# Allometric Equations for Estimating Carbon Stocks in Natural Forest in New Zealand

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}concentrations through the effects of land use change and forest management activities on the terrestrial biomass carbon pool [1]. A large number of biomass data sets and allometric equations have recently been compiled to predict above ground carbon in forest trees [2,3,4,5], motivated by the need to estimate carbon stocks for national greenhouse gas balance estimates [6]. It is important that accurate allometric equations are available to estimate carbon stocks from national plot inventory data. New Zealand has approximately 8.1 million ha of natural forest [7]. Most natural forest types in New Zealand consist of species rich mixtures of hardwood and softwood trees and tree ferns [8]. Allometric equations have been based on diameter at breast height (DBH), or DBH and height (H); however, DBH, H and density were included in allometric equations developed previously in New Zealand to estimate above ground biomass from plot data [9,10] because tree height H is not proportional to DBH [11] and wood density differs widely among species.

- 1. Above ground biomass (AGB) for stems ≥2.5 cm DBH;
- 2. Below ground biomass in live roots (BGB);
- 3. Dead wood (includes standing dead trees, spars, and dead roots, and coarse woody debris ≥10 cm in diameter on the forest floor);
- 4. Litter (includes dead leaves, reproductive parts, and woody debris <10 cm in diameter), and fermenting (F) and humus (H) material on the forest floor, and
- 5. Mineral soil organic carbon to 30 cm depth.

## 2. Methods

#### 2.1. Sites

#### 2.1.1. Whirinaki Forest

#### 2.1.2. Kaimanawa Forest

#### 2.1.3. Mt Maungatautari

#### 2.1.4. Taranaki

#### 2.1.5. Hunua Forest

^{2}/ha. Ring counts indicated a stand age of approximately 130 years. Over 90% of the stems and 93% of the basal area comprised kauri, the remaining trees being tanekaha (Phyllocladus trichomanoides D. Don) [20].

#### 2.1.6. Maimai

#### 2.1.7. Whakarewarewa Forest

#### 2.2. Biomass Procedures

#### 2.2.1. Trees

#### 2.2.2. Tree Ferns

#### 2.3. Supplementary Wood Density Data

_{stem}is the survey adjusted ratio estimator of mean whole stem and branch ≥10 cm density; d

_{stem,bio}is mean whole stem and branch ≥10 cm density of the biomass trees; d

_{ow,bio}is the mean breast height outerwood basic density of the biomass trees; and d

_{ow,all}is the mean breast height outerwood basic density of the national survey sample. Ratios (d

_{stem,bio}/d

_{ow,bio}) were developed by species and across species for both the outer 0–5 cm and 5–15 cm core depths.

- 1. If breast height basic density at 5–15 cm depth was measured, the ratio estimator was applied (following Beets et al. [17]).
- 2. If merchantable log basic density was measured, the ratio estimator was multiplied by 1.03 (1.03 was derived from the wood density database and converts breast height basic density at 0–5 cm depth to breast height basic density at 5–15 cm depth. It should be noted that merchantable log basic density and breast height basic density at 0–5 cm depth have a 1:1 conversion, based on the database).
- 3. If breast height outerwood (0–5 cm) basic density was measured, the ratio estimator was multiplied by 1.03.

#### 2.4. Allometric Equations for Live Trees and Tree Ferns

_{sp}replacing a when testing for species effects; X for trees is DBH (cm) or DBH

^{2}× H (cm

^{2}m); Y is stem plus large branch over bark volume

_{ob}(m

^{3}tree

^{−1}) or component carbon weights (kg tree

^{−1}).

_{sp}) and a common “b” parameter were fitted (which is equivalent to fitting separate intercepts and a common slope on the log-log scale). Species were compared using an approximate χ

^{2}test. This analysis helps to inform whether tree carbon should be estimated from allometric volume equations coupled with density or estimated directly from DBH and H.

