# Error Analysis of Measuring the Diameter, Tree Height, and Volume of Standing Tree Using Electronic Theodolite

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Technical principles

#### 2.1.1. Principle of Measuring the Tilt Distance

#### 2.1.2. Tree Height Measurement Principle

#### 2.1.3. Measuring the Diameter at Any Point

#### 2.1.4. Measurement of the Height at Any Segment

#### 2.1.5. Calculation Model of the Stem Volume

#### 2.2. Error Analysis

#### 2.2.1. Error Analysis of the Tilt Distance

#### 2.2.2. Error Analysis of Tree Height H

#### 2.2.3. Error Analysis of the Diameter ${D}_{i}$ at Any Position

#### 2.2.4. Error Analysis at Arbitrary Heights

#### 2.2.5. Variance of the Diameter ${D}_{i}$ and the Height ${h}_{i}$ at Any Point

#### 2.2.6. Error Analysis of Stem Volume

#### 2.3. Accuracy Verification

#### 2.3.1. Measurement of Felled Trees

#### 2.3.2. F Test Principle

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Number | Height/m | Diameter/m | Zenith | Horizontal Angle |
---|---|---|---|---|

D_{0} | - | 0.286 | ||

D_{1} | 1.30 | 0.208 | 91°15′42″ | 00°56′12″ |

D_{2} | 2.38 | 0.187 | 86°23′17″ | 00°50′31″ |

D_{3} | 3.43 | 0.175 | 81°44′27″ | 00°46′52″ |

D_{4} | 4.53 | 0.152 | 76°55′31″ | 00°39′55″ |

D_{5} | 5.61 | 0.134 | 72°24′23″ | 00°34′24″ |

D_{6} | 6.90 | 0.114 | 67°18′05″ | 00°28′21″ |

D_{7} | 8.41 | 0.092 | 61°45′51″ | 00°21′49″ |

D_{8} | 9.68 | 0.066 | 57°30′00″ | 00°15′01″ |

D_{9} | 12.18 | 0.038 | 50°11′12″ | 00°07′49″ |

D_{10} | 14.92 | 43°38′49″ |

Type | Diameter/cm | Height/m | Volume/m^{3} | Diameter System Error/% | Height System Error/% | Volume System Error/% |
---|---|---|---|---|---|---|

Large | 33.5 | 24.79 | 0.8852 | 0.44 | 0.29 | 0.31 |

Medium | 20.8 | 14.92 | 0.2053 | 0.51 | 0.44 | 0.46 |

Small | 8.4 | 5.13 | 0.0212 | 0.59 | 0.89 | 0.99 |

Type | Quantity | RoD | RoDr | Roh | Rohr | Rov | Rovr |
---|---|---|---|---|---|---|---|

Populus tomentosa | 36 | 10.10–49.60 | 0–5.26% | 13.85–30.24 | 0.12–5.35% | 0.057–2.39 | 0.31–9.13% |

Populus cathayana Rehd | 47 | 9.9–42.63 | 0–6.58% | 11.34–35.75 | 0.03–6.43% | 0.040–1.74 | 0.2–9.49% |

Populus canadensis Moench | 4 | 31.2–38.05 | 0–6.87% | 22.19–31.6 | 0–1.91% | 0.69–1.38 | 3.18–8.86% |

Type | Quantity | RoD | RoDr | Roh | Rohr | Rov | Rovr |
---|---|---|---|---|---|---|---|

Populus tomentosa | 11 | 25.10–49.60 | 0.11–5.10% | 18.36–30.24 | 0.26–4.43% | 0.4598–2.3906 | 1.15–8.23% |

Populus cathayana Rehd | 22 | 25.4–42.63 | 0–8.82% | 19.78–35.75 | 0.03–3.55% | 0.4205–1.7411 | 0.31–8.04% |

Populus canadensis Moench | 4 | 31.2–38.05 | 0–6.87% | 22.19–31.6 | 0–1.91% | 0.6973–1.3809 | 3.18–8.86% |

Type | Quantity | RoD | RoDr | Roh | Rohr | Rov | Rovr |
---|---|---|---|---|---|---|---|

Populus- tomentosa | 19 | 15.2–24 | 0–5.26% | 16.29–22.3 | 0.28–5.35% | 0.16–0.39 | 0.31–8.69% |

Populus cathayana Rehd | 15 | 15.75–24.7 | 0–4.33% | 16.44–26.94 | 0.04–4.79% | 0.15–0.51 | 1.42–7.81% |

