Amaro A., Reed D., Soares P. (editors) Modelling Forest Systems

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02Amaro Forests - Chap 01 25/7/03 11:04 am Page 6

6 H.E. Burkhart

two predictor variable model is best. In these spacing trial data, the variation in site

index is small; furthermore, site index was not observed but rather it was estimated,

with error, from the height measurements at age 16 years (the oldest observation

available). Inclusion of a rather ‘noisy’ (imprecise) variable deteriorated model per-

formance for predicting volume per hectare in the locations not used in ﬁtting.

Model complexity can also be increased by adding more terms involving the

basic variables chosen. The model utilizing age and tr

ees per hectare as predictors

(model 2) was expanded to include the ﬁrst-order interaction term for the regres-

sors; that is

ln(Y) =

(1/A) +

(N) +

(1/A)(N) (4)

The square root of the MSE values for models 1, 2 and 4 were computed as

befor

e, and the reciprocal taken to compute the deﬁned measure of accuracy, with

the following results:

Model R

√ MSE Accuracy

1 0.84 15.89 0.0629

2 0.90 6.44 0.1553

4 0.91 8.81 0.1135

The R

value for Equation 4 was 0.91 and all coefﬁcients were signiﬁcant (P <

0.07), but the simple two variable model (model 2) again performed best when pre-

dicting with independent data.

This simple illustration shows that, for these data, the pattern of stand volume

over time at varying densities can be described with a very parsimonious model.

Adding mor

e predictor variables or more terms with the age and density predictors

caused a loss in model accuracy. Although results from this example are speciﬁc to

the yield functions chosen and data used for ﬁtting, they do demonstrate that accu-

racy of prediction can be expected to increase with increasing model complexity, but

only up to a point (Ockham’s Hill).

Forest Models

Models of forests have been constructed for a host of management and research

objectives. Forest management decision making is predicated on accurate forecasting

of growth and yield. While growth and yield forecasts enter into virtually all deci-

sions, the primary uses of models can be categorized as: inventory updating, man-

agement planning, evaluation of silvicultural alternatives and harvest scheduling.

Growth and yield information is used for a variety of purposes; no single data-

base or modelling appr

oach can be optimal for all applications. As an example, if

one were primarily interested in evaluation of silvicultural alternatives, designed-

experiment type data with the relevant silvicultural treatments included would

probably be used. The model structure would probably be quite detailed in terms of

the component equations and the types of output produced in order that the full

range of treatments could be evaluated under varying assumptions. If, on the other

hand, one were primarily interested in inventory updating, the data should be

obtained from a representative sample of the types of stands to which the model is

going to be applied. The input to the model would necessarily need to be consistent

and compatible with the inventory data deﬁnitions and quantities available. The

component equations should be as simple and straightforward as possible for pro-

02Amaro Forests - Chap 01 25/7/03 11:04 am Page 7

7 Choosing Appropriate Level for Modelling

ducing the output (updated stand statistics) that is needed for and consistent with

the inventory database. What is ‘best’ depends primarily on the objective(s) for

developing the model; obviously, the objectives should be speciﬁed clearly before

any data collection or analyses are initiated.

Growth and yield models produced to date can be broadly grouped as follows:

(1) whole-stand models: (a) aggregated values only, (b) size class information; (2)

individual tr

ee models: (a) distance-independent, (b) distance-dependent.

In the whole-stand approach, quantities such as volume, basal area and/or

number of tr

ees per unit area are forecast. The basic input or predictor variables for

these models for even-aged stands are generally age, site index and stand density

(number of trees planted per unit area for plantations; some initial basal area for

natural stands). Often only aggregated volume growth and/or yield is predicted for

the total stand. As a variation on this approach, several researchers have applied

probability density functions to estimate the number of trees by diameter at breast

height (DBH) class, given that an estimate of the total number per unit area is avail-

able. This approach, commonly termed the ‘diameter distribution approach’, still

relies on overall stand values as the basic modelling unit.

Approaches to predicting stand growth and yield that use individual trees as

the modelling unit ar

e referred to as ‘individual-tree models’. The components of

tree growth (e.g. diameter increment, height increment) in these models are com-

monly linked through a computer program that simulates the growth of each tree

and then aggregates these to provide estimates of stand growth and yield.

