Height Increment

For every year of modeled growth, MGM predicts an annual height increment for each tree, relative to site index assumptions, local competition, and/or other variables. Different height increment models are used for trees ≥ 4cm DBH and trees < 4cm DBH.

Trees ≥ 4cm DBH

For trees ≥ 4cm DBH, the maximum potential height increment of each tree is defined using provincial site index curves (height-age-site index models). Then, trees are ranked by social status to determine competition-adjusted height increment. Under this approach, the largest diameter tree in each stand grows at a height increment defined by the site index curve (Equation H1; Equation H2 or H9; RF_ijk = 1). Smaller diameter trees grow at a slower rate relative to competition (Equation H1; Equations H2 or H9; RF_ijk < 1). Competition is applied using species-specific reduction factors in Equations H3 to H7.


HI_ijk = MaxHI_ijk × RF_ijk	(Equation H1)²
MaxHI_ijk = SiteHeight_j(SiteIndex_ij, BHage_ijk + 1) – SiteHeight_j(SiteIndex_ij, BHage_ijk)	(Equation H2)²

Where:

HI_ijk = Height increment (m/year) of tree k of species j in stand i

MaxHI_ijk = Maximum potential height increment (m/year) of tree k of species j in stand i

RF_ijk = Reduction factor of tree k of species j in stand i

SiteHeight_j = Height (m) defined by the provincial site index curve for species j, given SiteIndex_ij and BHage_ijk.

SiteIndex_ij = Site index (m@50 years breast height age) of species j in stand i

BHage_ijk = Breast height age (years) of tree k of species j in stand i

White Spruce Reduction Factor

RF_ijk = exp((a₀DDLT_ijk + a₁PDLT_ijk + a₂SwFDLT_ijk) / 10,000)

(Equation H3)²

Where:

RF_ijk = Reduction factor of tree k of species j in stand i

DBH = Diameter at breast height (cm)

TF = Tree factor (trees/ha)

DDLT_ijk = Sum of DBH × TF values for deciduous trees larger than subject tree k of species j in stand i

PDLT_ijk = Sum of DBH × TF values for pine trees larger than subject tree k of species j in stand i

SwFDLT_ijk = Sum of DBH × TF values for white spruce-fir trees larger than subject tree k of species j in stand i

a₀ = -0.119, a₁ = -0.229, a₂ = -0.115

Lodgepole Pine Reduction Factor

RF_ijk = exp(a₀rDBALT_ijk + a₁rCBALT_ijk)

(Equation H4)²

Where:

RF_ijk = Reduction factor of tree k of species j in stand i

rDBALT_ijk = DBALT_ijk / maxBA_i

rCBALT_ijk = CBALT_ijk / maxBA_i

DBALT_ijk = Basal area (m²/ha) of deciduous trees larger than subject tree k of species j in stand i

CBALT_ijk = Basal area (m²/ha) of conifer trees larger than subject tree k of species j in stand i

MaxBA_i = maxDen_i × AreaConvConst × QMD_i²

maxDen_i = ((1 / QMD_i + 0.00865) / 0.001244) ^{(1 / 0.5225)}

AreaConvConst = 7.8537E^-05

QMD_i = Quadradic mean diameter (cm) of stand i for all species

a₀ = -1.85, a₁ = -0.437

Jack Pine Reduction Factor

The jack pine reduction factor (Equation H5) was developed in Strimbu et al. (2017) using 422 plots across Alberta, Saskatchewan, and Manitoba.

RF_ijk = exp(a₀DDLT_ijk + a₁PDLT_ijk + a₂SFDLT_ijk)

(Equation H5)²

Where:

RF_ijk = Reduction factor of tree k of species j in stand i

DBH = Diameter at breast height (cm)

TF = Tree factor (trees/ha)

DDLT_ijk = Sum of DBH × TF values for deciduous trees larger than subject tree k of species j in stand i

PDLT_ijk = Sum of DBH × TF values for pine trees larger than subject tree k of species j in stand i

