*3.5. The Effect of Variation of Size*

As we can see, BRDF depends on the flake size (and its distribution) through the ratio *A*|*accurate A*|*LTE* , see (15). Strange as it may seem, the correction scale is nearly insensitive to the distribution of flake area being determined by the average area. To understand this effect, let us consider the exponential distribution of area

$$D(S) = \overline{DS}^{-1}e^{-\frac{S}{\mathfrak{T}}}$$

where *S* is the mean area and *D* is the "total density". This form of distribution allows to calculate *α* analytically.

We see that while the difference between fixed size flakes and LTE can be even twofold, the difference between fixed size flakes and variable size flakes can only reach about 3.5% for very special parameters.

According to (14), the main first-order component of BRDF is proportional to the integral of attenuation

$$A(t, \rho, F) \equiv \int\_0^{\frac{1}{2\tilde{\rho}}} \mathcal{e}^{-4(1-t)\rho F(\zeta - \frac{(1-t)}{\tilde{\pi}}\tilde{\alpha}(\zeta))} d\zeta$$

Therefore, the quantitative measure of the effect of the distribution of size on BRDF is the relative difference of that factor, i.e., *Avar*−*Afix Avar* . According to (13)

*<sup>α</sup>*(*Z*) <sup>=</sup> <sup>2</sup>*DS*−<sup>1</sup> *<sup>π</sup>*3/2 <sup>2</sup> 3 *s* 0 *e* − *S <sup>S</sup>*¯ *<sup>S</sup>*3/2*dS* <sup>+</sup> 1.6361√*<sup>s</sup>* <sup>∞</sup> *<sup>s</sup> e* − *S <sup>S</sup>*¯ *SdS* − 1.2722*s* <sup>∞</sup> *<sup>s</sup> e* − *S S*¯ <sup>√</sup>*SdS* <sup>+</sup> 0.30278*s*3/2 <sup>∞</sup> *<sup>s</sup> e* − *S <sup>S</sup>*¯ *dS* <sup>=</sup> <sup>2</sup>*DS*3/2 *<sup>π</sup>*3/2 <sup>2</sup> 3 *σ* <sup>0</sup> *<sup>x</sup>*3/2*e*<sup>−</sup>*xdx* <sup>+</sup> 1.6361√*<sup>σ</sup>* <sup>∞</sup> *<sup>σ</sup> xe*<sup>−</sup>*xdx* − 1.2722*<sup>σ</sup>* <sup>∞</sup> *σ* <sup>√</sup>*xe*<sup>−</sup>*xdx* <sup>+</sup> 0.30278*σ*3/2*e*−*<sup>σ</sup>* <sup>=</sup> <sup>2</sup>*DS*3/2 *<sup>π</sup>*3/2 1.3333 <sup>×</sup> <sup>10</sup>−5*<sup>σ</sup>* <sup>+</sup> 0.6361√*σe*−*<sup>σ</sup>* <sup>−</sup> 1.1275*<sup>σ</sup>* <sup>+</sup> (0.88623 <sup>+</sup> 1.1275*σ*)erf√*<sup>σ</sup>* <sup>=</sup> <sup>2</sup> *DS*3/2 *π*3/2 ∼ *α*(*ζ*) = <sup>2</sup> *π*3/2 *DSH*<sup>√</sup> *<sup>S</sup> <sup>H</sup>* ∼ *α*(*ζ*) = <sup>2</sup>*crF π* ∼ *α*(*ζ*) ∼ *α*(*ζ*) ≡ 1.3333 × <sup>10</sup>−5*ζ*<sup>2</sup> + 0.6361 *ζe*−*ζ*<sup>2</sup> − 1.1275*ζ*<sup>2</sup> + 0.88623 + 1.1275*ζ*<sup>2</sup> erf(*ζ*)

where *<sup>s</sup>* <sup>≡</sup> *<sup>π</sup>Z*2, *<sup>σ</sup>* <sup>≡</sup> *<sup>s</sup>*/*<sup>S</sup>* and *<sup>ζ</sup>* <sup>≡</sup> \$*<sup>π</sup> <sup>S</sup> <sup>Z</sup>* <sup>=</sup> <sup>√</sup>*σ*.

The case of fixed size flakes relates to the delta-function distribution

$$D(\mathcal{S}) = \overline{D}\delta(\mathcal{S} - \overline{\mathcal{S}}),$$

for which

$$\begin{array}{rcl} \mathfrak{a}(Z) &=& \frac{2\overline{D}\overline{S}^{3/2}}{\pi^{3/2}} \begin{cases} 1.6361\sqrt{\frac{s}{\mathfrak{T}}} - 1.2722\frac{s}{\mathfrak{T}} + 0.30278 \left(\frac{s}{\mathfrak{T}}\right)^{3/2}, & s \le \overline{S} \\ \frac{2}{\mathfrak{T}}, & s > \overline{S} \end{cases} \\ &=& \frac{2\overline{D}\overline{S}^{3/2}}{\pi^{3/2}} \Big( 1.6361\sqrt{\min(\sigma, 1)} - 1.2722\min(\sigma, 1) + 0.30278 \left(\min(\sigma, 1)\right)^{3/2} \Big)^{3/2} \Big) \\ \widetilde{\mathfrak{a}}(\zeta) & \equiv& 1.6361\min(\widetilde{\zeta}, 1) - 1.2722\left(\min(\widetilde{\zeta}, 1)\right)^{2} + 0.30278\left(\min(\widetilde{\zeta}, 1)\right)^{3} \end{array}$$

The plots of <sup>∼</sup> α(ζ) as a function of ζ for both cases are shown in Figure 4.

**Figure 4.** <sup>∼</sup> α(ζ) as a function of ζ for flakes with exponentially distributed area (red) and for the fixed-size flakes with the same mean area (green).

$$\begin{array}{c} \text{For } \zeta \to 0 \\\\ \widetilde{\omega} \end{array} \qquad \qquad \qquad \qquad \widetilde{\omega}$$

$$\begin{array}{rcl}\widetilde{\alpha}\_{var}(\zeta) & \approx & 1.6361\zeta - 1.1275\zeta^2 + O(\zeta^3),\\\widetilde{\alpha}\_{fixed}(Z) & \approx & 1.6361\zeta - 1.2722\zeta^2 + O(\zeta^3).\end{array}$$

So, the two functions converge.

For *ζ* → ∞ both functions saturate

$$\begin{array}{rcl} \widetilde{\alpha}\_{\text{true}}(Z) & \approx & \frac{\overline{DS}^{3/2}}{\pi\_{\text{3}}}\\ \widetilde{\alpha}\_{\text{fixed}}(Z) & \approx & \frac{4 \overline{DS}^{3/2}}{3 \pi^{3/2}} \end{array}$$
