What is 3PG?

Adaptation of a presentation from Peter Sands

An overview of Landsberg and Waring’s

model of forest productivity

3PG is:

A tree growth model based on Physiological

Principles that Predict Growth

Bridges the gap between mensuration-based

growth and yield models and process-based, C-

balance models

Provides fully dynamic predictions of biomass

pools, stand attributes, stocking and soil water

usage

Maintains an admirable level of simplicity

2

Comparison with empirical models

Advantages

based on a wide range of conditions

applicable under changing conditions,“at the

edges”, to novel situations

provides explanation, aids understanding

Disadvantages

not as widely understood as empirical growth

models

not necessarily as accurate, either

can require data not readily available

A quick summary of 3-PG

Attribute Comments

Model type Dynamic, includes process-based and empirical relationships

Time step Month

Processes NPP, biomass allocation, water usage and soil water balance,

stem mortality, litterfall and root turnover

Inputs Monthly climate data, soil texture and water capacity,

fertility

Outputs Biomass pools, stocking, available soil water, NPP and ET,

avegeare DBH and standard stand attributes, and others

Strengths Fully dynamic, can be adapted for a range of species,

provides management-related outputs

Weaknesses Naïve treatment of soil nutrition, allocation based largely on

size, poor predictor of canopy development and of mortality

Input data for 3-PG is …

…of a quality and quantity that is readily

obtained by the forest manager

mean monthly weather data

very basic physical site and soil factors

simple (naïve?) ranking of site fertility

Input data (continued)

Climate data

monthly mean temperature, radiation, rainfall, VPD

observed or long-term average data

Site and soil descriptors

latitude

soil texture and water capacity

fertility rating

Also need stand initialisation data

foliage, stem and root biomass

stocking

Main Components of 3-PG

Production of biomass – environmental modification of

light use efficiency; constant ratio of NPP to GPP

Biomass allocation – affected by growing conditions and

tree size

Stem mortality – probability of death; self-thinning

Soil water balance – single soil layer model; evapo-

transpiration determined from Penman-Monteith equation

Stand properties – from biomass pools and assumptions

about specific leaf area, branch+bark fraction, and wood

density

Structure and processes in

3-PG

8

9

Conceptual PBM of forest growth

Lightinterception

Assimilation,respiration

Litterfall

Nutrientuptake

Rainfall

Transpiration

Decomposition,mineralisation,OM cycling

Wateruptake

Understoryevaporation

Biomassallocation

Rootturnover

Defoliation,disease

Silviculture

Soilevaporation

Nutrientvolatilisation& leaching

Watertable access

3PG does not

include all these

processes...

Conceptual PBM… (continued)

McMurtrie and Wolf (1983) model is a

common basis for many implementations

of the conceptual model

3-PG follows this model

Wi – biomass, i - proportion allocated

i – turnovers (litterfall, death, root turn.)

l – leaves, s – stem, r - roots

Photosynthesis

LAI

CO2

Light

interception

Net primary

production

Roots

LeavesStem

Litter,turnover, etc

01 QeQ kLint

Q0

GPPYNPPPn

intcg QGPPP

1

rsl

rrnrr

snss

llnll

FfnFF

WPW

WsPW

WPW

WPW

Causal loop diagrams

Powerful tools to communicate and explore system behaviour

They summarize structure, causal influences and feedback loops

ABCA negative feedback (odd number of '-')

ABCDA positive feedback (even number of '-')

A B

C

D+ +

-

+

-

A influences B

causalinfluence

an increase causesan increase

an increase causesa decrease

3-PG as a carbon flow model

3-PG is essentially a McMurtrie

and Wolf (1983) carbon balance

model

radiation is intercepted by the

canopy

converted to assimilates

allocated to foliage, stem and

roots

lost to respiration, litterfall and

root turnover

NPP

Stem

Foliage

Roots

GPP

CO2

C,N

Litter

Assimilation and allocation

Assimilation and allocation are

based on simple, well

established principles and sound

observations

radiation interception via Beers

law

assimilation via light use

efficiency

simple foliage, stem and root

allocation ratios

foliage:stem allocation depends

on tree size

NPP

Stem

Foliage

Roots

GPP

CO2

C,N

Litter

++

+

+

DBH

F/SR

LAILUE

SLA

+

+

_

+

NPP

Stem

Foliage

Roots

GPP

CO2

C,N

Litter

Site and environmental effects

Site and environmental factors

affect growth (and water use)

