# Allometric Biomass Models for European Beech and Silver Fir: Testing Approaches to Minimize the Demand for Site-Specific Biomass Observations

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## Abstract

**:**

## 1. Introduction

_{i}) as a function of easy-to-measure independent variable x

_{i}(e.g., diameter at breast height, D, and/or tree height, H):

## 2. Materials and Methods

#### 2.1. Materials

#### 2.1.1. Study Site

#### 2.1.2. Biomass Datasets

#### Dataset #1

#### Dataset #2

#### Dataset #3

#### Dataset #4

#### 2.1.3. Inventory Plot

#### 2.2. Development of Allometric Biomass Models

#### 2.2.1. Linear Regression Model on Log-Transformed Data (LM)

#### 2.2.2. Random Intercept Models (RIM)

#### 2.2.3. Bayesian Models

#### 2.3. Evaluation of Calibration Approaches

^{th}tree, $\mathrm{ln}{\left(AGB\right)}_{i}$ is the observed ln(AGB) of i

^{th}tree, and n is the sample size.

- (1)
- For the k
^{th}replication (K = 5000, K is the total number of replications), a set of allometric model parameters and residuals were sampled from a multivariate normal distribution and a univariate normal distribution, respectively.- (a)
- Sampling a residual value from a normal distribution with the mean zero and standard deviation equal to residual standard error of the allometric model;
- (b)
- Sampling a set of model parameter values from a bivariate normal distribution (for models using only D as predictor of AGB) or a trivariate normal distribution (for models based on both D and H to predict AGB);
- (c)
- Calculate the predicted ln(AGB) for each tree within the 1 ha inventory plot (Section 2.1.3), based on the model parameters sampled at step 1.b and the residual sampled at step 1.a;
- (d)
- Back transform the predicted ln(AGB), using a correction factor (CF) calculated as in Section 2.2.1;
- (e)
- Calculate the total plot AGB by addition of individual tree predictions;

- (2)
- Steps (1.a) to (1.e) were repeated for a number of K = 5000 times to calculate:
- (a)
- Mean predicted plot biomass, as the mean of values obtained at step 1.e;
- (b)
- Standard error of the mean (values at step 1.e), which, because of using a single plot, equals the standard deviation of the sample mean.

_{rep}= 5000, to stabilize the mean and the standard error. The values further reported are (i) the mean of mean values at step 2.a and (ii) the mean of the standard error values at step 2.b.

#### 2.4. Data Processing

## 3. Results

#### 3.1. Allometric Biomass Models

^{2}) for models based on single predictor were 0.9972 for European beech and 0.9862 for silver fir, whereas for models based on D and H, the R

^{2}increased to 0.9986 for European beech and to 0.9921 for silver fir. The reason for these large coefficients of determination is related to the wide D range of the sample trees (Table 1).

#### 3.2. Comparison of Biomass Estimates on 1 ha Sample Plot

^{−1}for RIM using the reduced sample (Dataset #4) and based on single predictor and 542,758 kg ha

^{−1}for simple regression model based on D only (Table 6). Therefore, the plot AGB estimates varied more when a single predictor of AGB was used. On the other hand, the models based on both D and H produced more stable AGB estimates at plot level (i.e., differences between estimates resulted from different fitting approaches were smaller). In addition, the standard errors of mean AGB were smaller when both D and H were used as predictors. Overall, the Bayesian models resulted in larger standard errors compared to both RIM and LM.

^{−1}, whereas the Bayesian model (based on Dataset #2) predicted 460,279 kg ha

^{−1}. However, both values are smaller compared to AGB estimate resulted from LM, by approximately 9% (Table 6). At the species level, the differences were larger for silver fir (approximately 15%–17%) than for European beech (approximately 6%).

^{−1}). Nevertheless, at the species level, the situation was similar: the differences to LM estimate were larger for silver fir (approximately 4%–8%) than for European beech (approximately 0.5%–2%).

