2.7} Biasing

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 Although FLUKA is able to perform fully analogue particle transport
 calculations (i.e. to reproduce faithfully actual particle histories), in
 many cases of very non-uniform radiation fields, such as those encountered in
 shielding design, only a very small fraction of all the histories contributes
 to the desired response (dose, fluence) in the regions of interest, for
 instance behind thick shielding. In these cases, the user's concern is not to
 simulate exactly what occurs in reality, but to estimate in the most
 efficient way the desired response. This can be obtained by replacing the
 actual physical problem with a mathematically equivalent one, i.e. having the
 same solution but faster statistical convergence.
 A rigorous mathematical treatment of such variance-reduction techniques can
 be found in several textbooks (see for instance those of Lux and Koblinger
 [Lux91] or Carter and Cashwell [Car75]).
 In the present practical introduction we will only issue a few important
 warnings:
 1. In the limit of the number of histories tending to infinity, the value of
    all calculated quantities tend EXACTLY to the same average in the analogue
    and in the corresponding biased calculation. In other words, biasing is
    mathematically correct and implies no approximation. However, an
    acceleration of convergence in certain regions of phase space
    (space/energy/angle) will generally be paid for by a slower convergence in
    other regions.
    Because an actual calculation does not use an infinite number of
    particles, but is necessarily truncated after a finite number of
    histories, results must be considered with some judgment. For instance, if
    the number of histories is too small, it can happen that total energy is
    not conserved (check the energy budget summary at the very end of main
    output!)
 2. A bad choice of biasing parameters may have consequences opposite to what
    is desired, namely a slower convergence. A long experience, and often some
    preliminary trial-and-error investigation, are needed in order to fully
    master these techniques (but some biasing algorithms are "safer" than
    others).
 3. Because biasing implies replacing some distributions by others having the
    same expectation values but different variance (and different higher
    moments), biasing techniques in general do not conserve correlations and
    cannot describe correctly fluctuations.

 The simplest (and safest) biasing option offered by FLUKA is importance
 biasing, which can be requested by option BIASING. Each geometry region is
 assigned an "importance", namely a number between 1.0E-4 and 1.0E+4,
 proportional to the contribution that particles in that region are expected
 to give to the desired result. The ratio of importances in any two adjacent
 regions is used to drive a classical biasing algorithm ("Splitting" and
 "Russian Roulette"). In a simple, monodimensional attenuation problem, the
 importance is often set equal to the inverse of the expected fluence
 attenuation factor for each region.

 In electron accelerator shielding, two other biasing options are commonly
 employed: EMF-BIAS and LAM-BIAS. The first one is used to request leading
 particle biasing, a technique which reduces considerably the computer time
 required to handle high-energy electromagnetic showers. With this option, CPU
 time becomes proportional to primary energy rather than increasing
 exponentially with it. Option LAM-BIAS is necessary in order to sample with
 acceptable statistics photonuclear reactions which have a much lower
 probability than competing electromagnetic photon reactions, but are often
 more important from the radiological point of view.

 Other important options are those which set weight window biasing (WW-FACTOr
 WW-THRESh and WW-PROFIle) but their use requires more experience than assumed
 here for a beginner.

 Particle importances, weight windows and low-energy neutron biasing
 parameters are reported for each region in standard output. On user's request
 (expressed as 
SDUM
 = PRINT in a BIASING card), Russian Roulette and Splitting
 counters are printed for each region on standard output before the final
 summary. Such counters can be used for a better tuning of the biasing
 options.

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