Predicting Stand Volume by Number of Trees Automatically Detected in UAV Images: An Alternative Method for Forest Inventory

: In this study, we estimate the forest stock volume by multiplying the number of trees detected remotely by the estimated mean individual volume of the population (individual approach). A comparison was made with the conventional inventory method (area approach), which included 100 simulations of a simple random sampling process and a Bootstrap resampling. The study area included three stands: stand 1, 16-year-old pine; stand 2, 7-year-old pine; and stand 3, 5-year-old eucalyptus. A census was carried out in each stand for the variables diameter and total height. Individual volume was estimated by a ratio estimator, and the sum of all volumes was considered as the total parametric volume. The area approach presented parametric values within the conﬁdence interval for 91%, 94%, and 98% of the simulations for the three stands, respectively. The mean relative errors for the area approach were − 3.5% for stand 1, 0.3% for stand 2, and − 0.9% for stand 3. The errors in stands 1 and 3 were associated with the spatial distribution of the volume. The individual approach proved to be efﬁcient for all stands, and their respective parametric values were within the conﬁdence interval. The relative errors were 1% for stand 1, − 0.7% for stand 2, and 1.8% for stand 3. For stand 1 and 3, this approach yielded better results than the mean values obtained by the area approach simulations (Bootstrap resampling). Future research should evaluate other remote sources of data and other forest conditions. Â data curation, L.S. and Â writing—original preparation, L.S. and A.P.D.C.; writing—review and L.S., A.P.D.C., C.R.S., A.B., and R.d.L.E.; visualization, L.S.;


Introduction
Planted forests occupy approximately 294 million hectares or 7% of the global forest area [1] They constitute a sustainable alternative for the global demand for timber products [2,3] and their use has led to a decline in the deforestation of natural forests [4][5][6]. Planted forests may also mitigate the effects of climate change [7], maintain biodiversity, and sequester carbon [8,9].
Managing planted forests requires precise, reliable information to support the decisions made in operational planning [10][11][12]. Forest attributes are estimated through forest inventories, which usually apply sampling methods based on sampling units and statistical extrapolation techniques [10,11,13,14]. For this reason, a precise estimation of the forest area is fundamental, since the results of the sampling are multiplied by it to determine the final estimate of the whole forest [15]. The same author mentions that "poor area measurements are often one of the biggest sources of error in an inventory estimate". In the continuous forest inventories carried out in Brazil, discrepancies between the estimated

Field Data and Volume Estimation Modeling
Two types of field data were used in this study: a) census data from the three stands of the study area, and b) stem data obtained from Klabin's database, independent of this study area, but of the same study region, with similar characteristics of climate and geological material. This dataset was used to model the individual tree volume.
In the census, each tree was identified by its row and position in the row, we counted the number of trees and measured the total height (h) using a Haglöf Electronic Clinometer and the circumference with a tape measure at a height of 1.30 m, which was converted to the diameter at a height of 1.30 m (d).
The volume of each tree measured in the census was determined by Equation (1), where v is the individual volume (m 3 ), g is the cross-sectional area (m 2 ), h is the total height (m), and f is the form factor. = . ℎ.
(1) The form factor (Eq. 2) was determined by ratio estimators [36], whose ratio of the mean individual stem volume to the mean total stem volume from the scaling dataset, which resulted in a good estimate with low variance [37].
where = f = ratio estimator = mean form factor, is the mean individual stem volume obtained from the scaling dataset, and ̅ is the mean total stem volume of the scaled trees.
We obtained the total volume of each stand by summing the estimated volume of the trees measured in the census, which was treated as the total parametric volume in this study.
The scaling dataset (Table 1) used to obtain the ratio estimator was taken from a consolidated database of the same study region. We selected trees of the same species and approximate age without distinguishing between site classes. The trees were randomly selected in a database and maintained the same proportion found in the frequency distribution for the diameter and height classes determined in the census. We applied this method to ensure that the mean individual volume and form factor were representative

