## 1 Introduction
The total amount of noise embedded within a remotely sensed signal is the combination of noise from both environmental and sensor sources. Light energy received by a remote sensor is expressed as photon flux, \(\phi_{q}\), and defined as the average number of photons received by a sensor per unit time per unit area,  
\[\phi_{q}=\frac{\lambda}{hc}X_{sys}L_{sat}, \tag{1}\]  
where \(h\) is Planck's constant, \(c\) is the speed of light, \(X_{sys}\) accounts for system design attributes, including field of view, aperture and integration time, and \(L_{sat}\) is radiance [24]. Variations in \(\phi_{q}\) attributed tothe sensor include readout noise associated with errors in reading detector signals, digitization noise due to rounding errors in the conversion of analog signals to digitized values, dark noise resulting from electric current in the system even when no photons are incident on the detector, and random variations in the number of photons received, referred to as shot noise. Recent advances in sensor design have greatly reduced readout, digitization, and dark noise across the designed signal range so that shot noise forms the primary source of measurement uncertainty. Thus, our investigation assumes that all non-environmental signal uncertainty is in the form of shot noise. Shot noise forms a Poisson distribution in \(\phi_{q}\) with a standard deviation \(\sigma_{q}\) equal to the square root of the signal. It follows, therefore, that  
\[SNR=\phi_{q}/\sigma_{q}=\frac{\lambda}{hc}X_{sys}L_{sat}/\sqrt{\frac{\lambda }{hc}X_{sys}L_{sat}}. \tag{2}\]  
Since \(SNR\) is a function of the radiance received, sensor uncertainty is expressed as a reference signal to noise ratio, \(SNR^{*}\), reported for a typical at-sensor radiance, \(L_{typ}\), and representing a specified set of environmental conditions, including solar zenith angle and mean Earth-Sun distance. For aquatic applications, the reference condition typically includes a clear maritime atmosphere overlying a surface reflectance of 5%. With knowledge of the reference noise level \(SNR\) for any scene radiance, \(L_{sat}\), can be computed as  
\[SNR=SNR^{*}\sqrt{\frac{L_{sat}}{L_{typ}}}. \tag{3}\]  
It can be shown that while \(SNR\) is formally defined in terms of photon flux, it applies equally to radiance [25];  
\[SNR=\frac{\phi_{q}}{\sigma_{q}}=\frac{L_{sat}}{\sigma_{L}}, \tag{4}\]  
where \(\sigma_{L}\) is the standard deviation of radiance in units of radiance, e.g., W m\({}^{-2}\) nm\({}^{-1}\) sr\({}^{-1}\). Thus, the uncertainty in radiance resulting from sensor shot noise alone may be computed as \(\sigma_{L}=L_{sat}/SNR\). A recent model analysis of remotely sensed coral reef benthic cover concluded that under most conditions environmental variability dominates the total noise and that sensor design efforts to maximize \(SNR\) in order to retrieve more subtle benthic signatures are overemphasized [26]. This is an important point because relaxing \(SNR\) requirements within the engineering trade space can result in potential increases in spectral sampling and spatial resolution. The purpose of this work is to reexamine the problem posed by this earlier work and to extend the analysis to include uncertainty in the retrieval of bottom depth and water column properties that impact light attenuation. The analysis omits all noise imposed by the water surface and the atmosphere and focuses solely on the impacts of \(SNR\). Uncertainty is defined as the envelope of environmental parameter values imposed by sensor noise that bracket the true (reference) values. We consider continuous sampling at a resolution of 1 nm within the visible and near-infrared portions of the electromagnetic spectrum (400 nm \(\leq\lambda\leq\) 750 nm). This spectral sampling is not meant to represent a specific future satellite sensor, but rather an experimental design to avoid uncertainty imposed by under sampling the spectral domain.