## 4 Conclusions  
A number of conclusions from this work can be made that relate (1) to using Lambertian bottom boundary conditions and linear mixing models, and (2) to data collection of reflectances and imagery. In summary:  
* If the input reflectances for different benthic types can be considered 'pure' surface reflectances, then structural complexity implies the introduction of a shading scale factor or a black shade endmember in mixing models is required. In our data, that shading scale factor could be anywhere from 0.5 to 0.9, whereas for a genuinely flat substrate it would be 1.0.
* The magnitude of the shading factor is inversely related to surface rugosity; hence the shade factor value could be set based on the expected bottom types, or if derivable in an image analysis, it would give additional information on benthic type.
* Nadir viewing reflectance over the bottom (\(L_{\text{u}}/E_{\text{d}}\)) for individual corals varies as much as \(\sim\)20% (more only in exceptional cases) under the range of light environments found on reefs, under typical remote sensing conditions.
* Regarding view directions beyond a few 10s of degrees from nadir for dense canopies in shallow water shading, hotspot effects become relevant, leading to potentially a 50% difference (factor of 1.5) in above water reflectance. At narrower remote sensing geometries (view zenith angles < 20\({}^{\circ}\)), differences can be 20% (factor of 1.2).
* For all modelled canopy assemblages, there were only small BRDF effects at a relative viewing angle of 90\({}^{\circ}\) to the incident solar plane. The advice for minimizing cross-track BRDF effects in airborne imagery is therefore consistent with that for minimizing surface glint, i.e., fly in a direction close to the solar plane with a solar zenith angle greater than 30\({}^{\circ}\).
* Collection of in-situ spectra of benthic types requires consideration of what level of shading is already included, or whether the data is close to being a surface reflectance measurement. However, including shading in field measurements cannot fully accommodate the influence of canopy interactions, because there are effects at canopy scale involving different benthic types.
* These modelling results provide useful concepts and parameter ranges, which can assist in the interpretation of empirical data and the development of image processing algorithms.  
Conceptualization J.D.H., H.M.D.; Methodology, J.D.H., M.M., A.W., H.M.D.; Writing-Original draft preparation, J.D.H.; Writing-Review & Editing J.D.H., M.M., A.W., H.M.D.  
This work was funded by award #NNX15A32G of the NASA Biodiversity and Ecological Forecasting Program.  
The authors extend thanks to [PERSON] and [PERSON] for assistance in data acquisition and field work. Part of [PERSON]'s time was funded by the NASA Ocean Biology and Biogeochemistry program and the COral Reef Airborne Laboratory (CORAL) project. Coral collection was conducted under special activity permit (SAP) 2018-02 issued on 2 February 2017, by the State of Hawaii Department of Land and Natural Resources.  
The authors declare no conflict of interest. The funding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript and in the decision to publish the results.