Field Data and Volume Estimation Modeling
Two types of field data were used in this study: (a) census data from the three stands of the study area, and (b) stem data obtained from Klabin's database, independent of this study area, but of the same study region, with similar characteristics of climate and geological material. This dataset was used to model the individual tree volume.
In the census, each tree was identified by its row and position in the row, we counted the number of trees and measured the total height (h) using a Haglöf Electronic Clinometer and the circumference with a tape measure at a height of 1.30 m, which was converted to the diameter at a height of 1.30 m (d).
The volume of each tree measured in the census was determined by Equation (1), where v is the individual volume (m 3 ), g is the cross-sectional area (m 2 ), h is the total height (m), and f is the form factor. v = g·h· f (1) The form factor (Equation (2)) was determined by ratio estimators [36], whose ratio of the mean individual stem volume to the mean total stem volume from the scaling dataset, which resulted in a good estimate with low variance [37].
whereR j = f = ratio estimator = mean form factor, y i is the mean individual stem volume obtained from the scaling dataset, and x i is the mean total stem volume of the scaled trees. We obtained the total volume of each stand by summing the estimated volume of the trees measured in the census, which was treated as the total parametric volume in this study.
The scaling dataset (Table 1) used to obtain the ratio estimator was taken from a consolidated database of the same study region. We selected trees of the same species and approximate age without distinguishing between site classes. The trees were randomly selected in a database and maintained the same proportion found in the frequency distribution for the diameter and height classes determined in the census. We applied this method to ensure that the mean individual volume and form factor were representative of the stands. By using these pre-existing data, field sampling was not required for volume estimation modeling, allowing a practical and quick estimation method. Based on [36,38], we estimated the variance (Equation (3)), standard error (Equation (4)), confidence interval (Equation (5)), and relative error (Equation (6)) of the form factor, represented by the ratio estimator (R j ).
where n = number of scaled trees, t = student's t-distribution,R j = ratio estimator, x i = volume of the scaled log cylinders, y i = individual volume per scaled tree, x = mean volume of scaled trees cylinders.

UAV Data and TreeDetect Algorithm
In this study, we used the number of trees detected by a tree detection algorithm used in a previous study [39,40] as both studies have the same areas. The authors tested a toolbox (TreeDetect) built in ArcGIS with ArcPy, which was designed to automatically detect trees from high-resolution data obtained using UAV. Field data (census) and UAV data were collected on the same date.
The UAV we used was an eBee-Ag (Sensefly), with a different camera for each species: a near infrared (red, green, and near infrared (NIR)) Canon S110 NIR for the P. taeda stands and a multispectral camera (green, red, red edge, and NIR) Multispec 4C for the Eucalyptus stand. The NIR camera had a resolution of 12 MP, a sensor size of 6.23 × 4.69 mm (4048 × 3048 pixels), a focal length of 5 mm, and a pixel size of 1.54 µm. The multispectral camera had a resolution of 1.2 MP, a sensor size of 4.8 × 3.6 mm (1280 × 960 pixels), a focal length of 4 mm, and a pixel size of 3.75 µm [39,40]. Four ground control points were used in each stand, and their positions were collected by a RTK GPS (GPS Pathfinder ProXRT Receiver Trimble). The RMSE obtained for point geolocation was ±0.11 m for x, ±0.04 m for y, and ±0.02 m for z (stand 1), ±0.030 m for x, ±0.047 m for y, and ±0.048 m for z (stand 2) and ±0.026 m for x, ±0.046 m for y, and ±0.147 m for z (stand 3). The images were processed by the Pix4D Mapper (version 3.2.17). The NIR camera has the following processing settings: full keypoints; image scale 1; automatic number of keypoints; geometric verified matching and standard calibration in the initial step; multiscale; 1 2 image scale; high density; three minimum matches in the dense cloud step; and filter noise and no smooth surface in the DSM option. We obtained DSM and orthomosaic with a resolution of 5.33 cm/pixel and RMS error of 0.052 m by the NIR camera for the stand 1 and 6.22 cm/pixel and RMS error of 0.041 m for stand 2. For the Multispec camera, the Rapid keypoints and Alternative calibration in the initial step of processing were changed, and the optimal point cloud density in the dense cloud step was selected. The other parameters were the same as the ones from the NIR camera. In the Multispec the camera and sun irradiance were corrected by using the values of reflectance from a calibration target. The correction enables to generation of a reflectance map of each band from the Multispec camera, instead of one orthomosaic. The resolution of the reflectance maps of each band from the Multispec was 12.09 cm/pixel, and the RMSE error was 0.067 m [39,40].
The input raster file on the TreeDetect Algorithm included the canopy height model (CHM) normalized by a LiDAR cloud point for the 16-year-old P. taeda stand, a NIR image for the 7-year-old P. taeda stand, and a normalized difference vegetation index (NDVI) image for the Eucalyptus stand [39,40]. The following input data parameters were used for all stands in the algorithm: an input raster file with a cell size of 0.5 m and a conversion value of −1 to invert the structure of the raster (watershed segmentation method); a minimum size (crown tree area) of 3 m 2 for the 16-year-old P. taeda and Eucalyptus stands and 2 m 2 for the 7-year-old P. taeda stand; and a smoothing factor of 2 m 2 for the 16-year-old P. taeda and Eucalyptus stands and 1 m 2 for the 7-year-old P. taeda stand [39,40].

