General statistical model for geometrical splitting in Monte Carlo - I

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Abstract
Geometrical surface splitting is used extensively in deep penetration Monte Carlo calculations for the purpose of variance reduction. For many years this method has been used on a rather intuitive basis. No detailed statistical model existed to support the unbiasedness of the method, although it is quite obvious that it is unbiased, or to provide a starting point for analytic evaluation of the second moment which is essential for optimization of the method. In the following, we develop such a statistical model for a general case of any number of routes from the source to the detector and Russian Roulette. The model involves the description of a general independent source paricle event along with the probabilistic quantities accompanying such an event. The detector contribution of such an event is established together with its probability density. Folding the detector contribution with the probability density and averaging over all possible source particle events yields the first momqt or the detector response. We also briefly discuss the possibility of obtaining the second moment by this direct statistical approach.

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