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

Abstract

The ability to measure diameter, tree height, and tree volume is of great value in forestry investigations. These three factors are not considered together when error analyses of measurement results are conducted. The aim of this study was to quantify the system error analysis of the three factors mentioned above. Based on the principle of electronic theodolite non-destructive measurement of stem and error propagation theory, a mathematical model for calculating diameter, tree height, and volume error was derived to calculate the error in theoretical measurements. Moreover, a method utilizing F tests was proposed for analyzing the relative error based on the diameter, tree height, and volume factors. A total of 87 trees in Beijing were chosen as the experimental sample; the trees were divided into three sample groups according to size. The accuracy was assessed using the traditional method. The results showed that the error varied slightly with the size of the tree. The system error of the measured diameter was 0.44–0.59%; for tree height, this was 0.29–0.89%, and for the volume, it was 0.31–0.99%. There were no significant differences between the measured values and the true values of the resolved wood. The relative mean errors of volume, tree height, and diameter in practice were 4.47%, 1.18%, and 2.02%, respectively. The results suggest that the precision in electronic theodolite measurements in China is much higher than the accuracy requirement of relative error not exceeding 3–5%.

1. Introduction

Diameter tree height and stem volume are the most significant factors in forestry investigations. Improving the measurement accuracy of these three factors is an important topic in forest tree measurement. In 1855, Pressler proposed the Wang Gao method [1], Schliffel [2] proposed the chest height rate method, and Jonson [3] proposed the absolute shape rate method. On this basis, Xu Yuxiang of the Henan Forestry Institute proposed the point method in 1990. His analysis used 49 strains of 13 species of eucalyptus to compare the accuracy of the four methods [4]. Since the beginning of the 21st century, with the development of modern forestry techniques, forest measurement technology has received increasing attention. Zaman et al. [5,6,7,8,9,10] used a 3D laser scanner to measure standing volume and compared the precision with that of traditional measurements. Feng Zhongke et al. [11,12] introduced digital close-range photogrammetry and total station measurement technology into the field of forest surveying. Y. Liyan presented a nondestructive method using CCD to measure tree height and volume [13]. Xu Wenbing et al. [14] used a compass, a total station, and a hand-held device in a forest resource survey. Comparisons of the precision provided by the different methods indicated that traditional compass measurements are insufficient to meet the developing needs of the forestry disciplines, and thus modern measurement technology will play an increasingly important role. Song Yuhui [15] improved the method of calculating the stem volume by measuring the height of the tree with an electronic theodolite. The method finds the location of the point of view and applies the improved Wang Gao method to calculate the stem volume. Guo Baosheng et al. [16] measured the stem volume using a total station. Zhao Fang et al. [17] studied the method of measuring the volume under the canopy with a total station.

In addition, the common tree height measurement method measures the tree height by the triangle elevation standing height measurement principle [18] and an electronic tree gun [19]. The tree height can also be measured by photogrammetry techniques to obtain stereo image pairs, the tree height is then obtained through three-dimensional coordinates [20,21,22]; LIDAR data for single tree height measurement [23,24]; single-chip cameras via the tilt angle; measurements of displacement [25]; and many other methods. Among these, regarding the problem of canopy occlusion, Wang Xuefeng et al. [26] proposed dividing the standing tree height into the height of the branches and the height of the canopy and then using the stereoscopic reconstruction technique to obtain the tree height. Zhao Fang et al. [17] proposed a method for measuring tree height using a total station under canopy occlusion conditions. Brandtberg and Clark et al. [27,28] used small spot, high sampling density LIDAR data to estimate the understory tree height. St-Onge et al. [29] used a combination of stereophotometry and LIDAR to measure the height of a single tree with an average sampling density of approximately 0.1 echo point per square meter. Suarez et al. [30] used airborne LIDAR and aerial photography to estimate forest tree heights.

Other researchers have proposed error analysis methods for DBH, tree height, and volume measurements. Manuel et al. [31] conducted research to evaluate whether height and upper-stem diameter actually improve the accuracy of volume estimation and its effect on prediction when measurement error is considered. Maria [32] analyzed the standard error of tree height measurement, given the error caused by sight acuity of the operator at the creation of the line at the top of the tree. Ambrose et al. [33] analyzed the uncertainty in individual tree measurements when these values were determined during inventory measurements. Steven [34] analyzed the accuracy of DBH collected at a distance and performed a correlation analysis between light conditions and remote measurements. Because 3D laser scanners and total stations are expensive, bulky, and difficult to carry, the breast diameter ruler and electronic theodolite are still commonly used and were used in the actual tree measurement projects that were adopted for this paper. Although many studies have focused on improving the accuracy of diameter, tree height, and volume measurements as well as the assessment of measurement errors, there is at present no formal overall analysis concerning the accuracy of such measurements.

