## 3 Results and Discussion
### Canopy Assemblage BRDF Effects  
The discussion so far has concentrated on nadir view bottom of water column reflectance over individual corals under naturalistic light conditions, calculated based on the upward directed radiance and downward irradiance, i.e., \(\pi L_{\text{u}}(\lambda)/E_{\text{d}}(\lambda)\). Whilst the previous results were fairly insensitive to illumination conditions, such as solar zenith angle (Figures 4 and 6), view angle effects were not considered. Direct use of these results as a bottom boundary requires the assumption of Lambertian reflectance. As such, they are appropriate for incorporation into models where this assumption is implicit, for example, in model inversion techniques for image analysis [27]. However, to correctly model propagation of light through the water column in a physically exact model, such as HydroLight, the full BRDF of canopy assemblages is required (activities 5 and 6, Table 1). The BRDF function is calculated using light incident only for a specific direction in an otherwise black radiance field (Figure 1i), so in the model outputs, shading effects are very apparent (Figure 8). Visually, Figure 8 may appear to contradict Figure 7b, since shading as a function of incident zenith  
Figure 7: Median shading factor as a function of surface rugosity: (**a**) Over coral area only (Figure 4), and; (**b**) over coral area and surrounding substrate in a 50% areal mix (Figure 6) with data separated into solar zenith angles of \(10^{\circ}\) and \(50^{\circ}\). Lines are of the form \(y=(1-A)\times\exp[-S\times(x-1)]+A\) and are the least squares best fit, giving (**a**) \(A=0.51\), \(S=0.72\); (**b**) \(A=0.61\), \(S=0.61\). angle in Figure 8 appears stronger than implied by Figure 7b. To reconcile this first note that Figure 7b is expressed in terms of above-surface solar zenith angles of 10\({}^{\circ}\) and 50\({}^{\circ}\), and so most closely corresponds to Figure 8a,b (sub-surface incident angles of 10\({}^{\circ}\) and 30\({}^{\circ}\)), Figure 8c is a more extreme example. Furthermore, Figure 8 arises from unidirectional illumination, whereas Figure 7b arises from a more diffuse illumination, being the sub-surface propagated solar and sky illumination. A final point is that the increased shadow from Figure 8a to 8b, although visually very apparent, is probably not more than 10% of the actual image area, so even under unidirectional illumination it likely would not affect the shading factor by more than 10%, which is comparable to the differences seen for rugosities \(\sim\)3 in Figure 7b. One of the densest canopy assemblages, D74 (74% coral cover), also tended to have vertically higher coral structures and consequentially showed the strongest BRDF effect, with an almost linear effect of view angle in the incident plane (\(\Delta\varphi=0\)), when incident light was at a 50\({}^{\circ}\) zenith angle (Figure 9e). The reflected radiance decreased by a factor of seven when varying the view direction from almost horizontal in the same direction as the incident light, to horizontally toward the incident light (Figure 9e). In all canopy assemblages there is a clear hotspot effect [30], where brightness is at a maximum when the incident and view directions are the same. The BRDFs discussed here were spatially averaged, and the hot spot occurs because the viewing geometry determines the fraction of shadowed area detected. When the viewing and incident light directions are the same, shadows are maximally obscured by the illuminated surfaces. Interestingly, for the canopy structures modelled here, the hotspot effect is stretched out to larger view angles (Figure 9). This is likely a geometrical consequence of an increasingly horizontal view onto vertical structures illuminated from the side (Figure 8). In all canopy assemblages, there were only small BRDF effects at 90\({}^{\circ}\) to the incident plane (\(\Delta\varphi=90^{\circ}\)) (Figure 9b,d,f).