Forest inventory - a challenge for statistics Erkki Tompp o Finnish Forest Research Institute Unioninkatu 40 A FIN-00170 Helsinki, Finland erkki.tomppo@metla. Intro duction Statistically designed forest inventories were intro duced simultaneously in three Nordic countries, Norway, Finland and Sweden, in the b eginning of the 1920's घe.g. Ilvessalo 1927ङ. Estimating the forest area and the volume of growing sto ck as well as analysing of increment and drain of growing sto ck have been original ob jectives of inventories. The scop e of the rst inventories was, however, already much wider, including, e.g., information on site typ es, forest silvicultural state, structure of the growing sto ck, and applied and required silvicultural and cutting regimes. Later, new parameters related to forest health and forest bio diversity have app eared, e.g. sp ecies abundances and distributions. The information needs are increasing, esp ecially at the moment when the awareness of the forest health status and loss of biological diversity has arisen, the role of forests in pre- venting global warming has b een recognised and, at the same time, the pressure to increase timb er consumption is increasing. For instance, the pap er consumption has increased since the b eginning of 1970s from ab out 130 million tons to 276 million tons in 1995. It is exp ected to increase to 420-440 million tons by 2010. On the other hand, one half of the harvest timber is still used for co oking and heating causing wide area deforestation in the dry tropics. Forest inventories have traditionally provided information related the biological diversity of forests, such as the structure of growing sto ck, areas of site fertility classes and sometimes the distribution and abundance of plant sp ecies. An increasing concern ab out the loss of diversity, caused e.g. by deforestation, human induced environmental and climate changes as well as the extinction of sp ecies, has increased interest in the whole forest ecosystem and its biological diversity. The comp osition and structure of landscap e, fragmentation of forests or land typ es, the areas and spatial distributions of imp ortant habitat typ es are examples of characteristics which can be measured in the context of large area inventories, at least when multi-source information is utilised. Hanski घ1999ङ has presented mathematical mo dels that connect the dynamics of sp ecies to the structure of fragmented landscap es. Forests have also been seen as having a role in reducing the e ects of global warming by binding the increasing amount of carb on dioxide in the atmosphere. Global forest area is known reasonably well. However, the annual incrementplays an imp ortantroleincarbon ux and is not known globally. In order to be able to satisfy the increasing and diverse demands for scienti cally sub- stantiated information, ecient metho ds are needed to measure forest resources, their status and the comp onents of the whole forest ecosystem. Examples of spatial variation typ es present in forests Forest variables are commonly divided into groups describing an individual tree, a forest stand घor a sample plotङ and a forest region. Each variable has usually its own covariance structure which dep ends on the geographical scale. A single tree stem volume is assessed as R h an integral of a stem curve, i.e. diameter as a function of height, V = dघhङdh. Trees in a 0 same stand are of similar form while those further apart from each other di er more. Tree stem form dep ends also on the tree sp ecies and site variables. The within stand and between stand variation can be mo delled, e.g, with mixed mo dels, d घhङ = f घx; y ङ+ v घhङ+e घhङ; where ki k ki d घhङ is the diameter of a tree i in a stand k , f घx; y ङ is a function of tree variables x and stand ki variables y , v घhङ is a random stand e ect and e a random tree e ect घLappi 1996ङ. k ik The relative lo cations of trees, spatial pattern of trees, a ects for instance the eciency of sampling design, the growth of trees and can thus b e utilised in planning sampling metho ds, in assessing the naturalness of forests, e.g. Spatial p oint pro cesses, e.g. Gibbs pro cesses, have been used in mo delling spatial patterns of trees घSarkka and Tompp o 1998ङ. Variables, like growth factors, site fertility, nutrient availability,cumulative temp erature sum of growing season, e.g., are examples of variations of di erent scales. These can regarded as a realisations of sto chastic pro cesses on the plane. These, on the other hand, very much determine, the structure of