## 3 Results  
### Bottom Depth  
The maximum depth at which benthic cover could be detected, i.e., where \(\varepsilon>1\) when comparing the reference condition with the signal representing optically deep water, was affected by a combination of the concentration of water constituents (as the determinants of \(K\)), the benthic composition (that controls \(\rho_{b}\)), and the noise envelope surrounding the reference condition (defined by \(SNR\)). For example, considering \(SNR=500\), the extinction depths for benthic cover representing the three endmember bottom types were 18.4, 11.4, and 21.6 m for \(B^{\prime}_{c}\), \(B^{\prime}_{a}\), and \(B^{\prime}_{s}\), respectively (Figure 3a). The difference results from the relative contrast between the optically shallow and deep-water reflectance. The extinction depth for benthic algae was nearly half that of uncolonized sand because the spectrally averaged \(\rho_{b}\) for \(B^{\prime}_{a}\) was more similar to \(r_{rs,\infty}\) and, therefore, became indistinguishable at a shallower depth compared with \(B^{\prime}_{s}\), which was the least similar to optically deep water. For a given bottom type (results for 100% coral cover are presented as an example), the extinction depth decreased with \(SNR\) (Figure 3b), because the envelope of uncertainty increased with sensor noise (inversely with \(SNR\)). For example, given a bottom composed of 100% coral, the extinction depth was 20.9 m for \(SNR=1000\) and was more than 1.5 times the extinction depth (12.6 m) under the same environmental conditions when \(SNR=100\).  
Uncertainty in the detection of bottom depth increased with increasing reference depth, expressed as the dimensionless optical depth \(D_{o}\) (\(=D^{\prime}\ \overline{K}_{d}\)) and sensor noise, expressed as the difference between the upper and lower bounds of spectral similarity and the reference condition, \(\Delta D\) (Figure 4). Positive values indicate an overestimation of \(D^{\prime}\) and negative values indicate underestimation. The rate of increase in uncertainty with respect to the reference bottom depth was greatest for \(SNR=100\) and progressively diminished for \(SNR=500\) and 1000. Retrieval uncertainty was greatest when no assumptions were made about the environment, i.e., scenario \(S1\), where water constituent concentration was allowed to vary across the prescribed ranges. However, when water constituent concentration was constant and set equal to the reference condition (\(S2\)), uncertainty in bottom depth retrieval decreased for each of the \(SNR\) values (bottom panel in Figure 4), although uncertainty associated with \(SNR=100\) remained noticeably greater than for \(SNR=500\) and 1000.  
Figure 3: Average spectral difference parameter, \(\varepsilon\), as a function of depth, where \(L^{\prime}_{sat}\) representing shallow reef conditions is compared with optically deep water. The water column is relatively clear (\(C^{\prime}_{dil}=0.1\) mg m\({}^{-3}\), \(a^{\prime}_{g,450}\)= 0.2 m\({}^{-1}\), and \(C^{\prime}_{cal}=0.3\) g m\({}^{-3}\)) and the threshold of spectral separation is \(\varepsilon=1.0\) (gray line). Impacts of cover type (**A**) are shown for \(B^{\prime}_{c}\) (solid), \(B^{\prime}_{a}\) (dot-dash), and \(B^{\prime}_{s}\) (dash) and the response to \(SNR\) (B) is shown for \(B^{\prime}_{c}\), where \(SNR=100\) (dot-dash), 500 (solid), and 1000 (dash).