Fundamental limits to the precision in astrometry and photometry using array detectors through the Cramér-Rao minimum variance bound Jorge Silva, Marcos Orchard DIE Rene Mendez DAS U. de Chile Espinosa et al., 2018, A&A Bouquillon et al, 2017, A&A Echeverria et al., 2016, A&A Lobos et al., 2015, PASP Mendez et al., 2013, 2014, PASP
Motivation Mendez et al. (2010): First entirely CCD-based ground-based PM study of Fornax dsph (m-m) 0 20.8 PM few x 0.1 mas/yr 1.5 mas positional accuracy @ ESO-NTT (3.5m)+SuSI2 (0.08 /pix). Astrometry limited by positional precision on indiv. QSOs, not registration. HST (2.5m)+WFPC (0.1 /pix) result, for similar QSOs, reaches 0.25 mas (6 NTT!...) HST finer pixels, diffraction limited (undersampled though), low background Ground-based Wider PSF (oversampled), higher background, high S/N (in pple.) What are the main limitations to positional uncertainty for a point source, as a function of detector, source & background properties? Some common wisdom elements: Lennart Lindegren, 1997: GAIA Technical Note Astrometry improves as S/N increases Astrometry degrades as FWHM degrades Astrometry degrades for undersampled images (HST?)
mas as Paper by Erik Hög on arxiv:1707.01020
mas as Paper by Erik Hög on arxiv:1707.01020 prompts us to revisit the question of the maximum attainable astrometric precision Do practical estimators approach that limit?
Given a set of measurements that depends on parameter : (i=1 n) that follow and underlying distribution What is the minimum variance of the parameter? How do we estimate the parameter? Optimal parameter estimation can, in general, be formulated using decision-making theory (Cover & Thomas 2006). Claude Shannon, 1948 (1916-2001): A mathematical theory of communication (transmission of information over a noisy channel). This framework can also provide absolute lower bounds to the uncertainty of these estimators. The Cramer-Rao (CR) lower uncertainty bound determines the minimum theoretical variance achievable by any unbiased estimator (Stuart, Ord, & Arnold 2004).
Parameter estimation & the Cramér-Rao minimum variance bound Untractable, as it requires knowledge of the unknown performance bounds Harald Cramér (1946). A contribution to the theory of statistical estimation Scandinavian Actuarial Journal 1: 85-94 Calyampudi Radakrishna Rao (1945). "Information and accuracy attainable in the estimation of statistical parameters" Bulletin of the Calcutta Mathematical Society 37: 81 89 1893-1985 1920
Fisher information about
Why would you care about fundamental bounds? Can I reach my science goals with a given instrument and observing conditions? Observing proposal preparation Observational planning and strategy ETCs @ observatories How can I best design my instrument? Requirements on the site conditions to reach certain precision? Am I obtaining the most out of my observations? Calibration steps, reduction strategies System performance monitoring Is my reduction protocol adequate? Are my results biased?...
Data analysis on parametric statistics (incl. computing the CR bound!) specify a model of the observations Our model includes two ingredients: A deterministic component: The distribution of light consists of a point spread function which characterizes the source, plus a background (sky, detector). A stochastic component: The distribution of brightness follows a Poisson distribution (mass) function, driven by the expected flux @ each pixel.
Deterministic part: Source flux (photo-e ) per pixel All background (e ) per pixel Total source flux i = 1 n pixels (linear array) Normalized pixel response Point spread function i (X c ) i = X c NB: So far no intra-pixel response function, no FF (QE) uncertainties
Stochastic part: Poisson distribution (photon counting statistics), with mean - shot noise - One can verify that the CR conditions are met CR minimum variance of position Fisher information about x c
Cramér-Rao bound linear detector needs prescription for the flux (PSF) and the background (sky, detector)
Gaussian PSF (Inverse) gain of detector in e /ADU B, F in ADU Exact expression discrete! Background: Sky + Detector NB: RON as Poisson but really Gaussian
Astrometric Cramér-Rao bound as a function of S/N of the source FWHM = 1.0 arcsec Solid: f s = 2,000 ADU/arc-sec Dashed: f s = 0 ADU/arc-sec FWHM = 0.5 arcsec 20% gain in Xc @ S/N = 50 by suppressing background! x = 0.2 arcsec, D = 0 e-, RON = 5 e-, G = 2 e-/adu
Cramér-Rao limit as a function of detector & source parameters RON = 5 e, G = 2 e /ADU, f s = 2000 ADU/arc-sec, FWHM = 1 arc-sec F=1000 ADU, S/N=20 F=5000 ADU, S/N=74 F=50000 ADU, S/N=300 1/50th of a pix @ 0.5 /pix Pixel size
Comparisons to real astrometric data HST beats ground-based, despite being undersampled (mostly due to low-background & small image size) Ground based (NTT+SuSI R-band), F = 10000 ADU f s = 3000 ADU/arcsec, FWHM = 0.45 arcsec, x = 0.08 arcsec σ x (CR) = 1.7 mas 1.5 mas reported by Mendez et al. (2010)! HST WFPC2, F = 25000 ADU (scaling by aperture and exp. time) f s = 30 ADU/arcsec, FWHM = 0.1 arcsec, x = 0.1 arcsec σ x (CR)= 0.2 mas.0.25 mas reported by Piatek et al. (2002)!
