**AAS 197, January 2001**

*Session 44. Observations and Analysis of Stellar Atmospheres*

Display, Tuesday, January 9, 2001, 9:30am-7:00pm, Exhibit Hall
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## [44.03] Unbiased Weights and Error Estimation in Monte Carlo Radiation Transfer

*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 \chi^{2} 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

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