Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

235The general form of track-length estimators is obtained from the transformation inEquation (5.223) by puttingX(D 5D 1) = a if D 1> Dand zero otherwise. In this case, the transformed estimator in Equation (5.223) becomesf(D) = f T(P,P') =JoDdD,f(D,)T(D 1)/dD,T(D,)(5.230)Iu 1With the explicit form of the transition kernel, it is writtenf T(P,P') = I dD, f(r + D.w.EMr + D 1W 1E) (5.231)Again, in a homogeneous medium, if f(P,) = f for D 1< L and is zero otherwise, the tracklengthestimator scoresf T(P,P') - fa • min(D,L)Hence, the score is proportional to that part of the flight length which lies inside the regionof interest. Obviously, the track-length estimator gives a contribution whenever a part of aflight is traveled in the region, irrespective of whether the next collision is inside or outsidethe region.The estimators obtained through transformations of the simplest estimator f(P) in thisSection were all introduced by heuristic arguments during the long history of the transportMonte Carlo methods. Several other special estimators occasionally used in practice are alsospecial cases of the transformation (e.g., the special track length-type estimators proposedin Reference 42). The transformation, however, may also provide new estimators which donot follow from obvious heuristic reasoning. For example, consider the estimator which isthe track length-type transform of the expectation estimator. Inserting the expectation estimatorI(P) in place of f(P) in Equation (5.230), we obtain a new estimator of the formf(D) = f TX(P,P') = 1(P)T(D) (5.232)Here, t(D) is the optical distance between P and P'. This estimator was called "trexpectationestimator" in Reference 26 since it is a hybrid of the track-length and expectation estimators.One may contemplate whether it unifies the advantages or the disadvantages of these estimators.It turns out that for light absorbers in optically not-too-small regions, it results ina lower variance than both the track-length and expectation estimators. For optically thinregions, it is much worse than any of them; otherwise, it resembles the track-length estimatorA very important point should be emphasized here. The estimators introduced in thisChapter were derived under the assumption that the transition kernel is normalized to unity.This assumption has been repeatedly exploited in the various forms of the transformations.It has been seen in Chapter 5.1.C that assuming a vacuum-equivalent black absorber aroundthe domain of simulation, the transition kernel can always be normalized to unity. Whenapplying the above estimators, this normalization should always be performed or, if forsome special reason (e.g., for the introduction of some special nonanalog game) it is notconvenient, the estimators must be used in their proper form, which accounts for the finiteprobability of an endless free flight. The respective formulas are derived in Reference 26.The variety of partially unbiased estimators, of course, has its own theoretical interest.