In this work, a mathematical model for calculating diameter, tree height, and volume error was derived to calculate the error in theoretical measurements. A dataset from a Beijing stand of Populus used trees classified into three sets, i.e., large, medium, and small. Moreover, a method utilizing F tests was proposed for analyzing the overall error based on the diameter, tree height, and volume factors.

2. Materials and Methods

The experimental instrument used in this study was the Southern Electronic Theodolite DT-02 (Figure 1). The electronic theodolite consists of three parts: horizontal measurement, vertical measurement, and automatic vertical index. To establish a uniform angle origin, it is necessary to ensure that the vertical axis, the horizontal axis, and the telescope sight axis are set at the three-axis intersection. The electronic theodolite has the advantages of convenient operation, ease of measurement of the horizontal angle and zenith distance, and enhancement of on-site processing functions. The southern DT-02 electronic theodolite has the following characteristics: (1) an automatic vertical compensation device and system liquid electronic sensor bubble compensation, resolution 3”, magnification 30; (2) an output interface RS-; and (3) an instrument weight of 4.3 kg and the ability to work continuously for 10 h.

2.1. Technical principles

Non-destructive tree measurement is a method of manually measuring DBH and the ground diameter data using DBH tape. The method employs parsing the wood to make an approximate segmentation. The horizontal angle and the zenith distance of the corresponding position of the living tree trunk need to be measured by using the triangular elevation. The principle calculates the height of the tree and simulates the method of calculating the product of the average section of the wood by the cumulative method of cylindrical sections, as shown in Figure 2. In the figure, $\beta$ is the horizontal angle of the trunk diameter; v is the zenith distance of the observation target; D is the trunk diameter (m); h is the segment height (m); H is the tree height (m); and $S_0$ is the tilt distance from the center of the instrument to the DBH (m).

2.1.1. Principle of Measuring the Tilt Distance

As shown in Figure 3, the horizontal angle ${\beta }_0$ and the DBH $D_{13}$ are known. The tilt distance $S_0$ can be derived as follows:

$$S_0=D_{13}/2\sin (0.5{\beta }_0)$$

(1)

and the horizontal angle ${\beta }_0$ is generally a minimum value. Therefore, the tilt distance calculation formula can be simplified as in Equation (2):

$$S_0=\frac{D_{13}}{{\beta }_0}$$

(2)

2.1.2. Tree Height Measurement Principle

The height H of the tree is calculated by the tilt distance from the center of the instrument to the center of the trunk $S_0$ and the distance between the center of the instrument $D_{13}$ and the zenith $D_n$ of the treetop. The calculation formula is

$$H=S_0\frac{\sin (v_0-v_n)}{\sin v_n}+h_0$$

(3)

In the formula, the height $h_0$ of the first segment is 1.3 m.

2.1.3. Measuring the Diameter at Any Point

As shown in Figure 2, when the horizontal angle obtained by measuring different segment diameters is ${\beta }_1,{\beta }_2,\cdots {\beta }_n$, the calculation formula for the diameter at any segment is

$$D_i=S_0{\beta }_i\left(i=1,2,\cdots ,n-1\right)$$

(4)

Both the diameter of the ground $D_0$ and the diameter at breast height $D_1\left(D_1=D_{13}\right)$ are determined by using the DBH tape; this is why $i>1$.

2.1.4. Measurement of the Height at Any Segment

Similarly, when different segments are at different heights, the zeniths obtained are $v_0,v_1,\cdots v_n$, and the calculation formula for any segment height is

$$h_i=S_0\left[\frac{\sin (v_0-v_n)}{\sin v_i}-\frac{\sin (v_0-v_{i-1})}{\sin v_{i-1}}\right]$$

(5)

The first segment $h_0$ is 1.3 m, which needs to be measured with a breast diameter rule, so in the formula $i\ge 1$.

2.1.5. Calculation Model of the Stem Volume

As shown in Figure 4, in this paper, we consider the trunk as a cone and analyze the trunk according to the method of division. The stem volume of the trunk is the sum of several cylinders and the cones at the top of the trunk.