via simple empirical modifiers

temperature affects only LUE

VPD and soil water affect LUE

and root allocation

site fertility affects root

allocation and maybe LUE

++

+

+

DBH

F/SR

LAILUE

SLA

+

+

_

+

NPP

Stem

Foliage

Roots

GPP

CO2

C,N

Litter

+

__

Stress

VPD

T

FR

f

+

_

+

_

+

++

+

+

DBH

F/SR

LAILUE

SLA

+

+

_

+

NPP

Stem

Foliage

Roots

GPP

CO2

C,N

Litter

Soil water balance

Soil water balance via simple

single layer model with

transpiration determined using a

Penman-Monteith equation

canopy conductance scaled for

canopy LAI

and affected by VPD and soil

water

ET driven by radiation

feedback from soil water status

into growth modifiers

+

__

Stress

VPD

T

FR

f

+

_

+

_

+

++

+

+

DBH

F/SR

LAILUE

SLA

+

+

_

+

NPP

Stem

Foliage

Roots

GPP

CO2

C,N

Litter

Soil water balance

Soil water balance via simple

single layer model with

transpiration determined using a

Penman-Monteith equation

canopy conductance scaled for

canopy LAI

and affected by VPD and soil

water

ET driven by radiation

feedback from soil water status

into growth modifiers

+

__

Stress

VPD

T

FR

f

+

_

+

_

+

++

+

+

DBH

F/SR

LAILUE

SLA

+

+

_

+

NPP

Stem

Foliage

Roots

GPP

CO2

C,N

Litter

+

H20 Rain

gC

Soil H20

ET

+

+

+

_

+

__

Stress

VPD

T

FR

f

+

_

+

_

+

++

+

+

DBH

F/SR

LAILUE

SLA

+

+

_

+

NPP

Stem

Foliage

Roots

GPP

CO2

C,N

Litter

Stocking and mortality

Stocking is an essential component of 3-PG as it affects allocation

through stand-mean DBH

mortality model very simple-minded

probability of death age and (potentially) stress related

density dependent mortality implemented via self-thinning law

This is the final “picture”

State variables

Subsidiary variables

Climate & site Inputs

Losses

Material f low s

Influences

Carbon

Water

Trees

Key to colours & shapes

Subsidiary variables

+

H20 Rain

gC

Soil H20

ET

+

+

+

_

_

+

wSx

Deadtrees

Stocking

+

+

_

wS+

w S>w Sx

_ _

N

+

+

__

Stress

VPD

T

FR

f

+

_

+

_

+

++

+

+

DBH

F/SR

LAILUE

SLA

+

+

_

NPP

Stem

Foliage

Roots

GPP

CO2

C,N

Litter

+

3-PG in more detail

19

Light interception

Light is absorbed as it passes through canopy

Intercepted radiation varies with LAI via Beer’s

law:

LAI determined by specific leaf area (sla) and

foliage biomass

01 QeQ kLint

slaWLAI F

Light interception

Light is absorbed as it passes through canopy

Intercepted radiation varies with LAI via Beer’s

law:

LAI determined by specific leaf area (sla) and

foliage biomass

01 QeQ kLint

slaWLAI F

0

20

40

60

80

100

0 1 2 3 4 5 6

Canopy LAI

Inte

rce

pte

d r

ad

iati

on

(%

)