SFDLT_ijk = Sum of DBH × TF values for spruce-fir trees larger than subject tree k of species j in stand i

a₀ = -0.00007541086, a₁ = -0.00003089446, a₂ = -0.00004884201

Trembling Aspen Reduction Factor

RF_ijk = exp(1.7 × (exp(-1 × (((a₀DDLT_ijk + a₁SwFDLT_ijk + a₂PDLT_ijk) / 10,000)³)) – 1))

(Equation H6)²

Where:

RF_ijk = Reduction factor of tree k of species j in stand i

DBH = Diameter at breast height (cm)

TF = Tree factor (trees/ha)

DDLT_ijk = Sum of DBH × TF values for deciduous trees larger than subject tree k of species j in stand i

SwFDLT_ijk = Sum of DBH × TF values for white spruce-fir trees larger than subject tree k of species j in stand i

PDLT_ijk = Sum of DBH × TF values for pine trees larger than subject tree k of species j in stand i

a₀ = 0.400, a₁ = 0.669, a₂ = 0.230

Black Spruce Reduction Factor

The black spruce reduction factor (Equation H7) was developed in Oboite and Comeau (2021) using 4,139 plots, ranging from Alaska to Manitoba.

RF_ijk = exp(-1 × (a₀DBALT_ijk + a₁SFBALT_ijk + a₂PBALT_ijk))

(Equation H7)²

Where:

RF_ijk = Reduction factor of tree k of species j in stand i

DBALT_ijk = Basal area (m²/ha) of deciduous trees larger than subject tree k of species j in stand i

SFBALT_ijk = Basal area (m²/ha) of spruce-fir trees larger than subject tree k of species j in stand i

PBALT_ijk = Basal area (m²/ha) of pine trees larger than subject tree k of species j in stand i

a₀ = 0.02640888, a₁ = 0.08004063, a₂ = 0.03360386

Estimating Breast Height Age

MGM uses breast height age to calculate the maximum potential height increment for each tree (Equation H1; Equations H2 or H9). However, breast height age is often difficult to obtain, requiring invasive sampling techniques (e.g. increment coring) or detailed juvenile measurements. Breast height age is also difficult to obtain in uneven-aged stands where tree age varies by cohort and regeneration occurred over many years. MGM overcomes these issues by estimating breast height age using the site index curve. This is achieved by rearranging the site index curve and solving breast height age relative to tree height and site index (Equation H8). Similar age estimation techniques are used by tree-level models for other mixedwood forest types, including Lake States FVS (Dixon and Keyser 2008a), Northeast FVS (Dixon and Keyser 2008b), and MOSES (Thurnher et al. 2017). MGM estimates breast height age when the model is initialized and when new trees are added during a Regeneration Event. MGM also re-solves breast height age after thinning or partial harvesting, allowing small or suppressed trees to release. More information about MGM’s Age Solver routine can be found in the Age Solver page.

BHage_ijk = f(H_ijk, SiteIndex_ij)

(Equation H8)²

Where:

BHage_ijk = Breast height age (years) of tree k of species j in stand i

H_ijk = Height (m) of tree k of species j in stand i

SiteIndex_ij = Site index (m@50 years breast height age) of species j in stand i

Dominance Switching Correction (BC, SK, and MB)

MGM applies a “dominance switching correction” to site index curves that were developed using stem analysis data (Feng et al. 2006b; Equation H9). This correction addresses a bias introduced by stem analysis (SA) curves and their handling of “dominant tree dynamics”† (e.g. Raulier et al. 2003; Allen and Burkhart 2015). Due to data limitations, the “dominance switching correction” is only applied through breast height age 50. Currently, MGM applies the “dominance switching correction” to the SA curves in the British Columbia, Saskatchewan, and Manitoba variants.† MGM does not apply the “dominance switching correction” to the PSP-based curves in the Alberta variant.‡

MaxHI_ijk = [(SiteHeight_j(SiteIndex_ij, BHage_ijk + 1) × dsc1] – [SiteHeight_j(SiteIndex_ij, BHage_ijk) × dsc0]

(Equation H9)²

Where:

MaxHI_ijk = Maximum potential height increment (m/year) of tree k of species j in stand i

SiteHeight_j = Height (m) defined by the provincial site index curve for species j, given SiteIndex_ij and BHage_ijk.

