From: Greg Aldering (aldering@panisse.lbl.gov)
Date: Thu Jul 22 2004 - 16:49:03 PDT
Attached is the plot I just showed at the SCP group meeting which examines
the S/N and systematic error for faint-source aperture photometry using
different aperture sizes and different uncertainties in knowledge of the
PSF. The calculation assumes that all of the statistical noise on the flux
arises from the sky background.
A Gaussian PSF is assumed, characterized by a standard deviation sigma.
The object flux is S. The signal-to-noise is SNR. The fractional error in
the PSF width is sigma/dsigma.
The solid line plots the photon-noise SNR divided by the best possible
photon signal-to-noise, SNR_max. This shows that from a purely
photon-noise standpoint, the optimal SNR is obtained using an aperture
radius equal to 1.7*sigma.
The dashed curve is the relative flux error S/dS for a given PSF
uncertainty, divided by the PSF uncertainty sigma/dsigma. In this case
note that dS is the flux error from the PSF uncertainty arising from the
error in aperture correction.
An interesting observation is that at the radius giving the optimal
photon-limited SNR, the relative flux uncertainty from aperture correction
equals the relative PSF uncertainty ([S/dS]/[sigma/dsigma] = 1). Also,
note that the importance of the aperture correction error due to the PSF
uncertainty drops much more rapidly than the photon SNR. So, one can use a
slightly larger aperture to get more robustness against PSF uncertainties
without losing much SNR. For the standard SCP aperture photometry where
the aperture radius is about 2.3*sigma, there is a roughly 10% loss in SNR
but a small systematic uncertainty arising from PSF uncertainty.
Enjoy!
Greg
P.S. [S/dS]/[sigma/dsigma] = 0.02/0.05 just illustrates a typical case,
where one wants to limit the systematic flux error to 2% in the
face of a PSF width uncertainty of 5%
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