Individual-tree models are divided into two classes, distance-independent and dis-

tance-dependent, depending on whether or not individual tree locations are used.

Distance-independent models project tree growth either individually or by size

classes, usually as a function of present size and stand level variables (e.g. age, site

index, number of trees per unit area). In distance-dependent models, initial stand

conditions are input or generated and each tree is assigned a coordinate location.

Typically, the growth of each tree is predicted as a function of its attributes, the site

quality and a measure of competition from neighbours. Additional information

about the growth and yield modelling approaches in common use can be found in

the books by Vanclay (1994), Gadow and Hui (1999), Avery and Burkhart (2002,

Chapter 17), Clutter et al. (1983, Chapter 4) and Davis et al. (2001, Chapter 5).

Level for Modelling Forest Stands

The level at which forest stands can be modelled is often dictated by the data avail-

able. If, for instance, individual trees are not numbered and identiﬁed, individual-

tree-based approaches are not possible. Permanent plots established in the past have

sometimes had limited usefulness because of inadequacies in the measurements

taken. In permanent plots established for growth estimation purposes, the mini-

mum data measurements should include DBH, height, crown measures, stem qual-

ity assessment and tree spatial locations, to allow for ﬂexibility in modelling

approaches.

The typical approach taken in past growth and yield studies was to deﬁne a

population of inter

est, obtain a sample from the deﬁned population (the sample

could consist of temporary plots, permanent plots, or both), and estimate coefﬁ-

cients (usually with least squares) in speciﬁed equation forms. This approach pro-

duces satisfactory prediction tools for many purposes, but it may not be adequate in

circumstances where forest management practices and objectives are changing

rapidly. Given that growth and yield models are used to project the present forest

02Amaro Forests - Chap 01 25/7/03 11:04 am Page 8

8 H.E. Burkhart

resource and to evaluate alternative treatment effects, data both of the inventory

type (which describe operational stands of interest) and of the experimental or

research type (which describe response to treatment) are needed. The amount of

effort that should be devoted to each type of data collection is not immediately obvi-

ous. Nor is it at all clear whether the data should be combined and a single model

produced or the data kept separate and different models produced for different uses

or objectives.

There are some guiding principles that can aid in selecting an appropriate level

for modelling. One concept that has been advanced is to model at one level of detail

below the level desir

ed for prediction. Following that principle for forest stand mod-

elling, one would be led to individual-tree models. The use of individual-based

models in ecology has gained considerable acceptance in recent years (DeAngelis

and Gross, 1992), and individual-tree models have been developed for a number of

forest types (examples for North America are Mitchell (1975) for Douglas-ﬁr and

Burkhart et al. (1987) for loblolly pine). While individual-tree models offer a great

deal of ﬂexibility in describing stand structure and simulating silvicultural opera-

tions such as thinnings, they may not estimate overall stand values (volume and

basal area per hectare) as accurately as whole-stand equations.

Whether one should model at the tree level and aggregate for stand estimates

or model at an aggr

egated level depends on the speciﬁc objectives for modelling.

The use for which a growth model is intended, it is generally argued, should deter-

mine the resolution level at which one should operate. However, as Leary (1979) dis-

cussed, another consideration is the relationship between dimensionality of the

model (or resolution level) and the time horizon over which projections are to be

made. The following relationship from Kahne (1976) has been found to be useful in

a number of large-scale modelling efforts:

 k

(5)

d =









where h is the time horizon over which projections are to be made, d is the dimen-

sion of the model state vector and k

is a constant for a given accuracy or precision

level. This relationship indicates that the model dimension should be reduced for

long-term projections and increased for short-term projections, to give the same

level of accuracy.

The relationship between model dimensionality and projection length has been

fairly well accepted by scientists working in certain ar

eas with large-scale models,

but it has not received much attention by researchers involved in forest projection.

Forest projections are often made for any time horizon of interest without regard to

the dimensionality of the model.

Insight into the inﬂuence of dimensionality of growth and yield models and

performance over incr

easing projection length can be gained from the study of

Shortt and Burkhart (1996). They evaluated a whole-stand (Sullivan and Clutter,

1972) and an individual-tree (Amateis et al., 1989) growth and yield model for

loblolly pine for the purpose of updating forest inventory data. Both of the growth

and yield models were evaluated at varying projection periods by using permanent

plots measured at 0, 3, 6 and 9 years after initial plot establishment. Evaluations

were based solely on the capability of each model to predict merchantable volume.

