EFIMED Advanced course onMODELLING MEDITERRANEAN FOREST STAND

DYNAMICS FOR FOREST MANAGEMENT

SITE INDEX MODELLING

MARC PALAHIHead of EFIMED Office

20.8.20042

Forest stand development affected by

RegenerationGrowth of trees Mortality

Models should be able to predict these processes which are

affected by factors like• Productive capacity of an area• Degree to which the site is occupied• Point in time in stand development

20.8.20043

Site quality

Defined as the yield potential for specific tree species on a given growing site

key to explain and predict forest growth and yield and therefore for defining optimal forest management. Certain investments might be only justify in certain sites…

0

10

20

30

40

50

60

0 20 40 60 80 100 120 140 160

Age (years)

Ba

sa

l are

a (

m2 h

a-1

)

SI-13

SI-21

20.8.20044

Assesing site quality

Might be assessed directly or indirectly

Indirect methods: topographic descriptors, location descriptors, soil types, presence of plant species, etc

Direct methods: require the presence of the species at the location where site is evaluated

- Why not using the volume-age relationship? m3ha-1 at a given age

Site index, dominant height at an specified reference age; the height development of dominant trees in even-aged stands is not affected by stand density = in good sites height growth rates are high0

5

10

15

20

25

30

35

0 20 40 60 80 100 120 140 160 180

Stand age (years)

Vo

lum

e m

3 h

a-1

20.8.20045

Site index curves

A family of height development patterns with a qualitative

symbol or number associated with each curve

usually the height achieved at a reference age

Site index curves are the graphic representations of

mathematical equations obtained by applying regression

analysis to height age data

0

5

10

15

20

25

30

35

0 20 40 60 80 100 120 140 160 180

Stand age (years)

Do

min

an

t h

eig

ht

(m)

26

23

20

17

13

AGE HDOM

5 0,605395

10 2,489373

20 9,169787

30 16,46173

40 21,82125

50 25,10701

60 26,98154

70 28,01996

80 28,58086

90 28,8691

100 29

110 29,03919

120 29,0247

130 28,97905

140 28,91578

150 28,84314

160 28,76624

2

2

cba tt

tH

20.8.20046

Many equations used

d

k

Ae tH

c

1balnt

H

m1

1ke1A tH 2

2

cba tt

tH

Non-linear regression required

20.8.20047

Data for site index modelling

Derived from three sources:

1. Meaurement of height and age on temporary plots

- Inexpensive, full range should be represented

2. Measurement of height and age over time: permanent plots

- Many years, good dynamic data, expensive

3. Reconstruction of height/age through stem analysis

- Immediately, expensive, good dynamic data

20.8.20048

Methods for site index modelling1. The guide curve method

2. The difference equation method

3. The parameter prediction method

The guide curve method produces anamporphic site index

curves and is usually used when only temporary plots are

available

The difference equation method requieres permanent plots

or stem analysis data

20.8.20049

Amamorphic versus Polymorphic

0

5

10

15

20

25

30

0 20 40 60 80 100 120

Age (years)

Hdo

m (m

)

0

5

10

15

20

25

30

0 20 40 60 80 100 120

Age (years)

Hd

om

(m

)

20.8.200410

The guide curve method (1)

refAgeBS

1)ln( 1oiB

AGE HDOM

5 0,605

10 2,4893

20 9,169

30 16,461

40 21,821

50 25,107

60 26,981

70 28,019

80 28,580

90 28,860

100 29,000

110 29,039

120 29,024

130 28,979

140 28,915

150 28,843

160 28,766

AgeBHdom

1ln 1oiB

Boi= constant associate with the ith curveB1= constant for all curves

refAgeS

11)ln()ln( 1

AgeBHdom

20.8.200411

The guide curve method (2)

Produces a set of anamorphic curves (proportional curves)

Needs to be algebraically adjusted after fitting the equation,

- such site index equations varies depending on

which reference age is chosen

0

5

10

15

20

25

30

0 20 40 60 80 100 120

Age (years)

Hd

om

(m

)

20.8.200412

The difference equation method (1)

Requires permanent plots or stem analysis dataFlexible method, can be used with any equation to

produce anamorphic or polymorphic curvesFirst step: developing a difference form of the heigh/age

equation being fittedExpressing Height at remeasurement (H2) as a function

of remeasurement age (A2), initial measurement age (A1),

and heigh at initial measurement (H1)

20.8.200413

The difference equation method (4)

Makes direct use of the fact that observations in a give plot

should belong to the same site index curve

Difference equtions traditionally obtained through substituting

one parameter, which is site-specific, by dynamic information

Substitution of the asymptote = anamorphic curves

Substitution of other parameters = polymorphic curves

Different approaches to obtain them ADA, GADA, equating…

Dynamic equations representing a continuos four variable

prediction system directly interpreting three dimensional

surfaces without explicit knowledge of the third dimension

20.8.200414

The difference equation method (2)

A family of curves with a general mathematical form

m1

1ke1A tH

A = asymptotic parameter

K= growth rate parameter

m= shape parameter

Where each individual height/Age curve has its own unique value of A (but we could also do it for k or m depending on which we assume is the site dependent parameter)

20.8.200415

The difference equation method (3) - Example of obtaining the difference form, ADA approach

m1

1ke1A 11 t

iH m1

1ke1A 12 t

iH

m1

1k

1m1

1k e1e12

tt HH

m1

1ke1A 21

tHi m1

1ke1A 11

tHi

m1

1

k

k

12 1

2

e1

e1

t

t

HH

20.8.200416

Final remarks

Difference equation methods:

Can compute predictions directly from any age-dominant

height pair without compromising consistency of the

predictions, which are unaffected by changes in the base age

- Better than guide curve method

Evaluating site index models:

-Biological realism (asymptote, growth pattern, quality of

extrapolations out of the age and site range of the data)

- Fitting statistics (Mef, Mres, Amres, etc)

20.8.200417

Exercise I

2

2

cba tt

tH

1. Derive a difference equation from the Hossfeld model assuming that parameter is b is the site dependent one

20.8.200418

Exercise II

1. Open the SPSS file Site_stems and fit a non-linear regression

model using the difference form of the Hossfeld model.

Based on previous studies, initial values for a (between 10

and 10) and c (between 0,02 and 0,04).

The asymptote of the model is equal 1/c

2. Fit now the McDill-Amateis equation (M= asymptote)

- How we decide which one is better? Which model is better?