#### 2.5. Allometric Equations for Dead Stems and Spars

_{1}, b

_{2}and β

_{1}–β

_{5}are parameters; V

_{stem}is intact stem wood plus bark volume (excluding volume in large branches); Dob is stem diameter over bark at height L; and DBH and H are as defined previously. The parameters of the volume function (b

_{1}and b

_{2}) were fitted using log transformations with b

_{1}adjusted for logarithmic bias. The taper function (Equation 4) is a 4th order polynomial with an additional higher order term to account for butt swell [29]. The parameters of the taper function (β

_{1}–β

_{5}) were fitted subject to a constraint which ensured that when integrated over total tree height the function yielded the same volume as Equation 3 (see [28,29] for details). An equation for estimating the volume of the spar was then obtained by integrating Equation 4 from the base of the stem to a height representing the top of an intact dead tree or a truncated spar.

## 3. Results

#### 3.1. Allometric Equations

#### 3.1.1. Characteristics of Biomass Trees and Tree Ferns

**Table 1.**Tree and tree fern species sampled at biomass study sites throughout New Zealand, sample size (n for stem, crown, root), and mean diameter at breast height (DBH) and height (H).

Site | Species | Sample size | ||||
---|---|---|---|---|---|---|

Means | ||||||

n_{stem} | n_{crown} | n_{root} | DBH (cm) | H (m) | ||

Whirinaki | B. tawa | 10 | 0 | 36.9 | 26.5 | |

D. cupressinum | 18 | 0 | 95.8 | 50.1 | ||

D. dacrydioides | 2 | 0 | 140.0 | 52.0 | ||

P. taxifolia | 12 | 0 | 75.3 | 38.8 | ||

P. totara | 1 | 0 | 107.6 | 46.7 | ||

P. ferruginea | 7 | 0 | 49.2 | 34.3 | ||

Kaimanawa | N. fusca | 16 | 0 | 39.5 | 24.1 | |

N. menziesii | 12 | 1 | 59.3 | 27.2 | ||

Hunua | A. australis | 6 | 6 | 19.4 | 14.6 | |

Taranaki | A. australis | 20 | 10 | 33.2 | 23.1 | |

Maimai | N. fusca | 2 | 2 | 89.0 | 32.2 | |

N. truncata | 11 | 11 | 41.8 | 20.9 | ||

P. ferruginea | 1 | 1 | 32.0 | 23.9 | ||

W. racemosa | 36 | 36 | 8.6 | 7.8 | ||

Maungatautari/Manawahe | L. calicaris | 7 | 1 | 40.1 | 20.7 | |

H. arborea | 2 | 2 | 30.0 | 14.3 | ||

L. n. zealandiae | 6 | 0 | 40.0 | 18.0 | ||

K. excelsa | 4 | 0 | 35.5 | 19.5 | ||

B. tawa | 15 | 0 | 36.5 | 19.3 | ||

D. dacrydioides | 1 | 0 | 20.4 | 8.2 | ||

W. racemosa | 1 | 0 | 33.0 | - | ||

Maimai | C. smithii | 2 | 2 | 19.4 | 7.7 | |

Maungatautari | C. dealbata | 1 | 1 | 28.5 | 8.2 | |

Whakarewarewa | C. dealbata | 20 | 20 | 4 | 27.3 | 2.7 |

C. medullaris | 20 | 20 | 5 | 21.6 | 4.6 | |

C. smithii | 20 | 20 | 5 | 14.3 | 2.9 | |

D. squarrosa | 20 | 20 | 4 | 15.9 | 3.2 |

#### 3.1.2. Stem Volume and Carbon in Live Trees

^{2}coefficient.

**Table 2.**Mixed-species allometric equations for estimating stem and large branch volume and component carbon for natural forest trees, sample size (n), and deviance explained. The equation form is Y = aX

^{b}.

Y | Stem & Branch (≥10 cm) Volume_{ob} (m^{3}/tree) | Stem & Branch (≥10 cm) Carbon (kg/tree) | Branch (<10 cm) Carbon (kg/tree) | Foliage Carbon (kg/tree) |
---|---|---|---|---|

X | DBH^{2}·H (cm^{2}·m) | DBH^{2}·H (cm^{2}·m) | DBH (cm) | DBH (cm) |

a | 4.83 × 10^{−}^{5} | 1.62 × 10^{−2} | 1.75 × 10^{−2} | 1.71 × 10^{−2} |

b | 0.978 | 0.943 | 2.20 | 1.75 |

Deviance explained % | 98.9 | 98.8 | 96.7 | 85.2 |

Sample size n | 141 | 127 | 70 | 70 |

^{−5}for softwoods and 5.19 × 10

^{−5}for hardwoods, and will reflect the inclusion of large branches with stems. Species effects were not significant for estimating carbon in branches <10 cm; however, species effects were significant for foliage, due to the inclusion of naturally regenerated and planted pole kauri trees within the same regression equation as mature trees.