Production and solar radiation

Observation shows

above-ground and gross production are linearly related

to intercepted radiation

Slope of these relationships is a measure of light

use efficiency (LUE)

daily canopy-level LUE varies seasonally

annual stand-level LUE stable

Production and solar radiation

Observation shows

above-ground and gross production are linearly related

to intercepted radiation

Slope of these relationships is a measure of light

use efficiency (LUE)

daily canopy-level LUE varies seasonally

annual stand-level LUE stable

y = 4 .2 9 0 8 x - 0 .6 2 1 1

R 2 = 0 .9 8 3 9

0

2

4

6

8

1 0

1 2

1 4

1 6

0 1 2 3 4

Inte rc e pte d ra dia tion (GJ m-2

yr-1

)

Ab

ov

e-g

ro

un

d p

ro

du

cti

on

(t

ha

-1 y

r-1

)

As s o r te d s p e c ie s

fo u r s i te s

E s p e ra n c e , T a s .

y = 4 .3 6 7 3 x - 1 .4 3 6 6

R 2 = 0 .9 8 7 3

0

2

4

6

8

1 0

1 2

1 4

1 6

0 1 2 3 4

Inte rc e pte d ra dia tion (GJ m-2

yr-1

)

Ab

ov

e-g

ro

un

d p

ro

du

cti

on

(t

ha

-1 y

r-1

)

E . g lo b u lu s

a g e x n u t rit io n

G ip p s la n d , V ic .

y = 4 .2 9 0 8 x - 0 .6 2 1 1

R 2 = 0 .9 8 3 9

0

2

4

6

8

1 0

1 2

1 4

1 6

0 1 2 3 4

Inte rc e pte d ra dia tion (GJ m-2

yr-1

)

Ab

ov

e-g

ro

un

d p

ro

du

cti

on

(t

ha

-1 y

r-1

)

As s o r te d s p e c ie s

fo u r s i te s

E s p e ra n c e , T a s .

y = 4 .3 6 7 3 x - 1 .4 3 6 6

R 2 = 0 .9 8 7 3

0

2

4

6

8

1 0

1 2

1 4

1 6

0 1 2 3 4

Inte rc e pte d ra dia tion (GJ m-2

yr-1

)

Ab

ov

e-g

ro

un

d p

ro

du

cti

on

(t

ha

-1 y

r-1

)

E . g lo b u lu s

a g e x n u t rit io n

G ip p s la n d , V ic .

Light use efficiency

Light use efficiency (LUE) is a powerful,

simplifying concept

annual stand-level LUE quite stable

species-specific

varies with climatic and site factors through use of

simple modifiers

early use of this concept by Fitzpatrick and Nix (1970)

in GROWEST, and by Monteith (1972)

Light use efficiency

Light use efficiency (LUE) is a powerful,

simplifying concept

annual stand-level LUE quite stable

species-specific

varies with climatic and site factors through use of

simple modifiers

early use of this concept by Fitzpatrick and Nix (1970)

in GROWEST, and by Monteith (1972)

0

5

10

15

20

25

30

0 1 2 3 4

Intercepted radiation (GJ yr-1)

Ab

ove-g

rou

nd

NP

P (

t h

a-1

yr-1

)

Esp 2, 3 species

WA, E. glo, 3 spacings

Vic, E.glo, 4 fertilities

= 0.43 g MJ-1

= 0.68 g MJ-1

= 0.55 g MJ-1

Gross primary production

Use of LUE is a key simplification in 3-PG

also known as “canopy quantum efficiency” denoted by

c

GPP proportional to intercepted radiation:

c depends on site and climatic conditions

0(1 ) kL

g CP e Q

min{ , }C T F N D age C xf f f f f f

Gross primary production

Use of LUE is a key simplification in 3-PG

also known as “canopy quantum efficiency” denoted by

c

GPP proportional to intercepted radiation:

c depends on site and climatic conditions

0(1 ) kL

g CP e Q

min{ , }C T F N D age C xf f f f f f

Gross primaryproduction

LAI

CO2

Interceptedlight

Photosynthesis

Light interception

T, VPD,

H2O, N

Net primaryproduction

Respiration

Net primary production

3-PG assumes constant fraction Y (=0.47) of GPP

is lost as construction and maintenance

respiration

Net primary production is then

Y probably varies seasonally with temperature

this would be an issue for a daily version of 3-PG

01 QeYPYP kLcgn

3-PG growth modifiers

Each environmental factor is represented by a growth

modifier, i.e. a function of the factor which varies

between 0 (total limitation) and 1 (no limitation)

