Modeling Tree Growth Under Varying Silvicultural Prescriptions

Leah RathbunUniversity of British Columbia

Presented at Western Mensurationists 2010

Location• Coastal Western Hemlock

(CWH) BEC zone• Temperatures - 5.2 to 10.5˚

C • Second growth uneven-

and even-aged multi-species stands regenerated naturally and from plantings.

• Western hemlock, Douglas-fir, western redcedar, red alder, Sitka spruce and yellow cedar

Box-Lucas Model

Developed from Von Bertalanffy function

Flexible model form

dbhfdbhfff

fkDinc 12

21

11 expexpˆ

Database for Diameter Growth• Permanent Sample Plot (PSP) data • 1,455 untreated plots • 0.008 to 0.806 ha • 1932 to 2003• Varying intervals - 1 to 17 years, average of 5.0

years• 13 to 11,750 live trees per hectare• Site Index values from 6.2 to 52.8 m• Average basal area per hectare value of 38.7

m2/ha

Parameter Prediction Approach

• Data was subset to obtain estimates for f1 and f2 for a species

• Subset defined from untreated data containing at least 4 measurement periods.

• Different combinations of predictor variables were used, based upon correlation values

• Linear, log-linear, and nonlinear equations• AIC values were used to select model forms• Box-Lucas model was refit using the resultant

parameter estimates as starting values.

Variable Selection

Tree size and stage of development, site productivity, and inter-tree competition were considered:

•dbh, height, and diameter increment•Site index and growth effective age •Curtis’ relative density and basal area per hectare•Basal area of larger, stems per hectare, relative dbh, crown competition factor of larger

Selected Diameter Model

Douglas fir:

Cedar:

Hemlock:

dbhfdbhfff

fkDinc 12

21

11 expexpˆ

)lnexp 8654101 SpHaRDBHaGEAaSIa(dbh)a(af

))ln(exp( 876543102 SpHbCurtisRDbRDBHbGEAbSIbBALbdbhbbf

))ln(exp( 6532101 RDBHaGEAaBALaGadbhaaf

))ln(exp( 65102 RDBHbGEAbdbhbbf

))ln(exp( 653101 RDBHaGEAaBALadbhaaf

)SpHbRDBHbGEAbGb)dbhln(bbexp(f 8652102

Untreated trees

Douglas-fir

Hemlock

Cedar

Fertilization Effects• 85 plots received fertilization• Majority received one application of

nitrogen ranging in concentration from 50 to 400 kg/ha

• A few plots additionally received ammonium phosphate.

Fertilization Effects

• All three parameters were modified individually • Each possible combination of the three

parameters were modified • Akaike’s Information Criteria (AIC) values used

to select final model

dbhfdbhfff

ffDinc 3443

35 expexpˆ

F924 time

Fbexpff

FtimeFaff 913 exp

FtimeFckf 15

Fertilization Effects

For Douglas fir and cedar:

And for hemlock:

FtimeFaff 913 exp

FtimeFbff 924 exp

15 kf

FtimeFckf 15

Fertilized

Douglas-fir

Hemlock

Cedar

Thinning Effects

• 419 plots received a thinning event• Average basal area cut was 8.2 m2/ha• Average of 892 trees per hectare was

removed• Original model was fit using parameter

estimates found from untreated data • State variables changed immediately

following a thin

Thinned

Douglas-fir

Hemlock

Cedar

Fertilized and Thinned Effects

• 93 plots received a combination of a thinning followed by fertilization

• Fertilization was applied within the measurement period following thinning

• Fertilized model was fit using parameter estimates found from fertilized data

• State variables changed immediately following a thin

Fertilized & Thinned

Douglas-fir

Hemlock

Cedar

Database for Height Growth• 1,316 untreated plots • 0.008 to 0.253 ha • 1932 to 1996• Varying intervals - 1 to 30 years, average of 4.8

years• 222 to 11,750 live trees per hectare• Site Index values from 10.7 to 52.8 m• Average basal area per hectare value of 38.1

m2/ha

Potential Height Model• Data selection

– Binning by dbh (90th and 99th percentiles)– Predicted values approach

• Box-Lucas model fit using dbh and a transformation of dbh in a linear model for f1 and f2

