Small area estimation of forest attributes in the Norwegian National Forest Inventory

Abstract

The Norwegian National Forest Inventory (NNFI) provides estimates of forest parameters on national and regional scales by means of a systematic network of permanent sample plots. One of the biggest challenges for the NNFI is the interest in forest attribute information for small sub-populations such as municipalities or protected areas. Frequently, too few sampled observations are available for such small areas to allow estimates with acceptable precision. However, if an auxiliary variable exists that is correlated with the variable of interest, small area estimation (SAE) techniques may provide means to improve the precision of estimates. The study aimed at estimating the mean above-ground forest biomass for small areas with high precision and accuracy, using SAE techniques. For this purpose, the simple random sampling (SRS) estimator, the generalized regression (GREG) estimator, and the unit-level empirical best linear unbiased prediction (EBLUP) estimator were compared. Mean canopy height obtained from a photogrammetric canopy height model (CHM) was the auxiliary variable available for every population element. The small areas were 14 municipalities within a 2,184 km² study area for which an estimate of the mean forest biomass was sought. The municipalities were between 31 and 527 km² and contained 1–35 NNFI sample plots located within forest. The mean canopy height obtained from the CHM was found to have a strong linear correlation with forest biomass. Both the SRS estimator and the GREG estimator result in unstable estimates if they are based on too few observations. Although this is not the case for the EBLUP estimator, the estimators were only compared for municipalities with more than five sample plots. The SRS resulted in the highest standard errors in all municipalities. Whereas the GREG and EBLUP standard errors were similar for small areas with many sample plots, the EBLUP standard error was usually smaller than the GREG standard error. The difference between the EBLUP and GREG standard error increased with a decreasing number of sample plots within the small area. The EBLUP estimates of mean forest biomass within the municipalities ranged between 95.01 and 153.76 Mg ha⁻¹, with standard errors between 8.20 and 12.84 Mg ha⁻¹.

Appendix

Asymptotic variances and covariances

The asymptotic variances and covariances $\bar{V}$ of $\hat{\sigma}^2_\varepsilon$ and $\hat{\sigma}^2_\upsilon$ are elements of the inverse of the information matrix M (Rao 2003, p. 140):

$$ \bar{V}_{\upsilon\upsilon} = {\bf M}^{-1}_{11}, \bar{V}_{\varepsilon\varepsilon} = {\bf M}^{-1}_{22}, \bar{V}_{\varepsilon\upsilon} = \bar{V}_{\upsilon\varepsilon} = {\bf M}^{-1}_{12} = {\bf M}^{-1}_{21}. $$

The information matrix M is given by

$$ {\bf M}_{11} = \frac{1}{2} \sum_i n_i^2 \alpha_i^{-2}, $$

$$ {\bf M}_{22} = \frac{1}{2} \sum_i([n_i-1] \hat{\sigma}^{-4}_\varepsilon + \alpha_i^{-2}) $$

and

$$ {\bf M}_{12} = {\bf M}_{21} = \frac{1}{2} \sum_i n_i \alpha_i^{-2} $$

with

$$ \alpha_i = \hat{\sigma}^2_\varepsilon + n_i \hat{\sigma}^2_\upsilon. $$