**Table 3.**Allometric equations for estimating stem and large branch volume and component carbon for natural forest trees with species specific parameters. Values in a column followed by the same letter do not differ significantly (least significant difference test with p = 0.05).

Y | Stem & Brch (≥10 cm) Volume_{ob} (m^{3}/tree) | Stem & Brch (≥10 cm) Carbon (kg/tree) | Branch (<10 cm) Carbon (kg/tree) | Foliage Carbon (kg/tree) | |||
---|---|---|---|---|---|---|---|

X | DBH^{2}·H (cm^{2}·m) | DBH^{2}·H (cm^{2}·m) | DBH (cm) | DBH (cm) | |||

b | 0.968 | 0.936 | 0.936 | 1.595 | |||

a_{sp} parameters for: | |||||||

A. australis | 5.39 | bc | 0.0157 | de | 0.0156 | 0.0541 | a |

B. tawa | 5.46 | ab | 0.0190 | bc | |||

D. cupressinum | 5.46 | b | 0.0182 | bcd | |||

D. dacrydioides | 4.90 | bc | 0.0105 | f | |||

H. arborea | 4.65 | bc | 0.0141 | def | 0.0274 | 0.0045 | d |

K. excelsa | 4.67 | bc | 0.0170 | bcde | |||

L. calicaris | 4.33 | ac | 0.0135 | ef | 0.0147 | 0.0071 | cd |

L. n. zealandiae | 4.84 | bc | 0.0116 | f | |||

N. fusca | 5.47 | b | 0.0189 | abcd | 0.0147 | 0.0090 | cd |

N. menziesii | 6.18 | a | 0.0256 | a | 0.0220 | 0.0474 | a |

N. truncata | 5.14 | bc | 0.0196 | ab | 0.0219 | 0.0167 | bc |

P. ferruginea | 5.53 | ab | 0.0208 | ab | 0.0138 | 0.0132 | bc |

P. taxifolia | 5.58 | ab | 0.0222 | ab | |||

P. totara | 5.45 | abc | 0.0129 | cdef | |||

W. racemosa | 5.15 | bc | 0.0168 | cde | 0.0172 | 0.0186 | b |

a_{sp} parameters ×10^{−5} | |||||||

Deviance explained % | 99.1 | 99.3 | 99.3 | 95.0 | |||

Sample size n | 141 | 127 | 70 | 70 | |||

Test for species differences | F_{14,125} = 1.94, p = 0.028 | F_{14,111} = 5.91, p ≤ 0.0001 | F_{14,111} = 1.35, p = 0.22 | F_{7,61 }= 17.13, p ≤ 0.0001 |

#### 3.1.3. Stem Volume in Standing Dead Trees and Spars

**Table 4.**Sample size (n) and dimensions of biomass trees with stem sectional measurements suitable for developing volume/taper functions.

n | Minimum | Mean | Maximum | Standard Deviation | |
---|---|---|---|---|---|

DBH | 115 | 5.3 | 52.9 | 142.0 | 34.7 |

H | 115 | 7.7 | 27.9 | 59.2 | 12.7 |

Stem volume_{ob} | 115 | 0.011 | 5.46 | 32.1 | 7.91 |

Form factor | 115 | 0.254 | 0.485 | 0.691 | 0.078 |

_{stem}of an intact stem of height H was estimated as:

_{dead}was obtained by integrating the fitted Equation 4 along the stem length up to the measured height:

_{dead})/H.

_{dead}, x is set to zero.

#### 3.1.4. Caudex Volume and Carbon in Tree Ferns

**Table 5.**Mixed-species allometric equations for estimating caudex volume and component carbon for tree ferns, sample size (n), and variance explained. The equation form is Y = aX

^{b}.

