# Allometric Equations for Predicting Culm Surface Area of Three Bamboo Species (Phyllostachys spp.)

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{2}emissions from these forests. The surface area (S) of bamboo culms (stems) is important for estimating culm respiration, a major component of carbon cycling in bamboo forests. However, few studies have attempted to formulate predictive equations for S. In this study, we developed allometric equations for predicting S in three bamboo species grown in Kyushu Island, western Japan: Phyllostachys pubescens Mazel ex Houz., P. bambusoides Sieb. et Zucc. and P. nigra var. henonis. We used a power equation between S and diameter at breast height (D) and a linear equation between S and D × total culm length (H). The results indicated that P. bambusoides and P. nigra shared common site-independent equations. In contrast, P. pubescens required species-specific equations due to interspecific variation in culm slenderness and tapering. We also found that D was a better predictive variable than DH when quantifying S because of its satisfactory predictive performance and simplicity. These findings will be beneficial for evaluating the contribution of bamboo forest ecosystems to carbon cycling.

## 1. Introduction

_{2}emissions from bamboo forests [14].

_{2}into the atmosphere through respiration of the culm (stem), leaves, branches and soil, including roots [10,14,15,16,17]. In particular, culm respiration (R

_{C}) is a major component of carbon cycling in bamboo forests, accounting for nearly 30% of aboveground respiration [18,19]. R

_{C}can be quantified by scaling the point-measured respiration rate per unit area (R

_{AREA}) up to the individual- or stand-level with the culm surface area (S), i.e., R

_{C}= S × R

_{AREA}. This convenient approach is widely used to quantify the stem respiration of a tree as well as R

_{C}[20,21,22,23].

_{AREA}can increase errors in the R

_{C}estimate [24]. For bamboo, a reliable R

_{AREA}must consider vertical variation, culm size and age, and response to temperature change [14,19]. By contrast, previous studies have not focused on the quantification of S. A previous study assumed cone geometry for culms when estimating R

_{C}in a bamboo forest [19]. However, this simplistic assumption is violated when quantifying S [25]. Because small errors in S could lead to large errors in scaled-up R

_{C}[24], S must be characterized accurately and precisely.

_{C}prediction.

- (1)
- Are the allometric equations site-specific?
- (2)
- Are the allometric equations species-specific?
- (3)
- What predictive variable is optimal for S prediction?

## 2. Materials and Methods

#### 2.1. Study Sites

#### 2.2. Field Measurements

#### 2.3. Analyses Methods

^{β},

^{2}), root mean square error (RMSE) and Akaike’s Information Criteria (AIC).

^{δ},

^{δ}H

^{ε},

_{predicted}and S

_{observed}are predicted and observed S, respectively; n is the number of culm samples; and S

_{average}is the average S

_{observed}values. Differences between S

_{predicted}and S

_{observed}were examined with a Wilcoxon signed rank test.

_{b}) [57] were calculated from modeling data. The former is known as a slenderness coefficient, whereas the latter is a measure of culm tapering. λ

_{b}was calculated as:

_{b}= 4V/πD

^{2}H.

_{b}are more slender and non-tapering, respectively. These values were compared using Tukey’s HSD test. All statistical analyses were performed using R software version 3.4.3 [58], and p < 0.05 was considered significant.

## 3. Results and Discussion

#### 3.1. Are the Allometric Equations Site-Specific?

^{2}> 0.94). The relationship between S and DH can also be expressed by a linear equation (r

^{2}> 0.99; Figure 2). For all species and sites, %AE of the site-specific equations was small (less than ±3%; Table 4). Hence, Equations (1) and (2) are acceptable as the predictive equations for every species and sites examined in this study.

^{2}and RMSE of the power equation were, respectively, 0.98 and 0.131 m

^{2}for P. bambusoides and 0.95 and 0.051 m

^{2}for P. nigra. The relationship between S and DH was fitted by a single linear equation for both species, with r

^{2}= 0.99 for both species (Figure 4). RMSE was 0.069 m

^{2}for P. bambusoides and 0.017 m

^{2}for P. nigra. Site- and species-specific equations for both species had comparable fit statistics and predictive performance, suggesting that separate equations are not necessary for different P. bambusoides and P. nigra sites. Therefore, the answer to the first question is that allometric equations for P. bambusoides and P. nigra are not site-specific. However, further studies are necessary to verify whether equations for P. pubescens are site-specific, as our data were insufficient to clarify this point.

