Bias properties of logarithmic and asinh magnitudes at very low flux levels

The observation of flux sources near the limit of detection requires a careful evaluation of possible biases in magnitude determination. Both the traditional logarithmic magnitudes and the recently proposed inverse hyperbolic sine (asinh) magnitudes are considered. Formulae are derived for three different biasing mechanisms: the statistical spread of the observed flux values arising from e.g. measurement error; the dependence of these errors on the true flux; and the dependence of the observing probability on the true flux. As an example of the results, it is noted that biases at large signal-to-noise ratios R , at which the two types of magnitude are similar, are of the order of −( p +1)/ R ², where the exponent p parametrizes a power-law dependence of the probability of observation on the true flux.