# Allometric Equations for Shrub and Short-Stature Tree Aboveground Biomass within Boreal Ecosystems of Northwestern Canada

^{*}

## Abstract

**:**

^{2}= 0.79, p < 0.001), as opposed to the commonly used one-dimensional variable (basal diameter) measured on the longest and thickest stem (R

^{2}= 0.23, p < 0.001). Short-stature tree AGB was most accurately predicted by stem diameter measured at 0.3 m along the stem length (R

^{2}= 0.99, p < 0.001) rather than stem length (R

^{2}= 0.29, p < 0.001). Via the two-dimensional variable cross-sectional area, small-stature shrub AGB was combined with small-stature tree AGB within one single allometric model (R

^{2}= 0.78, p < 0.001). The AGB models provided in this paper will improve our understanding of shrub and tree AGB within rapidly changing boreal environments.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Shrub Measurements, Destructive Sampling, and Processing

^{2}) and summed this to total cross-sectional area per shrub individual. For the 3D variable, we measured the extent of the uppermost foliage layer perpendicular to the transect (herein “width” (m)) and parallel to the transect (herein “line-intercept cover” (m)) using a tape measure. The 3D shrub volume (m

^{3}) (1) was then calculated as follows:

^{3}) = max height (m) × line-intercept cover (m) × width (m).

#### 2.3. Tree Measurements, Sampling, and Processing

^{2}) per tree individual to provide a 2D variable analogous to that for shrubs and thus offer the potential for a joint shrub and juvenile/low productive tree allometric equation. 3D volume was not measured for trees because tree AGB can best be predicted with diameter, stem length, or both variables combined (e.g., [14]). Following measurements in situ, trees were cut as close to the ground surface as possible and packed into large paper bags to be transported back to the University of Lethbridge. In the laboratory, trees were separated into stem, branch, and leaf components after air drying of up to four months. Branches were cut off directly at the stem. Twigs and fruits were included as leaf parts, while bark was included as part of the stem. Dead branches were not included in the analysis. Oven drying and biomass derivation was completed using the methods described above for shrubs.

#### 2.4. Derivation of Aboveground Biomass Allometric Equations

^{2}), and regression residual analysis. Residual analysis was performed using visual inspection of the relationships between dependent and independent variables as well as the total percentage error (%) derived via (8):

#### 2.5. Biomass Allometric Models

## 3. Results and Discussion

#### 3.1. Comparison of 1D, 2D, and 3D Variables for Shrub Total AGB Prediction

^{2}(0.005 ≤ R

^{2}≤ 0.325) compared to Shepherdia canadensis (0.433 ≤ R

^{2}≤ 0.809, Table S6). In addition, Dasiphora fruticosa was the only species where stem length was not significantly related (p > 0.05) to the dependent variable of measured total AGB. For the multispecies shrub models (pooled for all shrub genera and species), the use of 1D predictor variables yielded the lowest model fits (RMSE) ranging from 262 (NLS) to 318 (LLRC) g for max stem length and 252 (NLS) to 388 (LRC) g for max basal diameter (Table 2). These results are in contrast with previous allometric models for boreal [11,16,17] or subtropical [27] shrubs, where the 1D variable basal diameter of the longest stem had provided the most accurate prediction of total AGB. However, 1D field variables, although related to the dependent variable (p < 0.001) (with the exception of max stem length of Dasiphora fruticosa), did not explain total AGB variability when considering all stems of the entire plant (0.228 ≤R

^{2}≤ 0.335, Table 2, Figure 3a,b). This was true for each genus/species model as well as the multispecies equation, with the exception of Sheperdia canadensis. For this species, max basal diameter was a similarly good predictor variable (R

^{2}= 0.809) to cross-sectional area (R

^{2}= 0.738) and volume (R

^{2}= 0.765, Table S6). The performance of total AGB models increased for all other genera and species as well as for all genera and species combined using the 3D predictor variable of volume, with R

^{2}ranging between 0.684 (Betula spp. and Dasiphora fruticosa) and 0.882 (Alnus spp.). The RMSEs for the multispecies models ranged from 141 (NLS) to 144 (LLR and LLRC) g with R

^{2}of ~0.790 using any of the three models (LLR, LLRC, and NLS; Table 2). Figure 3a–d shows the relationship between the three models for 1D, 2D, and 3D variables for the multispecies shrub AGB.