## 4. Discussion

- (a)
- The H-D ratio, which should be checked in advance. As we observed in our analysis with silver fir trees, the H-D ratio can affect the parameter estimates, which affect further the performance of small trees sample approach. Therefore, the user should check whether the H-D ratio decreases relatively linearly with the increase in tree size.
- (b)
- Either a random intercept model or a Bayesian model can be used with the reduced sample approach. Preference to one of the methods can be decided based on the raw data availability. Nevertheless, access to raw observations should not be an issue given the increasing trend in publication of biomass datasets, e.g., [44,55,56,57,58].
- (c)
- The generic biomass sample should contain as many species-specific observations as possible, including very large trees (D-range should match that of the local population for which the models are developed).
- (d)
- The reduced sample of small trees should contain a large enough number of trees to calibrate mainly the intercept; at least 6–7 trees should be used (the greater the number, the better the result). It is recommended that trees with D < 5 cm should not be used with the reduced sample approach, since the allometry of very small trees can be affected by the competition with herbaceous plants.
- (e)
- Using the reduced sample approach should always be performed using no less than D and H as predictors; other additional predictors can be used, because using both variables, the biomass estimates were more precise.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The observed ln(AGB) against the predicted ln(AGB). Note: AGB is the aboveground biomass; the values are presented in log-scale for two species, two predictor combination, and five fitting approaches. The data used for validation is from Dataset #1. LM is the linear regression model on log-transformed data (Section 2.2.1); RIM is the random intercept model (Section 2.2.2); ln(D) is the logarithm of diameter at breast height; ln(H) is the logarithm of tree height; ln(AGB) is the logarithm of aboveground biomass; RMSD is the root mean squared deviation (Section 2.3).

**Figure 2.**The Bland-ltman plots comparing individual tree AGB predictions. Note: the mean differences are calculated as the mean of relative individual differences (not differences between sum of individual AGB estimates).

**Figure 3.**The H-D ratio by D (H is the tree height; D is the diameter at breast height) (

**a**) and the relationship between H and D in log-scale (

**b**), for European beech and silver fir (site-specific biomass datasets). Note: the plotted trees are from Dataset #1.

**Table 1.**The characteristics of site-specific sample trees (full sample). Note: D is the diameter at breast height; H is the tree height; AGB is the aboveground biomass.