The individual-tree model produced the best results until the 6-year period, at

which time it was approximately equal to the whole-stand model. After 6 years, the

whole-stand model produced the most reliable results. Only the whole-stand model

appeared to be unbiased. Both models displayed rapidly increasing error of predic-

tion with increasing projection length. As expected from the general relationship of

02Amaro Forests - Chap 01 25/7/03 11:04 am Page 9

9 Choosing Appropriate Level for Modelling

model dimension to projection length, for short-term projections the more detailed indi-

vidual-tree model performed best, but for long-term projections the simple whole-stand

model performed best.

Conclusions

From experience and information reported in the scientiﬁc literature, the following

conclusions can be drawn.

● There is no easy answer to the question of what is the appropriate modelling

level but, within cost constraints, data should be collected to allow for model-

ling at the highest level of r

esolution that might be needed.

● Relatively simple, parsimonious models are generally adequate.

● In determining the level of resolution for modelling, one should also consider

the time horizon for pr

ojection; as a rule, when the projection period increases

the model dimensionality should decrease.

Although the science of modelling has advanced greatly, and continues to

advance, ther

e is still a great deal of art involved. It is impossible to overemphasize

the necessity of having clearly stated objectives for modelling, because no approach

can satisfy all purposes. Regardless of the modelling objective, one should strive to:

(i) ﬁt as parsimonious a model as possible to describe the population trends of inter-

est, and (ii) adjust the dimensionality of the model to suit the pr

ojection length.

Acknowledgements

The support of the Loblolly Pine Growth and Yield Research Cooperative at Virginia

Tech and the assistance of Dr Mahadev Sharma with analyses of the Cooperative’s

spacing trial data reported here are gratefully acknowledged.

References

Amateis, R.L., Burkhart, H.E. and Zedaker, S.M. (1988) Experimental design and early analyses

for a set of loblolly pine spacing trials. In: Forest Growth Modelling and Prediction. North

Central Forest Experiment Station General Technical Report NC-120, pp. 1058–1065.

Amateis, R.L., Burkhart, H.E. and Walsh, T.A. (1989) Diameter increment and survival equa-

tions for loblolly pine trees growing in thinned and unthinned plantations on cutover,

site-prepared lands. Southern Journal of Applied Forestry 13, 170–174.

Avery, T.E. and Burkhart, H.E. (2002) Forest Measurements, 5th edn. McGraw-Hill, New York,

456 pp.

Burkhart, H.E., Farrar, K.D., Amateis, R.L. and Daniels, R.F. (1987) Simulation of Individual Tree

Growth and Stand Development in Loblolly Pine Plantations on Cutover, Site-prepared Areas.

Virginia Polytechnic Institute and State University, Publication FWS-1-87, 47 pp.

Clutter, J.L., Fortson, J.C., Pienaar, L.V., Brister, G.H. and Bailey, R.L. (1983) Timber Management:

a Quantitative Approach. John Wiley & Sons, New York, 333 pp.

Davis, L.S., Johnson, K.N., Bettinger, P.S. and Howard, T.E. (2001) Forest Management, 4th edn.

McGraw-Hill, New York, 804 pp.

DeAngelis, D.L. and Gross, L.J. (eds) (1992) Individual-based Models and Approaches in Ecology.

Chapman and Hall, New York, 525 pp.

Gadow, K. von and Hui, G. (1999) Modelling Forest Development. Kluwer Academic Publishers,

Dordrecht, The Netherlands, 213 pp.

02Amaro Forests - Chap 01 25/7/03 11:04 am Page 10

10 H.E. Burkhart

Gauch, H.G. Jr (1993) Prediction, parsimony and noise. American Scientist 81, 468–478.

Kahne, S. (1976) Model credibility for large-scale systems. IEEE Transactions on Systems, Man

and Cybernetics

6(8), 53–57.

Leary, R.A. (1979) Design. In: A Generalized Forest Growth Projection System Applied to the Lake

States Region. US Forest Service, North Central Forest Experiment Station General

Technical Report NC-49, pp. 5–15.

Mitchell, K.J. (1975) Dynamics and simulated yield of Douglas-ﬁr. Forest Science Monograph 17,

39 pp.

