In this chapter we shall design optimal sampling schemes. We saw in Section 3.2 that in sampling theory the concept of optimality is not as straightforward as it seems. Meaningful optimality criteria must rely somehow on a super-population model. The actual population to be surveyed is viewed as one realization of many similar ones. Design-based variance, i.e. under hypothetical repetition of the samples, is fixed for the given realization at hand. The average of that variance under the super-population model is called the anticipated variance. Optimal sampling schemes are those which minimize the anticipated variance for given costs or, conversely, minimize the costs for a given anticipated variance. This concept has been used successfully for many standard problems when sampling finite populations. The Yi are assumed to be random variables, usually described by a linear multiple regression model with uncorrelated errors (Sa¨rndal et al. 2003). However, that approach is not well suited for forest inventory. Instead, we shall assume that the Yi (e.g. tree volume) are fixed but that the location of those trees has been determined stochastically. We have seen that the troublesome term for the variance of the Horvitz-Thompson estimator is due to the pair-wise inclusion probabilities πij of any two trees in a forest. The solution is very simple: we assume that those locations are generated by a so-called local Poisson model: The forest is partitioned into Poisson strata, in which the trees are independently and uniformly distributed. Then, by calculating the anticipated variance under this model we can eliminate the πij . This is, of course, a very crude approximation because we do not account for repulsion or aggregation mechanisms. To do so would require the use of marked stochastic point processes for which the calculation of the anticipated variance is an extremely difficult problem and, so far, unsolved. On the other hand, the local Poisson model leads to simple and very intuitive results, that have been validated already by simulations. Though the basic ideas are very simple there are some technical difficulties with boundary effects. Here, we shall provide the main results and sketch the proofs for simple random sampling. Likewise, only some of the results will be stated for cluster-sampling, which are technically speaking more difficult to

been outlined previously in dallaz (1997, 2001, 2002), Mandallaz and Ye (1999) and Mandallaz and Lanz (2001).