficant measurement as the2179 signal-to-noiseincreases with the combination of more data.2213 measurements result in a signficant measurement as the signal-to-noise 2214 increases with the combination of more data. 2180 2215 2181 2216 Individual warp images are processed independently with separate … … 2192 2227 \label{sec:galaxy.forced.fit} 2193 2228 2194 The convolved galaxy models are also re-measured on the 2195 \ippstage{warp} images by the \ippstage{fullforce} stage analysis. In 2196 this analysis, the galaxy models determined by the 2197 \ippstage{staticsky} photometry analysis are used to seed the analysis 2198 in the individual \ippstage{warp} images. The motivation of this 2199 analysis is the same as the \ippstage{fullforce} PSF photometry: the 2200 PSF of the \ippstage{stack} image is poorly determined due to the 2201 masking and PSF variations in the inputs. Without a good PSF model, 2202 the PSF-convolved galaxy models are of limited accuracy. 2203 2204 In the \ippstage{fullforce} galaxy model analysis, we assume that the 2205 galaxy position and position angle, along with the Sersic index if 2206 appropriate, have been sufficiently well determined in the 2207 \ippstage{staticsky} analysis. In this case, the goal is to determine 2208 the best values for the major and minor axis of the elliptical contour 2209 and at the same time the best normalization corresponding to the best 2229 The convolved galaxy models are also re-measured on the warp images by 2230 the \ippprog{psphotFullForce} analysis. In this analysis, the galaxy 2231 models determined from the stack image analysis are used to seed the 2232 analysis in the individual warp images. The motivation of this 2233 analysis is the same as the forced PSF photometry: the PSF of the 2234 stack image is poorly determined due to the masking and PSF variations 2235 in the inputs. Without a good PSF model, the PSF-convolved galaxy 2236 models are of limited accuracy. 2237 2238 In the forced galaxy model analysis, we assume that the galaxy 2239 position and position angle, along with the S\'ersic index if 2240 appropriate, have been sufficiently well determined in the analysis of 2241 the stack image. In this case, the goal is to determine the best 2242 values for the major and minor axis of the elliptical contour and at 2243 the same time the best normalization corresponding to the best 2210 2244 elliptical shape, and thus the best galaxy magnitude value. 2211 2245 2212 For each \ippstage{warp} image, the \ippstage{staticsky} values for2213 the major and minor axis are used as the center of a $5 \times 5$ grid 2214 search of the major and minor axis parameter values. The grid spacing 2215 is defined as a function of the signal-to-noise of the galaxy in the 2216 stack image so that bright galaxies are measured with a much finer 2217 g rid spacing than faint galaxies. For both the major and minor axis2218 directions, values of ($1 - \frac{3.0}{S/N}$, $1 - \frac{1.5}{S/N}$, 2219 1.0, $1 + \frac{1.5}{S/N}$, $1 + \frac{3.0}{S/N}$) times the dimension 2220 are tested. For each grid point, the major and minor axis values at 2221 that point are used to generate the model. The model is then 2222 convolved with the PSF model for the \ippstage{warp} image at that 2223 point. The resulting convolved model is then compared to the 2224 \ippstage{warp} pixel data values and the best fit normalization value 2225 is determined. The integrated flux, flux error, and the $\chi^2$ 2226 value for each grid point are recorded.2246 For each warp image, the stack values for the major and minor axis are 2247 used as the center of a grid search of the major and minor axis 2248 parameter values. The grid spacing is defined as a function of the 2249 signal-to-noise of the galaxy in the stack image so that bright 2250 galaxies are measured with a much finer grid spacing than faint 2251 galaxies. For the PV3 $3\pi$ analysis, a $5 \times 5$ grid was used; 2252 values in both the major and minor axis directions of ($1 - 2253 \frac{3.0}{S/N}$, $1 - \frac{1.5}{S/N}$, 1.0, $1 + \frac{1.5}{S/N}$, 2254 $1 + \frac{3.0}{S/N}$) times the dimension are tested. For each grid 2255 point, the major and minor axis values at that point are used to 2256 generate the model. The model is then convolved with the PSF model 2257 for the warp image at that point. The resulting convolved model is 2258 then compared to the warp pixel data values and the best fit 2259 normalization value is determined. The integrated flux, flux error, 2260 and the $\chi^2$ value for each grid point are recorded. 