Factor Modifier Parameters

Vapor pressure deficit fD(D) kD

Soil water f() max , c , n

Temperature fT(Tav) Tmin , Topt , Tmax

Frost fF(df) kF

Site nutrition fN(FR) fN0

Stand age fage(t) nage , rage

Effects on production

All modifiers affect canopy production:

Where C is maximum canopy quantum efficiency

In 3-PG the combination of modifiers called

“PhysMod “

also affects canopy conductance

min{ , }C T F N D age Cxf f f f f f

min{ , }D agef f f

Temperature growth modifier

where

Ta = mean monthly daily temperatire

Tmin = minimum temp. for growth

Topt = optimum temp. for growth

Tmax = maximum temp. for growth

( ) ( )

( )

max opt opt minT T T T

a min max aT a

opt min max opt

T T T Tf T

T T T T

Temperature growth modifier

where

Ta = mean monthly daily temperatire

Tmin = minimum temp. for growth

Topt = optimum temp. for growth

Tmax = maximum temp. for growth

( ) ( )

( )

max opt opt minT T T T

a min max aT a

opt min max opt

T T T Tf T

T T T T

Tmin = 7.5,

Topt = 15,

Tmax = 35

0.0

0.2

0.4

0.6

0.8

1.0

5 10 15 20 25 30

Mean temperature

Tem

per

atu

re m

od

ifie

r (f

T)

Frost growth modifier

where

dF = number of frosty days in month

kF = number of days of production lost for each day of

frost

( ) 1 ( /30)F F F Ff d k d

Frost growth modifier

where

dF = number of frosty days in month

kF = number of days of production lost for each day of

frost

( ) 1 ( /30)F F F Ff d k d

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25 30

Days of frost in month

Fro

st

mo

dif

ier

(fF)

kF = 1

Soil-water growth modifier

where

= current available soil water

x = maximum available soil water

c = relative water deficit for 50% reduction.

n = power determining shape of soil water response

1( )

1 (1 / ) /n

x

fc

Soil-water growth modifier

where

= current available soil water

x = maximum available soil water

c = relative water deficit for 50% reduction.

n = power determining shape of soil water response

1( )

1 (1 / ) /n

x

fc

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Relative available soil water

So

il w

ate

r g

row

th m

od

ifie

r (f

SW

)

Sand

Sandy-loam

Clay-loam

Clay

VPD growth modifier

where

D = current VPD

kD = strength of VPD response

( ) Dk D

Df D e

VPD growth modifier

where

D = current VPD

kD = strength of VPD response

( ) Dk D

Df D e

kD = 0.05

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10

Vapor pressure deficit (mBar)

VP

D g

row

th m

od

ifie

r (f

VP

D)

Age-related growth modifier

where

t = current stand age

tx = likely max. stand age

rage = relative stand age for 50% growth reduction

nage = power determining strength of growth reduction

1( )

1 ( / ) ageage n

age x

f tt r t

Age-related growth modifier

where

t = current stand age

tx = likely max. stand age

rage = relative stand age for 50% growth reduction

nage = power determining strength of growth reduction

1( )

1 ( / ) ageage n

age x

f tt r t

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Relative stand age

Ag

e-r

ela

ted

mo

dif

ier

nage = 4,

rage = 0.95

Biomass Partitioning (1/2)

NPP is partitioned into biomass pools (tDM ha-1)

foliage (Wl)

above-ground woody tissue (Ws)

roots (Wr)

Partitioning rates (l, r, s) depend on site and

growth conditions, and stand DBH

Litter-fall (l), root-turnover (r) and wood

mortality (s) also taken into account

Biomass Partitioning (2/2)

