# Deriving Merchantable Volume in Poplar through a Localized Tapering Function from Non-Destructive Terrestrial Laser Scanning

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

^{2}and RMSE. A modified Schumacher equation was the most suitable taper function. Volume estimates from the TLS-derived taper function were better than those derived using the stem-analysis data. Finally, regression analysis showed that predictions of stem size were similar when data were based on TLS versus stem analysis. Its high accuracy and efficiency indicates that TLS technology can play an important role in forest inventory assessment.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Field Measurements

^{2}hybrid poplar (P. × canadensis Moench cv. ‘I-72/58’) plantation near Huai′an, Jiangsu Province, China (33°40′ N, 119°23′ E). The plantation was established progressively between 1990 and 1995; tree age at the time of measurement was 20–24 years.

#### 2.2. Laser Scanning

#### 2.3. Stem Analysis

#### 2.4. Scanning Analysis

#### 2.5. Taper Functions

$${d}^{2}={D}^{{a}_{0}+{a}_{1}H}{(H-h)}^{{a}_{2}+{a}_{3}D}/H$$Meng, 1982 [8]

$${d}^{2}={a}_{0}{D}^{{a}_{1}}{\frac{(H-h)}{{H}^{{a}_{3}}}}^{{a}_{2}}$$Schumacher and Hall, 1933 [23]

$${d}^{2}={a}_{0}D{(\frac{H-h}{H-1.3})}^{{a}_{1}}$$Yan, 1992 [24]

$${d}^{2}={D}^{2}{(\frac{H-h}{H-1.3})}^{{a}_{0}}$$Ormerod, 1971 [25]

$${d}^{2}={D}^{2}({a}_{0}+{a}_{1}(\frac{h}{H})+{a}_{2}{(\frac{h}{H})}^{2})$$Kozak, 1966 [26]

where D is DBH (cm), H is total tree height (m), and a$${d}^{2}={D}^{2}({a}_{0}+{a}_{1}\frac{H-h}{h})$$Nagashima, 1980 [27]

_{0}, a

_{1}, a

_{2}, and a

_{3}are fitted parameters.

#### 2.6. Evaluation of the Taper Equations

^{2}), a representation of the degree of association between the dependent and independent variables, and root mean square error (RMSE), an index of the variance between observed and predicted values. The objective at this stage was to determine which equations best fit the TLS and stem analysis data. This was accomplished by scoring equations from 1 to 6 with the lowest score assigned to the highest R

^{2}and the lowest RMSE. Scores were then simply added.

#### 2.7. Plot Merchantable Volume

_{Lk}) can thus be calculated as:

_{total}) can be calculated from the definite integral of total tree height:

## 3. Results

#### 3.1. Summary Data

^{2}·ha

^{−1}.

#### 3.2. Taper Equation Comparison

^{2}and lowest RMSE; Table 6). Equations (1), (3), (4), and (6) were therefore removed from further consideration.

#### 3.3. Merchantable Volume Predictions

_{k}) according to accepted timber standards (Table 2):

_{k}is the upper trunk diameter according to timber standards (Table 2), and a

_{i}are fitted parameters. The merchantable volume (MV) can now be calculated from the definite integral of Equation (2):

_{2}is the breast height diameter without bark, and all other variables are as described above; a

_{i}are fitted parameters.

## 4. Discussion

^{2}as the dependent variable. After model evaluations, predictions of upper stem diameters were compared between the two best models, Equations (2) and (5). Model 5 has a variable exponent which changes continuously along the stem. The stem analysis data, however, showed that the upper diameters were not normally distributed. That likely reduced its predictive accuracy relative to Model 2.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**An illustration of tree stem, scanning stations, and reference target locations in a 15-m circular plot. S0 to S4 represent the four scanning locations, and the triangles show the three reference points.

**Figure 2.**Predicted stem diameter in relation to tree height, as derived using stem analysis (sta) and laser scanning data (tls), from Model 2 (

**a**) and Model 5 (

**b**). Also shown are the corresponding measured (true) values.

Specifications | Leica Scanstation C10 |
---|---|

Field-of-View (Horizontal × Vertical) | 360° × 270° |

Range | 300 m @90%;134 m @18% albedo (minimum range 0.1 m) |

Scan Rate | Up to 50,000 points/second |

Type | Pulsed; proprietary microchip |

Log Length | Minimum Top Diameter under Bark (cm) | Minimum Length (m) | Use |
---|---|---|---|

Large | >26 | ≥2 | Special timber, plywood, ship-building wood |

Mid-length | >20–26 | ≥2 | Normal timber, pit wood, pile timber |

Short | 6–20 | ≥2 | Civil timber, pulpwood |

DBH (cm) | 9–14 | >14–19 | >19–24 | >24–29 | >29–34 | >34–39 | >39–44 |
---|---|---|---|---|---|---|---|

Frequency | 7 | 19 | 33 | 52 | 54 | 30 | 3 |

Plot Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

Number of trees | 16 | 28 | 32 | 27 | 28 | 18 | 26 | 23 | |

Basal area (G) (m^{2}/ha) | 12.1 | 17.4 | 15.1 | 13.4 | 15.6 | 14.4 | 13.1 | 15.9 | |

