# Diameter Increment

For every year of modeled growth, MGM predicts an annual diameter increment for each tree > 1.3m tall, relative to height increment, local competition, and/or other variables. Different diameter increment models are used for trees ≥ 4cm DBH and trees < 4cm DBH.

## Trees ≥ 4cm DBH

For trees ≥ 4cm DBH, annual diameter increment is predicted using 1 of 2 approaches:

1. Compatible Diameter Increment Models – Diameter increment models that include height increment as a variable.
2. Non-Compatible Diameter Increment Models – Diameter increment models that do not include height increment as a variable.

#### Compatible Diameter Increment Models

The annual diameter increment models for lodgepole pine and black spruce include height increment and competition as variables (Equations D1 and D2). As a result, trees with a large height increment and low competition experience more diameter growth. Trees with a small height increment and high competition experience less diameter growth. These models apply to all lodgepole pine and black spruce ≥ 4cm DBH.

Lodgepole Pine

 DIijk = HIijk / exp(a0 + a1rDBALTijk + a2rSwFBALTijk + a3rPBALTijk) (Equation D1)2

Where:

DIijk = Diameter increment (cm/year) of tree k of species j in stand i

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

rDBALTijk = DBALTijk / maxBAi

rSwFBALTijk = SwFBALTijk / maxBAi

rPBALTijk = PBALTijk / maxBAi

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

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

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

MaxBAi = maxDeni × AreaConvConst × QMDi2

maxDeni = ((1 / QMDi + 0.00865) / 0.001244) (1 / 0.5225)

AreaConvConst = 7.8537E-05

QMDi = Quadratic mean diameter (cm) of stand i for all species

a0 = -0.331, a1 = 1.263, a2 = 3.34, a3 = 2.061

Black Spruce

For black spruce, the annual diameter increment model (Equation D2) was developed in Oboite and Comeau (2021) using 4,139 plots, ranging from Alaska to Manitoba. Equation D2 considers the impact of height increment, diameter (i.e. tree size), and competition on annual diameter increment.

 DIijk = HIijk × (a0 + exp(a1DBHijka2) × exp(-1 × (a3DBALTijk + a4SFBALTijk + a5PBALTijk))) (Equation D2)2

Where:

DIijk = Diameter increment (cm/year) of tree k of species j in stand i

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

DBHijk = Diameter at breast height (cm) of tree k of species j in stand i

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

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

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

a0 = 0.9285128, a1 = -0.2663871, a2 = 0.0218894, a3 = 0.0076929, a4 = 0.0910156, a5 = 0.0027382

#### Non-Compatible Diameter Increment Models

The annual diameter increment models for white spruce, trembling aspen, and jack pine include maximum diameter increment and competition (Equation D3). After defining maximum diameter increment, each tree is ranked by social status to determine competition-adjusted diameter increment. As a result, the largest diameter tree in each stand grows at the maximum diameter increment (Equation D3; RFijk = 1). Smaller diameter trees grow at a slower rate relative to competition (Equation D3; RFijk < 1). For mid-rotation trees, maximum diameter increment is defined using a relationship between maximum height increment, DBH, and a height-diameter curve (Equation D7). For old growth trees, maximum diameter increment is defined using a constant basal area increment model (Equation D9) and top height DBH. Competition is applied using species-specific reduction factors in Equations D10 to D13.

 DIijk = MaxDIijk × RFijk (Equation D3)2

Where:

DIijk = Diameter increment (cm/year) of tree k of species j in stand i

MaxDIijk = Maximum diameter increment (cm/year) of tree k in species j in stand i (Equations D7 or D9)

RFijk = Reduction factor of tree k of species j in stand i (Equations D10, D11, D12, or D13)

Mid-Rotation Maximum Diameter Increment

For mid-rotation trees, maximum height increment, maximum diameter increment, and DBH are linked using Equation D4 and a Weibull height-diameter curve (Equation D6). Then, Equation D4 is rearranged to solve for maximum diameter increment by multiplying the maximum potential height increment and the slope of the height-diameter curve (Equation D7). The Weibull height-diameter curve uses the parameters in Table D1.