2227 2261 2228 2262 For a given galaxy, the result is a collection of $\chi^2$ values, 2229 2263 fluxes, and flux errors for each of the grid points spanning all 2230 \ippstage{warp}images. A single $\chi^2$ grid can then be made by2264 warp images. A single $\chi^2$ grid can then be made by 2231 2265 combining each grid point across the inputs. The combined $\chi^2$ 2232 2266 for a single grid point is simply the sum of all $\chi^2$ values at 2233 that point. If, for a single \ippstage{warp}image, the galaxy model2267 that point. If, for a single warp image, the galaxy model 2234 2268 is excessively masked, then that image will be dropped for all grid 2235 2269 points for that galaxy. The reduced $\chi^2$ values can be determined … … 2240 2274 axis values for the interpolated minimum $\chi^2$ value. The errors 2241 2275 on these two parameters is then found by determining the contour at 2242 which the \note{reduced?} $\chi^2$ increases by 1. 2243 2244 In this way, the \ippstage{fullforce} galaxy analysis uses the PSF 2245 information from each \ippstage{warp} to determine a best set of 2246 convolved galaxy models for each object in the \ippstage{skycal} 2247 catalog. 2276 which the $\chi^2$ increases by 1. 2277 2278 In this way, the forced galaxy model analysis uses the PSF information 2279 from each warp image to determine a best set of convolved galaxy 2280 models for each galaxy model measured for the stack image. 2248 2281 2249 2282 \section{Difference Image Photometry} 2250 2283 2251 \note{need an intro paragraph or so} 2252 2253 The variance map for a difference image must be generated from the two 2254 images used to construct the difference. Otherwise, the low sky level 2255 will automatically result in inconsistent interpretation of the variance. 2284 Among the primary science drivers for Pan-STARRS are the detection of 2285 moving objects (e.g., asteroids) and explosive transient sources 2286 (e.g., supernovae). For both of these situations, difference images 2287 are commonly used to remove the clutter of the static stars and 2288 galaxies. In the Pan-STARRS system, difference images are generated 2289 using the PSF-matching technique described by 2290 \citep[e.g.,][]{1998ApJ...503..325A}. The description of the 2291 Pan-STARRS implementation is given by \cite{price2017}. The analysis 2292 of the sources detected in these difference images uses a portion of 2293 the \ippprog{psphot} code embedded in the program, \ippprog{ppSub}, 2294 which generates those image. 2295 2296 The analysis of the difference image follows the same basic steps as 2297 other \ippprog{psphot} versions with some minor modifictions (see 2298 Table~\ref{tab:measurements}), as follows. The background subtraction 2299 is performed before the PSF matching and image subtraction is 2300 performed. The PSF model construction stage is not possible in the 2301 difference image due to the lack of valid sources. Instead, the PSF 2302 model from is generated from the positive image, after PSF-matching 2303 but before the subtraction is performed. Because we do not expect to 2304 have a large number of sources, only a single source detection pass is 2305 performed, and at the lowest signal-to-noise threshold. Only linear 2306 PSF model fitting is performed using the centroid determined from the 2307 analysis of the source moments. 2308 2309 For the difference images, the galaxy model analysis is not relevant. 2310 In a properly-constructed difference image, galaxies are unlikely to 2311 remain behind as significant sources. Most real sources in the 2312 difference image will be PSF-like and will consist of photometrically 2313 variable sources (flare stars, supernovae, etc) or astrometrically 2314 variable sources (high-proper motion stars or solar-system bodies). 2315 There are three likely classes of sources which will not be well 2316 represented by the PSF model, as discussed below. 2317 2318 Fast-moving solar-system objects will appear as short streaks. For 2319 example, a fast solar system object may have an apparent rate of 0.5 2320 degrees per hour, translating to 15 arcseconds in a 30 second 2321 exposure. Even a main belt asteroid at roughly 1 AU has reflex motion 2322 of approximately 1 degree per day, equivalent to 1.25 arcsec in a 30 2323 second exposure, and may be noticeably smeared and non-PSF-like. In 2324 \ippprog{psphot}, we use a trailed-star model to characterize these 2325 types of sources. This model is fitted in the same portion of the 2326 code which performs the unconvolved galaxy model analysis. 2327 \note{describe the trailed analytical model}. 2328 2329 In some cases, the stars in the two images may be somewhat offset. 