DBH (cm) | Max | 40.8 | 38.0 | 33.9 | 34.8 | 41.3 | 38.2 | 34.6 | 36.8 |

Min | 17.8 | 14.8 | 13.6 | 10.7 | 10.7 | 23.9 | 9.1 | 23.4 | |

Mean | 29.5 | 29.0 | 24.7 | 24.9 | 25.1 | 33.1 | 25.6 | 30.7 | |

Tree Height (m) | Max | 25.9 | 29.8 | 26.5 | 26.6 | 26.9 | 27.6 | 27.8 | 29.2 |

Min | 15.0 | 24.9 | 12.5 | 12.5 | 14.7 | 19.8 | 11.3 | 21.2 | |

Mean | 21.3 | 27.6 | 21.6 | 21.8 | 22.2 | 24.4 | 23.5 | 26.3 |

Tree | Plot Number | Age (Years) | DBH (cm) | Height (m) |
---|---|---|---|---|

1 | 1 | 18 | 35.2 | 26.6 |

2 | 1 | 21 | 35.6 | 28.2 |

3 | 2 | 24 | 31.9 | 29.8 |

4 | 2 | 24 | 37.5 | 30.0 |

5 | 3 | 17 | 27.2 | 21.3 |

6 | 3 | 17 | 24.2 | 22.4 |

7 | 4 | 20 | 26.1 | 24.7 |

8 | 4 | 21 | 22.6 | 20.6 |

9 | 5 | 22 | 24.5 | 22.4 |

10 | 5 | 21 | 21.7 | 22.8 |

11 | 6 | 23 | 34.2 | 24.9 |

12 | 6 | 22 | 37.2 | 27.6 |

13 | 7 | 20 | 27.6 | 23.7 |

14 | 7 | 21 | 34.6 | 27.2 |

15 | 8 | 19 | 31.9 | 29.2 |

16 | 8 | 21 | 36.8 | 26.9 |

Model | Stem Analysis | TLS Data | ||
---|---|---|---|---|

RMSE (cm^{2}) | R^{2} | RMSE (cm^{2}) | R^{2} | |

1 | 61.2 | 0.96 | 62.1 | 0.96 |

2 | 60.9 | 0.96 | 61.8 | 0.96 |

3 | 99.1 | 0.88 | 130.7 | 0.80 |

4 | 65.3 | 0.95 | 67.5 | 0.95 |

5 | 61.1 | 0.96 | 55.2 | 0.97 |

6 | 142.4 | 0.65 | 84.3 | 0.90 |

**Table 7.**Comparison of Equation (2) predictions when calibrated with stem analysis versus terrestrial laser scanning (TLS) data.

Stem Analysis | TLS | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Parameter | a_{0} | a_{1} | a_{2} | a_{3} | R^{2} | a_{0} | a_{1} | a_{2} | a_{3} | R^{2} | |

Estimate | 1.10 | 2.63 | 1.70 | 2.32 | 0.955 | 1.62 | 2.01 | 1.31 | 1.47 | 0.96 | |

std. error | 0.12 | 0.11 | 0.02 | 0.11 | 0.13 | 0.01 | 0.01 | 0.03 | |||

95% confidence interval | lower bound | 0.86 | 2.41 | 1.65 | 2.10 | 1.37 | 1.98 | 1.29 | 0.41 | ||

upper bound | 1.33 | 2.85 | 1.74 | 2.54 | 1.89 | 2.04 | 1.34 | 1.52 |

Average DBH in the plot (cm) | 23.9 | 24.6 | 24.9 | 26.2 | 29.3 | 33.4 | 37.5 | 37.9 |

Local volume tables (m^{3}) | 0.38 | 0.41 | 0.42 | 0.49 | 0.66 | 0.95 | 1.30 | 1.34 |

TLS (m^{3}) | 0.44 | 0.48 | 0.48 | 0.55 | 0.76 | 0.97 | 1.43 | 1.35 |

Percent difference | 15 | 18 | 13 | 13 | 15 | 2 | 10 | 1 |

Stem analysis (m^{3}) | 0.45 | 0.47 | 0.48 | 0.56 | 0.77 | 1.08 | 1.57 | 1.55 |

Percent difference ^{1} | 18 | 14 | 12 | 13 | 17 | 14 | 21 | 15 |

^{1}Expressed as the difference from the local volume table predictions [31].

**Table 9.**Regression analysis for TLS (x-axis) versus stem analysis (y-axis) predictions from Model 2.

Log Types | Small | Medium | Large | |||
---|---|---|---|---|---|---|

Variable | Slope | R^{2} | Slope | R^{2} | Slope | R^{2} |

Accumulated log length | 1.01 | 0.99 | 0.98 | 0.99 | 0.90 | 0.99 |

Merchantable volume | 0.89 | 0.99 | 0.70 | 0.98 | 1.16 | 0.99 |

Percent of merchantable volume | 0.99 | 0.97 | 0.77 | 0.99 | 0.82 | 0.99 |

© 2016 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/).

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**MDPI and ACS Style**

Sun, Y.; Liang, X.; Liang, Z.; Welham, C.; Li, W. Deriving Merchantable Volume in Poplar through a Localized Tapering Function from Non-Destructive Terrestrial Laser Scanning. *Forests* **2016**, *7*, 87.
https://doi.org/10.3390/f7040087

**AMA Style**

Sun Y, Liang X, Liang Z, Welham C, Li W. Deriving Merchantable Volume in Poplar through a Localized Tapering Function from Non-Destructive Terrestrial Laser Scanning. *Forests*. 2016; 7(4):87.
https://doi.org/10.3390/f7040087

**Chicago/Turabian Style**

Sun, Yuan, Xinlian Liang, Ziyu Liang, Clive Welham, and Weizheng Li. 2016. "Deriving Merchantable Volume in Poplar through a Localized Tapering Function from Non-Destructive Terrestrial Laser Scanning" *Forests* 7, no. 4: 87.
https://doi.org/10.3390/f7040087