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Double sampling for stratification (2SS) is a sampling design that is widely used for forest inventories. We present the mathematical derivation of two appropriate variance estimators for mean growth from repeated 2SS with updated stratification on each measurement occasion. Both estimators account for substratification based on the transition of sampling units among the strata due to the updated allocation. For the first estimator, sizes of the substrata were estimated from the second-phase sample (sample plots), whereas the respective sizes in the second variance estimator relied on the larger first-phase sample. The estimators were empirically compared with a modified version of Cochran’s well-known 2SS variance estimator that ignores substratification. This was done by performing bootstrap resampling on data from two German forest districts. The major findings were as follows: (i) accounting for substratification, as implemented in both new estimators, has substantial impact in terms of significantly smaller variance estimates and bias compared with the estimator without substratification, and (ii) the second estimator with substrata sizes being estimated from the first-phase sample shows a smaller bias than the first estimator.

Volume growth is a key indicator in forest management and planning and, accordingly, an integral part of the estimation procedure of forest resources from sample based inventories. Growth estimation from successive double sampling for stratification (2SS) is somewhat challenging and has not been sufficiently addressed in the pertinent literature. Applying 2SS on successive occasions, with updated stratification on each occasion, may lead to fluctuation of sampling units among the strata and to a certain number of sample plots that have to be discarded or that have to be newly established on the second occasion, to obtain the required per-strata sampling proportions, which are stipulated in advance. After presenting a notation to implement growth estimation into 2SS standard formulas, the question of strata shifts and the occurrence of discarded and of new sample plots in the context of growth estimation is addressed. Although growth, unlike net change, can only be estimated from direct observations on remeasured sample plots, it was shown that ignoring discarded or new plots might lead to severely biased estimators. Modified estimators for mean growth and variances are provided and their application is illustrated using data from a repeated survey in a central German forest district.

The effort required for an extensive litterfall measurement campaign can be prohibitive. We propose a double sampling approach, in which a large set of traps is used in each stand to estimate total litterfall, and only a subset of these traps is sorted to the relevant components. We examine its feasibility using data from a regional litterfall study of eastern white pine ( Pinus strobus L.), in which the variables of interest were biomass of foliar litterfall from pine and nitrogen content of foliar litterfall from all vegetation. Double sampling was more efficient than simple random sampling but only if every trap received a rapid presorting to remove twigs and cones. The optimal strategy when pine foliar litterfall biomass was the target variable was to conduct full sorting on 33% of the traps. When foliar litterfall N was the target, sorting only 20% of the traps was optimal. Holding time costs constant, the variance of estimated pine foliar litterfall biomass could be reduced by 18%, whereas that for foliar litterfall N could be reduced by 49%. Alternately, when variance was held constant, the time cost could be reduced by 17% for pine foliar litterfall biomass or 44% for foliar litterfall N.

An approximation to the variances of the regression and mean of ratios estimators for double sampling is presented that can be used to determine how many basal area points are needed to augment an existing inventory to achieve a desired precision. The approximation may also be used to determine the effective regular sample size to achieve equal precision to a double sample.

Formulas are derived for determining the total number of sample points and the number of volume points for a point, double sample with a ratio of means estimator to replace a point sample and achieve the same variance. A minimum ratio of the cost of measuring volume to the cost of measuring basal area at a point is determined for which the point, double sample will be less costly, in terms of time required to measure points, than the point sample.

Une methode est proposee pour etablir les tables de stock et de peuplement par la mise en oeuvre d'un echantillonnage a deux phases impliquant la relation entre le grand et le petit echantillons. L'estimateur utilise est le rapport de moyennes. En utilisant les proprietes de l'estimateur, l'ajustement s'assure que les totaux des tables de stock et de peuplement egalent ceux du grand echantillon et preserve le rapport volume/surface terriere par classe de diametre avant et apres l'ajustement. L'estimation du biais et de la variance de l'ajustement est presentee