The measured stem volume of a complete tree is

$$\begin{array}{l}V=V_1+V_2+\cdots +V_n=1.3\pi (D_0^2+D_0{D}_{13}+D_{13}^2)/12\\ +\pi h_2(D_{13}^2+D_{13}{D}_2+D_2^2)/12+\cdots +\pi h_n\cdot D_{n-1}^2/12\end{array}$$

(6)

The diameter of the bottom of the trunk and the diameter of the trunk at 1.3 m are obtained by field measurement, and the data obtained by the electronic theodolite are used to calculate values using several mathematical models listed in this paper.

Taking a medium-size tree as an example, segment information summarized from the tree sample is shown in

Table 1. Segment information summarized from a medium-size tree sample.

Number	Height/m	Diameter/m	Zenith	Horizontal Angle
D0	-	0.286
D1	1.30	0.208	91°15′42″	00°56′12″
D2	2.38	0.187	86°23′17″	00°50′31″
D3	3.43	0.175	81°44′27″	00°46′52″
D4	4.53	0.152	76°55′31″	00°39′55″
D5	5.61	0.134	72°24′23″	00°34′24″
D6	6.90	0.114	67°18′05″	00°28′21″
D7	8.41	0.092	61°45′51″	00°21′49″
D8	9.68	0.066	57°30′00″	00°15′01″
D9	12.18	0.038	50°11′12″	00°07′49″
D10	14.92		43°38′49″

.

2.2. Error Analysis

By analyzing the error propagation mechanism of the electronic theodolite, we can conclude that the accuracy of the technique is mainly affected by the angle measurement and the tilt distance of the electronic theodolite. If it is assumed that the trunk is ideal and the error is unknown, the accuracy of the technique is mainly affected by the angular deviation of the electronic theodolite, the error of the tilt distance measurement, and the deviation of the breast diameter gauge [30].

2.2.1. Error Analysis of the Tilt Distance

The accuracy of the tilt distance $S_0$ measurement is affected by the error in the measurement of the chest diameter $D_{13}$ and the horizontal measurement angle ${\beta }_0$ of the electronic theodolite.

According to the law of error propagation $D_{13}$ and ${\beta }_0$ are independent variables; the errors are ${\delta }_{D_{13}}$ and ${\delta }_{{\beta }_0}$. Then, the system error of the tilt distance is

$$\frac{{\delta }_{S_0}}{S_0}=\sqrt{{\left(\frac{{\delta }_{D_{13}}}{D_{13}}\right)}^2+{\left(\frac{{\delta }_{{\beta }_0}}{{\beta }_0}\right)}^2\frac{1}{{\rho }^2}}$$

(7)

In the formula, $\rho$ is the conversion factor of the arc system and the angle system, where $\rho$ = (180/π) ° × 60′ × 60″ = 206,264.80624″, taking 206,265″. The error ${\delta }_{S_0}$ is the mid-range error of the tilt distance; ${\delta }_{D_{13}}$ is the mid-error of the DBH, and ${\delta }_{{\beta }_0}$ is the mid-error of the horizontal angle.

2.2.2. Error Analysis of Tree Height H

The accuracy of the tree height measurement H is affected by the error in Equation (2) for the slant distance, the zenith distance measured by the electronic theodolite at the DBH, and the zenith distance of the treetops. The formula is derived according to the law of error propagation.

The system error of the tree height H is

$$\frac{\delta H}{H}=\sqrt{\begin{array}{l}\left[\frac{\sin (v_0-v_n)}{S_0\sin (v_0-v_n)+h_0\sin v_n}\right]{\delta }_{S_0}^2\\ +{\left[\frac{h_0\cos v_n-S_0\cos (v_0-v_n)}{S_0\sin (v_0-v_n)+h_0\sin v_n}-\cot v_n\right]}^2{\left(\frac{{\delta }_{angle}}{\rho }\right)}^2\\ +{\left[\frac{\sin v_n}{S_0\sin (v_0-v_n)+h_0\sin v_n}\right]}^2{\delta }_{h_0}^2\end{array}}$$

(8)

In the formula, ${\delta }_H$ is the mean error of the tree height, ${\delta }_{\mathrm{angle}}$ is the angular deviation of the electronic theodolite, and ${\delta }_{h_0}$ is the mid-error of the first segment $h_0$.

