Path Length Distribution

The probability distribution for the path length x a photon travels before it is scattered or absorbed is given in Section 5.1, rewritten here as p(x) = i exp(—x/t), (6.52)

where x is the physical path length and t the total mean free path. It is more convenient here to express this distribution in terms of the optical path length r = x/t p(r) = exp(-t ), (6.53)

the integral of which from 0 to to is 1, as it must be if this is a proper probability distribution. As we did previously for the probability distribution 2x, we find r as a function of £, a random number between 0 and 1 with a uniform probability distribution, such that

Figure 6.2: The continuous probability distribution p = 2x (solid curve) for the variable x can be approximated by dividing the x-axis into equal intervals (bins); 100 bins were used here. Values are assigned to each bin with a random number generator. The number of values in each bin divided by the total number is the probability of a value lying within a bin. Divide by the bin width to obtain the probability density. The discrete probability densities more closely approximate the continuous distribution the greater the number of values (indicated in the upper left corner of each plot).

Figure 6.2: The continuous probability distribution p = 2x (solid curve) for the variable x can be approximated by dividing the x-axis into equal intervals (bins); 100 bins were used here. Values are assigned to each bin with a random number generator. The number of values in each bin divided by the total number is the probability of a value lying within a bin. Divide by the bin width to obtain the probability density. The discrete probability densities more closely approximate the continuous distribution the greater the number of values (indicated in the upper left corner of each plot).

Figure 6.3: The same calculations shown in Fig. 6.2 for 1000000 random numbers but done two times. The probability densities for each run are shown with open circles and open triangles.

The solution to this differential equation, subject to the boundary condition t = 0 when £ = 1, is

With this transformation, the random number £ is uniformly distributed between 0 and 1 and the optical path length t is distributed between 0 and to according to the probability distribution Eq. (6.53).

To show this the t-axis was divided into bins of width 0.01,106 random numbers £ chosen, the corresponding values of t calculated fromEq. (6.55) and assigned to appropriate bins, and the fraction of values in each bin divided by the bin width 0.01 and plotted. As expected, this set of discrete probability densities (Fig. 6.4) lies close to the continuous probability distribution exp(-T).

Given that a photon travels an optical path t before something happens to it, the probability it is scattered is the single-scattering albedo w, and hence the probability it is absorbed is 1 - w. Thus to determine if a photon is scattered or absorbed, compute a random number. If it is less than or equal to w, the photon is scattered; if not, the photon is absorbed, its death duly recorded, another photon launched, and so on into the long hours of night. If a photon is scattered, the direction of its next path is determined by a probability distribution for scattering directions.

6.3.2 Scattering Direction Distribution

The probability distribution for scattering in a particular direction per unit solid angle is specified by the inaptly-named phase function p(d, y) (see Sec. 6.1.2), normalized in that

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