Modeling of the height–diameter relationship using an allometric equation model: a case study of stands of Phyllostachys edulis

Abstract

Understanding the relationship between tree height (H) and diameter at breast height (D) is vital to forest design, monitoring and biomass estimation. We developed an allometric equation model and tested its applicability for unevenly aged stands of moso bamboo forest at a regional scale. Field data were collected for 21 plots. Based on these data, we identified two strong power relationships: a correlation between the mean bamboo height (H _m) and the upper mean H (H _u), and a correlation between the mean D (D _m) and the upper mean D (D _u). Simulation results derived from the allometric equation model were in good agreement with observed culms derived from the field data for the 21 stands, with a root-mean-square error and relative root-mean-square error of 1.40 m and 13.41 %, respectively. These results demonstrate that the allometric equation model had a strong predictive power in the unevenly aged stands at a regional scale. In addition, the estimated average height–diameter (H–D) model for South Anhui Province was used to predict H for the same type of bamboo in Hunan Province based on the measured D, and the results were highly similar. The allometric equation model has multiple uses at the regional scale, including the evaluation of the variation in the H–D relationship among regions. The model describes the average H–D relationship without considering the effects caused by variation in site conditions, tree density and other factors.

Appendix

During model evolution, uncertainty on the variables H _m and D _m in Eqs. 1 and 2 can certainly propagate into Eq. 14. As addressed in the main text, we first obtained Eq. 6 by combining Eq. 1 with Eq. 5 and eliminating H _u, and by substituting Eq. 2 into Eq. 6, we obtained Eq. 7:

$$ \alpha \left( {pD_{\text{m}}^{q} } \right)^{\beta } = aH_{\text{m}}^{b} $$

(7)

The general form of Eq. 14 was expressed using Eq. 15:

$$ H = \alpha D^{\beta } $$

(15)

Replacing α with Eq. 7, we obtained the Eq. 16:

$$ H = \frac{{aH_{\text{m}}^{\text{b}} }}{{\left( {pD_{\text{m}}^{q} } \right)^{\beta } }}D^{\beta } $$

(16)

We assumed that the expected e _Hm and e _Dm deviate by small amounts δH _m and δD _m, normally distributed with zero mean and with standard deviations σH _m and σD _m. The uncertainty on the estimate of H (culm height) associated with the measurement is measured by the standard deviation. Thus, the model error (σ _H ) can be written as:

$$ \frac{{\sigma_{\text{H}}^{2} }}{{H^{2} }} = \left( {\frac{\delta\,\hbox{ln}\,f}{{\delta D_{\text{m}} }}\sigma_{{D_{\text{m}} }} } \right)^{2}\,+\,\left( {\frac{\delta\,\hbox{ln}\,f}{{\delta H_{\text{m}} }}\sigma_{{H_{\text{m}} }} } \right)^{2}\,+\,\left( {\frac{\delta\,\hbox{ln}\,f}{\delta D}\sigma_{D} } \right)^{2} + 2\frac{{\sigma_{{D_{\text{m}} H_{\text{m}} }} \delta\,\hbox{ln}\,(f)\delta\, \hbox{ln}\,(f)}}{{H_{\text{m}} D_{\text{m}} \delta H_{\text{m}} \delta\, D_{\text{m}} }} $$

(17)

H _m, D _m, $\sigma_{{H_{\text{m}} }}^{{}}$, $\sigma_{{D_{\text{m}} }}^{{}}$ and $\sigma_{{H_{\text{m}} D_{\text{m}} }}^{{}}$ were estimated as 10.41 m, 7.74, 1.52 cm, 1.59 m and 25.32 m cm, respectively. If one assumes a 10 % uncertainty on the measurement of D, the model error (σ _H ) is thus 33.17 % of H.