2.2.3. Error Analysis of the Diameter $D_i$ at Any Position

The accuracy of the diameter $D_i$ at any point of the trunk is mainly affected by the error in the slant distance $S_0$ and the accuracy of the horizontal angle at the height ${\beta }_i$ measured by the electronic theodolite. Then, the system error of the diameter is

$$\frac{{\delta }_{D_1}}{D_i}=\sqrt{{\left(\frac{{\delta }_{S_0}}{S_0}\right)}^2+{\left(\frac{{\delta }_{{\beta }_1}}{\rho }\right)}^2\frac{1}{{\rho }^2}}$$

(9)

where ${\delta }_{D_1}$ is the mean error of the diameter at any point of the trunk and ${\delta }_{{\beta }_1}$ is the mean error of the horizontal angle.

2.2.4. Error Analysis at Arbitrary Heights

The accuracy of the arbitrary segment trunk height $h_i$ is affected by the tilt distance $S_0$ and the measurement error of the zenith distance ${\gamma }_i$ at any point.

The relative error of any segment height is

$$\frac{{\delta }_{h_1}}{h_i}=\sqrt{\begin{array}{l}{\left(\frac{{\delta }_{S_0}}{S_0}\right)}^2+{\left[\frac{(\sin v_{i-1}-\sin v_i)\cos (v_0-v_i)-\sin (v_0-v_{i-1})\cos v_i}{\sin v_{i-1}\sin (v_0-v_i)-\sin v_i\sin (v_0-v_{i-1})}+\cot v_i\right]}^2{\delta }_{v_i}^2\\ +{\left[\frac{\sin (v_0-v_{i-1})\cos v_{i-1}}{\sin v_{i-1}\sin (v_0-v_i)-\sin v_i\sin (v_0-v_{i-1})}-\cot v_{i-1}\right]}^2{\delta }_{v_{i-1}}^2\end{array}}$$

(10)

In the formula, ${\beta }_{h_1}$ is the mean error of an arbitrary segmentation; ${\delta }_{v_i}$ and ${\delta }_{v_{i-1}}$ are the mean errors for measuring the zenith distance.

2.2.5. Variance of the Diameter $D_i$ and the Height $h_i$ at Any Point

The covariance between $D_i$ and $h_i$ is

$$\begin{array}{l}{\delta }_{D_i{h}_i}={\beta }_i\left[\frac{\sin (v_0-v_i)}{\sin v_i}-\frac{\sin (v_0-v_{i-1})}{\sin v_{i-1}}\right]{\delta }_{S_0}^2\\ -\frac{S_0\left[\cos (v_0-v_i)\sin v_i+\cos v_i\sin (v_0-v_i)\right]}{{\sin }_{v_1}^2}{\left(\frac{{\delta }_{angle}}{\rho }\right)}^2\end{array}$$

(11)

2.2.6. Error Analysis of Stem Volume

The accuracy of the stem volume V is affected by the measurement error in $D_0$, the DBH, the height $h_i$ of the arbitrary section of the trunk, the error of the diameter $D_i$ at that point, the covariance of the diameter $D_i$ at that point, and the height $h_i$ at that point.

The system error of the stem volume is

$$\frac{{\delta }_v}{V}=\frac{{\delta }_{v_1}}{V_1}+\cdots +\frac{{\delta }_{vi}}{V_i}+\cdots +\frac{{\delta }_{v_{n_0}}}{V_{tip}}$$

(12)

In the formula, ${\delta }_v$ is the mean error of the total stem volume of the trunk, and ${\delta }_{v_{n_0}}$ is the mean error of the trunk tip volume.

The data collection site was located at Xinggezhuang Village, Yucheng Town, Tongzhou District, Beijing, at 39°59′ north latitude and 116°55′ east longitude. The collection date was 1–28 March 2014. The sample tree species were Populus tomentosa, Populus cathayana Rehd, and Populus canadensis Moench. To further analyze the error of the electronic theodolite in measuring the volume, height, and diameter of individual trees of different sizes, the sample of standing trees were classified by the tree height and the diameter (large, medium, and small size) of the representative average size tree. The criteria for selecting large, medium, and small sample woods were large size (breast diameter 25–50 cm, tree height 16–37 m), medium size (diameter 15–25 cm, tree height 13–23 m), and small size (15 cm and below for the breast diameter, 10 m highest tree height).

Normally, the standard deviation of the breast diameter (±1 mm) and the standard deviation of the electronic theodolite are ± 2.0′′, where the DBH $D_{13}$, the ground diameter $D_0$, and the height from the ground $h_0$, the DBH are measured by the tape. The volume is calculated based on the method described above.