SiteIndex_ij = Site index (m@50 years breast height age) of species j in stand i

BHage_ijk = Breast height age (years) of tree k of species j in stand i

dsc0 = 1 / (0.818 + 0.0036 × BHage_ijk)

dsc1 = 1 / (0.818 + 0.0036 × BHage_ijk + 1)

† SA curves are developed using stem analysis data on present-day dominant trees. This approach assumes that present-day dominant trees are lifelong dominants. However, dominant-tree social status can change over time and skew the height-age-site index relationships of SA curves (Raulier et al. 2003; Allen and Burkhart 2015). In MGM, the Saskatchewan Provincial Site index Curves and British Columbia Provincial Site Index Curves were developed using SA data.

‡ The Alberta site index curves (Huang et al. 2009) were developed using Permanent Sample Plot (PSP) data. This PSP-based approach tracks a population of dominant trees over time (Raulier et al. 2003). As a result, PSP-based curves are considered representative of “dominant tree dynamics” (Raulier et al. 2003; Allen and Burkhart 2015).

Trees < 4cm DBH

For trees with a DBH < 4cm, annual height increment is modeled by MGM species group (i.e. white spruce, pine, trembling aspen, black spruce).

White Spruce Group

For white spruce < 4cm DBH, the annual height increment model is based on a 2004 analysis of the Alberta SDS (May 2004), WESBOGY LTS (December 2000), and Alberta regenerated permanent sample plot (December 2002) datasets (Equations H10 and H11). In Equation H10, the exponential adjustment for spruce-fir competition (SFBALT) is based on work by Krebs (2016), as described by Comeau and Bokalo (2020).

HI_ijk = a₀(SFDec^a₁) × (1 – exp(-H_ijk / a₂)) × exp(a₃SFBALT_ijk)	(Equation H10)²
a₀ = (1.15 × SIHI6m) / ((2^a₁)(1 – exp(-6 / a₂))	(Equation H11)

Where:

HI_ijk = Height increment (m/year) of tree k of species j in stand i

SIHI6m = Height increment given by the white spruce site index curve (Huang et al. 1997) at 6m (m / year).

SFDec = 1 / [MHtall10D_i × (DDen_i / 10000)^0.5]; Maximum Value = 2; If DDen_i = 0 then SFDec = 2

MHtall10D_i = Mean height (m) of the tallest 10% of deciduous trees (i.e. deciduous canopy height) in stand i

DDen_i = Deciduous tree density (trees/ha) of stand i

H_ijk = Height (m) of tree k of species j in stand i

SFBALT_ijk = Basal area (m²/ha) of spruce-fir trees larger than subject tree k of species j in stand i

a₁ = 0.0461, a₂ = 1.5212, a₃ = -0.09366

Pine Group

Pine ≤ 1.3m

For pine < 1.3m tall, the annual height increment model is based on preliminary relationships fitted to the Alberta LFS, SDS, and MP data (Yao 1997). This function predicts height growth using primarily total age; only slight impacts for density and species composition were present. Site index has no effect on the predictions (Equations H12 and H13).

HI_ijk = jHtpl(Tage_ijk + 1) – jHtpl(Tage_ijk)	(Equation H12)²
jHtpl = a₀ × exp(a₁Health_i + a₂TDen_i + a₃rPDen_i + a₄SQ_i + a₅DW_i) × (exp(a₆Tage_ijk) – 1) / 100	(Equation H13)

Where:

HI_ijk = Height increment (m/year) of tree k of species j in stand i

Health_ijk = Tree health of tree k of species j in stand i; (Health_ijk = 1 if healthy; Health_ijk = 0 if damaged; Defaults to 1)

TDen_i = Total density (trees/ha) of stand i

rPDen_i = Relative pine density (%) of stand i

SQ_i = Site quality of stand i; (SQ_i = 0 if site index is ≤ 18; SQ_i = 1 if site index > 18)

DW_i = Drainage of stand i; (DW_i = 1 if well drained; DW_i = 0 if poorly drained; Defaults to 1)

Tage_ijk = Total age (years) of tree k of species j in stand i

a₀ = 1.385364, a₁ = 0.164135, a₂ = -0.0000039035, a₃ = 0.623597, a₄ = 0.243658, a₅ = 0.626597, a₆ = 0.355587

Pine > 1.3m and < 4cm DBH

For pine > 1.3m tall and < 4cm DBH, annual height increment is based on the site index curve and the breast height age of the tree. No competition reduction factor is applied (Equation H14).

