Assessing the performance of NDVI as a proxy for plant biomass using non-linear models: a case study on the Kerguelen archipelago

Abstract

Numerous ecological studies, including of the polar environment, are now using the remotely sensed normalized difference vegetation index (NDVI, e.g. PAL-NDVI or MODIS-NDVI) as a proxy of vegetation productivity rather than performing direct vegetation assessments. Even though previous data strongly suggested a saturation of NDVI at high biomass values, few studies have explicitly included this characteristic in the modelling process. Here, we developed a generalized non-linear model to explicitly model the relationship between temporal variations of NDVI (Pathfinder AVHRR Land 8 km dataset) and empirical field data. We illustrated our approach on the Kerguelen archipelago by using a green biomass index (point-intercept protocol) sampled at a small scale relative to PAL-NDVI data, and in presence of spatial (water) and temporal (cloud contamination, snow) heterogeneity, i.e. field conditions encountered in many ecological studies. We showed a strong relationship (r _pred.obs = 0.89 [0.77; 0.95]_95%) between this index and the seasonal component of NDVI time series (NDVI_comp). Despite the absence of lignified species in the stand, the NDVI_comp reached an asymptote (0.54 ± 0.05) for high values of green biomass index stressing the need to account for non-linearity when relating NDVI and plant measurements. We provided here a new methodological framework to standardize comparisons between studies assessing performance of NDVI as a proxy of vegetation data.

Appendix

This appendix describes the 3 steps of the statistical procedure used to compute prediction intervals of the model presented in this study (see Eqs. 2, 3),

Step 1: Computation of the mean model predictions

We obtained maximum likelihood estimation of {β₀, β₁, β₁, θ} noted $\{\hat{\beta_{0}},\hat{\beta_{1}},\hat{\beta_{1}}, \hat{\theta}\}$ and computed the means $(\hat{\mu}_{i})$ predicted by the fixed part of the model:

$$ y_i \sim {\text {NegBin}}(\hat{\mu}_{i},\hat{\theta}) $$

(6)

where

$$ \hat{\mu}_{i} = \hat{\beta}_{0} + {\frac{\hat{\beta}_{1}{\rm NDVI}_{i}} {\hat{\beta}_{2}-{\rm NDVI}_{i}}} $$

(7)

Step 2: Parametric bootstrapping of the fixed part of the model

We computed 1,000 vectors of bootstrap observations of length n = 27,

$$ Q^{{\ast}j} [\mu^{{\ast}j}_{1},\mu^{{\ast}j}_{i},\cdots,\mu^{{\ast}j}_{n},]_{(1 \leq j \leq 1,000)} $$

(8)

where each μ ^*j _i was sampled in

$$ {\text {NegBin}}(\hat{\mu}_{i},\hat{\theta}) $$

(9)

We fitted the non-linear model (see Eq. 2) on each Q ^*j to obtain 1,000 bootstrap vectors of parameter estimations,

$$ \{\hat{\beta}^{{\ast}j}_{0},\hat{\beta}^{{\ast}j}_{1},\hat{\beta}^{{\ast}j}_{2},\hat{\theta}^{{\ast}j}\}_{(1 \leq j \leq 1,000)} $$

(10)

Using Eq. 7, we computed the 1,000 bootstrap vectors of predictions,

$$ \hat{P}^{{\ast}j} [\hat{\mu}^{{\ast}j}_{1},\hat{\mu}^{{\ast}j}_{i},\cdots,\hat{\mu}^{{\ast}j}_{n}]_{(1 \leq j \leq 1,000)} $$

(11)

Then, to compute the 95% bootstrap confidence intervals of the ith predicted mean $(\hat{\mu}_{i}),$ we have taken the 0.025 and 0.975 quantiles of the corresponding $\hat{\mu}^{{\ast}}_{i}$ bootstrap distribution.

Step 3: Monte Carlo generation of the posterior distribution of individual predictions

In order to obtain the 95% prediction intervals, we included the random part of the model in the bootstrap procedure. Thus, we had a noise (Negative Binomial) to the 1000 bootstrap vectors of predicted means $(\hat{P}^{{\ast}j},$ see Eq. 11) to take into account the individual variability,

$$ \hat{P}^{{\ast}{\ast}j} [\hat{y}^{{\ast}j}_{1},\hat{y}^{{\ast}j}_{i},\cdots,\hat{y}^{{\ast}j}_{n}]_{(1 \leq j \leq 1,000)} $$

(12)

where each $\hat{y}^{{\ast}j}_{i}$ (see Eq. 2) was sampled in

$$ {\text {NegBin}}(\hat{\mu}^{{\ast}j}_{i},\hat{\theta}^{{\ast}j}) $$

(13)

Then, to compute the 95% bootstrap prediction interval of the ith predicted observation $(\hat{y}_{i}),$ we took the 0.025 and 0.975 quantiles of the corresponding $\hat{y}^{{\ast}}_{i}$ bootstrap distribution.