^{1}

^{*}

The aim of this paper is to give an overview of some recent proposals to support national forest inventories. The reviewed literature is strictly of design- based nature,

The analysis of forest ecology and the wise management of forest resources requires accurate monitoring of forestlands at regular time intervals. In turn, the monitoring process involves sample surveys to estimate extents and standing volumes for a wide set of forest types and districts. These surveys are usually referred to as forest inventories (

The early forest inventories were performed toward the end of 15^{th} century in the form of censuses of oaks in the Republic of Venice (^{th} century. During the 20^{th} century, methods for inventorying forests were rapidly improved and the possibility of reducing costs by adopting sampling methods were recognized. The first forest inventory at country level, henceforth referred to as national forest inventory (NFI), was performed in Finland from 1922 to 1924 and from now repeated every ten years. The first Italian NFI was performed from 1982 to 1985, while the second started in 2003 and concluded at the end of 2006 (see http:// www.infc.it).

This paper provides a review of some recent methodological contributions to support NFIs. All the reviewed articles are of design-based nature, in the sense that uncertainty only stems from the sampling scheme adopted to perform the inventory. As

In the next section NFIs are viewed as two-phase sample surveys to estimate at the same occasion the extent of the continuous population of points constituting the forest cover and the total of a forest attribute (

Consider a delineated study area A partitioned into two land cover classes: forest and non forest. Denote by F ⊂ A the forest portion of A and by U the population of forest trees within F. Suppose that F is partitioned into _{1}, …, F_{K} corresponding to _{1}, …, U_{K} of trees within the

Generally speaking, a forest inventory is a sampling strategy to estimate the extent of the

and the total of a forest attribute

for each _{k }(_{k }(_{k }(_{j} is the amount of

An essential requirement of any NFI is that extents and totals are estimated in the same survey, using the same aerial and field investigations.

Despite its simplicity, the completely random placement of _{1}, …, R_{N}, and such that R_{i} ∩ A ≠ Ø for all

If each first-phase point is visited on the ground and the indicator variable of each type/district is recorded at the point, the first-phase Monte-Carlo integration estimator of A_{k} turns out to be (

where _{k} = _{k} / _{k} is the number of sample points lying in A_{k}. Moreover, if for each first-phase point _{k} a plot of fixed size _{ki} is reckoned, the first-phase Monte-Carlo integration estimator of _{k} turns out to be (

where _{ki} = 0 if _{k}.

It is well-known from Monte Carlo integration (^{-1} (^{-1}. Accordingly, for large

It is worth noting that some edge effects may be present owing to forest trees positioned near the edge of the study region, which have inclusion probabilities smaller than the inner trees. A long list of correction methods has been proposed in order to avoid the negative bias induced by edge effects (

Owing to costs and time involved, in real situations the

Regarding the second phase, the collection of the _{F} of the _{F} points lying in the forest area F, and the sub-set P-P_{F} of the remaining _{F} points lying outside. It is worth noting that the partition is performed by satellite imagery of aerial photos, without field work. Obviously, because the plots centered at the points of P-P_{F} lie completely or partially outside forest, no or very few forest trees are likely to be found in these plots. Hence, it is customary to assume _{ki} = 0 for any _{F}, in such a way that the sampling effort can be completely devoted to P_{F}.

The procedure of neglecting non-forest points in the second phase is adopted in most NFIs but it is not sufficiently focused in most familiar textbooks (

If non-forest points are discarded, denote by S ⊂ P_{F} the second-phase sample of size _{F} by means of a fixed-size scheme inducing first- and second-order inclusion probabilities π_{i} and π_{ih} (_{F}). Suppose that π_{ih} > 0 for any _{F}, in such a way that the second-phase variance can be unbiasedly estimated. Suppose also that no classification errors between forest and non-forest points occur, and that Pr(_{ki} = 0 ∀ _{F}) = 1. Using the double-expansion estimation (

and (

respectively, where S_{k} ⊂ S denotes the sub-sample of second-phase points lying in the

Under the above-mentioned assumptions, estimators (5) and (6) turn out to be unbiased with sampling variances which in the case of (5) can be estimated by (

and in the case of (6) by (

Under TSS, (7) and (8) are proven to be conservative estimators of the actual variances. The conservative nature of (7) and (8) stems from TSS, owing to the independence among first-phase points (

Most of NFIs involve two phases of sampling. Relevant examples are the NFIs of Canada (

As already pointed out, NFIs usually require estimates of extents and totals for several forest types, for several regions defined by political subdivisions, for other domains such as ownership categories and silvicultural types and for combinations of them. Practically speaking, thousands of estimates are produced as the output of a NFI. In this framework, statisticians have neither time nor resources to select

On the other hand,

During the FAO Expert Consultation on Global Forest Resources Assessment 2000 (Kotka, Finland 1996), the importance of trees outside forests (TOF) and the need for complete and detailed information about these stands were underlined for the first time. NFIs are currently requested to broaden their scopes to include the assessment of TOF attributes (