• Predict 90th and 99th percent confidence interval, upper value

• Fit Box-Lucas model using dbh and a transformation of dbh in a linear model for f1 and f2

Potential Height Model

Binning approach:- f1: dbh0.5 for Douglas-fir and ln(dbh) for cedar and hemlock- f2: dbh2 for Douglas-fir and cedar and ln(dbh) for hemlockPredicted values approach:- f1: dbh2 for Douglas-fir and ln(dbh) for cedar and hemlock- f2: dbh2 for Douglas-fir and cedar and ln(dbh) for hemlock

heightfheightfff

fkH 1incP 1221

1 expexp

)SIadbhadbha(aexpf 3dtransforme2101 )SIbdbhbdbhb(bexpf 3dtransforme2102

Potential height

Douglas-fir

Western hemlock

Western redcedar

• Douglas-fir

• Cedar

• Hemlock

Average Height ModelGrowth Potential Dependent

incPinc H*MH

CurtisRDcRDBHcGEAcSIcBALcGccM 7654320

CurtisRDcSIcBALcGcdbhccM 743210

CurtisRDcRDBHcGEAcSIcBALcGcdbhccM 76543210

Average Height ModelGrowth Potential Independent

Douglas fir:

Cedar:

Hemlock:

dbhfdbhfff

fkDinc 12

21

11 expexpˆ

)lnexp 8654101 SpHaRDBHaGEAaSIa(dbh)a(af

))ln(exp( 876543102 SpHbCurtisRDbRDBHbGEAbSIbBALbdbhbbf

))ln(exp( 6532101 RDBHaGEAaBALaGadbhaaf

))ln(exp( 65102 RDBHbGEAbdbhbbf

))ln(exp( 653101 RDBHaGEAaBALadbhaaf

)SpHbRDBHbGEAbGb)dbhln(bbexp(f 8652102

Average height

Douglas-fir

Hemlock

Cedar

Fertilization Effects

• 82 plots received fertilization• All three parameters were modified individually • Each possible combination of the three

parameters were modified • Akaike’s Information Criteria (AIC) values used

to select final model

dbhfdbhfff

ffDinc 3443

35 expexpˆ

F924 time

Fbexpff

FtimeFaff 913 exp

FtimeFckf 15

FertilizedDouglas-fir

a)Low site quality, low density

b)High site quality, low density

c)High site quality, high density

FertilizedHemlock

a)Low site quality, low densityb)Low site quality, high densityc)High site quality, low densityd)High site quality, high density

Thinning Effects

• 388 plots received a thinning event• Average basal area cut was 7.6 m2/ha• Average of 988 trees per hectare was

removed• Original model was fit using parameter

estimates found from untreated data • State variables changed immediately

following a thin

ThinnedDouglas-fir

a)Low site quality, low densityb)Low site quality, high densityc)High site quality, low densityd)High site quality, high density

ThinnedCedar

a)Low site quality, low density

b)High site quality, low density

c)High site quality, high density

ThinnedHemlock

a)Low site quality, low density

b)High site quality, low density

c)High site quality, high density

Fertilized and Thinned Effects

• 92 plots received a combination of a thinning followed by fertilization

• Fertilization was applied within the measurement period following thinning

• Fertilized model was fit using parameter estimates found from fertilized data

• State variables changed immediately following a thin

Fertilized & ThinnedDouglas-fir

a)Low site quality, low densityb)Low site quality, high densityc)High site quality, low densityd)High site quality, high density

a)Low site quality, low densityb)Low site quality, high densityc)High site quality, low densityd)High site quality, high density

Fertilized & Thinned

Hemlock

Conclusions• Box-Lucas model works well • Silvicultural treatments applied to managed

stands, not experimentally designed study• The adjustments added for fertilization

worked well • Modification of state variables only modeled

the effects of thinning well for Douglas-fir • Separate models are recommended for

potential and average height

Acknowledgements

• Dr. Valerie Lemay• Dr. Nick Smith• Dr. Peter Marshall• Dr. Lori Daniels• Ken Epps and Island Timberlands• FVS group at the U.S. Forest Service