Small-pixel (high-res) approximation (Δx / σ 1) L. Lindegren, 1978 x FWHM/(S/N), no background I. King, 1983, LS x B ½ /F @ low S/N x 1/F ½ @ high S/N G. Gatewood, 1985, MP x 1/F ½ @ moderate/high S/N Captures the astrometric common wisdom Quite resilient to high-res approximation
Can the Cramér-Rao bound be reached? If we can write (N&S condition!): Function of alone Then θ(i 1,,I n ) is an unbiased estimate of, with a variance that reaches the CR bound Unfortunately Thus, theoretically, neither Maximum-likelihood nor least-squares satisfy this condition directly numerical experiments
LS exhibits a large performance gap!..while ML seems to be very tight
Practical estimators can approach the CR bound very tightly! ML Adaptive WLS S/N = 10 S/N = 30 S/N = 10 S/N = 30 S/N = 120 S/N = 230 S/N = 120 S/N = 230
What about photometry? (2x2 Fisher matrix, determinant cross-correlation) High-resolution approximation, Source-dominated, B/F << 1 : Background-dominated, F/B << 1 : Astrometry & photometry become decoupled!
Star trackers (ST) are currently the best devices for attitude determination of spacecrafts that require high pointing stability (imaging & spectroscopic applications), Requirement imply sub-arcsec accuracy 1/100 of a pix on the detectors, Various sources of error arise which limit this accuracy: device noise, intra-pixel non-uniformity, optical distortions, temperature-induced changes in focal-length, spacecraft jitter we need to: (1) quantify these sources of error, 2 2 and (2) define some benchmark. The benchmark is, precisely, the CR bound: σ Inst σ CR Zhang & colleagues present a very detailed analysis of all relevant sources of error & compare them to the CR bound due to uncertainties in brightness, position & spread ( FWHM) of the source, Are able to identify parameters of ST that are critical to approach the CR bound design, calibration & telemetry control parameters of ST (confirmed by real ground-based test observations with STs in the Lab). An enhanced Cramér-Rao bound weighted method for attitude accuracy improvement of a star tracker RSI, 87, 063112, 2016
Paranal/VST+OmegaCAM trailed image of Gaia. GBOT program (60s, r 21) Gaia-GBOT experiment (http://gbot.obspm.fr/) Gaia needs its orbit good to 2.5 mm/s and 150 m (20 mas) at any given time Ground Based Optical Tracking implemented (Heidelberg-Paris) Satellite fainter than 2 mag than expected!... VST 2.6m @ ESO/Paranal alliance ESA/ESO Cramer-Rao predictions & observations
The CR bound is a powerful maximum precision benchmark indicator Use it! or ask us to help you on how to adapt it to your needs Range of applications: Instrument design, observational planning (CRC@Observatories), data analysis benchmarking, instrument performance (e.g., STs, Gaia focus), data quality control (LSST), Limitation: Does not specify how to construct (unbiased) estimators that reach the bound! Beware: A biased estimator may have a variance < than the CR bound!, Our theoretical studies & numerical experiments demonstrate that practical estimators perform tightly, We can avoid some of the losses in precision found on the classical LS estimators (AWLS, ML) Comparison to astrometric measurements on real observations confirm our CR bounds
http://www.das.uchile.cl/ rmendez/documents/book.htm www.amazon.com
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