 MaxHIijk / MaxDIijk = f ‘ (HtDbhCurve(HDijk)) (Equation D4)2 MaxHIijk = SiteHeightj(SiteIndexij, BHageijk + 1) – SiteHeightj(SiteIndexij, BHageijk) (Equation D5) PHijk = 1.3 + a0(1 – exp(-1 × ((DBHijk / a2)a1))) (Equation D6) MaxDIijk = MaxHIijk × f ‘ (HtDbhCurve(HDijk)) (Equation D7)

Where:

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

MaxDIijk = Maximum diameter increment (cm/year) of tree k of species j in stand i

HtDbhCurve = Weibull height-diameter curve in Equation D6

HDijk = Height (m) for conifer species or diameter at breast height (cm) for deciduous species of tree k of species j in stand i

SiteHeightj = Height (m) defined by the provincial site index curve for species j, given SiteIndexij and BHageijk

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

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

PHijk = Predicted height (m) of tree k of species j in stand i

a0 – a2 = Parameter estimates in Table D1.

Table D1. Parameters for the Weibull height-diameter curve (Equation D6) by species, regional variant, and subregion. The white spruce and trembling aspen parameters are from height-diameter curves fit to PSP top height trees. The jack pine parameters are from the “grand mean model” in Huang et al. (2009); Huang et al. (2009) parameter a2 is expressed as 1 / x to match the model form of Equation D6.

 Species Regional Variant Subregion a0 a1 a2 White Spruce Alberta Montaine, Lower Foothills, Foothills Parkland 28.54 1.91 24.34 Alpine, Subalpine, Upper Foothills 25.72 1.68 24.93 Other Subregions 25.86 2.23 21.47 British Columbia, Saskatchewan, Manitoba All 25.86 2.23 21.47 Trembling Aspen All All 34.74 1.14 29.50 Jack Pine All All 23.54 1.23 14.36

Old Growth Maximum Diameter Increment

For old growth trees, maximum diameter increment cannot be solved using mid-rotation Equation D7. In old growth trees, maximum diameter increment approaches zero as trees approach the height-diameter asymptote (Equation D7; MaxDIijk → 0 as f ‘ (HtDbhCurve(DBHijk) → 0). However, large old growth trees may persist and accrue diameter increment in late life.

To address this issue, a constant basal area increment (BAI) model (Equation D8) and top height DBH are used to define maximum diameter increment for old growth trees (Equation D9). Old growth trees are taller than > 80% of the height-diameter asymptote. Tree-level variables are assigned relative to a hypothetical site tree at 80% of the height-diameter asymptote. Under this approach, a constant BAI is applied to an increasing stem perimeter. This produces a maximum diameter increment that is > 0 and decreases with increasing tree diameter.

 BAI80% = (π / 4) × ((DBH80% + MaxDI80%)2 – DBH80%2) (Equation D8)2 MaxDIijk = sqrt(4 / π × (BAI80% + π / 4 × TopHtDBHij2)) – TopHtDBHij (Equation D9)

Where:

BAI80% = Basal area increment (cm²/year) for a hypothetical tree at 80% of the height-diameter asymptote

DBH80% = Diameter at breast height (cm) for a hypothetical tree at 80% of the height-diameter asymptote; Solved using Equation D6; If DBH80% > TopDBHij then DBH80% = TopDBHij.

TopHtDBHij = Mean top height DBH (cm) of stand i of species j

MaxDI80% = Maximum diameter increment (cm/year) for a hypothetical top height tree at 80% of the height-diameter asymptote; Solved using Equation D7 and DBH80%.

White Spruce Reduction Factors

White spruce uses a mid-rotation reduction factor (Equation D10) and an old growth reduction factor (Equation D11). Old growth trees are taller than > 80% of the height-diameter asymptote in Equation D6.

 RFijk = exp((a0DDLTijk + a1SwFDLTijk + a2PDLTijk) / 10,000) (Equation D10)2 RFijk = exp((b0DDLTijk + b1SwFDLTijk + b2PDLTijk) / 10,000) (Equation D11)2

Where:

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

DBH = Diameter at breast height (cm)

TF = Tree factor (trees/ha)

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

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

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

a0 = -0.15, a1 = -0.39, a2 = -0.159

b0 = -0.07449, b1 = -0.41993, b2 = -0.07046

Trembling Aspen Reduction Factor

Trembling aspen uses one reduction factor for mid-rotation and old growth (Equation D12). Old growth trees are taller than > 80% of the height-diameter asymptote in Equation D6.