Predicting Stand Volume: Area Versus Number of Trees
We evaluated two approaches for estimating the total volume of the stand: the first was the conventional method for forest inventories, where the total volume of the forest is estimated by extrapolating the volume (m 3 ha −1 ) of the sampling units for the stand area (method 1); the second approach estimates the total volume by the number of trees detected remotely (method 2). Both methods are explained in detail below: (a) Method 1 (area): We performed 100 simulations for each stand using a simple random sampling process [41] with circular sampling units with a fixed area (600 m 2 ), n = 5, to represent a sampling fraction of 9.7% for stand 1, 10.7% for stand 2, and 13% for stand 3. We adopted these sampling fractions to ensure the accuracy of the volume estimation of the stands. The sampling units were randomly distributed in the ArcGis software with parcel replacement. For each sampling unit, the total volume (sum of estimated individual volumes-see Section 2.2, Equation (1)) was extracted and converted to volume per hectare (m 3 ha −1 ). The estimators (mean, variance, standard error, population total, and confidence intervals) were calculated according to previous studies [41].
We determined the RMSE% and mean difference (Bias%) (Equations (7) and (8)) based on the simulation of the stands using the area approach. Y corresponds to the total volume based on the field data (census),Ŷ represents the estimated total volume, and n refers to the number of sampling units.
A Bootstrap resampling (with replacement and 5000 replications) was made for the 100 simulations to obtain the mean volume (m 3 ha −1 ) and the confidence interval for this approach. This analysis was performed with the Boot package of the R software (Version 4.1.0).
(b) Method 2 (individual): The total volume of stands (Equation (9)) was estimated by multiplying the mean individual stem volume of the scaled trees (ŷ) by the number of trees in each stand (N).Ŷ = Nŷ Forests 2021, 12, 1508 6 of 14 The number of trees (N) in each stand was determined by the automatic detection of trees with the TreeDetect algorithm [39,40]. The overall accuracy of TreeDetect algorithm was computed by comparing the trees identified in field data inventory (census) and by the TreeDetect algorithm. The trees were plotted in the orthomosaic by using the field data.
We used Equation (10) to obtain the mean individual stem volume (ŷ), where x i is the volume of the scaled log cylinders (m 3 ), obtained with the data referenced in Table 1,R j is the ratio estimator, and n is the number of scaled trees.
We estimated the variance, standard error, confidence interval, and relative error for the mean individual volume (ŷ) obtained from the scaling dataset (Equations (11) to (14) and for the estimated total volume (Ŷ) (Equations (15) to (18)) based on a previous study [38]. Where N is the number of trees in the stand, n is the number of scaled trees, t is Student's t-distribution,R j = ratio estimator, x i = is the volume of the scaled log cylinders, y i = individual volume per scaled tree, and x = mean volume of scaled log cylinders.

Validation and Comparison of the Two Approaches for Predicting Forest Stand Volume
To compare the estimates from the area and individual approaches, we calculated the absolute (Equation (19)) and relative differences (Equation (20)) at the stand level between the volume from the census and the mean volume based on Bootstrap resampling of the simulations (area approach) or the estimated volume based on the TreeDetect algorithm (individual approach).
absolute difference = Y −Ŷ We also evaluated the confidence intervals for all estimates. The method resulting in the smallest error and parametric values within the confidence interval (CI) was considered to be satisfactory.