Shortt, J.S. and Burkhart, H.E. (1996) A comparison of loblolly pine plantation growth and

yield models for inventory updating. Southern Journal of Applied Forestry 20, 15–22.

Sullivan, A.D. and Clutter, J.L. (1972) A simultaneous growth and yield model for loblolly pine.

Forest Science 18, 76–86.

Vanclay, J.K. (1994) Modelling Forest Growth and Yield: Applications to Mixed Tropical Forests. CAB

International, Wallingford, UK, 312 pp.

Vanclay, J.K., Skovsgaard, J.P. and Hansen, C.P. (1995) Assessing the quality of permanent sam-

ple plot databases for growth modelling in forest plantations. Forest Ecology and

Management 71, 177–186.

03Amaro Forests - Chap 02 25/7/03 11:04 am Page 11

2 Mapping

Lodgepole Pine Site Index

in Alberta

Robert A. Monserud

and Shongming Huang

Abstract

We demonstrate methods and results for broad-scale mapping of forest site productivity for

the Canadian province of Alberta. Data are observed site index (SI) for lodgepole pine (Pinus

contorta var. latifolia Engelm.) based on stem analysis (observed height at an index breast height

age of 50 years). A total of 2624 trees at nearly 1000 site locations were available for the analy-

sis. Mapping methods are based on ANUSPLIN, Hutchinson’s thin-plate smoothing spline in four

dimensions (latitude, longitude, elevation, site index). Although this approach is most often

used for modelling climatic surfaces, the high density of the site productivity network in

Alberta made this an appropriate application of the method. Maps are presented for lodgepole

pine, the major forest species of Alberta. Although map patterns are highly complex, predicted

SI decreases regularly and continuously as elevation increases from the parklands, through the

foothills, to the mountains, conforming to ﬁeld observations and a shorter growing season. In

the high mountains, SI predictions are the lowest (<10 m), again conforming to ﬁeld observa-

tions. Thus, the map correctly represents the inverse relationship between SI and elevation

exhibited in the data. Analysis of residuals revealed no bias in the predictions. Furthermore,

residuals were homogeneous and had no apparent pattern. The standard deviation of the

observed site index values was 3.23 m, and the root mean squared error of the spline surface

predictions was 1.16 m.

Introduction

Lodgepole pine is Alberta’s provincial tree. It is the most common tree species in the

Rocky Mountains and Foothills regions (Alberta Environmental Protection, 1994),

occurring on the eastern slopes of the Canadian Rocky Mountains (Fig. 2.1). It also

occurs in a large zone to the north in boreal regions where it hybridizes with jack

pine (Pinus banksiana Lamb.). Although lodgepole pine comprises about 20% of the

USDA Forest Service, Paciﬁc Northwest and Rocky Mountain Research Stations, USA

Correspondence to: rmonserud@fs.fed.us

Forest Management Branch, Ministry of Sustainable Resource Development, Government of Alberta,

Canada

03Amaro Forests - Chap 02 25/7/03 11:04 am Page 12

12 R.A. Monserud and S. Huang

Fig. 2.1. The distribution of lodgepole pine in Alberta.

mature standing timber in Alberta, it accounts for approximately 40% of the annual

harvest in the province (Huang et al., 2001).

Lodgepole pine grows well on a wide range of both edaphic and climatic site

conditions, a quality that Wheeler and Critchﬁeld (1985) describe as r

emarkable. In

Alberta, it has optimal growth on sites similar to those of aspen (Populus tremuloides

Michx.) and white spruce (Picea glauca (Moench) Voss): moist, rich, well-aerated sites

with friable soils and long, warm growing seasons. Unlike aspen and white spruce,

however, lodgepole pine can tolerate sites that have short and irregular growing

seasons, poor to very poor nutrient regimes and a distinct moisture deﬁcit (Huang et

al., 2001).

Lodgepole pine is a major forest component in four of the 20 subregions of

Alberta (Alberta Envir

onmental Protection, 1994): the Upper and Lower Foothills

subregions, the Montane subregion and the Subalpine subregion. It is a minor com-

ponent in the Boreal Forest region. A brief description of these regions follows:

● The Upper Foothills subregion occurs on strongly rolling topography along the

eastern edge of the Rocky Mountains (Alberta Envir

onmental Protection, 1994).