Quoting from _{1}, …, _{M} be the sizes of the _{i} be the value of a physical attribute of the

and/or the population mean _{W} = _{W} /

turns out to be an approximately unbiased estimators of _{W}. It is worth noting that (10) avoids the troublesome quantification of the portion of the selected units lying in adjacent quadrats, as would be requested by the genuine Horvitz-Thompson estimator. Moreover (

is proven to be a conservative estimator for the variance of _{(1)W}. For _{j} invariably equal to 1, _{W} coincides with the population abundance _{(1)}. Thus a very natural estimator of _{W} = _{W} /

which is approximately unbiased with variance estimator (

The validity of these estimators is empirically checked by a simulation study (

Regarding isolated trees, their abundance can be estimated from the aerial information acquired during NFIs, even if a further aerial sampling phase is necessary in this case. _{1}, …, P_{L} denote the _{1}, …, _{L} denote the stratum sizes, S_{1}, …, S_{L} denote the samples of points selected from each stratum and _{1} , …, _{L} the sample sizes, the two-phase aerial estimator of

where _{l} = _{l} / _{i} denotes the number of isolated trees aerially counted within the plot _{l} is the average of the _{i}s for _{l}. The estimator (10) is unbiased with variance which can be unbiasedly estimated by (

where _{l}^{2} is the sample variance of the _{i}s for _{l}.

Obviously, if totals or averages of some biophysical attributes such as tree volume and biomass are of interest (rather than size or length), subsequent sampling phases must be performed on the field.

Non-response is often a problem in NFIs. Non-response occurs for two main causes: (a) plots selected in the second phase are located in difficult terrains and cannot be reached by survey crews or, even if reached, the steep slope of the terrain does not allow the recording activities within; (b) plots selected in the second phase are inaccessible because of cultural prohibitions or because landowners deny field crews access. The reasons for non-response are not necessarily the same all over the world. In the USA, (a) is the primary reason for non-response. Procedures for treating non-response due to (a) are proposed by

In this framework, non-response problems manly concern the estimation of totals, which involves the recording of forest attributes within the plots centered at second-phase points. On the other hand, as to the estimation of extents, in most situations the forest type at plot centers can be determined some distance away from the points or by means of high resolution aerial image, while the district in which plot centers fall can be even determined from the map.

A vast literature deals with the problem of non-response adjustment by means of several techniques, but most of them are judged unfeasible by

Alternatively, unit non-response could be handled by a plethora of imputation techniques. Generally speaking, imputation is a procedure in which non response values are replaced by substitutes and estimation is performed on the completed data. As pointed out by

Because both response and prediction modeling are not sufficiently convincing for non-response treatment in forest inventories,

The theoretical and empirical design-based findings achieved by

As to the choice of auxiliary information to be used in NCW, it should be mainly guided by cautionary, practical considerations about the nature of the auxiliary variables and their relationships with the survey variable. In presence of non-response, the crucial point is to eliminate non-response bias, because “if an estimator is greatly biased, it is poor consolation that its variance is low” (

Noteworthy, in recent years laser scanning (and, namely, airborne laser scanning, henceforth referred to as ALS) is increasingly being applied to support forest inventories, providing assessment of the height of upper canopy for the surveyed area. A close relationship has been proven between the timber volume (or standing biomass) in inventory plots and the canopy height from ALS surveys (

Quoting from _{i} = [1,_{i}]^{T} the vector where _{i} denotes the sum of CHM values for all the pixels belonging to the inventory plot centered at the _{i}s are available for each point _{F}, the mean vectors (

are known for each _{F,k} is the set of the _{F,k} first-phase points lying in the _{k} is the mean of the CHM values within the plots centred at these points. Then X_{k} can be used in the calibration estimator of _{k}, say (

where _{F,k}_{F,k}

and R_{k} ⊂ R denotes the sub-sample of second-phase points reached in the field among the second-phase points lying in the _{F,k}, then _{(2)CAL,k} constitutes an approximately unbiased estimator of _{k}. The authors also propose three conservative estimator of the variance of _{(2)CAL,k}.

The simulation of a two-phase forest inventory performed by

NFIs based on two-phase sampling strategies with first-phase points selected by means of stratified or systematic schemes and second-phase points selected ignoring non-forest points ensure a statistically sound estimation of extents and totals for all variables and domains. From a design-based point of view, the resulting estimators are indeed approximately unbiased and conservative estimators of their variances are available. Moreover, information achieved in the first inventory phase can be used to estimate totals and averages of physical attributes of TOF without any additional field work. Inexpensive auxiliary data from remote sensing and aerial sources can be used at estimation level,

Even if canopy height is often available at low or even no cost from ALS surveys carried out on large territories for purposes other than forestry applications,

The author thanks Gherardo Chirici from the University of Molise, Isernia, Italy, and Piermaria Corona from