Y | Caudex Volume (m^{3}/tree) | Caudex & Frond Carbon (kg/tree) | Caudex Carbon (kg/tree) | Root Carbon (kg/tree) |
---|---|---|---|---|

X | DBH^{2}·H (cm^{2}·m) | DBH^{2}·H (cm^{2}·m) | DBH^{2}·H (cm^{2}·m) | DBH^{2}·H (cm^{2}·m) |

a | 1.34 × 10^{−5} | 2.70 × 10^{−3} | 1.10 × 10^{−3} | 6.99 × 10^{−4} |

b | 1.22 | 1.19 | 1.25 | 1.14 |

Deviance explained % | 93.1 | 91.6 | 91.9 | 81.0 |

n | 80 | 80 | 80 | 18 |

**Table 6.**Allometric equations for estimating caudex volume and component carbon for tree ferns with species specific parameters. Values in a column followed by the same letter do not differ significantly (least significant difference test with p = 0.05).

Y | Caudex Volume (m^{3}/tree) | Caudex & Frond Carbon (kg/tree) | Caudex Carbon (kg/tree) | Root Carbon (kg/tree) |
---|---|---|---|---|

X | DBH^{2}·H (cm^{2}·m) | DBH^{2}·H (cm^{2}·m) | DBH^{2}·H (cm^{2}·m) | DBH^{2}·H (cm^{2}·m) |

b | 1.26 | 1.06 | 1.23 | 1.20 |

a_{sp} parameters for: | ||||

C. medullaris | 1.06 × 10^{−5} a | 8.45 × 10^{−3} a | 1.32 × 10^{−3} ab | 4.05 × 10^{−4} b |

C. dealbata | 8.15 × 10^{−6} b | 7.59 × 10^{−3} a | 1.36 × 10^{−3} ab | 4.37 × 10^{−4} b |

C. smithii | 1.01 × 10^{−5} a | 5.31 × 10^{−3} c | 1.13 × 10^{−3} b | 3.38 × 10^{−4} b |

D. squarrosa | 1.03 × 10^{−5} a | 6.33 × 10^{−3} b | 1.48 × 10^{−3} a | 7.90 × 10^{−4} a |

Deviance explained % | 94.1 | 93.6 | 92.7 | 90.4 |

Sample size n | 80 | 80 | 80 | 18 |

Test for species differences | F_{3,75} = 4.12, p = 0.0093 | F_{3,75} = 7.87, p = 0.0001 | F_{3,75} = 2.80, p = 0.046 | F_{3,13} = 4.29, p = 0.026 |

#### 3.1.5. Carbon in Below Ground Biomass

#### 3.2. Ratio Estimators

**Table 7.**Number of biomass trees with the requisite data (N) and the calculated mean ratios of whole stem and branch density to breast height outerwood basic density by species and over all species. Values in a column followed by the same letter do not differ significantly (least significant difference test with p = 0.05).

Species | N | Ratio of whole stem to 0–5 cm BH basic density | Ratio of whole stem to 5–15 cm BH basic density | ||||
---|---|---|---|---|---|---|---|

Mean | s.e. | Mean | s.e. | ||||

A. australis | 20 | 0.969 | cde | 0.017 | 0.938 | a | 0.015 |

B. tawa | 5 | 0.891 | def | 0.035 | 0.898 | ac | 0.029 |

D. cupressinum | 10 | 1.083 | ab | 0.024 | 0.933 | ab | 0.021 |

D. dacrydioides | 2 | 0.909 | cdef | 0.055 | 0.807 | ac | 0.047 |

H. arborea | 2 | 0.831 | ef | 0.055 | 0.785 | ac | 0.047 |

K. excelsa | 2 | 0.976 | bcde | 0.055 | 0.993 | a | 0.047 |

L. calicaris | 7 | 0.846 | f | 0.029 | 0.844 | c | 0.025 |

L. n. zealandiae | 5 | 0.940 | cde | 0.035 | 0.913 | a | 0.029 |

N. fusca | 2 | 0.869 | def | 0.055 | 0.879 | ac | 0.047 |

N. menziesii | 2 | 1.010 | bcd | 0.055 | 0.886 | abc | 0.047 |

P. ferruginea | 6 | 1.018 | bc | 0.032 | 0.904 | ab | 0.027 |

P. taxifolia | 4 | 0.999 | bcd | 0.039 | 0.891 | ab | 0.033 |

P. totara | 1 | 1.234 | a | 0.077 | 0.806 | b | 0.066 |

All species | 68 | 0.968 | 0.013 | 0.905 | 0.009 | ||

Test for species differences | F_{12,55} = 5.92, p ≤ 0.0001 | F_{12,55} = 2.52, p = 0.010 |