#### 3.2. Are the Allometric Equations Species-Specific?

^{2}= 0.98 for the former and r

^{2}= 0.99 for the latter. Common equations had intermediate RMSEs compared to site- and species-specific equations. The relationships of S to D and DH for all three species are shown in Figure 6. Although the generic power equation for all three species was significant (r

^{2}= 0.97), S and D scatterplots indicate a clear separation of P. pubescens and the other two species; for the same D, P. pubescens had a smaller S than P. bambusoides and P. nigra (Figure 6a). Similarly, the relationship between S and DH for the three species could be fitted by the single linear equation (r

^{2}= 0.98). However, for the same DH, S was slightly smaller in P. pubescens than in the other two species (Figure 6b). In both generic equations, %AE was positive for P. pubescens, but negative for P. bambusoides and P. nigra. Therefore, when applying the generic equations, R

_{C}will be overestimated for P. pubescens and underestimated for the other two species. Moreover, the generic equations had larger %RMSEs than the other equations. These results indicate that generic equations are inappropriate when estimating S, and the predictive equations for P. pubescens should be species-specific. For these reasons, the optimal equations for predicting S are as follows:

^{1.49},

^{1.70},

_{predicted}and S

_{observed}. Generally, S

_{predicted}of P. bambusoides from power and linear common equations did not differ significantly from S

_{observed}, with %AE less than ±1.2% for all sites and equations except for S

_{predicted}in the TO from the power equation (%AE = −2.3%) and in the HI from the linear equation (−5.7%). For P. nigra, a significant difference was found between S

_{predicted}and S

_{observed}at both sites; %AE from the power equation was relatively large for IT (9.1%), but less than ±5% for other sites and equations. It should be noted that the common power equation in MU had smaller %AE than the species-specific power equation. Therefore, the predictive performance of common equations is not inferior to that of species-specific equations. However, common equations have broader application than species-specific equations, even if they are not relevant for P. pubescens.

_{b}(Figure 7b), indicating a more tapered culm. A tapered culm results in smaller S for the same D and H. These facts indicate that interspecific variation in culm slenderness and tapering explains differences in allometric power equations, while only variation in culm tapering explains differences in the linear equation. Therefore, the answer to the second question is that species-specific predictive equations are required for P. pubescens, but not for P. bambusoides and P. nigra, because of interspecific variation in culm characteristics.

#### 3.3. What Predictive Variable Is Optimal for S Prediction?

^{2}and smaller RMSE and AIC than the D-based equation, Equation (1) (Table 3). However, %AE and %RMSE magnitudes exhibited inter-site and inter-specific variation (Table 4), suggesting that the inclusion of the additive variable H does not improve the predictive performance. Diameter at breast height and tree height are easy measureable dendrometric variables of trees [28]. D of bamboo is also the most easily measured culm attribute. In contrast, it is difficult to measure H of standing bamboo culm, since the culm tip is generally bent askew [35,52,59]. Equations (10) and (12) clearly demonstrated that inaccurate measurement of H causes errors in S prediction, which results in uncertain R

_{C}prediction. Similar arguments are found in estimating V or biomass of tree species [60,61,62]. For these reasons, our answer to the third question is that D is the optimal variable for estimating S of the three examined bamboo species.

## 4. Conclusions

_{AREA}up to R

_{C}at individual- or stand-levels efficiently through simple measurement of D. Therefore, the equations developed here will be beneficial for evaluating R

_{C}, and bring a consensus regarding the contribution of bamboo forest ecosystems to the carbon cycle. In future, it is necessary to examine whether the equations vary with culm sizes, climatic conditions and soil properties.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Relationship between culm surface area and diameter at breast height for each species and site. The solid lines represent the power equations detailed in Table 3.

**Figure 2.**Relationship between culm surface area and product of diameter at breast height and total culm length for each species and site. The solid lines represent the linear equations detailed in Table 3.

**Figure 3.**Relationship between culm surface area and diameter at breast height in (

**a**) Phyllostachys bambusoides and (

**b**) P. nigra. The solid lines represent the power equations detailed in Table 3.

**Figure 4.**Relationships between culm surface area and product of diameter at breast height and total culm length in (

**a**) Phyllostachys bambusoides and (

**b**) P. nigra. The solid lines represent the linear equations detailed in Table 3.

**Figure 5.**Relationships of culm surface area to (

**a**) diameter at breast height and (

**b**) product of diameter at breast height and total culm length for Phyllostachys bambusoides and P. nigra. The solid lines represent the power and linear equations detailed in Table 3.

