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.

The k-nn metho d is a exible and practical way to combine eld measurements and supple-

mentary data into an op erational inventory system, to obtain much more detailed information

ab out forests with very low additional costs compared to the inventory metho ds employing

sampling and eld measurements only. The metho d is more statistically oriented than the

old classi cation-based approach to the use of satellite images. The key feature concerning

the routine op erational use is, that the image pro cessing phase do es not dep end 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 b ecause the weights are the same for all

variables, the metho d preserves the natural dep endence structures between forest parameters.

Further advantages are the applicabilitytovery di erenttyp es of forests and the p ossibilityto

use di erent kinds of remote sensing material, b oth with only minor mo di cations.

For the k-nn metho d, the RMSE of the pixel level estimates can b e statistically assessed by

cross validation. However, the error in the estimate of a forest parameter in one pixel is highly

dep endent on the true value there, and thereby the errors are spatially correlated. The error

structure is made even more complex by the spatial dep endencies in the image itself and the

errors of p ossible other data sources like maps. Developing an op erationally usable statistical

error assessment technique is a highly challenging task, and a fully satisfactory solution is yet

to b e found.

Conclusions

Forest inventories have motivated pioneering work in statistics, esp ecially in spatial statis-

tics. Global forestry information needs are increasing at the moment when the requirements to

increases timber pro duction, and at the same, to preserve forest ecosystems exist. Increasing

amount of versatile supplementary data makes it p ossible to increase the eciency of inven-

tories. Analysis of temp orally and spatially correlated, multi-temp oral, multi-resolution and

multi-source data sets of future forest inventories is achallenging task for statistics.

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