^{2}= 0.769, RMSE 148.6 g) compared to LLRC or NLS (Table 2). Decreasing exponents (holding all else equal) can be explained by the nature of allometric scaling between 1D, 2D, and 3D measurements of a plant to its mass (a 3D attribute) via a power function. Assuming no change in the multiplier (β) (which might be considered analogous to a density attribute), scaling from a 1D measurement to a 3D property requires a higher exponent (α) compared to scaling from a 2D or 3D measurement (Table 3). Therefore, predictions that are extrapolated from lower to higher dimensions contain more inherent model-based uncertainty than predictions requiring no dimensional extrapolation. However, field volume observations consisted of three single measurements and therefore might contain a high overall measurement uncertainty compared to a single 1D measurement. The exact quantity of model vs. field measurement error propagation is unknown, but the net outcome of the tests performed shows that 3D volume produced the highest AGB model accuracies, followed by 2D and then 1D models.

#### 3.2. Comparison of Regression Models for Shrub Total AGB Prediction

^{2}= 0.790 (NLS), RMSE 144.08 g, R

^{2}= 0.788 (LLR), RMSE = 144.12 g, R

^{2}= 0.788 (LLRC); Table 2). However, although NLS produces slightly better model fits, nonlinear models require an even variance of errors across the domain of the predictor variable in order to perform valid comparisons of model uncertainties and regression coefficients amongst datasets (e.g., [23]). Residual analysis of our models showed that errors were free of heteroscedasticity. However, when models are transferred to different areas and data, we recommend using the LLRC-based models. This is because biomass data can contain natural heteroscedastic variation. Heteroscedasticity needs to be accounted for in the model development to ensure that model results do not contain bias [25]. For example, Mascaro et al. [25] reported a bias of ~100% overestimation when extending predicted small tree (diameter at breast height (DBH) range 2–12 cm) aboveground biomass to stand level biomass using NLS. For model transfer purposes, we have provided regression coefficients and error statistics not only for our best models based on NLS but also for our LLRC models (Supplementary Material 2, Table S3).

#### 3.3. Comparison of 1D and 2D Variables for Tree Total AGB Prediction

^{2}, Table 4, Figure 4), while inclusion of stem diameter at 0.3 m, alternatively cross-sectional area at 0.3 m, improved predictions by 87% using LLRC or NLS (Table 4).

#### 3.4. Comparison of Regression Models for Tree Total AGB Prediction

^{2}, and lowest total percentage error compared to measured biomass (<−5%–4%), while the dependent and independent variables were significantly related (p < 0.001, Table 4 and Table S7). LLR predictions resulted in the highest RMSE, similar R

^{2}, and highest total percentage error relative to the measured biomass compared to LLRC and NLS for each genus/species and for all data combined. The single exception was for predicting total AGB for Picea spp. based on stem length (Table S7). Using LLR, the prediction based on stem length resulted in an underestimation of total AGB of −51% and an underestimation of −20% when using diameter or cross-sectional area, respectively, as input variable for the multispecies models. LLRC had comparably lower RMSE, similar R

^{2}, and underestimated total multispecies AGB by −6% (stem length) to <−10% (diameter at 0.3 m, cross-sectional area at 0.3 m). In order to address potential heteroscedasticity effects, we have provided the regression coefficients for both NLS and LLRC models with cross-sectional area (measured at 0.3 m stem length) as predictor variable (Supplementary Material 2, Table S4).