Thus

1

rsl

rrnrr

snss

llnll

FfnFF

WPW

WsPW

WPW

WPW

Allocation in 3-PG

A simple-minded approach reproduces well-

established responses to site conditions

root allocation determined by fertility and ASW

poor conditions favour below-ground growth

foliage:stem allocation determined by tree size

large trees have more allocation to stem wood

Dynamic changes in allocation typically observed

in thinning or pruning responses are not

reproduced because allocation depends on tree

size

Allocation in 3-PG

A simple-minded approach reproduces well-

established responses to site conditions

root allocation determined by fertility and ASW

poor conditions favour below-ground growth

foliage:stem allocation determined by tree size

large trees have more allocation to stem wood

Dynamic changes in allocation typically observed

in thinning or pruning responses are not

reproduced because allocation depends on tree

size

Net primaryproduction

H2O, FR

DBH

F/S

StemFoliageRoots

Stocking

Root allocation

Root allocation is affected by growth conditions

through and by soil fertility through m

where

m = m0 + (1-m0)FR

Rx = root allocation under poor conditions

Rn = root allocation under optimal conditions

( )

Rx RnR

Rn Rx Rn m

Root allocation

Root allocation is affected by growth conditions

through and by soil fertility through m

where

m = m0 + (1-m0)FR

Rx = root allocation under poor conditions

Rn = root allocation under optimal conditions

( )

Rx RnR

Rn Rx Rn m

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Growth conditions

Ro

ot

part

itio

nin

gRx = 0.8

Rn = 0.23

Foliage and stem allocation

Above-ground allocation is based on foliage:stem

partitioning ratio

B is diameter at breast height determined from an

allometric relationship between stem mass and B

ap, bp are coefficients determined from pFS at B = 2

and 20 cm

Then

/ pn

FS F S pp a B

1 ,

1

RS F FS S

FS

pp

Tree-size and allocation

Increasing DBH decreases foliage allocation and

increases stem allocation. Graphs show response

when pFS(2) = 1, pFS(20) = 0.2, hR = 0.4

0 .0

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0 10 20 30

S tem d iam eter

Ab

ov

e-g

rou

nd

pa

rtit

ion

ing

S tem

F o liage

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

1 .2

0 10 20 30

S tem d iam eter

Ra

tio

of

foli

ag

e:s

ho

ot

pa

rtit

ion

ing

Litter-fall and root-turnover

Litter-fall is an age-dependent fraction of

foliage biomass

F0 = litter-fall rate at age 0

Fx = maximum litter-fall rate (may be stress-related)

tF = age when F=½(F0+ Fx)

Root-turnover is a constant fraction of root

biomass (R=0.015 month-1)

0

0 0 0

1( ) , ln 1

( )

Fx F FxF kt

F Fx F F F

t ke t

Litter-fall and root-turnover

Litter-fall is an age-dependent fraction of

foliage biomass

F0 = litter-fall rate at age 0

Fx = maximum litter-fall rate (may be stress-related)

tF = age when F=½(F0+ Fx)

Root-turnover is a constant fraction of root

biomass (R=0.015 month-1)

0

0 0 0

1( ) , ln 1

( )

Fx F FxF kt

F Fx F F F

t ke t

0.00

0.01

0.02

0.03

0 1 2 3 4

Stand age (years)

Lit

ter-

fall

rate

/mo

nth

F0 = 0.002

Fx = 0.027

t F = 12

Summary: the basic C balance equations

These are equations for

the 3-PG carbon balance

submodel

includes light

interception, assimilation,

biomass allocation and

mortality

C and R determined

from site conditions

, F, S, R and N possibly

age-dependent and/or

stress-related

(1 )

(1 ) (1 )

(1 ) (1 )

F

s

FS

k W

N C

F F N F F

R R N R R

S S N

N

S R FS

F FS R FS

n

s S s

n

FS FS

P e YQ

W P W

W P W

W P

N N

p

p p

w W N a B

p a B

Water balance (1/3)

Soil water balance model has a single soil-layer

Simple balance between rainfall (R), irrigation

(I) and evapotranspiration (ET)