Bedrock outcrops of marine shales and non-marine sandstones are frequent.

This subregion has the greatest amount of summer precipitation in Alberta at

about 340 mm and a mean annual precipitation of about 540 mm. Winter is

03Amaro Forests - Chap 02 25/7/03 11:04 am Page 13

13 Mapping Lodgepole Pine Site Index in Alberta

colder than in the Lower Foothills subregion. Upland forests are nearly all

coniferous and are dominated by lodgepole pine, white spruce, black spruce

(Picea mariana (Mill.) B.S.P.) and subalpine ﬁr (Abies lasiocarpa (Hook.) Nutt.).

● The Lower Foothills subregion is a rolling landscape created by deformed sand-

stone and shale along the edge of the Rocky Mountains, as well as er

osional

remnants with ﬂat-lying bedrock (Alberta Environmental Protection, 1994).

Although this subregion is somewhat cooler in summer than the adjacent,

lower-elevation Boreal Forest subregions, it is warmer in winter because it is

inﬂuenced less often by cold Arctic air masses and more frequently moderated

by chinook winds. Mixed forests reﬂect the transitional nature of this subregion.

Lodgepole pine forests occupy extensive portions of the uplands, especially fol-

lowing ﬁre.

● The Montane subregion occurs in the lower elevations of the Rocky Mountain

egion in southwestern Alberta, above the Foothills region. This subregion has

two components. The ﬁrst is warm and dry, a structural montane landscape of

grassland communities interspersed with dry forest communities (Alberta

Environmental Protection, 1994). Lodgepole pine is present but is not abundant;

aspen and interior Douglas-ﬁr (Pseudotsuga menziesii var. glauca (Biessn.)

Franco) are more frequent, with minor admixtures of limber pine (Pinus ﬂexilis

James). The second montane component is also mild but is moister, and consists

of closed-canopy forests. They are composed of a large proportion of lodgepole

pine and white spruce, with lesser amounts of aspen, limber pine and occasion-

ally subalpine ﬁr. This closed-canopy montane forest is often quite productive.

● The Subalpine subregion occupies a band between the Montane and Alpine

subr

egions in the south and between the Upper Foothills and Alpine subre-

gions in the north (Alberta Environmental Protection, 1994). Freezing tempera-

tures occur in all months and the frost-free period probably lasts for less than 30

days. Winter precipitation is greater in this subregion than in any other part of

Alberta, often with more than 200 cm of snowfall. Soils are highly variable

because of complex topography and parent materials. Closed forests of lodge-

pole pine, Engelmann spruce (Picae engalmannii Parry ex Engelm.) and sub-

alpine ﬁr are characteristic of lower elevations within this subregion, and open

forests are typical of higher elevations near the treeline.

● Lodgepole pine is also a minor component of the Boreal Forest Natural Region,

which consists of br

oad lowland plains and discontinuous hill systems (Alberta

Environmental Protection, 1994). Extensive wetlands are characteristic of this

natural region. Climatic conditions reﬂect a strong boreal inﬂuence. Typically,

the winters are long and cold, the summers are short and cool, and the majority

of precipitation falls in the summer. The vegetation is typically dominated by

aspen, with mixed or coniferous forests at higher elevations or in wetlands. Jack

pine outcompetes lodgepole on dry, nutrient-poor sites. Lodgepole pine also

occurs in association with black spruce in fringes around bog margins on wetter

sites before disappearing in the boreal lowlands.

The increased values and pressures placed on the timber resources in Alberta

call for accurate information r

elevant to management of the forest resource. A key

ﬁrst step is accurate estimation of site productivity for a given location. To choose

the best management strategy, it is necessary to have a quantitative means for esti-

mating site productivity for a range of management objectives.

Over the last few decades, both the government and the industry in Alberta

have allocated a sizeable amount of time and r

esources to the collection and analysis

of data on productivity, growth and yield of lodgepole pine stands. Thousands of

felled-tree stem analyses have been conducted. One product has been the develop-

03Amaro Forests - Chap 02 25/7/03 11:04 am Page 14

14 R.A. Monserud and S. Huang

ment of site index (SI)/height growth curves useful for indexing site productivity

(Huang, 1997; Huang et al., 1997, 2001). We concentrate on the observed SIs from

these stem analyses. In particular, we set out ﬁrst to map them, and then to develop

isoclines of SI within the range of lodgepole pine in Alberta.

