Uncertainty characterisation
===

Here we consider quantification of factors which impact on the derived reflectance values and uncertainties from iSPEX 2. To do this, we first need to consider the reflectance measurement equation and the propagation of uncertainties

Reflectance measurement equation
--

The measurement equation for above-water remote-sensing reflectance, $R_{rs}$, using a handheld and reference plaque (card) is

$R_{rs}(\lambda)=\frac{(L_{t}(\lambda)-\rho L_{s}(\lambda))}{\left(\frac{\pi}
{R_{ref}(\lambda)}\right)L_{c}(\lambda)} 
$

where $L_{t}(\lambda)$, $L_{s}(\lambda)$ and $L_{c}(\lambda)$ are the radiance measurements of total upwelling radiance, sky, and card, $R_{ref}(\lambda)$ is the reflectance of the card, and $\rho$ is the reflectance factor of the air-water interface (Leeuw and Boss, 2018). Due to the restricted wavelength range of iSPEX 2 this equation does not include an offset term and wavelength dependence of $\rho$ is neglected. $L_{t}$, $L_{s}$ and $L_{c}$ are all in relative units.

Reflectance uncertainty equation
--

The propagated reflectance uncertainty, is given by
$\sigma_{R_{rs}}=\sqrt{
\left(\frac{\partial R_{rs}}{\partial L_t}\right)^{2}\sigma^2_{L_{t}} +\left(\frac{\partial R_{rs}}{\partial L_s}\right)^{2}\sigma^2_{L_{s}}
+\left(\frac{\partial R_{rs}}{\partial L_c}\right)^{2}\sigma^2_{L_{c}}
+\left(\frac{\partial R_{rs}}{\partial R_{ref}}\right)^{2}\sigma^2_{R_{ref}}+\left(\frac{\partial R_{rs}}{\partial \rho}\right)^{2}\sigma^2_{\rho}}$

where $\sigma_{L_{t}}$, $\sigma_{L_{s}}$, $\sigma_{L_{c}}$, $\sigma_{R_{ref}}$ and $\sigma_{\rho}$ are the uncertainties of each variable. The partial derivative terms in the uncertainty equation are given by:
$
\left(\frac{\partial R_{rs}}{\partial L_t}\right)=\frac{1}{{(\frac{\pi}
{R_{ref}})L_{c}}},
$

$
\left(\frac{\partial R_{rs}}{\partial L_s}\right)=\frac{-\rho}{{(\frac{\pi}
{R_{ref}})L_{c}}},
$

$
\left(\frac{\partial R_{rs}}{\partial L_c}\right)=\frac{-L_{t}+\rho L_{s}}{{(\frac{\pi}
{R_{ref}})L_{c}^{2}}},
$

$
\left(\frac{\partial R_{rs}}{\partial R_{ref}}\right)=\frac{L_{t}-\rho L_{s}}{\pi L_{c}},
$
and
$
\left(\frac{\partial R_{rs}}{\partial \rho}\right)=\frac{-\rho L_{s}}{(\frac{\pi}{R_{ref}})L_{c}}.
$
(For now), this analysis neglects co-variance between each uncertainty component.

Conversion from digital counts to relative radiance
--
When deriving iSPEX reflectance, $L_{t}(\lambda)$, $L_{s}(\lambda)$ and $L_{c}(\lambda)$ are all measured using the same phone with fixed exposure and ISO-normalization
settings. The uncertainty associated with each radiance measurement can therefore be treated in
the same way, dropping the subscript notation, and using 𝐿 to notate relative radiance.

The relationship between relative radiance, 𝐿, and measured digital counts (per pixel), 𝑀, is
of the form:

$
L=Kg(M-B),
$

where B is the bias (dark pixel offset), g is the flat field correction, 
and K is a proportionality factor that incorporates multiple quantities that are either fixed in the app between measurements (ISO normalization
exposure time, focal length), fixed sensor properties (spectral response function, pixel area), or
physical constants (planks constant, speed of light) (Burggraaff et al. 2019). Sensor non-linearity and subtraction of dark
current are also neglected. 

To estimate $\sigma_L$ we therefore need to consider characterisation of g, M and B.

Spectral dependence of grey card reference reflectance
--


References
--

Leeuw T, Boss E. The HydroColor App: Above Water Measurements of Remote Sensing Reflectance and Turbidity Using a Smartphone Camera. Sensors. 2018; 18(1):256. https://doi.org/10.3390/s18010256.

Olivier Burggraaff, Norbert Schmidt, Jaime Zamorano, Klaas Pauly, Sergio Pascual, Carlos Tapia, Evangelos Spyrakos, and Frans Snik, "Standardized spectral and radiometric calibration of consumer cameras," Opt. Express 27, 19075-19101 (2019)