Results
Estimating the form factor and individual volume for the scaled trees using the ratio estimator proved to be an appropriate method. The values were similar to those found in other studies that used volume estimation models for pine and eucalyptus in Brazil [42][43][44][45][46][47], and they presented a high level of accuracy (relative error under 2%). The 7-year-old Pinus taeda in stand 2 had the best performance (Table 2). By using the ratio estimator to process the census data, we could determine the parametric values of the mean individual stem volume, and total volume of the stands (Table 3), where d is the diameter at breast height (1.30 m), h is the total height (m), v is the individual volume (m 3 ), N is the number of trees and CV% is the coefficient of variation. The estimate for the total stand volume simulated by method 1 (area approach), indicated that the total parametric volume was within the confidence interval for 91% of the simulations in stand 1, 94% of the simulations in stand 2, and 98% of the simulations in stand 3 (Figure 2).
In the simulations for stand 1, the estimated total volume ranged from 836.8 to 1049.9 m 3 , resulting in a bias between 7.1% and −16.6% and an RMSE between 3.8% and 19.1% (Figure 3a). The estimated total volume in stand 2 ranged from 428.9 to 505.7 m 3 , the bias varied from 9.6% to −6.6%, and the RMSE was between 4.0% and 13.4% (Figure 3b). The estimated total volume in stand 3 ranged from 540.3 to 615.1 m 3 , the bias varied between 6.1% and −6.9%, and the RMSE was between 3.1% and 10.9% (Figure 3c). The estimates were biased in stands 1 and 3 and tended to overestimate the volume of the stands. In the simulations for stand 1, the estimated total volume ranged from 836.8 to 1049.9 m 3 , resulting in a bias between 7.1% and -16.6% and an RMSE between 3.8% and 19.1% (Figure 3a). The estimated total volume in stand 2 ranged from 428.9 to 505.7 m 3 , the bias varied from 9.6% to −6.6%, and the RMSE was between 4.0% and 13.4% (Figure 3b). The estimated total volume in stand 3 ranged from 540.3 to 615.1 m 3 , the bias varied between 6.1% and −6.9%, and the RMSE was between 3.1% and 10.9% (Figure 3c). The estimates were biased in stands 1 and 3 and tended to overestimate the volume of the stands. Using method 2, the TreeDetect algorithm detected a total of 2681 individual trees in stand 1, 3962 individual trees in stand 2, and 2111 individual trees in stand 3, corresponding to detection rates of 102.7%, 94.1%, and 98.3%, respectively, including commission  Using method 2, the TreeDetect algorithm detected a total of 2681 individual trees in stand 1, 3962 individual trees in stand 2, and 2111 individual trees in stand 3, corresponding to detection rates of 102.7%, 94.1%, and 98.3%, respectively, including commission (nontrees classified as trees by the algorithm) and omission (undetected trees) errors. The total volume estimated by method 2, was 891.5 m 3 for stand 1, 477.8 m 3 for stand 2 and 565.4 m 3 for stand 3 (Table 4). Upon comparing the two approaches (Figure 4), we noted that the parametric value was included in the confidence interval (CI) for all stands in the individual approach (method 2). Using method 2, the TreeDetect algorithm detected a total of 2681 individual trees in stand 1, 3962 individual trees in stand 2, and 2111 individual trees in stand 3, corresponding to detection rates of 102.7%, 94.1%, and 98.3%, respectively, including commission (non-trees classified as trees by the algorithm) and omission (undetected trees) errors. The total volume estimated by method 2, was 891.5 m 3 for stand 1, 477.8 m 3 for stand 2 and 565.4 m 3 for stand 3 (Table 4).  (Figure 4), we noted that the parametric value was included in the confidence interval (CI) for all stands in the individual approach (method 2). Based on the Bootstrap resampling (area approach) the estimate was satisfactory only for stand 2 with the smallest difference relative to the volume of the census (Table 5) in Based on the Bootstrap resampling (area approach) the estimate was satisfactory only for stand 2 with the smallest difference relative to the volume of the census (Table 5) in which the total parametric volume was included in the confidence interval ( Figure 4). In the case of stand 1 and stand 3, the best approach was to use the number of individual trees, because the total parametric volume was included in the confidence interval ( Figure 4). The percentual differences (Table 5) show values below 5% when comparing the estimated averages with the parametric value, for both approaches. That value is considered appropriate for forest inventories in plantations. However, by looking at the CI and the relative differences, we can infer that the approach using the number of trees in the stand and the individual tree volume is appropriated to estimate the total population volume and this method's result is comparable to the traditional method (area approach).

Discussion
We verified that the parametric value remained within the confidence interval for most simulations of the conventional method of forest inventory. However, it is important to highlight that the stand areas in this study were determined by the limits established by the spatial location of the trees, which contributed to the high precision of the stand areas used to estimate the total population. If it is assumed that the stand area was determined correctly, then the estimation errors in this study (area approach) can be attributed to sampling errors arising from the sampling process, the sample size, or a lack of representativeness due to forest variability [48].
The trend presented by the negative bias (overestimation of the volume) observed in the simulations for stands 1 and 3 can be explained by an analysis of maps with spatialization (inverse distance weighted [IDW]) of the volume per hectare and of the sampling grid used in data interpolation ( Figure 5). In this case, there was a predominance of areas with volumes per hectare greater than the averages of 296.4 m 3 ha −1 in stand 1 (Figure 5a) and 254.7 m 3 ha −1 in stand 3 (Figure 5c).