the growing sto ck and its variability. Present silvicultural practice has led to forests which are mosaics of stands of di erent age classes and tree sp ecies comp osi- tions with a sp eci c tree form and spatial patterns of trees. The distribution of stands can be regarded as an output of mosaic mo dels घStoyan et al. 1995ङ. All these variations should be taken into account when planning sampling design of a forest inventory, in parameter estimation and in deriving con dence limits of estimators. On the other hand, practical questions, such as moving between sampling units a ects the costs and should b e considered in minimising standard errors with given costs. Parameters estimation with eld data Forest inventories have motivated much of the pioneering work on the general theory of line surveys, systematic sampling, and spatial statistics. Spatial auto correlation of trees, stand and regional variables often leads to multi-stage sampling. An increasing utilisation of supplementary data, e.g. remote sensing data or other georeferenced data, leads to two-phase घdoubleङ sampling घCo chran 1963ङ. Multi-stage sampling are commonly used in tree measurements. Few parameters, which vary much also between trees and which are usually easy to measure, are measured from each tree to be sampled. A smaller sub-sample is taken of the rst sample for measuring addi- tional variables which usually vary less within stand or within a eld plot. Statistical questions are, how many stages should be used, what are sizes of all samples, what variables should be measured in each stage, and how to estimate the variables which are not measured. Tree level volume and increment estimates are usually derived from the most intensive measure- ments. Statistical mo dels, e.g. non-parametric regression analysis, are applied in estimating the variables for trees of less intensive samples. R R The interest in forest inventory is often in the quantity M = z घtङ dt= y घtङ dt; where A A 2 A is an inventory area, z घtङ;t 2 R the variable of interest, e.g., an indicator of a land use 2 class, volume of timber assortment and y घtङ;t 2 R is an indicator function of the stratum घe.g, forestry landङ. After estimating all variables for each sampling unit, e.g. volumes for each tree, estimation of area and volume parameters of a forest region leads to a ratio estimator P P n n 2 m = z = y = z=ऌ yऌ. A natural reliability measure for the estimator is E घm M ङ . Unbi- i i i i ased estimator for systematic sampling is not known. Conservative estimators can be derived utilising the prop erties of second order stationary pro cesses घMatउern 1960ङ. The parameter estimation of spatial pattern mo dels with Gibbs pro cesses can be based on the prop erties of Palm distributions of the pro cess and a chosen test function घFiksel 1988ङ. Another p ossibility is to utilise approximative maximum likeliho o d approaches, for a review, see Geyer घ1998ङ. Estimation of parameters with multi-source inventory The increasing availability of supplementary data, e.g. remote sensing data, has changed the requirements for statistical metho ds in forest inventories. Supplementary data is usually cheap but much less accurate than eld measurements. A prop er use of the data can, however, make the inventories more ecient. Some practical questions, like availability of data, due to weather conditions, e.g., still prevent the full utilisation of data. The estimation with supplementary data could be done in the framework of two-phase घdoubleङ sampling. A non- parametric k-nn metho d, adopted in the Finnish national forest inventory, an be considered as an extension of double sampling घKilkki and Paivinen 1987, Tompp o 1996ङ. An essential prop erty is that all inventory variables, typically 100 to 400, can b e estimated at the same time. The pro cedure utilises a distance measure de ned in the feature space of the supplemen- tary data, denoted here by d, and de nes new area weights for each eld plot by computation units. The weight of eld plot i to pixel p is de ned as k X 1 1 = ; घ1ङ w = i;p 2 2 d d p ;p p ;p j =1 i घj ङ if pixel p is among the k nearest to p, otherwise w = 0. Here, k is a prede ned xed i i;p numb er. The weights w are summed over pixels p by computation units u घfor example by i;p municipalitiesङ yielding the weight of plot i to computation unit u X c = w : घ2ङ i;u i;p p2u The sum c can be interpreted as that area घin pixelsङ of unit u, which is most similar to i;u sample plot i. The plots outside u may also receive p ositiveweights घsynthetic estimationङ.