# Age Solver

MGM uses breast height age and/or total age to calculate height increment. Unfortunately, tree age is often difficult to obtain, requiring invasive sampling techniques (e.g. increment coring) or detailed juvenile measurements. Tree age is also difficult to obtain in uneven-aged stands where age varies by cohort and regeneration occurred over many years. MGM overcomes these issues by estimating breast height age and total age relative to tree height and site index.

MGM’s Age Solver estimates tree age under 4 cases:

1. Trees > 1.3m are initialized with an observed total age (>0) and without a breast height age (-1 placeholder). Breast height age is estimated using Equation A1.
2. Trees > 1.3m are initialized with an observed breast height age (>0) and without a total age (-1 placeholder). Total age is estimated using Equation A2.
3. Trees > 1.3m are initialized without a total age (-1 placeholder) or a breast height age (-1 placeholder). Breast height age is estimated using Equation A3. Total age is estimated using Equation A2.
4. Trees ≤ 1.3m are initialized without a total age (-1 placeholder). Total age is estimated using Huang et al. (2009) for Alberta (Equation A4) and a quadratic age-height relationship for Saskatchewan, Manitoba, and northeastern British Columbia (Equation A5).

 BHageijk = Tageijk – RegYearsToBHij (Equation A1) Tageijk = BHageijk + RegYearsToBHij (Equation A2) BHageijk = SiteCurvej(Hijk, SiteIndexij) (Equation A3) Tageijk = Huang2009Tagej(Hijk, SiteIndexij) (Equation A4) Tageijk = (RegYearsToBHij / 1.32) × (Hijk)2 (Equation A5)

Where:

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

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

RegYearsToBHij = Regional years to breast height assumption (years) for species j in stand i. RegYearsToBH scales with site index, using Equations A6 or A7.

SiteCurvej = Site index curve for species j that is rearranged to solve for BHAgeijk. Equivalent to height increment Equation H8.

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

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

Huang2009Tagej = The Huang et al. (2009) site index curve for species j that is rearranged to solve for Tageijk, given Hijk and SiteIndexij. The Huang et al. (2009) site index curves are based on total age.

Trees ≤ 1.3m that are initialized with an observed total age are not modified by the Age Solver. Trees > 1.3m that are initialized with an observed breast height age and an observed total age are not modified by the Age Solver.

## Years to Breast Height

“Years to breast height” is the number of years that a tree requires to reach breast height (1.3m). MGM’s regional years to breast height assumptions are defined in Equations A6 and A7. These years to breast height assumptions scale with site index. For example, years to breast height decreases as site index increases, or years to breast height increases as site index decreases.

Alberta

 RegYearsToBHij = Huang2009YearsToBHj(SiteIndexij) (Equation A6)

Where:

RegYearsToBHij = Regional years to breast height assumption (years) for species j in stand i

Huang2009YearsToBHj = The calculated Huang et al. (2009) years to breast height assumption for species j, given SiteIndexij. The Huang et al. (2009) total age site index curves include integrated years to breast height assumptions.

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

Saskatchewan, Manitoba, and Northeastern British Columbia

 RegYearsToBHij = a0 + a1 / SiteIndexij (Equation A7)

Where:

RegYearsToBHij = Regional years to breast height assumption (years) for species j in stand i

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

a0 – a1 = Region and species-specific parameter estimates in Table A1

 Table A1. Region and species-specific parameters for Equation A6. The British Columbia parameters match Site Tools 4.1 (BCMFLNRO 2017). Regional Variant Species a0 a1 Saskatchewan and Manitoba White Spruce 2.16 110.76 Jack Pine 3.60 42.64 Trembling Aspen 1.33 38.56 Black Spruce 4.04 61.08 British Columbia White Spruce* 4.16 110.76 Lodgepole Pine 2.55 69.40 Trembling Aspen 1.33 38.56 Black Spruce 11.04 61.08 * Parameters for planted white spruce.

## Using the Age Solver

In most situations, trees should be initialized in MGM using Case 3 (Tage = -1; BHage = -1) or Case 4 (Tage = -1). When submitting observed total ages or observed breast height ages under Case 1 (Tage > 0; BHage = -1) or Case 2 (Tage = -1; BHage > 0), these values should be obtained through increment coring or detailed juvenile measurements. Assigning the block age or AVI age to individual trees is not recommended.

## Age Solver Limits

When large trees are submitted to MGM with low site index values, the Age Solver (Equation A3) can produce unreasonable breast height age estimates. For example, a 32m white spruce that is initialized with a 5m site index could return a breast height age over 1500 years. To prevent unreasonable solutions, the Age Solver limits breast height age estimates to 300 years.

Saskatchewan and Manitoba – Hu and Garcia (2010)

Special Age Solver limits are applied to the Hu and Garcia (2010) site index curve for white spruce in Saskatchewan and Manitoba. This site index curve approaches its asymptote quickly at high site index values, producing irregular Age Solver solutions at tree heights above Equation A8.

 Hijk = a0 × (SiteIndexij)a2 (Equation A8)

Where:

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

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

a0 = 10.33, a1 = 0.43

Trees taller than Equation A8 are assigned special breast height age limits by site index:

• Site index < 10m. Maximum breast height age = 300 years.
• Site index ≥ 10m and < 15m. Maximum breast height age = 250 years.
• Site index ≥ 15m and < 25m. Maximum breast height age = 200 years.
• Site index ≥ 25m and < 30m. Maximum breast height age = 150 years.