**Figure 6.**Relationships of culm surface area to (

**a**) diameter at breast height and (

**b**) product of diameter at breast height and total culm length for three Phyllostachys species. The solid lines represent the power and linear equations detailed in Table 3.

**Figure 7.**Comparisons of (

**a**) height-diameter ratio and (

**b**) breast height form-factor for each species and site. Error bars indicate the standard deviation. Values labeled with the same lower case letter are not significantly different according to Tukey’s HSD test.

Species | Site ^{1} | Number of Samples ^{2} | Soil Type ^{3} | Total Culm Height (m) ^{4} | Diameter at Breast Height (cm) ^{4} | Culm Surface Area (m^{2}) ^{4} |
---|---|---|---|---|---|---|

P. pubescens | TO | 200 (100) | Ando soil | 14.1 ± 2.5 | 9.2 ± 2.7 | 2.46 ± 1.01 |

P. bambusoides | TO | 200 (100) | Ando soil | 11.1 ± 3.0 | 5.0 ± 1.7 | 1.26 ± 0.70 |

KI | 102 (51) | Brown forest soil or ando soil | 12.8 ± 3.4 | 6.2 ± 2.1 | 1.77 ± 1.00 | |

FU | 122 (61) | Brown forest soil or ando soil | 11.8 ± 2.5 | 5.7 ± 1.7 | 1.50 ± 0.81 | |

HI | 96 (48) | Brown forest soil or ando soil | 9.0 ± 2.1 | 4.1 ± 1.2 | 0.87 ± 0.46 | |

P. nigra | MU | 60 (30) | Brown forest soil | 7.3 ± 2.0 | 2.9 ± 1.1 | 0.50 ± 0.30 |

IT | 110 (55) | Gley soil or brown forest soil | 6.2 ± 1.5 | 2.7 ± 0.8 | 0.37 ± 0.19 |

^{1}TO: Toshima, KI: Kitakyushu, FU: Fukuoka, HI: Hisayama, MU: Munakata, IT: Ito;

^{2}Values in parentheses are sample sizes of modeling and validation data;

^{3}The soil type classification is based on website information [46];

^{4}Average ± S.D.

Equation | Species | Site ^{1} | Number of Samples ^{2} |
---|---|---|---|

Site-specific | P. bambusoides | TO | 200 |

KI | 102 | ||

FU | 122 | ||

HI | 96 | ||

P. nigra | MU | 60 | |

IT | 110 | ||

Species-specific | P. pubescens | TO | 200 |

P. bambusoides | TO, KI, FU and HI | 520 | |

P. nigra | MU and IT | 170 | |

Common | P. bambusoides and P. nigra | TO, KI, FU, HI, MU and IT | 690 |

Generic | P. pubescens, P. bambusoides and P. nigra | TO, KI, FU, HI, MU and IT | 890 |

^{1}TO: Toshima, KI: Kitakyushu, FU: Fukuoka, HI: Hisayama, MU: Munakata, IT: Ito;

^{2}The number of sample culms used for both modeling and validation data.

Equation | Species | Site ^{1} | Power Equation | Linear Equation | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

α | β | r^{2} | RMSE (m^{2}) | AIC | γ | r^{2} | RMSE (m^{2}) | AIC | |||

Site-specific | P. bambusoides | TO | 159.03 | 1.65 | 0.97 | 0.155 | −83.25 | 2.08 | 0.99 | 0.051 | −306.68 |

KI | 178.44 | 1.68 | 0.99 | 0.147 | −46.26 | 2.04 | 0.99 | 0.073 | −120.05 | ||

FU | 184.19 | 1.70 | 0.98 | 0.089 | −115.35 | 2.07 | 0.99 | 0.068 | −150.95 | ||

HI | 242.78 | 1.78 | 0.98 | 0.073 | −106.07 | 2.23 | 0.99 | 0.055 | −136.63 | ||

P. nigra | MU | 143.52 | 1.63 | 0.97 | 0.060 | −79.96 | 2.14 | 0.99 | 0.025 | −134.44 | |

IT | 263.16 | 1.83 | 0.94 | 0.039 | −191.73 | 2.12 | 0.99 | 0.011 | −322.93 | ||

Species-specific | P. pubescens | 83.21 | 1.49 | 0.98 | 0.174 | −58.68 | 1.80 | 0.99 | 0.101 | −168.49 | |

