In forest inventories, regression models are often applied to predict quantities such as biomass at the level of sampling units. In this paper, we propose a model-based inference framework for combining sampling and model errors in the variance estimation. It was applied to airborne laser (LiDAR) data sets from Hedmark County, Norway, where the model error proportion of the total variance was found to be large for both scanning (airborne laser scanning) and profiling LiDAR when biomass was estimated. With profiling LiDAR, the model error variance component for the entire county was as large as 71% whereas for airborne laser scanning, it was 43% of the total variance. Partly, this reflects the better accuracy of the pixel-based regression models estimated from scanner data as compared with the models estimated from profiler data. The framework proposed in our study can be applied in all types of sample surveys where model-based predictions are made at the level of individual sampling units. Especially, it should be useful in cases where model-assisted inference cannot be applied due to the lack of a probability sample from the target population or due to problems of correctly matching observations of auxiliary and target variables.

Airborne scanning LiDAR (Light Detection and Ranging) has emerged as a promising tool to provide auxiliary data for sample surveys aiming at estimation of above-ground tree biomass (AGB), with potential applications in REDD forest monitoring. For larger geographical regions such as counties, states or nations, it is not feasible to collect airborne LiDAR data continuously ("wall-to-wall") over the entire area of interest. Two-stage cluster survey designs have therefore been demonstrated by which LiDAR data are collected along selected individual flight-lines treated as clusters and with ground plots sampled along these LiDAR swaths. Recently, analytical AGB estimators and associated variance estimators that quantify the sampling variability have been proposed. Empirical studies employing these estimators have shown a seemingly equal or even larger uncertainty of the AGB estimates obtained with extensive use of LiDAR data to support the estimation as compared to pure field-based estimates employing estimators appropriate under simple random sampling (SRS). However, comparison of uncertainty estimates under SRS and sophisticated two-stage designs is complicated by large differences in the designs and assumptions. In this study, probability-based principles to estimation and inference were followed. We assumed designs of a field sample and a LiDAR-assisted survey of Hedmark County (HC) (27,390 km 2 ), Norway, considered to be more comparable than those assumed in previous studies. The field sample consisted of 659 systematically distributed National Forest Inventory (NFI) plots and the airborne scanning LiDAR data were collected along 53 parallel flight-lines flown over the NFI plots. We compared AGB estimates based on the field survey only assuming SRS against corresponding estimates assuming two-phase (double) sampling with LiDAR and employing model-assisted estimators. We also compared AGB estimates based on the field survey only assuming two-stage sampling (the NFI plots being grouped in clusters) against corresponding estimates assuming two-stage sampling with the LiDAR and employing model-assisted estimators. For each of the two comparisons, the standard errors of the AGB estimates were consistently lower for the LiDAR-assisted designs. The overall reduction of the standard errors in the LiDAR-assisted estimation was around 40-60% compared to the pure field survey. We conclude that the previously proposed two-stage model-assisted estimators are inappropriate for surveys with unequal lengths of the LiDAR flight-lines and new estimators are needed. Some options for design of LiDAR-assisted sample surveys under REDD are also discussed, which capitalize on the flexibility offered when the field survey is designed as an integrated part of the overall survey design as opposed to previous LiDAR-assisted sample surveys in the boreal and temperate zones which have been restricted by the current design of an existing NFI.