Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

390 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsLet us now consider two simple examples of partially unbiased point estimators. It wasshown in Section 5.VI.B that choosing X(D,D 1) = 1, the transformed estimator is theexpected score over the next flightf(P,P') = I(P)as also discussed at the beginning of this section. In this case, according to Equation (6.184)g(D,D,|r,E) = T(D,Sr,w,E)f Es(r4-D iw,E)|r4-D 1aj^r H!j 2If f Fsis sampled with the aid of the next-event estimator in Equation (6.169), the score froma flight that starts at P readss, = ~bP'1(0,(8)^,103)^(0)^00^^,3-3)1(0^^^10*^')/^^^')where r, - r D 1to and r* = r + D* *. Alternatively, by puttingX(D.D 1) = 1 for D 1> Dand zero otherwise, the track-length estimatorft P.P') = j o dD,o-( r f D,w.E) f, Jr + D,to.E)is obtained. The corresponding g function isg(D.D,|r.E) = (T(P 1)UP 1)Ir 4- D 1W - r*| 2 ; P 1= (r + D.w.E)This function is bounded (except for the degenerate case of a = + 0 0 ), and therefore itfollows from Equation (6.185) that the track-length point estimator has a 1/b singularity.The interval of the angular variable B is determined from the relationsD 1(B 1) = 0, D 1O 2) = Dwhere D is the distance of the next collision point P' from P. The score by the track-lengthestimator in a collision at P followed by a flight to P' isS 1= —- 1 1 — JbC.(6^w*|r'.E'.E)T(D*|r.,.oj*,Il')rr(r 1.E)/tr(r H,.E')provided the next-event estimator is used to sample f Es.by Rief et al. 15 - 68 - 69 .This estimator was first proposedTwo comments are to be made here. Let us first note that the estimation procedureabove is based on the one-sample Monte Carlo evaluation of the integral in Equation (6.185).Obviously, steps 1 through 4 can be repeated several times for a given flight, and the averageof the scores so obtained will be a better estimate of the integral than the single-trial sample.Execution of the estimation steps, however, requires approximately the same computingtime as the generation of a new flight. Therefore, it is not at all clear, in general, whetherthe accuracy of the estimate from a given flight or the number of different flights (histories)