P. bambusoides | 179.50 | 1.69 | 0.98 | 0.131 | −315.58 | 2.07 | 0.99 | 0.069 | −649.89 | ||

P. nigra | 239.91 | 1.79 | 0.95 | 0.051 | −260.28 | 2.13 | 0.99 | 0.017 | −441.90 | ||

Common | P. bambusoides and P. nigra | 187.66 | 1.70 | 0.98 | 0.118 | −495.65 | 2.08 | 0.99 | 0.061 | −950.25 | |

Generic | All species | 116.01 | 1.57 | 0.97 | 0.294 | −3.27 | 1.92 | 0.98 | 0.142 | −471.77 |

^{1}TO: Toshima, KI: Kitakyushu, FU: Fukuoka, HI: Hisayama, MU: Munakata, IT: Ito.

**Table 4.**Accuracy (%AE) and precision (%RMSE) of the culm surface area estimated by the developed allometric equations.

Equation | Species | Site ^{1} | Power Equation | Linear Equation | ||
---|---|---|---|---|---|---|

%AE (%) | %RMSE (%) | %AE (%) | %RMSE (%) | |||

Site-specific | P. bambusoides | TO | −1.45 | 12.04 | −0.55 | 4.04 |

KI | 2.06 | 6.17 | −1.95 | 2.73 | ||

FU | −0.43 | 7.91 | −1.17 | 5.54 | ||

HI | 0.95 | 9.44 | 1.12 | 7.86 | ||

P. nigra | MU | −1.94 | 12.76 | −1.73 | 3.80 | |

IT | −3.09 | 11.06 | −1.31 | 3.33 | ||

Species-specific | P. pubescens | TO | 0.30 | 8.70 | −0.51 | 4.74 |

P. bambusoides | TO | −1.42 | 11.30 | −0.88 | 4.10 | |

KI | 0.05 | 5.78 | −0.28 | 3.16 | ||

FU | 0.91 | 8.03 | −1.21 | 5.53 | ||

HI | 0.73 | 9.19 | −5.77 | 9.94 | ||

P. nigra | MU | −10.45 | 14.47 | −2.15 | 3.99 | |

IT | 0.51 | 11.13 | −0.83 | 3.41 | ||

Common | P. bambusoides | TO | −2.32 | 11.41 | −0.80 | 4.08 |

KI | −0.46 | 5.91 | −0.20 | 3.21 | ||

FU | 0.31 | 8.13 | −1.13 | 5.56 | ||

HI | −0.48 | 9.25 | −5.70 | 9.89 | ||

P. nigra | MU | −3.28 | 11.17 | −4.76 | 6.01 | |

IT | 9.08 | 14.06 | −3.47 | 4.01 | ||

Generic | P. pubescens | TO | 16.54 | 21.75 | 5.81 | 8.41 |

P. bambusoides | TO | −8.39 | 19.40 | −8.48 | 10.30 | |

KI | −9.59 | 15.81 | −7.92 | 7.77 | ||

FU | −8.32 | 14.42 | −8.78 | 9.06 | ||

HI | −4.62 | 14.10 | −12.99 | 16.34 | ||

P. nigra | MU | −2.28 | 15.18 | −12.12 | 14.02 | |

IT | −10.98 | 13.13 | −10.94 | 11.37 |

^{1}TO: Toshima, KI: Kitakyushu, FU: Fukuoka, HI: Hisayama, MU: Munakata, IT: Ito.

© 2018 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 (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Inoue, A.; Miyazawa, Y.; Sato, M.; Shima, H. Allometric Equations for Predicting Culm Surface Area of Three Bamboo Species (*Phyllostachys* spp.). *Forests* **2018**, *9*, 295.
https://doi.org/10.3390/f9060295

**AMA Style**

Inoue A, Miyazawa Y, Sato M, Shima H. Allometric Equations for Predicting Culm Surface Area of Three Bamboo Species (*Phyllostachys* spp.). *Forests*. 2018; 9(6):295.
https://doi.org/10.3390/f9060295

**Chicago/Turabian Style**

Inoue, Akio, Yoshiyuki Miyazawa, Motohiro Sato, and Hiroyuki Shima. 2018. "Allometric Equations for Predicting Culm Surface Area of Three Bamboo Species (*Phyllostachys* spp.)" *Forests* 9, no. 6: 295.
https://doi.org/10.3390/f9060295