Discussion
We verified that the parametric value remained within the confidence interval for most simulations of the conventional method of forest inventory. However, it is important to highlight that the stand areas in this study were determined by the limits established by the spatial location of the trees, which contributed to the high precision of the stand areas used to estimate the total population. If it is assumed that the stand area was determined correctly, then the estimation errors in this study (area approach) can be attributed to sampling errors arising from the sampling process, the sample size, or a lack of representativeness due to forest variability [48].
The trend presented by the negative bias (overestimation of the volume) observed in the simulations for stands 1 and 3 can be explained by an analysis of maps with spatialization (inverse distance weighted [IDW]) of the volume per hectare and of the sampling grid used in data interpolation ( Figure 5). In this case, there was a predominance of areas with volumes per hectare greater than the averages of 296.4 m 3 ha -1 in stand 1 (Figure 5a) and 254.7 m 3 ha -1 in stand 3 (Figure 5c). The characteristics of the spatial volume in stands 1 and 3 likely contributed to the relatively higher estimation errors that resulted from random sampling, which is a disadvantage of the inference method. Both the accuracy and reliability of forest inventories are related to the quality and quantity of the sampled units [49] and the level of data dispersion to ensure that sampling is representative of the spatial variability found in a forest [50]. The characteristics of the spatial volume in stands 1 and 3 likely contributed to the relatively higher estimation errors that resulted from random sampling, which is a disadvantage of the inference method. Both the accuracy and reliability of forest inventories are related to the quality and quantity of the sampled units [49] and the level of data dispersion to ensure that sampling is representative of the spatial variability found in a forest [50].
In method 2, we observed that the detection rates of the number of trees found in this study (102.7% for stand 1, 94.1% for stand 2, and 98.3% for stand 3) along with the mean individual volume of the scaled trees led to differences between the parametric and estimated volumes. Nevertheless, the method proved to have a high estimation capacity as the parametric volume was within the confidence intervals for all the stands, with a maximum error of 1.8% relative to the census volume.
The number of trees detected by the algorithm was lower than the number determined by the census in stands 2 and 3 and higher in stand 1. Trees in stands 2 and 3 may have gone undetected due to difficulties in visualizing the canopy (dominant, forked trees), thus resulting in lower detection accuracy [19].
It is important to note that counting algorithms occasionally detect forked trees as two or more separate trees, especially if the canopies are very close to each other [19,51,52], which leads to errors in determining the number of trees. The counting algorithm (TreeDetect) reported a higher number of trees in stand 1, which was primarily due to the presence of trees with side branches (14.7%), which resulted in peculiar geometric figures indicating the presence of more than one canopy. These branches form due to the actions of capuchin monkeys, who break tree branches. Excessive lateral branches were also responsible for an overestimation of the number of trees in a previous study [52], which evaluated two tree-detection methods using the LiDAR point cloud.
Regardless of the method used to acquire data (remote or field) or the approach (individual or area), this study revealed that it is important to control the estimation of the individual stem volume as its value directly influences estimations of the total stand volume [53].
The individual stem volume is usually determined by adjusted models based on data collected from scaled trees [37,[44][45][46][47]. However, other methods with terrestrial [14,[54][55][56] airborne [11,[57][58][59] or 3-D point clouds derived from UAV imagery [60][61][62][63][64] have also been investigated. In this study, we estimated the mean individual stem volume using the ratio estimator with a database independent of the study area and consequently reduced field sampling for volume estimation modeling and produced results with high levels of precision. Nonetheless, a prior diagnosis of the frequency distribution of the diameters and heights was required to obtain a representative selection of the trees in the stands.
The potential to make inferences about a total population based on the number of trees in a stand appears to be an alternative to the conventional forest inventory method, demonstrating promising results for automatic tree detection. Further studies should explore the application of this method in other forests with different characteristics and other remote sources of data collection.

Conclusions
Stand volume of planted forests can be estimated based on the number of individual trees and by the mean individual stem volume estimated by ratio estimates with a satisfactory result. The Bootstrap resampling (area approach) exhibited better results for stand 2 (Pinus taeda-7 years old), while the individual approach displayed better results for stand 1 (Pinus taeda-16 years old) and 3 (Eucalyptus urograndis-5 years old). The simulations using the area approach presented biased results for stands 1 and 3 that tended to overestimate the stand volumes.