Characteristic | European Beech | Silver Fir |
---|---|---|

Sample size | 15 | 14 |

D range (cm) | 5.7–86.3 | 6.3–92.6 |

D mean (standard deviation) (cm) | 32.8 (26.1) | 35.8 (27.0) |

H range (m) | 6.2–40.3 | 3.5–43.5 |

H mean (standard deviation) (cm) | 22.8 (12.5) | 21.2 (13.8) |

AGB range (kg) | 6.0–8447.1 | 4.2–4042.9 |

AGB mean (standard deviation) (kg) | 1561.4 (2382.7) | 1034.6 (1316.9) |

Characteristic | European Beech | Silver Fir |
---|---|---|

Sample size | 7 | 7 |

D range (cm) | 5.7–20.0 | 6.3–27.9 |

D mean (standard deviation) (cm) | 11.1 (5.8) | 14.2 (7.9) |

H range (m) | 6.2–19.8 | 3.5–19.8 |

H mean (standard deviation) (m) | 11.4 (5.2) | 9.3 (5.8) |

AGB range (kg) | 6.0–182.2 | 4.2–358.9 |

AGB mean (standard deviation) (kg) | 65.1 (73.9) | 84.2 (128.8) |

Characteristic | Generic Dataset | Dataset #3 | ||
---|---|---|---|---|

European Beech | Silver Fir | European Beech | Silver Fir | |

Sample size | 144 | 102 | 159 | 116 |

D range (cm) | 5.2–62.1 | 5.1–64.0 | 5.2–86.3 | 5.1–92.6 |

H range (m) | 9.2–33.0 | 4.1–28.9 | 6.2–40.3 | 3.5–43.5 |

AGB range (kg) | 6.6–3116.2 | 7.0–1652.3 | 6.0–8447.1 | 4.2–4042.9 |

Number of sites | 10 | 10 | 11 | 11 |

References | [35,44] | [35,44] and this study |

Characteristic | European Beech | Silver Fir |
---|---|---|

Sample size | 151 (i.e., 144 + 7) | 109 (i.e., 102 + 7) |

D range (cm) | 5.2–62.1 | 5.1–64.0 |

H range (m) | 6.2–33.0 | 3.5–28.9 |

AGB range (kg) | 6.0–3116.2 | 4.2–358.9 |

Number of sites | 11 | 11 |

**Table 5.**The parameters of allometric biomass models predicting tree aboveground biomass (AGB), by fitting approach and predictor combination. Standard errors of model parameters are presented in parenthesis. RSE is the residual standard error; CF is the correction factor and was calculated as in Equations (4) or (5); LM is the linear regression model on log-transformed data (Section 2.2.1); RIM is the random intercept model (Section 2.2.2); ln(D) is the logarithm of diameter at breast height; ln(H) is the logarithm of tree height.

Fitting Approach | Dataset | Predictors | Model Form | ${\mathit{\beta}}_{0}\left(\mathit{S}\mathit{E}\right)$ | ${\mathit{\beta}}_{1}\left(\mathit{S}\mathit{E}\right)$ | ${\mathit{\beta}}_{2}\left(\mathit{S}\mathit{E}\right)$ | RSE | CF |
---|---|---|---|---|---|---|---|---|

European Beech | ||||||||

LM | #1 | ln(D) | Equation (2) | −2.6634 (0.1254) | 2.6368 (0.0384) | N.A. | 0.1337 | 1.0089 |

ln(D), ln(H) | Equation (3) | −3.1632 (0.1761) | 2.1468 (0.1489) | 0.6909 (0.2060) | 0.1000 | 1.0050 | ||

RIM | #3 | ln(D) | Equation (2) | −2.1312 (0.0901) | 2.4714 (0.0253) | N.A. | 0.1712 | 1.0148 |

ln(D), ln(H) | Equation (3) | −3.0039 (0.1389) | 2.1151 (0.0495) | 0.6733 (0.0845) | 0.1450 | 1.0106 | ||

#4 | ln(D) | Equation (2) | −2.1625 (0.0997) | 2.4368 (0.0284) | N.A. | 0.1683 | 1.0143 | |

ln(D), ln(H) | Equation (3) | −2.9793 (0.1593) | 2.1191 (0.0511) | 0.6512 (0.0916) | 0.1474 | 1.0109 | ||

Bayesian model | #1 | ln(D) | Equation (2) | −2.1768 (0.1148) | 2.4884 (0.0345) | N.A. | 0.2042 | 1.0211 |

ln(D), ln(H) | Equation (3) | −3.0637 (0.1076) | 2.1497 (0.0445) | 0.6553 (0.0605) | 0.1091 | 1.0060 | ||

#2 | ln(D) | Equation (2) | −2.1456 (0.1115) | 2.4349 (0.0398) | N.A. | 0.2363 | 1.0283 | |

ln(D), ln(H) | Equation (3) | −2.9856 (0.1508) | 2.1347 (0.0525) | 0.6324 (0.0680) | 0.1475 | 1.0109 | ||

Silver Fir | ||||||||

LM | #1 | ln(D) | Equation (2) | −3.4141 (0.3106) | 2.6997 (0.0922) | N.A. | 0.2965 | 1.0449 |

ln(D), ln(H) | Equation (3) | −2.9687 (0.2907) | 1.3301 (0.4839) | 1.4460 (0.5051) | 0.2344 | 1.0278 | ||

RIM | #3 | ln(D) | Equation (2) | −2.4756 (0.1106) | 2.4219 (0.0346) | N.A. | 0.2033 | 1.0209 |

ln(D), ln(H) | Equation (3) | −2.8079 (0.0984) | 1.7737 (0.0708) | 0.8745 (0.0920) | 0.1624 | 1.0133 | ||