No understory or bare soil

Excess over maximum storage lost as runoff or

drainage

Canopy interception a % of rainfall and depends

on LAI up to a maximum

max ,x TW W W R I E

Water balance (1/3)

Soil water balance model has a single soil-layer

Simple balance between rainfall (R), irrigation

(I) and evapotranspiration (ET)

No understory or bare soil

Excess over maximum storage lost as runoff or

drainage

Canopy interception a % of rainfall and depends

on LAI up to a maximum

max ,x TW W W R I ERainfall

Transpiration

Understoryevaporation

Soilevaporation

Watertable access

Water balance (2/3)

Evapotranspiration determined from a Penman-

Monteith equation and canopy conductance

driven by incident solar radiation

driven by LAI through canopy conductance

conductance affected by site and environmental

factors through growth modifiers

Water balance (2/3)

Evapotranspiration determined from a Penman-

Monteith equation and canopy conductance

driven by incident solar radiation

driven by LAI through canopy conductance

conductance affected by site and environmental

factors through growth modifiers

RelativeASW

Rain

VPD

ActualET

+

_+

+

_Conduct-ance

PhysMod

_

+

+

Water balance (3/3)

Boundary layer conductance is constant (0.2 m s-1)

Canopy conductance affected by VPD, soil water

and stand age through , and increases with

canopy LAI

Where

= min{fVPD, fSW} fage

gCx= maximum canopy conductance

LgC= LAI at maximum conductance

gC = gCx min{L/LgC , 1}

Water balance (3/3)

Boundary layer conductance is constant (0.2 m s-1)

Canopy conductance affected by VPD, soil water

and stand age through , and increases with

canopy LAI

Where

= min{fVPD, fSW} fage

gCx= maximum canopy conductance

LgC= LAI at maximum conductance

gC = gCx min{L/LgC , 1}C

an

op

y c

on

du

cta

nce (

m s

-1)

Canopy leaf area index (L)

gCx

LgC

Stem mortality in 3-PG

3-PG includes density independent mortality

through probability of death

potentially age and stress related

Stem mortality in 3-PG is based on the self-

thinning law

driven by stocking via single-tree stem mass

Stem mortality in 3-PG

3-PG includes density independent mortality

through probability of death

potentially age and stress related

Stem mortality in 3-PG is based on the self-

thinning law

driven by stocking via single-tree stem mass

Mortality

Live stems

Stocking

Dead stems

max stem

massmean stem

mass

Growth

+

+ _+

_ _

_

+

Modelling mortality

Two types of mortality:

density independent

density dependent or self thinning

Density-independent mortality

due to random or stress-related effects

modelled by probability of death N so that

N = - NN

N increased in times of stress, e.g. in response to low

long-term average f

60

Self thinning mortality

Self thinning-line gives maximum single-tree

stem mass (kg/tree) at current stocking

wSx(N) = wSx0(1000/N)3/2

where wSx0 is max. stem mass at 1000 trees ha-1

When wS > wSx(N), stocking is reduced. Thus

at 1: no mortality

Self thinning mortality

Self thinning-line gives maximum single-tree

stem mass (kg/tree) at current stocking

wSx(N) = wSx0(1000/N)3/2

where wSx0 is max. stem mass at 1000 trees ha-1

When wS > wSx(N), stocking is reduced. Thus

at 1: no mortality

Avera

ge s

tem

mass (

kg

tre

e-1

)

Stocking (trees ha-1

)

1

2 2'wSx(N')

wSx(N)

N' N

self-thinningline

no mortality

mortality reducespopulation to self-thinning line

Calculation of stem volume

Stem volume calculated either from

allometric relationship with stand DBH,

or from wood density using

V = (1 - pBB)WS/r

where pBB is fraction of stem biomass in branch and

bark and r is stem density

Note that pBB and r can vary widely across a site

and calculation of stem volume from WS can be

error prone

63

Final comments

Some areas are not covered in detail, e.g.

details of Penman-Monteith equation

rainfall interception by canopy

an attempt to account for partial canopies (which does

not work well)

However, the above gives a good

coverage of the basics of 3-PG

3PG

Theend!