## 3 Results and Discussion
### Linear Mixing of Reflectance over Coral Shapes and Surrounding Substrate  
As with the reflectance over the coral shape only, reflectance over a 50% areal mix of coral and surrounding substrate was consistently darker than a 50% linear mix of the surface reflectances, but could be well represented in wavelengths less than 690 nm by a uniformly scaled version of that linear mix (Figure 6). The shading factor when substrate was included was a consistently higher value (less shading) than the values over coral shape only, where the median for each coral ranged from 0.59 (coral 16) to 0.88 (coral 40) (Table 3). Individual values were from 4% to 20% higher than the corresponding values over the coral area only. This occurred because the flat substrate does not introduce any shading itself but is shaded by the coral structure (Figure 3). Including surrounding substrate introduces relatively less shading than would occur within the area of the structure. Therefore, both in the experimental set-up and the physical world, the shading factor is likely dependent on the ratio of coral cover to substrate. The previous suggestion of including a variable black shade endmember in a linear mixing model for bottom reflectance would seem to be further supported by these findings. The shading factor is dependent on the areal cover of flat substrate included, and the difference was variable from 4% to 20%. A-priori inclusion of the shading factor, even by careful collection of in-situ spectra to include shadows, cannot account for the variability due to structural context. However, again the figures presented here assume that the basic input reflectances are surface reflectances, where for flat substrates such as sand or dead coral rock this is likely to be the case, whilst for corals some a-priori inclusion of shading is likely. The possible range of shading factors would have to be modified accordingly. Figure 7 shows that median shading factors were a clear function of surface rugosity (coral surface area divided by projected top-down area) for the reflectance over the coral area only (Figure 7a), and for coral shape and substrate mixes (Figure 7b). Intuitively, the shading factor should be 1 when rugosity is 1 (flat surface), and very high rugosities should tend to show some positive shading factor greater than zero. To express these constraints, a simple two-factor negative exponential model was fit to the plots in Figure 7, of which \(y=(1-A)\times(\exp[-S\times(x-1)]+A\), fit for \(S\) (slope) and, \(A\), the asymptotic shading factor. This function gives a good fit in both cases but is just indicative of the general shape of the relationship, since the validity of the concept of an asymptotic shading factor is debatable. The slightly reduced shading effect when flat substrate is included, can be seen in the smaller magnitude slope (\(S\)) of the best-fit line in Figure 7b. Note the surface rugosity value refers only to the coral shape. If rugosity in Figure 7b were calculated incorporating the flat substrate, the rugosity values would be lower and the relationship would be closer to Figure 7a. Solar zenith angle of 10\({}^{\circ}\) vs. 50\({}^{\circ}\) only had a small effect on the shading factor (Figure 7b), but as expected the effect increased with higher rugosity. Several factors were relevant to the observed small effect of the solar zenith angle, i.e.,  
Figure 5: (**a**) High-resolution model versus; (**b**) low-resolution model of coral 35, scale bar is 5 cm. due to refraction the corresponding sub-surface zenith angles are less, at approximately 7\({}^{\circ}\)-35\({}^{\circ}\), and side-ward illumination reduces the irradiance on some surface facets, but at the same time increases it on others. Considering Figure 2, produced with \(\theta_{\mathrm{s}}=50^{\circ}\), for most corals, only a small part of the area is actually visibly in shadow.