#### 3.5. Comparison of Regression Models for General Shrub and Tree Total AGB Prediction

^{2}≥ 0.770, p < 0.001). This was determined using the 2D independent variable of cross-sectional area, measured at the base for shrubs and at 0.3 m stem length for trees (Table 6, Figure 5). Relating modeled total AGB to measured total AGB showed no evident bias in the 2D-based prediction of combined shrub and tree AGB in comparison to 3D multispecies shrub and 2D multispecies tree AGB models. This is depicted in Figure 6, which shows modeled AGB in relation to measured AGB of the 2D general shrub and tree AGB model (Figure 6e,f) in comparison to the 3D multispecies shrub (Figure 6a,b) and 2D multispecies tree (Figure 6b,c) AGB models. Similar to the multispecies shrub models, NLS achieved the lowest RMSE and highest R

^{2}(RMSE = 94.80 g, R

^{2}= 0.776). The RMSE of model LLR increased by 53% (RMSE = 202.17 g, R

^{2}= 0.770) and by 54% for the LLRC model (RMSE = 206.37 g, R

^{2}= 0.770). Compared to Ali et al. [27], who derived best model fits for combined shrub and tree AGB prediction using diameter of the longest stem and total plant height combined, our model results show that AGB of boreal plants can be predicted with a simpler one-variable model using cross-sectional area. For our AGB models based on cross-sectional area, the exponent $\alpha $ was greater for the 1D model based on stem length and close to unity for the 2D model cross-sectional area at 0.3 m (Table 7). However, stem length of shrubs and trees was also weakly related to measured total AGB (p < 0.001, Table 6) and thus represents an alternative to cross-sectional area, which has potential for use in rapid field measurement or non-invasive observation situations (e.g., remote sensing via airborne lidar) where it may be acceptable to trade accuracy at the individual sample-level for greater overall population representation. For model transfer purposes, we recommend the use of LLRC regression coefficients and correction factor in order to address heteroscedasticity.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Field locations of destructive sampling of shrub and short-stature tree AGB in peatlands and upland forest ecosystems. Field locations are distributed across the (

**b**) mid-boreal Taiga Plains and high-boreal Taiga Shield ecoregions of the Northwest Territories, Canada.

**Figure 2.**(

**a**) Example of a transect traversing a burned upland forest into peatland (fire year 2015); (

**b**,

**c**) illustration of a Betula glandulosa shrub individual growing along the transect before and after destructive sampling, respectively.

**Figure 3.**LLRC, LLR, and NLS model fits for each 1D (

**a**,

**b**), 2D (

**c**), and 3D (

**d**) predictor variable utilized to model total AGB for multispecies shrubs.

**Figure 4.**LLRC, LLR, and NLS model fits for each 1D (

**a**,

**b**), and 2D (

**c**) predictor variable utilized to model total AGB for multispecies trees.

**Figure 5.**LLRC, LLR, and NLS model fits for the 2D predictor variable utilized to model total AGB for general shrubs and trees: (

**a**) stem length and (

**b**) cross-sectional area.

**Figure 6.**Measured total AGB related to modeled total AGB using LLRC and NLS for the 3D-based equations for multispecies shrubs (

**a**,

**b**) and the 2D-based equations for multispecies trees (

**c**,

**d**) as well as general shrubs and trees (

**e**,

**f**).

Ecozone | Transects (Samples) |
---|---|

Shrub Samples | |

Taiga Plains | 31 (127) |

Taiga Shield | 15 (79) |

Tree Samples | |

Taiga Plains | 20 (105) |

Total | 66 (311) |

Model | Dimension | Input Variable | Total Percentage Error (%) | RMSE (g) | R^{2} |
---|---|---|---|---|---|

Linear logarithmic regression (LLR) | 1D | Max stem length | −32 | 276.12 | 0.237 |

1D | Max basal Diameter | −26 | 293.20 | 0.228 | |

2D | Cross-sectional Area basal | −4 | 233.46 | 0.534 | |

3D | Volume | −11 | 144.08 | 0.788 | |

Linear logarithmic regression with correction (LLRC) | 1D | Max stem length | 23 | 317.95 | 0.237 |

1D | Max basal Diameter | 24 | 388.00 | 0.228 | |

2D | Cross-sectional Area basal | 13 | 263.55 | 0.534 | |

3D | Volume | 4 | 144.12 | 0.788 | |

Nonlinear least squares regression (NLS) | 1D | Max stem length | 4 | 262.22 | 0.273 |

1D | Max basal Diameter | 8 | 251.68 | 0.335 | |

2D | Cross-sectional Area basal | 10 | 190.75 | 0.621 | |

3D | Volume | 1 | 141.04 | 0.790 |

**Table 3.**Regression coefficients for each 1D, 2D, and 3D model to predict total AGB for multispecies shrubs with LLRC and NLS.