2.3. Accuracy Verification

2.3.1. Measurement of Felled Trees

To test the accuracy of the calculated system error from the electronic theodolite, we deemed the measured value of felled trees as the true value of tree diameter, height, and volume. The standard sectional method was used to measure the felled trees [28]. The method theoretically divides the tree stem into a number of (mostly) standard length sections. All the sections except the tip are assumed to be second-degree paraboloids. The tip is assumed to be a cone. The volumes for these assumed shapes are easily calculated from simple measurements of diameter and length. Then, stem volume is obtained by summation.

The length of each section can be measured by a rod or tape. Length may be measured to the nearest 5 or 10 cm.

The diameter of a section can be measured by a caliper or diameter tape. Measurement is made at the middle of the length of the log. To reduce error, two measurements at right angles may be taken, and the mean diameter may be used for the calculation of volume. Diameter may be measured to the nearest centimeter.

The volume of a section is determined by using Huber’s formula, which is calculated as follows:

$$V=H·AE$$

(13)

where $V$ is the stem volume, Am is the basal area at the mid section and h is the length of the log. The values of h and Am are obtained from the measurement of length and diameter as described above.

2.3.2. F Test Principle

The significance test of the regression equation is a common method to test whether there is a significant difference between two sets of data. For the purpose of this paper, it was necessary to check whether there was a statistically significant difference between the measurement data from the electronic theodolite and the measurement data of the felled wood. In this paper, the F test was used to test the significance of the regression equation. It is any statistical test in which the test statistic has an F-distribution under the null hypothesis and is calculated by the following formula [29]:

$$F=\frac{{\displaystyle \sum }{\left(a{X}_i-\overline{Y}\right)}^2}{{\displaystyle \sum }{\left(Y_i-a{X}_i\right)}^2\frac{1}{n-2}}$$

(14)

where $X_i,Y_i$ denotes the $i$-th predicted value and actual value, $\overline{Y}$ denotes the overall mean of the actual value, $n$ denotes the number of observations, and $a$ is a constant.

Based on the statistical software SPSS 27.0, the running of the F test includes four steps as follows:

(1) State the null hypothesis and the alternate hypothesis. Here, the null hypothesis is assumed to be no significant difference between the two groups of data, and the alternative hypothesis is that the two groups are significantly different.

(2) Calculate the F value based on the formula above.

(3) Find the F statistic (the critical value for the test) in the F-table. The quantile of F text in this study was defined as 0.95.

(4) Support or reject the null hypothesis according to the F statistic. In this study, when F ≤ F 0.05, there was no significant difference between the two groups.

3. Results

The theoretical values of the system errors in measuring the large-, medium-, and small-sized wood samples by theodolite are shown in

Table 2. Analysis of errors in the measurement of stems using electronic theodolite.

Type	Diameter/cm	Height/m	Volume/m3	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

and

Table 3. The number and ranges of diameter, height, and volume of sample trees.

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%

Notation: RoD (range of tree diameter); RoDr (relative error range of tree diameter); Roh (range of tree height); Rohr (relative error range of tree height); Rov (range of tree volume); Rovr (relative error range of tree volume).

. In the table, representative trees of the same tree species with large, medium, and small sizes are given. It can be noted from Table 2 that when the electronic theodolite measures different sizes of wood, the error will vary slightly with the size of the tree. The system error of the measured tree height was 0.29–0.89%, while the system error of the measured diameter was 0.44–0.59%, and the system error of the measured volume was 0.31–0.99%.

Table 4. Large-gauge wood diameter, tree height, and volume error.

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%

shows that on the large-scale sample wood, the relative error of diameter of Populus cathayana Rehd was higher than that of Populus tomentosa, and the relative tree height error of Populus tomentosa was higher than those of the other two species. The relative error of the volume of Populus canadensis Moench was higher than those of Populus cathayana Rehd and Populus tomentosa. In

Table 5. Medium-gauge wood diameter, tree height, and volume error.

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%

, there is no medium-sized wood for the Populus canadensis Moench. Therefore, compared with Populus cathayana Rehd or Populus tomentosa, for medium-sized wood, the relative error of diameter and tree height of Populus tomentosa is higher than that of Populus cathayana Rehd, and the volume error is 3.92%, slightly smaller than that of Populus cathayana Rehd.

Table 6. Small-gauge wood diameter, tree height, and volume error.

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%

shows that on the small-scale sample wood, the relative error of diameter and tree height of Populus cathayana Rehd is higher than that of Populus tomentosa, and the volume error of Populus cathayana Rehd is slightly smaller than that of Populus tomentosa. Data of all tested trees are listed in

Table 7. Data of all tested trees.

