HR: 1330h
AN: B22A-0804    [PDF]
TI: Measurement uncertainty in spectral indices from field spectroradiometers
AU: * Dungan, J L
EM: Jennifer.L.Dungan@nasa.gov
AF: NASA Ames Research Center, MS 242, Moffett Field, CA 94035-1000 United States
AU: Steffey, D
EM: steffey@math.sdsu.edu
AF: San Diego State University, Department of Mathematics and Statistics, San Diego, CA 92182-7720 United States
AB: Numerous vegetation indices have been used in remote sensing, all functions of reflectance in a small number of wavebands. Reflectance spectra of a vegetation target exhibit variation due to random and systematic effects. Uncertainty due to these effects is rarely propagated and represented in the calculated values of vegetation indices. We have applied a well-known approximation (referred to in the statistical literature as the ``delta method'') to the variance of a function of random variables to propagate uncertainty in several recognized spectral indices. Let $\rho_{1}$ and $\rho_{2}$ denote the measured reflectance in two wavebands and let $\mu_{i}$ and $\sigma_{i}$ denote the mean and standard deviation of reflectance measured in the $i$-th waveband. Also, let $\sigma_{12}$ denote the covariance of the two reflectance variables. Consider the case in which the index of interest is a function $g(\cdot)$ of $\rho_{1}$ and $\rho_{2}$. The variance of this function can be approximated using \begin{equation} Var[g(\rho_1,\rho_2)] \doteq \sigma_{1}^{2} \left(\frac{\partial g}{\partial \rho_1} \right)^{2} + \sigma_{2}^{2} \left(\frac{\partial g}{\partial \rho_2} \right)^{2} + 2 \sigma_{12} \left(\frac{\partial g}{\partial \rho_1} \right) \left(\frac{\partial g}{\partial \rho_2} \right), \end{equation} where the partial derivatives of $g$ are evaluated at $(\mu_1,\mu_2$). The two-variable case treated here includes some commonly used indices including the simple ratio (SR) and the normalized difference vegetation index (NDVI). With the delta method, we obtain the respective approximations \begin{equation} Var(SR) \doteq \frac{1}{\mu_1^{2}} \left[ \sigma_1^{2} \left( \frac{\mu_2}{\mu_1} \right)^{2} + \sigma_2^{2} - 2 \sigma_{12} \left( \frac{\mu_2}{\mu_1} \right) \right] \end{equation} and \begin{equation} Var(NDVI) \doteq \frac{4 (\sigma_1^{2} \mu_2^{2} + \sigma_2^{2} \mu_1^{2} - 2 \sigma_{12} \mu_1 \mu_2)}{(\mu_2 + \mu_1)^{4}} \end{equation} The result can be confirmed by bootstrapping. Estimates of means and variances can be made from experimental measurements. Variance from instrument noise is likely to be smaller than variance due to other exogeneous factors such as changing illumination conditions. These sources of uncertainty need to be understood in order to detect signals from ecophysiological processes in plant spectra.
DE: 0400 Biogeosciences
DE: 1640 Remote sensing
DE: 1694 Instruments and techniques
SC: Biogeosciences [B]
MN: 2003 Fall Meeting