Variable | LLRC β | NLS β | LLRC α | NLS α | |
---|---|---|---|---|---|

Multispecies shrubs | Max stem length | 100.988 | 192.186 | 2.262 | 1.530 |

Max basal diameter | 100.484 | 203.796 | 2.642 | 1.486 | |

Cross-sectional area | 106.911 | 177.614 | 1.075 | 0.723 | |

Volume | 233.224 | 272.116 | 0.829 | 0.778 |

**Table 4.**Model performance for multispecies tree total AGB prediction using 1D and 2D input parameters (all p values < 0.001).

Model | Dimension | Input Variable | Total Percentage Error (%) | RMSE (g) | R^{2} |
---|---|---|---|---|---|

LLR | 1D | Stem length | −51 | 529.08 | 0.285 |

1D | Diameter at 0.3 m | −20 | 188.71 | 0.987 | |

2D | Cross-sectional area at 0.3 m | −20 | 188.41 | 0.987 | |

LLRC | 1D | Stem length | −6 | 484.83 | 0.285 |

1D | Diameter at 0.3 m | −10 | 134.00 | 0.987 | |

2D | Cross-sectional area at 0.3 m | −9 | 133.68 | 0.987 | |

NLS | 1D | Stem length | −1 | 482.77 | 0.286 |

1D | Diameter at 0.3 m | 2 | 62.25 | 0.988 | |

2D | Cross-sectional area at 0.3 m | 2 | 62.25 | 0.988 |

Variable | LLRC β | NLS β | LLRC α | NLS α | |
---|---|---|---|---|---|

Multispecies trees | Stem length | 28.962 | 45.8224 | 2.132 | 2.380 |

Diameter at 0.3 m | 57.111 | 59.0721 | 2.272 | 2.479 | |

Cross-sectional area at 0.3 m | 75.189 | 79.6818 | 1.136 | 1.240 |

**Table 6.**Model performances for combined general shrub and tree total AGB prediction (all p values < 0.001).

Model | Dimension | Input Variable | Total Percentage Error (%) | RMSE (g) | R^{2} |
---|---|---|---|---|---|

LLR | 1D | Stem length | −44.15 | 378.72 | 0.249 |

2D | Cross-sectional area | −11.17 | 202.17 | 0.770 | |

LLRC | 1D | Stem length | 0.03 | 381.27 | 0.249 |

2D | Cross-sectional area | 0.02 | 206.37 | 0.770 | |

NLS | 1D | Stem length | 22.53 | 292.51 | 0.500 |

2D | Cross-sectional area | 2.28 | 94.80 | 0.776 |

**Table 7.**Regression coefficients to predict total AGB for general shrubs and trees with LLRC and NLS.

Variable | LLRC β | NLS β | LLRC α | NLS α | |
---|---|---|---|---|---|

General shrubs and trees | Stem length | 63.944 | 142.299 | 1.816 | 1.447 |

Cross-sectional area | 94.917 | 128.802 | 1.084 | 0.979 |

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**MDPI and ACS Style**

Flade, L.; Hopkinson, C.; Chasmer, L.
Allometric Equations for Shrub and Short-Stature Tree Aboveground Biomass within Boreal Ecosystems of Northwestern Canada. *Forests* **2020**, *11*, 1207.
https://doi.org/10.3390/f11111207

**AMA Style**

Flade L, Hopkinson C, Chasmer L.
Allometric Equations for Shrub and Short-Stature Tree Aboveground Biomass within Boreal Ecosystems of Northwestern Canada. *Forests*. 2020; 11(11):1207.
https://doi.org/10.3390/f11111207

**Chicago/Turabian Style**

Flade, Linda, Christopher Hopkinson, and Laura Chasmer.
2020. "Allometric Equations for Shrub and Short-Stature Tree Aboveground Biomass within Boreal Ecosystems of Northwestern Canada" *Forests* 11, no. 11: 1207.
https://doi.org/10.3390/f11111207