## 4. Discussion

^{2}× H) and wood specific gravity, whereas the natural forest “b” parameter applies only to the volume index. Model error for our natural forest allometric equation and the wet tropical forest equation averaged +3.0% versus +2.0%, respectively for trees ≥2.5 cm DBH with complete above ground biomass data (Figure 1). Errors tended to be larger using specific subsets of our biomass dataset, for example +8.7% versus +6.8% for 13 beech trees, −6.6% versus −7.1% for 29 trees of the subcanopy species Weinmannia racemosa, +3.8% versus +1.9% for 45 hardwood trees, +2.4% versus +6.1% for 16 pole kauri. Model error for small trees ≥2.5 to ≤10 cm averaged −11.4% using the natural forest allometric equation and −8.3% using the wet tropical forest allometric equation; however, the impact on forest carbon stock prediction will likely be small. It is clear that both equations provided reasonable predictions of above ground biomass carbon across a range of groups of species and tree size classes.

**Figure 1.**Measured versus predicted above ground carbon for temperate hardwood/softwood trees (DBH range 2.8–142 cm) in New Zealand obtained using the natural forest equation for New Zealand and wet pan-tropical forest equation of Chave et al. [9]. Independent variables were breast height diameter, total height, and wood density.

#### 4.1. Natural Forest Allometric Equations for Live Trees and Tree Ferns

_{stem+br}

_{≥10}(m

^{3}/tree) is estimated from DBH (cm) and total height H (m) using the mixed-species allometric equation:

_{agtree}(kg/tree) is obtained by multiplying the estimated volume by the corresponding density assuming 50% of the mass is carbon, to which the mass of carbon in branches <10 cm in diameter over bark and foliage are added:

_{stem}(kg.m

^{−3}) is the species-specific stem plus branch ≥10 cm diameter density over bark from the ratio estimator.

_{agtrfn}(kg/tree) is therefore estimated directly from DBH (cm) and H (m) as follows:

#### 4.2. Equations for Standing Dead Stems and Logs

_{dead}(m) in two steps using compatible volume and taper equations. Firstly, over-bark volume of an intact live stem V

_{stem}(m

^{3}/tree) (i.e., excluding volume in large branches) with the same DBH and expected total height H as the dead stem is estimated as follows:

_{spar}(m

^{3}/tree), is estimated from the estimated intact live stem volume up to the measured spar height using:

_{dead})/H.

_{spar}is calculated as follows:

_{CWD}) from the large and small end diameters and lengths of log and branch sections, using the formula for a truncated cone, which when converted to carbon is as follows:

## 5. Conclusions

## Appendix 1

## Appendix 2

## Acknowledgments

## Conflict of Interest

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© 2012 by the authors; licensee MDPI, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Beets, P.N.; Kimberley, M.O.; Oliver, G.R.; Pearce, S.H.; Graham, J.D.; Brandon, A. Allometric Equations for Estimating Carbon Stocks in Natural Forest in New Zealand. *Forests* **2012**, *3*, 818-839.
https://doi.org/10.3390/f3030818

**AMA Style**

Beets PN, Kimberley MO, Oliver GR, Pearce SH, Graham JD, Brandon A. Allometric Equations for Estimating Carbon Stocks in Natural Forest in New Zealand. *Forests*. 2012; 3(3):818-839.
https://doi.org/10.3390/f3030818

**Chicago/Turabian Style**

Beets, Peter N., Mark O. Kimberley, Graeme R. Oliver, Stephen H. Pearce, J. Doug Graham, and Andrea Brandon. 2012. "Allometric Equations for Estimating Carbon Stocks in Natural Forest in New Zealand" *Forests* 3, no. 3: 818-839.
https://doi.org/10.3390/f3030818