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%

Notation: Vbt (Volume measured by electronic theodolite); Vopt (Volume of parse tree); Aeov (Absolute error of volume); Reov (Range of tree height relative error); hbt (Tree height measured by electronic theodolite); hopt (Tree height of parse tree); Aeoh (Absolute error of tree height); Reoh (Relative error of tree height).

. In this paper, the regression test was used to analyze the overall error.

The electronic latitude and longitude measuring data of the three tree species (groups) are used to calculate the linear regression equation, and the F statistic is calculated according to formula (1). The F test was used to measure the overall accuracy, which shows the difference between the measurement data from the electronic theodolite and the measurement data of the felled wood. As shown in

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

	F	F0.05
Diameter	0.810	3.953
Tree height	0.953	3.953
Stem volume	0.828	3.953

, all F values are smaller than the critical value at the 0.05 level, indicating that there was no significant difference between the electronic theodolite measurement data and the actual measurements.

Figure 5, Figure 6 and Figure 7 illustrate the comparisons between the tree height, diameter, and volume data of the measurement of the electronic theodolite and the actual measurements of the poplars. In Figure 5, the F value is outside the rejection domain. We rejected the alternative hypothesis and accepted the null hypothesis, i.e., there was no significant difference between the two groups of volume data.

Table 9. Mean value of error.

	Diameter Relative Error	Height Relative Error	Volume Relative Error
Value	2.02%	1.18%	4.47%

shows that tree surveying by common precision electronic theodolite in China is more accurate than the requirement of error not exceeding the 3–5% standard.

4. Discussion

Comparing this study with earlier studies in China, there were some significant improvements. Cao et al. [30] carried out research on measuring tree height and volume with theodolites. Five kinds of theodolites with different accuracies were applied to measure different species. The average relative error of tree height and volume was 1.20% and 4.27%, respectively, which is consistent with our result. However, it did not explain the experimental tree species, which would have a different effect on the error. For that, we provided a more detailed explanation of the influence of different tree species on the error. Du [31] conducted an experiment on the height and volume of three tree species (Platycladus orientalis, larch, and poplar) with an electronic theodolite in Beijing. He found that the average relative errors of tree height and volume were 1.71% and 1.58%, respectively, passing the verification of the F test. Even though the accuracy of the result was higher than ours, the analysis of tree diameter and its influence on tree volume was lacking in his study. Considering the tree diameter as an important factor for volume calculation, we analyzed the error of tree diameter and applied it in calculating the system error of tree volume. Zhang [32] analyzed the system error of tree height and volume measurements in continuous forest inventories based on the law of error propagation. However, it was only deduced from the theoretical model and not verified by specific experimental data. In contrast, we took a total of 87 stand trees in Beijing as the experimental sample to verify the theoretical value of the system error of tree diameter, height, and volume. The trees were divided into three sample groups according to size. The study from Gao et al. [33] focused on how to better measure the upper diameter by sections and then calculate the volume based on 198 larch trees using electronic theodolite. According to the finding that a tree divided into 10 sections can better simulate the trunk shape and calculate the volume, we divided all sample trees into 10 segments for observation and measurement in our study. Yu et al. [34] measured the tree DBH and height of 10 larch trees using a total station and calculated the volume. This improved the measurement accuracy through a more accurate measurement instrument with average relative DBH, height, and volume errors of 0.070%, 0.023%, and 0.235%, respectively. However, the amount of data was not enough with the study sample of only one tree species and 10 trees. In our study, 87 poplars of three species (Populus tomentosa, Populus cathayana Rehd, Populus canadensis Moench) were measured by electronic theodolite to verify the error. The result that the average relative errors of volume, height, and diameter, i.e., 4.47%, 1.18%, and 2.02%, respectively, were verified to be similar to those of other similar instruments.

5. Conclusions

A comparative measurement study was performed based on a Beijing sample of 87 trees from the species Populus tomentosa, Populus cathayana Rehd and Populus canadensis Moench. Diameter, height, and volume measurement data were obtained. A differential significance test was performed on the measurement data from the electronic theodolite and the measurements from the felled wood. The regression test results showed that the differences between the diameter, tree height, and the volume data as measured by the electronic theodolite were not significantly different from the measured data of the felled wood. The theoretical error in tree height measurement was 0.29–0.89%, and the relative error of the measured volume in theory was 0.31–0.99%. Measuring the volume, tree height, and diameter with the electronic theodolite, the mean relative errors were 4.47%, 1.18%, and 2.02%, respectively, which are below the 5% standard for the measurements.