Type | Quantity | RoD | RoDr | Roh | Rohr | Rov | Rovr |
---|---|---|---|---|---|---|---|

Populus tomentosa | 7 | 10.1–14.7 | 0–2.780% | 13.85–18.39 | 0.65–4.73% | 0.0569–0.1360 | 0.85–7.82% |

Populus cathayana Rehd | 10 | 9.3–14.9 | 0–6.45% | 11.34–17.89 | 0.06–6.43% | 0.05–0.1244 | 0.2–6.96% |

Number | Vbt | Vopt | Aeov | Reov | hbt | hopt | Aeoh | Reoh |
---|---|---|---|---|---|---|---|---|

TZ01 | 0.0564 | 0.0569 | 0.0005 | 0.89% | 14.10 | 13.86 | 0.24 | 1.71% |

TZ02 | 0.1009 | 0.0999 | 0.0010 | 0.97% | 15.27 | 14.58 | 0.69 | 4.73% |

DJ03 | 0.1360 | 0.1372 | 0.0012 | 0.85% | 17.39 | 17.50 | 0.11 | 0.65% |

HJ01 | 0.1620 | 0.1615 | 0.0005 | 0.31% | 19.56 | 19.18 | 0.38 | 1.98% |

TZ03 | 0.2320 | 0.2332 | 0.0013 | 0.55% | 20.36 | 20.01 | 0.35 | 1.73% |

DJ11 | 0.0500 | 0.0500 | 0.0001 | 0.20% | 13.49 | 13.35 | 0.14 | 1.06% |

DJ13 | 0.1173 | 0.1162 | 0.0011 | 0.91% | 17.78 | 17.89 | 0.12 | 0.64% |

DJ14 | 0.1188 | 0.1180 | 0.0008 | 0.70% | 16.20 | 16.08 | 0.12 | 0.75% |

YJ04 | 0.1140 | 0.1102 | 0.0038 | 3.48% | 14.71 | 14.81 | 0.10 | 0.69% |

MJ01 | 0.2235 | 0.2180 | 0.0055 | 2.54% | 20.63 | 20.64 | 0.01 | 0.04% |

YJ06 | 0.1759 | 0.1677 | 0.0082 | 4.89% | 17.15 | 16.71 | 0.44 | 2.63% |

PJ07 | 0.2515 | 0.2691 | 0.0177 | 6.57% | 19.73 | 19.28 | 0.45 | 2.33% |

FJ01 | 0.4066 | 0.3713 | 0.0353 | 9.49% | 22.68 | 22.71 | 0.03 | 0.14% |

MJ02 | 0.3094 | 0.2861 | 0.0233 | 8.16% | 20.21 | 19.90 | 0.31 | 1.56% |

…… | ||||||||

PJ06 | 0.3567 | 0.3791 | 0.0224 | 5.91% | 21.04 | 20.92 | 0.12 | 0.55% |

CJ02 | 0.5136 | 0.4947 | 0.0188 | 3.81% | 26.00 | 26.07 | 0.07 | 0.28% |

CJ03 | 0.5492 | 0.5133 | 0.0358 | 6.98% | 26.38 | 26.94 | 0.56 | 2.09% |

**Table 8.**Adaptive test results of regression between two sets of measurements using the electronic theodolite and destructive felling.

F | F_{0.05} | |
---|---|---|

Diameter | 0.810 | 3.953 |

Tree height | 0.953 | 3.953 |

Stem volume | 0.828 | 3.953 |

Diameter Relative Error | Height Relative Error | Volume Relative Error | |
---|---|---|---|

Value | 2.02% | 1.18% | 4.47% |

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**MDPI and ACS Style**

Yao, Q.; Wang, J.; Zhang, J.; Xiong, N.
Error Analysis of Measuring the Diameter, Tree Height, and Volume of Standing Tree Using Electronic Theodolite. *Sustainability* **2022**, *14*, 6950.
https://doi.org/10.3390/su14126950

**AMA Style**

Yao Q, Wang J, Zhang J, Xiong N.
Error Analysis of Measuring the Diameter, Tree Height, and Volume of Standing Tree Using Electronic Theodolite. *Sustainability*. 2022; 14(12):6950.
https://doi.org/10.3390/su14126950

**Chicago/Turabian Style**

Yao, Qing, Jia Wang, Junping Zhang, and Nina Xiong.
2022. "Error Analysis of Measuring the Diameter, Tree Height, and Volume of Standing Tree Using Electronic Theodolite" *Sustainability* 14, no. 12: 6950.
https://doi.org/10.3390/su14126950