#4 | ln(D) | Equation (2) | −2.5086 (0.1205) | 2.3561 (0.0375) | N.A. | 0.1824 | 1.0168 | |

ln(D), ln(H) | Equation (3) | −2.7987 (0.1126) | 1.8076 (0.0751) | 0.8040 (0.0992) | 0.1548 | 1.0121 | ||

Bayesian model | #1 | ln(D) | Equation (2) | −2.3284 (0.1598) | 2.3859 (0.0471) | N.A. | 0.3965 | 1.0818 |

ln(D), ln(H) | Equation (3) | −2.7679 (0.1609) | 1.9264 (0.0771) | 0.6847 (0.0897) | 0.2788 | 1.0396 | ||

#2 | ln(D) | Equation (2) | −2.3553 (0.1591) | 2.3316 (0.0552) | N.A. | 0.4064 | 1.0861 | |

ln(D), ln(H) | Equation (3) | −2.6917 (0.1707) | 1.9063 (0.0810) | 0.6305 (0.0969) | 0.3192 | 1.0523 |

**Table 6.**Mean biomass per hectare and its standard error by model type and predictor combinations, presented for both species and for each individual species.

Fitting Approach | Biomass Dataset | Model Type | Both Species | European Beech | Silver Fir | |||
---|---|---|---|---|---|---|---|---|

Mean (kg/ha) | SE (kg/ha) | Mean (kg/ha) | SE (kg/ha) | Mean (kg/ha) | SE (kg/ha) | |||

LM | #1 | Equation (2) | 542,758 | 71,118 | 410,860 | 57,906 | 131,881 | 41,260 |

Equation (3) | 505,277 | 49,181 | 378,396 | 39,159 | 126,881 | 29,754 | ||

RIM | #3 | Equation (2) | 455,408 | 68,745 | 350,901 | 64,759 | 104,514 | 23,054 |

Equation (3) | 484,465 | 59,552 | 368,402 | 56,126 | 115,934 | 19,398 | ||

#4 | Equation (2) | 370,296 | 55,126 | 293,328 | 52,952 | 76,972 | 15,297 | |

Equation (3) | 459,387 | 57,947 | 354,853 | 55,409 | 104,537 | 16,992 | ||

Bayesian | #1 | Equation (2) | 487,053 | 100,743 | 364,564 | 77,672 | 122,523 | 64,197 |

Equation (3) | 497,354 | 76,448 | 376,447 | 68,229 | 120,921 | 34,488 | ||

#2 | Equation (2) | 438,556 | 175,137 | 318,630 | 129,117 | 119,831 | 118,287 | |

Equation (3) | 460,279 | 76,274 | 352,865 | 61,428 | 107,410 | 45,214 |

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**MDPI and ACS Style**

Dutcă, I.; Zianis, D.; Petrițan, I.C.; Bragă, C.I.; Ștefan, G.; Yuste, J.C.; Petrițan, A.M.
Allometric Biomass Models for European Beech and Silver Fir: Testing Approaches to Minimize the Demand for Site-Specific Biomass Observations. *Forests* **2020**, *11*, 1136.
https://doi.org/10.3390/f11111136

**AMA Style**

Dutcă I, Zianis D, Petrițan IC, Bragă CI, Ștefan G, Yuste JC, Petrițan AM.
Allometric Biomass Models for European Beech and Silver Fir: Testing Approaches to Minimize the Demand for Site-Specific Biomass Observations. *Forests*. 2020; 11(11):1136.
https://doi.org/10.3390/f11111136

**Chicago/Turabian Style**

Dutcă, Ioan, Dimitris Zianis, Ion Cătălin Petrițan, Cosmin Ion Bragă, Gheorghe Ștefan, Jorge Curiel Yuste, and Any Mary Petrițan.
2020. "Allometric Biomass Models for European Beech and Silver Fir: Testing Approaches to Minimize the Demand for Site-Specific Biomass Observations" *Forests* 11, no. 11: 1136.
https://doi.org/10.3390/f11111136