HI_ijk = SiteHeight_j(SiteIndex_ij, BHage_ijk + 1) – SiteHeight_j(SiteIndex_ij, BHage_ijk)

(Equation H14)²

Where:

HI_ijk = Height increment (m/year) of tree k of species j in stand i

SiteHeight_j = Height (m) defined by the provincial site index curve for species j, given SiteIndex_ij and BHage_ijk.

SiteIndex_ij = Site index (m@50 years breast height age) of species j in stand i

BHage_ijk = Breast height age (years) of tree k of species j in stand i

Trembling Aspen Group

For trembling aspen < 4cm DBH, the annual height increment model is based on a 2004 analysis of the Alberta SDS (May 2004), WESBOGY LTS (December 2000), and Alberta regenerated permanent sample plot (December 2002) datasets (Equation H15).

HI_ijk = (a₀ × a₁) + a₀(1 – exp(-H / (12a₀ + a₂DDenAbove_ijk)))

(Equation H15)²

Where:

H_ijk = Height (m) of tree k of species j in stand i

DDenAbove_ijk = Density of deciduous trees taller than (trees/ha) tree k of species j in stand i

SiteIndex_ij = Site index (m@50 years breast height age) of species j in stand i

a₀ = a₃ + a₄SiteIndex_ij

a₁ = 0.25, a₂ = 0.00175, a₃ = -0.4, a₄ = 0.0615

Black Spruce Group

Black Spruce ≤ 1.3m

For black spruce < 1.3m tall, annual height increment is predicted based on preliminary relationships fitted to the Alberta LFS, SDS, and MP data (Yao 1997). This function predicts height growth using primarily total age; only slight impacts for density and species composition were present. Site index has no effect on the predictions (Equations H16 and H17).

HI_ijk = jHtsb(Tage_ijk + 1) – jHtsb(Tage_ijk)	(Equation H16)²
jHtsb = a₀ × exp(a₁Bareroot_ijk + a₂Container_ijk + a₃Health_ijk + a₄TDen_i + a₅rSbDen_i + a₆DW_i) × (exp(a₈Tage_ijk) – 1 + a₈Container_ijk) / 100	(Equation H17)

Where:

HI_ijk = Height increment (m/year) of tree k of species j in stand i

Bareroot_ijk = Bareroot stock of tree k of species j in stand i; (Bareroot_ijk = 1 if bareroot stock; Default is 0)

Container_ijk = Container stock of tree k of species j in stand i; (Container_ijk= 1 if container stock; Default is 0)

Health_ijk = Tree health of tree k of species j in stand i; (Health_ijk = 1 if healthy; Health_ijk = 0 if damaged; Defaults to 1)

TDen_i = Total density (trees/ha) of stand i

rSbDen_i = Relative black spruce density (%) of stand i

DW_i = Drainage of stand i; (DW_i = 1 if well drained; DW_i = 0 if poorly drained; Defaults to 1)

Tage_ijk = Total age (years) of tree k of species j in stand i

a₀ = 22.167321, a₁ = 0.710512, a₂ = 0.607727, a₃ = 0.225849, a₄ = -0.0000068565, a₅ = -0.766986, a₆ = 0.096612, a₇ = 0.096272, a₈ = 0.333583

Black Spruce > 1.3m and < 4cm DBH

For black spruce > 1.3m tall and < 4cm DBH, annual height increment is based on Oboite and Comeau (2021). See Equation H1, Equations H2 or H9, and Equation H7 for more details.

¹ This function resides in the MGM21 workbook (VBA)
² This function resides in the MGM21 Growth Engine (DLL)

Last Modified: May 10, 2021