 RFijk = exp(-1 × (a0DDLTijk + a1SwFDLTijk + a2PDLTijk)) (Equation D12)2

Where:

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

DBH = Diameter at breast height (cm)

TF = Tree factor (trees/ha)

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

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

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

a0 = 0.00002854, a1 = 0.00010712, a2 = 0.00001291

Jack Pine Reduction Factor

Jack pine uses one reduction factor for mid-rotation and old growth (Equation D13). Old growth trees are taller than > 80% of the height-diameter asymptote in Equation D6. The jack pine reduction factor (Equation H13) was developed in Strimbu et al. (2017) using 422 plots across Alberta, Saskatchewan, and Manitoba.

 RFijk = exp(-1 × (a0DDLTijk + a1SFDLTijk + a2PDLTijk)) (Equation D13)2

Where:

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

DBH = Diameter at breast height (cm)

TF = Tree factor (trees/ha)

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

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

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

a0 = 0.0005743204, a1 = 0.0002949319, a2 = 0.00004629732

## Trees < 4cm DBH

For trees with a DBH < 4cm, annual diameter increment occurs under 3 cases:

1. Case 1 – A tree begins the year under 1.3m tall and remains under 1.3m after the current year’s height growth.
2. Case 2 – A tree begins the year under 1.3m tall and exceeds 1.3m after the current year’s height growth.
3. Case 3 – A tree begins the year above 1.3m tall.

All trees under Case 1 receive a null DBH, identified with a “-1” in the stand worksheet. For Cases 2 and 3, annual diameter increment is modeled by MGM species group (i.e. white spruce, pine, trembling aspen, black spruce).

#### White Spruce Group

For white spruce > 1.3m tall and < 4cm DBH, annual diameter increment is modeled using Equation D14 for Case 2 and Equations D14 to D19 for Case 3. Equations D14 to D19 implement a basal area increment model. In Equation D15, the exponential adjustment for spruce-fir competition (SFBALT) is based on work by Krebs (2016), as described by Comeau and Bokalo (2020).

Case 2

 DBHijk = a0 + a1H1ijk (Equation D14)2

Where:

DBHijk = Diameter at breast height (cm) of tree k of species j in stand i

H1ijk = Height (m) of tree k of species j in stand i after the current year’s height growth

a0 = -2.22, a1 = 1.9877

A minimum DBH of 0.3m is applied to any prediction < 0.3cm DBH (Equation D14).

Case 3

 BAIijk = a0 × DBHijk × viewfactora1 × exp(a2SFBALTijk) (Equation D15)2 viewfactorijk = arctan(1 / sqrt(DDenAboveijk / 10,000) × DH) / 2π (Equation D16) DHijk = MHtall10Di – 0.4 × H1ijk (Equation D17) BAnewijk = BAijk + BAIijk (Equation D18) DIijk = sqrt(BAnewijk / (π / 4)) – DBHijk (Equation D19)

Where:

BAIijk = Basal area increment (cm²/year) of tree k of species j in stand i

DBHijk = Diameter at breast height (cm) of tree k of species j in stand i

viewfactorijk = View factor of tree k of species j in stand i; If viewfactorijk > 0.25 then viewfactorijk = 0.25.

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

DHijk = Equation D17; If DHijk ≤ 0.1 then DHijk = 0.1.

MHtall10Di = Mean height (m) of the tallest 10% of deciduous trees (i.e. deciduous canopy height) in stand i

H1ijk = Height (m) of tree k of species j in stand i after the current year’s height growth

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

BAnewijk = New basal area (cm²) of tree k of species j in stand i for the current year

BAijk = Basal area (cm²) of tree k of species j in stand i from the previous year

DIijk = Diameter increment (cm/year) of tree k of species j in stand i

a0 = 1.609776, a1 = 0.208911, a2 = -0.0366

#### Pine Group

For pine > 1.3m tall and < 4cm DBH, annual diameter increment is modeled using Equation D20 for Case 2 and Equations D21 to D23 for Case 3. Equations D22 and D23 apply a diameter increment reduction for high density pine stands.