We are faced with two practical constraints in this mapping project. The ﬁrst is

the natural range of lodgepole pine (Fig. 2.1). This excludes two lar

ge agricultural

areas: the prairie and parklands in the southeast quarter of Alberta (too dry for

forests), and the agricultural lands in the Peace River area (west central Alberta).

Although lodgepole pine could certainly survive in favourable locations outside its

natural range, our intention for now is to stay inside this range. A climate-change sce-

nario analysis in a subsequent study will remove this constraint. Secondly, destruc-

tive stem analysis sampling is not allowed in National Parks (e.g. Jasper, Banff),

many of which include large numbers of lodgepole pine (Fig. 2.2). Both constraints

exclude the enormous Wood Buffalo National Park in northeast Alberta (Fig. 2.2).

Data

Site quality describes the carrying capacity of the environment and is determined by

the biogeoclimatic conditions of a site. We used a standard measure in forestry, SI,

deﬁned as dominant tree height at 50 years breast height age.

In the 1980s, Alberta began a vigorous programme of measuring stand growth

and site pr

oductivity. Thousands of trees were felled and the stems were sectioned

(stem analysis). Guidelines called for destructively sampling approximately three

dominant/codominant trees per location. Most of the felled trees were sectioned at

stump height (0.3 m), breast height (1.3 m), 1.5 m above breast height, and at equal

lengths of 2.5 m thereafter to the top of the tree (Huang et al., 2000). A small propor-

tion of the felled trees was sectioned more intensively at 0.3, 0.8, 1.3, 2.0 and 2.8 m

above ground and then at equal lengths of 1.0 m thereafter to the top of the tree.

Ring count and height were the variables recorded for each section that we used in

this study. By 1988, over 11,000 trees had been sectioned throughout the inventoried

areas of the Province (Alberta Forest Service, 1988). A large number of these sec-

tioned trees were sampled in the buffer zone around permanent sample plots (see

Alberta Forest Service, 1988). A second set was obtained on temporary sample plots.

Recently, the Foothills Growth and Yield Cooperative has conducted additional stem

analyses on the Foothills Model Forest (Forestry Corp., 2002).

We combined the lodgepole pine in these three data sets for a common analysis,

after determining the longitude, latitude and elevation of each sample stand. The

eakdown by data set is n

= 1225 from permanent sample plot buffers, n

= 1129 from

temporary sample plots and n

= 270 from the Foothills Model Forest. This left us with

n = 2624 observations of lodgepole pine SI in Alberta (Fig. 2.2). A summary of the data

by natural subregions of Alberta (Alberta Environmental Protection, 1994) is provided

in Table 2.1. The variation in number of sample trees by subregions is an indication of

the relative abundance and importance of lodgepole pine in that subregion.

Site index determination

Much of this stem analysis data was used by Huang et al. (1997) to develop species-

speciﬁc height gr

owth and SI curves for Alberta. Recall that the stem analysis data

consist of multiple pairs of observations of height and age for each tree. We began by

estimating SI for each pair using the appropriate SI curve from Huang et al. (1997,

03Amaro Forests - Chap 02 25/7/03 11:04 am Page 15

15 Mapping Lodgepole Pine Site Index in Alberta

Fig. 2.2. Sample location of lodgepole pine stem analysis trees. N = 2624 trees at nearly 1000 site

locations. National parks: Banff and Jasper in the Rocky Mountains in the southwest, Wood Buffalo in the

northeast, and Elk Island in central Alberta near Edmonton. Agricultural lands: prairie grasslands in the

southeast and Peace River area in the west-northwest.

2001). SI for each of the 2624 stem analysis trees was determined as the mean of the

individual SI estimates from the sectional height–age observations. We averaged

these estimates from each of the three site trees at each location to obtain plot SI for

995 locations. Because Huang et al. (1997) found that SI estimates below age 20 years

had much higher variability (Fig. 2.3), we conﬁned our estimates to observations that

were at least 20 years old. This method for estimating SI had the advantage of repli-

cating the ﬁeld application of the site curves of Huang et al. (1997), and of giving

equal importance to all stem sections beyond age 20.