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Forest inventory - a challenge for statistics
Erkki Tomppo
Finnish Forest Research Institute
Unioninkatu 40 A
FIN-00170 Helsinki, Finland
erkki.tomppo@metla.
Introduction
Statistically designed forest inventories were introduced simultaneously in three Nordic
countries, Norway, Finland and Sweden, in the beginning of the 1920's (e.g. Ilvessalo 1927).
Estimating the forest area and the volume of growing stock as well as analysing of increment
and drain of growing stock have been original objectives of inventories. The scope of the rst
inventories was, however, already much wider, including, e.g., information on site types, forest
silvicultural state, structure of the growing stock, and applied and required silvicultural and
cutting regimes. Later, new parameters related to forest health and forest biodiversity have
appeared, e.g. species abundances and distributions.
The information needs are increasing, especially 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 been recognised and, at the same time, the pressure to increase
timber consumption is increasing. For instance, the paper consumption has increased since the
beginning of 1970s from about 130 million tons to 276 million tons in 1995. It is expected to
increase to 420-440 million tons by 2010. On the other hand, one half of the harvest timber is
still used for cooking 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 stock, areas of site fertility classes and sometimes the
distribution and abundance of plant species. An increasing concern about the loss of diversity,
caused e.g. by deforestation, human induced environmental and climate changes as well as the
extinction of species, has increased interest in the whole forest ecosystem and its biological
diversity. The composition and structure of landscape, fragmentation of forests or land types,
the areas and spatial distributions of important habitat types are examples of characteristics
which can be measured in the context of large area inventories, at least when multi-source
information is utilised. Hanski (1999) has presented mathematical models that connect the
dynamics of species to the structure of fragmented landscapes.
Forests have also been seen as having a role in reducing the e ects of global warming
by binding the increasing amount of carbon dioxide in the atmosphere. Global forest area is
known reasonably well. However, the annual incrementplays an importantroleincarbon
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 methods are needed to measure forest resources, their status
and the components of the whole forest ecosystem.
Examples of spatial variation types present in forests
Forest variables are commonly divided into groups describing an individual tree, a forest
stand (or a sample plot) and a forest region. Each variable has usually its own covariance
structure which depends on the geographical scale. A single tree stem volume is assessed as
an integral of a stem curve, i.e. diameter as a function of height, V = Rhd(h)dh. Trees in a
0
same stand are of similar form while those further apart from each other di er more. Tree stem
form depends also on the tree species and site variables. The within stand and between stand
variation can be modelled, e.g, with mixed models, d (h) = f(x;y)+v (h)+e (h); where
ki k ki
dki(h) is the diameter of a tree i in a stand k, f(x;y) is a function of tree variables x and stand
variables y, vk(h) is a random stand e ect and eik a random tree e ect (Lappi 1996).
The relative locations of trees, spatial pattern of trees, a ects for instance the eciency
of sampling design, the growth of trees and can thus be utilised in planning sampling methods,
in assessing the naturalness of forests, e.g. Spatial point processes, e.g. Gibbs processes, have
been used in modelling spatial patterns of trees (Sarkka and Tomppo 1998).
Variables, like growth factors, site fertility, nutrient availability,cumulative temperature
sum of growing season, e.g., are examples of variations of di erent scales. These can regarded
as a realisations of stochastic processes on the plane. These, on the other hand, very much
determine, the structure of the growing stock and its variability. Present silvicultural practice
has led to forests which are mosaics of stands of di erent age classes and tree species composi-
tions with a speci c tree form and spatial patterns of trees. The distribution of stands can be
regarded as an output of mosaic models (Stoyan et al. 1995).
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 be 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 autocorrelation 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
(double) sampling (Cochran 1963).
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 models, e.g. non-parametric regression analysis, are applied in estimating
the variables for trees of less intensive samples.
The interest in forest inventory is often in the quantity M = R z(t)dt=R y(t)dt; where
A A
A is an inventory area, z(t);t 2 R2 the variable of interest, e.g., an indicator of a land use
class, volume of timber assortment and y(t);t 2 R2 is an indicator function of the stratum
(e.g, forestry land). 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= izi= i yi =z= y. A natural reliability measure for the estimator is E(m M) . Unbi-
ased estimator for systematic sampling is not known. Conservative estimators can be derived
utilising the properties of second order stationary processes (Matern 1960). The parameter
estimation of spatial pattern models with Gibbs processes can be based on the properties of
Palm distributions of the process and a chosen test function (Fiksel 1988). Another possibility
is to utilise approximative maximum likelihood approaches, for a review, see Geyer (1998).
Estimation of parameters with multi-source inventory
The increasing availability of supplementary data, e.g. remote sensing data, has changed
the requirements for statistical methods in forest inventories. Supplementary data is usually
cheap but much less accurate than eld measurements. A proper 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 (double) sampling. A non-
parametric k-nn method, adopted in the Finnish national forest inventory, an be considered
as an extension of double sampling (Kilkki and Paivinen 1987, Tomppo 1996). An essential
property is that all inventory variables, typically 100 to 400, can be estimated at the same time.
The procedure 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
w = 1 =X 1 ; (1)
i;p 2 2
d d
pi;p j=1 p(j);p
if pixel p is among the k nearest to p, otherwise w = 0. Here, k is a prede ned xed
i i;p
number. The weights w are summed over pixels p by computation units u (for example by
i;p
municipalities) yielding the weight of plot i to computation unit u
c =Xw : (2)
i;u i;p
p2u
The sum c can be interpreted as that area (in pixels) of unit u, which is most similar to
i;u
sample plot i. The plots outside u may also receive positiveweights (synthetic estimation).
Thek-nnmethodisa
exibleandpracticalwaytocombine eldmeasurementsandsupple-
mentary data into an operational inventory system, to obtain much more detailed information
about forests with very low additional costs compared to the inventory methods employing
sampling and eld measurements only. The method is more statistically oriented than the
old classi cation-based approach to the use of satellite images. The key feature concerning
the routine operational use is, that the image processing phase does not depend on the forest
variables to be estimated. After computing the sample plot weights c for each computation
i;u
unit u of interest, the image data is no longer used, and in principle, all parameters can be
estimated as weighted averages of eld plot data. Also because the weights are the same for all
variables, the method preserves the natural dependence structures between forest parameters.
Further advantages are the applicabilitytovery di erenttypes of forests and the possibilityto
use di erent kinds of remote sensing material, both with only minor modi cations.
Forthek-nnmethod,theRMSEofthepixellevelestimatescanbestatisticallyassessed by
cross validation. However, the error in the estimate of a forest parameter in one pixel is highly
dependent on the true value there, and thereby the errors are spatially correlated. The error
structure is made even more complex by the spatial dependencies in the image itself and the
errors of possible other data sources like maps. Developing an operationally usable statistical
error assessment technique is a highly challenging task, and a fully satisfactory solution is yet
to be found.
Conclusions
Forest inventories have motivated pioneering work in statistics, especially in spatial statis-
tics. Global forestry information needs are increasing at the moment when the requirements to
increases timber production, and at the same, to preserve forest ecosystems exist. Increasing
amount of versatile supplementary data makes it possible to increase the eciency of inven-
tories. Analysis of temporally and spatially correlated, multi-temporal, multi-resolution and
multi-source data sets of future forest inventories is achallenging task for statistics.
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