Case 2

 DBHijk = ln(1 – ((Hijk – 1.3) / a0) (1 / a2)) / a1 (Equation D20)2

Where:

DBHijk = Diameter at breast height (cm) of tree k of species j in stand i

H1ijk = Height (m) of tree k of species j in stand i after the current year’s height growth

a0, a1, and a2 are subregional parameters from the Huang (1994) height-diameter model

A minimum DBH of 0.1m is applied to any prediction < 0.1cm DBH (Equation D20).

Case 3

 DIijk = [ln(1 – ((H1ijk – 1.3) / a0) (1 / a2)) / a1] – [ln(1 – ((Hijk – 1.3) / a0) (1 / a2)) / a1] (Equation D21)2

Where:

DIijk = Diameter increment (cm/year) of tree k of species j in stand i

H1ijk = Height (m) of tree k of species j in stand i after the current year’s height growth

Hijk = Height (m) of tree k of species j in stand i from the previous year

a0, a1, and a2 are subregional parameters from the Huang (1994) height-diameter model

A minimum DBH of 0.1m is applied to any prediction < 0.1cm DBH (Equation D21).

Case 3 – High Density Stands

For pine under Case 3, a reduction factor is applied to high density stands above 20,000 trees/ha. This reduction factor reduces the diameter increment by a factor of 1.0 to 0.5 as density increases from 20,000 to 70,000 trees/ha (Equations D22 and D23). The reduction factor is fixed at 0.5 at densities above 70,000 trees/ha.

 DIhdijk = DIijk × RFijk (Equation D22)2 RFijk = 1 – (0.5 × (TDeni – 20,000) / 50,000) (Equation D23)

Where:

DIhdijk = Diameter increment (cm/year) of tree k of species j in stand i under the pine high density reduction factor

DIijk = Diameter increment (cm/year) of tree k of species j in stand i from Equation D21

RFijk = Reduction factor of tree k of species j in stand i; RFijk may not exceed 0.5 in Equation D22.

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

#### Trembling Aspen Group

For trembling aspen > 1.3m tall and < 4cm DBH, annual diameter increment is modeled using Equation D24 for Case 2 and Equations D25 to D28 for Case 3. Equations D25 to D28 implement a basal area increment model.

Case 2

 DBHijk = a0 + a1H1ijk (Equation D24)2

Where:

DBHijk = Diameter at breast height (cm) of tree k of species j in stand i

H1ijk = Height (m) of tree k of species j in stand i after the current year’s height growth

a0 = -1.83, a1 = 1.444

A minimum DBH of 0.3m is applied to any prediction < 0.3cm DBH (Equation D24).

Case 3

 BAIijk = a0 × (BAijk)a1 × (H1ijk / MHtall10Di)a2 × (SFDec)a3 (Equation D25)2 SFDec = 1 / [MHtall10Di × (DDeni / 10,000)0.5] (Equation D26) BAnewijk = BAijk + BAIijk (Equation D27) DIijk = sqrt(BAnewijk / (π / 4)) – DBHijk (Equation D28)

Where:

BAIijk = Basal area increment (cm²/year) of tree k of species j in stand i

BAijk = Basal area (cm²) of tree k of species j in stand i from the previous year

H1ijk = Height (m) of tree k of species j in stand i after the current year’s height growth

MHtall10Di = Mean height (m) of the tallest 10% of deciduous trees (i.e. deciduous canopy height) in stand i

SFDec = Deciduous spacing factor from Equation D26; Maximum Value = 2; If DDeni = 0 then SFDec = 2.

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

BAnewijk = New basal area increment (cm²) of tree k of species j in stand i

DIijk = Diameter increment (cm/year) of tree k of species j in stand i

DBHijk = Diameter at breast height (cm) of tree k of species j in stand i

a0 = 0.7898, a1 = 0.6653, a2 = 0.3452, a3 = 0.2844

#### Black Spruce Group

Case 2

For black spruce under Case 2, the initial diameter above breast height is predicted using Equation D20. This equation uses the black spruce subregional parameters in Huang (1994).

Case 3

For black spruce under Case 3, annual diameter increment is modeled using Equation D2 (Oboite and Comeau 2021). This relationship was developed using 4,139 plots, ranging from Alaska to Manitoba. See Equation D2 and Oboite and Comeau (2021) for more details.

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