Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

388 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsi.e., the expected score over a flight has a 1/b-type singularity in contrast to the 1/jr' —r*| 2 singularity of the eventwise estimators. I(P) can be easily sampled by selecting a (3value uniformly in [(3,,(3,], i.e., it may also serve as an estimator which has better propertiesthan the next-event estimator. On the other hand, it was seen in Chapter 5.Vl that theexpectation estimator f F(P) — I(P) is a member of a broad class of partially unbiasedestimators, and there exist several other estimators with similar properties. By analogy tothe class of partially unbiased reaction-rate estimators, a similar class of point estimatorscan also be defined. We shall show that all estimators belonging to the class have a singularityat least as severe as 1/b, and therefore it is pointless to seek bounded variance estimatorsamong the commonly used ones.Before doing so, let us discuss in some detail the variance of estimators with 1/bsingularities. Taking the example of the expectation estimator I(P), the dominant term ofthe total second moment for small b isOne is tempted to believe that the 1/b 2singularity drops out, as has happened to thel/|r — r*| 2singularity in the expectation of the next-event estimator. This, however, is notthe case because jr — r*j and b are essentially different quantities. The former is the distanceof a collision point (spatial integration point in the second-moment integral) from the detectorpoint r*, while the latter is the distance of a possible flight direction from r*. Their probabilitydensities are different, as follows from the following simplified reasoning. The collisionpoints are assumed to be uniformly distributed in some volume around r* and therefore theirdensity function is proportional to 3r 2 dr. The product of the probability density of r and thesingularity 1/r 2 is bounded, thus resulting in a finite expectation. As for the density of d,let us consider a given flight direction and a plane perpendicular to it that contains the pointr*. The crossing points of the. possible flights in the given direction with the plane areapproximately uniformly distributed over a surface on the plane around r*. Therefore, thedensity function of b is proportional to the area element of the surface, i.e., to 2rdr. and,again, only the expected score remains bounded. This also means that Theorem 6.5 appliesto the estimators considered in this section, and an estimator with a 1/b singularity providesa total score that converges to its expectation at a rate of (log rn/n U2 .Let us now consider the class of partially unbiased point estimators generated by thetransformation of the expected scattering point estimator f Es(P') according to the formulasin Section 5.VLB. Again. P = (r,E). P' = (r + D w,E), P 1= (r + D,w.E), and P 2=(r + D 2co.E). Then, according to Equation (5.223), the transformed estimatorf(P,P') dD, T(D,|r,w,E)X(D,D I)f Es(r + D,w,E)/(6.183)is also partially unbiased, with an arbitrary function X(D 1,D). Here againT(D|r,w,E) = cr(r + Doj,E)exp - dto(r Mw.t-3