2330 For specific stars, this offset may be due to differential chromatic 2331 aberration from the atmosphere or the optics, or from modest proper 2332 motion. If the astrometric solution for one of the two images is 2333 insufficiently accurate, all stars in large portions of the images may 2334 be noticably displaced. In both of these situations, the stars will 2335 appear as PSF dipoles in the difference images. The positive and the 2336 negative images will have stellar profiles, but they will be offset 2337 and will not subtract well. The two components may not have the same 2338 amplitude. In theory, a PSF-dipole model could be used to fit these types of 2339 sources, with free parameters of the two centroids and the two 2340 fluxes. In practice in \ippprog{psphot}, we use a number of non-parametric 2341 pixel-level statistics in an attempt to detect these cases. 2342 2343 \note{list the parameters} 2344 2345 Comets appear in the difference images as a non-PSF sources. Their 2346 2-D structure includes both the flux from the coma (with a typical 2347 power-law profile) and flux from the tail (with a more complex flux 2348 distribution). We use the Kron magnitudes to identify possibly 2349 extended objects which may be cometary in nature. \note{need some 2350 info from MOPS folks on what is used} 2256 2351 2257 2352 For a difference image, both positive and negative sources will be 2258 2353 present. The basic peak detection algorithm will only trigger for the 2259 positive sources. One solution is to simply apply \code{psphot} to 2260 both the difference image and its negative value. 2261 2262 In the case of a difference image, the PSF model construction stage 2263 will probably fail for lack of valid sources. It is better in these 2264 cases to provide PSF model from some other source. For example, the 2265 two images which are combined to generate the difference image 2266 represent the PSF. Presumably, one or both have been convolved with a 2267 PSF-matching kernel. The images which result from the convolution 2268 should be used to measure the PSF model. \note{this is what we 2269 actually do, so remove hypothetical wording.} 2270 2271 The source classification scheme defaults to the galaxy models for 2272 sources which are not well represented by the PSF model. In a 2273 properly-constructed difference image, galaxies are unlikely to remain 2274 behind as significant sources. Most real sources in the difference 2275 image will be PSF-like and will consist of photometrically variable 2276 sources (flare stars, supernovae, etc) or astrometrically variable 2277 sources (high-proper motion stars or solar-system bodies). There are 2278 three likely classes of sources which will not be well represented by 2279 the PSF model. 1) Fast-moving solar-system objects will appear as 2280 short streaks. For example, a fast solar system object would have an 2281 apparent rate of 0.5 degrees per hour, translating to 15 arcseconds in 2282 a 30 second exposure. Even a main belt asteroid at roughly 1 AU would 2283 have reflect motion of approximately 1 degree per day, equivalent to 2284 1.25 arcsec in a 30 second exposure, and could be noticeably smeared 2285 and non-PSF-like. A trailed-star model can be used to characterize 2286 these types of sourcess. 2) Small offset stars, either due to 2287 atmospheric / color effects or modest proper motion will appear as PSF 2288 dipoles in the difference images. The positive and the negative 2289 images will have stellar profiles, but they will be significantly 2290 offset and will not subtract well. The two components may not have 2291 the same amplitude. A PSF-dipole model can be used to fit these types 2292 of sources, with free parameters of the two centroids and the two 2293 fluxes. 3) Comets will appear in the difference images as a non-PSF 2294 sources. Their 2-D structure includes both the flux from the coma 2295 (with a typical power-law profile) and flux from the tail (with a more 2296 complex flux distribution). A comet flux model can be used to 2297 characterize these sources in difference images. A major difficulty 2298 in applying these three types of models is in making a robust test of 2299 which model should be used. This problem is akin to the issue of 2300 selecting and distinguishing between multiple galaxy models, as 2301 discussed in the section on Galaxy models. 2354 positive sources. In the \ippprog{ppSub} program, both the $A - B$ 2355 and the $B - A$ images are sent to the \ippprog{psphot} routine for 2356 source detection and characterization. 2357 2358 Note that the variance image for a difference image must be generated 2359 from the two positive images used to construct the difference. It is 2360 possible to run \ippprog{psphot} as an external program on a 2361 difference image generated previously. In this case, the variance 2362 image and the PSF model must be supplied as well as the difference 2363 image. 2302 2364 2303 2365 \section{Examples and Tests} 2366 2367 \note{to be added} 2304 2368 2305 2369 \acknowledgments
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