We consider the small-area estimation problem in forest inventories with two-phase sampling schemes. We propose an improvement of the synthetic estimator, when the true mean of the auxiliary variables over the small-area is unknown and must be estimated, likewise for the residual corrected small-area estimator. We derive the asymptotic designbased variances of these new estimators, the pseudo-synthetic and pseudo smallarea estimators, by incorporating also the design-based variance of the regression coefficients. We then propose a very simple mathematical device that transforms pseudo small-area estimators into pseudo-synthetic estimators, which is very convenient to derive asymptotic variances. The results are extended to cluster and two-stage sampling at the plot level. To illustrate the theory we consider the case of post-stratification and a case study. RésuméNous considèrons le problème de l'estimation pour petits domaines dans le contexte d'inventaires forestiers en deux phases. Nous proposons une amélioration simple de l'estimateur synthétique quand la moyenne des variables auxiliaires dans le petit domaine doitêtre estimée en premier lieu , de même pour l'estimateur pour petit domain basé sur les résidus. Nous calculons la variance sous le plan de sondage de ces nouveaux estimateurs en tenant compte de la variance des coefficients de régression. De plus, nous proposons un artifice mathématique qui permet de transformer un estimateur pour petit domaine en un estimateur synthétique, ce qui simplifie le calcul de la variance asymptotique. L'extension aux sondages par satellites et deux degrés au niveau de la placette est aussi traitée. La théorie est illustrée par la post-stratification et par uneétude de cas.

We consider two-phase sampling schemes where one component of the auxiliary information is known in every point ("wall-to-wall") and a second component is available only in the large sample of the first phase, whereas the second phase yields a subsample with the terrestrial inventory. This setup is of growing interest in forest inventory thanks to the recent advances in remote sensing, in particular, the availability of LiDAR data. We propose a new two-phase regression estimator for global and local estimation and derive its asymptotic design-based variance. The new estimator performs better than the classical regression estimator. Furthermore, it can be generalized to cluster sampling and two-stage tree sampling within plots. Simulations and a case study with LiDAR data illustrate the theory.Résumé : Cet article propose un nouvel estimateur pour les inventaires forestiers utilisant des plans de sondage à deux phases pour lesquels l'information auxiliaire consiste en une première composante exhaustive connue en chaque point et une seconde composante connue seulement aux points du grand échantillon de la première phase. La deuxième phase consiste en un sous-échantillon des points de la première phase dans lesquels l'inventaire terrestre est effectué. Ce contexte est appelé à jour un rôle croissant grâce aux développements récents dans l'aquisition de données par télédétection. Nous proposons un nouvel estimateur par regression, aussi bien pour l'estimation globale que locale, et nous donnons sa variance asymptotique sous le plan de sondage. Le nouvel estimateur peut être adapté aux inventaire par satellites et avec tirage double des arbres au niveau de la placette terrestre. Un exemple utilisant des données LiDAR et des simulations illustrent la théorie. Le nouvel estimateur a de meilleures performance que l'estimateur de régression classique.

In 2009, the Swiss National Forest Inventory (NFI) turned from a periodic into an annual measurement design in which only one-ninth of the overall sample of permanent plots is measured every year. The reduction in sample size due to the implementation of the annual design results in an unacceptably large increase in variance when using the standard simple random sampling estimator. Thus, a flexible estimation procedure using two- and three-phase regression estimators is presented with a special focus on utilizing updating techniques to account for disturbances and growth and is applied to the second and third Swiss NFIs. The first phase consists of a dense sample of systematically distributed plots on a 500 m × 500 m grid for which auxiliary variables are obtained through the interpretation of aerial photographs. The second phase is an eightfold looser subgrid with terrestrial plot data collected from the past inventory, and the third and final phase consists of the three most recent annual subgrids with the current state of the target variable (stem volume). The proposed three-phase estimators reduce the increase in variance from 294% to 145% compared with the estimator based on the full periodic sample while remaining unbiased.

We consider three-phase sampling schemes in which one component of the auxiliary information is known in the very large sample of the so-called null phase and the second component is available only in the large sample of the first phase, whereas the second phase provides the terrestrial inventory data. We extend to three-phase sampling the generalized regression estimator that applies when the null phase is exhaustive, for global and local estimation, and derive its asymptotic design-based variance. The new three-phase regression estimator is particularly useful for reducing substantially the computing time required to treat exhaustively very large data sets generated by modern remote sensing technology such as LiDAR.Résumé : Nous proposons un nouvel estimateur pour les inventaires forestiers utilisant des plans de sondage à trois phases pour lesquels l'information auxiliaire consiste en une première composante connue en chaque point de la phase « nulle » et une seconde composante connue seulement en chaque point de la première phase, alors que la deuxième phase consiste en l'inventaire terrestre. Nous proposons une nouvelle version de l'estimateur par regression, aussi bien pour l'estimation globale que locale, et nous donnons la variance asymptotique sous le plan de sondage. Le nouvel estimateur est particulièrement utile pour réduire substantiellement le temps de calcul requis pour un traitement exhaustif de très grandes bases de données obtenues par les moyens modernes de télédétection tels que LiDAR.

This thesis investigates the performances of various estimators in one-phase (purely terrestrial) two-stage forest inventories, where trees in the first-stage are selected by concentric circles (approximate PPS) and a subset thereof are selected by Poisson sampling for further measurements to get an accurate estimation of the timber volume. Poisson sampling is used because it is easy to implement in field work. However, this comes with the drawback of a random second-stage sample size that can drive up the variance. The widely accepted remedy in analogous situations in survey sampling is to add a stabilizing factor to the estimator that compensates for this randomness and presumably lowers the variance. In this paper the effectiveness of three formulations of such stabilizing factors are examined in the context of timber density estimation. These factors are applied to the residual component of a generalized two-stage density estimator and tested using data from the 3rd Swiss National Forest Inventory taken in 2003. These factors introduce a negligible bias. Contrary to empirical findings in general survey sampling and asymptotic results, the adjusted estimators did not perform really better than the original unadjusted two-stage estimator.i