In this chapter we illustrate the optimization technique, using data based on the ﬁrst Swiss National Forest Inventory (SNFI1;1983-1985) and on the second Swiss National Forest Inventory (SNFI2;1993-1995). SNFI1 involved a one-phase two-stage simple sampling scheme comprising plots with two concentric circles: 200m2 and 500m2. DBH thresholds were set at 12cm and 36cm. In all, n2 = 10′974 plots were available from a 1km × 1km grid with m1 ≈ 11.7. Its second-stage procedure samples with essentially equal probability pi ≈ 0.33 for DBHi < 60cm and pi = 1 for DBHi ≥ 60cm. This resulted in m2 ≈ 4. SNFI2 consisted of a two-phase two-stage sampling scheme with the same concentric plots, but with only half the original plots from the sub-grid√ 2km × √2km. Its ﬁrst phase relied on aerial photographs and was based
on a 0.5km × 0.5km grid, resulting in n1 = 51′296, n2 = 6′412, m1 ≈ 12. The second-stage procedure was changed to implement PPE with m2 ≈ 2. Switzerland (CH) is divided into D = 5 domains: Jura (JU), Swiss Plateau (SP), Pre-Alps (PA), Alps (AL) and Southern Alps (SA). Here, those regions will be identiﬁed by the index g. The working strata were based on semiautomatic interpretation of aerial photographs to determine the average tree height (with 25 points per plot). This led to four working strata within each domain: 0m− 10m, 10m− 20m, 20m− 30m and > 30m. The corresponding R2’s fell into the range 0.12−0.24, which is not as good as the usual stand-map stratiﬁcation with R2 ≈ 0.4. However, it required only the aerial photographs available from the Swiss Topographic Survey. The post-stratiﬁcation procedure analyzed the four working strata individually within each domain, which explains why the R2 = 0.27 for (CH) is higher than the R2’s in each domain. Cost parameters, as well as the pg, Y g were based on the ﬁrst inventory, while the estimation of lack of ﬁt and pure error had to be based on the second. Some of the results presented below relied on a more general optimization procedure with terrestrial sampling density varying in the D = 5 regions in such a way as to minimize the resulting anticipated variance for Switzerland as a whole. There, the general ideas remained the same but, of course, the formulae were more complicated (Mandallaz, 2001, 2002). However, one can still intuitively understand the results without having to study the technical details involved in those calculations. The results from D = 1 (Switzerland as a single region) were obtained via the technique given in Section 9.6.