While most of the BRDF models contained in the SCATMECH library are based upon physical principles, many are approximations that have a limited range of accuracy. It is up to the user to assess the accuracy of each model to determine if the results are of quantitative value or are just qualitative guides to data trends.
In order to facilitate the integration of Eq. (1.2), it helps to make a change of variables, so that the coverage of sampled points is more uniform over the scattering hemisphere. There are a number of options for such a change of variables. If we let $\xi_1=\theta_{\rm s}\cos\phi_{\rm s}$ and $\eta_1=\theta_{\rm s}\sin\phi_{\rm s}$, then Eq. (1.2) becomes \begin{equation}\rho(\Omega) = \int_\Omega f_{\rm r}(\theta_{\rm i},\theta_{\rm s},\phi_{\rm s}) \cos\theta_{\rm s}\; {\rm sinc}\;\theta_{\rm s}\;{\rm d}\xi_1\;{\rm d}\eta_1. \end{equation}
Versions of MIST before 4.00 used Eq. (11.1) for the integration. However, the change of variables $\xi_2=\sqrt{1-\cos\theta_{\rm s}}\cos\phi_{\rm s}$ and $\xi_2=\sqrt{1-\cos\theta_{\rm s}}\sin\phi_{\rm s}$ allows for a uniform coverage of sampled points in solid angle. In that case, Eq. (1.2) becomes \begin{equation}\rho(\Omega) = 2\int_\Omega f_{\rm r}(\theta_{\rm i},\theta_{\rm s},\phi_{\rm s}) \cos\theta_{\rm s}\;{\rm d}\xi_2\;{\rm d}\eta_2. \end{equation}
The change of variables $\xi_3=\sin\theta_{\rm s}\cos\phi_{\rm s}$ $\eta_3=\sin\theta_{\rm s}\cos\phi_{\rm s}$ allows for uniform coverage of sampled points in projected solid angle, $\cos\theta_{\rm s}\;{\rm d}\Omega$; in this case, Eq. (1.2) becomes \begin{equation}\rho(\Omega) = \int_\Omega f_{\rm r}(\theta_{\rm i},\theta_{\rm s},\phi_{\rm s})\;{\rm d}\xi_3\;{\rm d}\eta_3. \end{equation}
MIST Version 4.00 now uses Eq. (11.2) by default, but allows the user to use any of Eqs. (11.1) — (11.3), with a uniformly sampled grid or with Monte Carlo sampling.
Any numerical integration introduces errors from the finite sampling of the integrand. For two-dimensional integration with a fixed sampling grid and a complex boundary, these errors are compounded by the sampling at the edges. For this reason, integration is performed twice, once with the desired model, yielding $\rho(\Omega)$, and once with a perfectly reflecting diffuser (PRD), yielding $\rho_{\rm PRD}(\Omega)$. The projected solid angle $\Omega_{\rm proj}$ is also calculated from the nominal boundary. Since $\rho_{\rm PRD}(\Omega)=\Omega_{\rm proj}/\pi$, the results of $\rho(\Omega)$ are adjusted by the factor $\Omega_{\rm proj}/[\pi\rho_{\rm PRD}(\Omega)]$, where $\rho_{\rm PRD}(\Omega)$ determined numerically using the same sampled points and $\Omega_{\rm proj}$ is determined from a line integral around irregular boundaries, \begin{equation} \Omega_{\rm proj}=\int_\Omega {\rm d}\xi_3\;{\rm d}\eta_3 = \oint_{\partial\Omega}(\eta_3\;{\rm d}\xi_3-\xi_3\;{\rm d}\eta_3), \end{equation}
or, using an exact expression for circular boundaries.
Aside from the intrinsic accuracy of the particular model, the accuracy of the integration depends upon the values of the variables SOLIDANGLE and MINSAMPLES, any structure in the scattering within the integration solid angle, and the shape of the integration solid angle. The program chooses the number of sampled points by dividing the solid angle of a right circular cone that circumscribes the integration solid angle by the variable SOLIDANGLE, assuring that this value is above MINSAMPLES. It then samples directions on a square grid within this right circular cone. If the point is outside of the integration solid angle, it returns zero intensity for that direction. The results are adjusted by the ratio of the calculated projected solid angle (as determined by the sampled points) and the actual projected solid angle of the desired shape, as determined by a path integral around the perimeter.
If the user creates a variable integrationmode, the integration method can be set. The following table gives the permitted values and their meaning:
| integrationmode | Interpretation | Normalization |
|---|---|---|
| 1 | Use Eq. (11.1), sampling $\xi_1$ and $\eta_1$ on a uniform grid | Normalize to PRD |
| 2 (default) | Use Eq. (11.2), sampling $\xi_2$ and $\eta_2$ on a uniform grid | Normalize to PRD |
| 3 | Use Eq. (11.3), sampling $\xi_3$ and $\eta_3$ on a uniform grid | Normalize to PRD |
| 4 | Use Eq. (11.1), sampling $\xi_1$ and $\eta_1$ with uniform Monte Carlo sampling | Normalize to PRD |
| 5 | Use Eq. (11.2), sampling $\xi_2$ and $\eta_2$ with uniform Monte Carlo sampling | Normalize to PRD |
| 6 | Use Eq. (11.3), sampling $\xi_3$ and $\eta_3$ with uniform Monte Carlo sampling | Normalize to PRD |
| 11 | Use Eq. (11.1), sampling $\xi_1$ and $\eta_1$ on a uniform grid | Not normalized |
| 12 | Use Eq. (11.2), sampling $\xi_2$ and $\eta_2$ on a uniform grid | Not normalized |
| 13 | Use Eq. (11.3), sampling $\xi_3$ and $\eta_3$ on a uniform grid | Not normalized |
| 14 | Use Eq. (11.1), sampling $\xi_1$ and $\eta_1$ with uniform Monte Carlo sampling | Not normalized |
| 15 | Use Eq. (11.2), sampling $\xi_2$ and $\eta_2$ with uniform Monte Carlo sampling | Not normalized |
| 16 | Use Eq. (11.3), sampling $\xi_3$ and $\eta_3$ with uniform Monte Carlo sampling | Not normalized |
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Latest MIST Version: 4.10 (October 2017)