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I.B. Mihaylov, J.E. Bjorkman (University of Toledo)
We present a general procedure for determining photon weights and measurement errors in Monte Carlo simulation of the transfer of polarized radiation. We simulate the radiation transfer by tracing monochromatic photon packets that propagate outward in a stellar atmosphere, undergoing a number of scatterings before they finally escape. Usually one uses equal energy photon packets because the variance (error estimate) of any given measurement is minimized in that case. Although the introduction of weights increases the variance, and it no longer obeys simple binomial/Poisson statistics, there are many situations where for computational efficiency it is advantageous to use weighted photons. We distinguish two kinds of weights: 1) the energy content of the packet, and 2) the sampling weight. Since any measurement in the Monte Carlo simulation is obtained by sampling, the expectation value of the measured quantity is given by an integral of that function over a probability distribution. Thus this function becomes the sample weight for the discrete samples in the simulation.
In this paper we develop a formalism for systematically determining photon weights (both energy and sampling), and we derive formulae for estimating the errors of any measured quantity. The error formulae are derived by finding unbiased estimators of the variance of the measured quantity. We test our error estimation and weight definitions by comparing our simulation results with the analytic solution for a plane parallel gray stellar atmosphere and demonstrate that our error formulae produce the appropriate \chi2 statistics. In one case (forced final scattering), these tests revealed a subtle bias in the results, which arises from the choice of the sampling weight. Although we show how to correct for this bias, the method of forced final scattering has rather undesirable statistical properties.
This work has been supported under NASA grants NAG5-3447 (IBM) and NAG5-3248 (JEB) to the University of Toledo.
The author(s) of this abstract have provided an email address for comments about the abstract: imihail@physics.utoledo.edu