În mod obişnuit, silvicultorii estimează suprafaţa de bază a unui arboret prin măsurarea diametrelor de bază ale arborilor în suprafeţe de probă, deoarece măsurarea tuturor arborilor din arboret este o operaţie impracticabilă. Expansiunea de la suprafaţe de probă la arboret este complicată deoarece arborii nu sunt în mod obişnuit uniform aleator răspândiţi în arboret (adesea fiind grupaţi), diametrele lor nu urmează o repartiţie uniformă iar densitatea arboretului poate juca un rol semnificativ în procesul de estimare, deoarece aceasta afectează direct omogenitatea arboretului.
Obiectivul acestui articol este de a prezenta un algoritm care poate fi utilizat în generarea de arborete pure având parametri prestabiliţi şi care descriu atributele investigate, în acest caz diametrul de bază. Algoritmul generalizează procedura dezvoltată de Wang et al.(2009) prin incorporarea distribuţiei Weibull şi a gradului de grupare a arborilor in procesul de calcul. Rezultatele obţinute arată că, generând arborete utilizând procedura propusă, se produc date valide, care pot fi folosite pentru cercetări ulterioare.
Foresters routinely estimate basal area of a stand by measuring diameters at breast height of trees within a set of sample plots since measuring every tree in the forest is impractical. The central limit theorem provides the expansion framework from sample based estimates to stand (i.e., population) parameters. The expansion from sample to population is complicated as trees are seldom uniformly random distributed through the stand, often showing a clumping pattern. Similarly, tree diameters do not follow a uniform random pattern either. Different diameter curves are being developed to represent the number of trees in each diameter class, namely the stand table. Furthermore, the forest structure varies according to the management goals from uneven-aged, for which the diameter distribution can be described in the extreme case by the exponential distribution, to even-aged, for which the diameter distribution follow a curve close to the normal distribution, commonly the Weibull distribution. Finally, the stand density can play a significant role in the expansion from the sample to population, as stand density seems to directly impacts stand homogeneity, some studies indicating that the increase in number of trees increases the stand variability (Ohlson and Schellhaas, 1999; Ozcelik et al., 2008). Consequently, one could ask whether or not the variation of diameter and tree spatial distribution have an effect on the accuracy of the sample based estimates. Furthermore, are sample based estimates influenced by the shape or size of sample plots, besides the trees biometry and location?
The traditional method to answer the above questions would be to find representative stands (i.e., from spatial, density, structural and diameter distribution perspective), establish a sampling scheme, measure the trees and compare the results to a complete inventory of the stand. However, natural stands do not have a homogeneous tree distribution throughout the stand consequently the sample could supply evidence for a different distribution than the actual stand distribution. Also, measuring tree locations and mapping the spatial pattern is difficult and time consuming. Finally, finding representative stands from both distributional and spatial perspective can be challenging and could make difficult the assessment of all the density/spatial association/diameter combinations of practical interest. With the advent of high speed computers having large memory, the development of spatially explicit tree dependent forest representations has become possible on personal computers, without the need of accessing large mainframes. Virtual forests can be created to represent and simulate any natural forest situation, specifically diameter distribution and spatial association. Overlaying the virtual forests with polygons representing the shape and size of the sampling unit one can perform a virtual sampling on the computer generated forests. Therefore, to investigate the effect of the shape and size of the sampling plot on parameter estimates and the impact of the density, spatial and diameter distribution of trees on these estimates the authors embraced the computer generation of stands. The objective of the paper is to present a computer algorithm that could be used to generate pure stands having preset parameters as well as different sampling method used for estimation, more specifically the size, shape and layout of the sampling design.
To model pure stands having different densities, diameter distributions and spatial repartition we have used a factorial design with three factors. The first factor, representing the density, expressed as number of trees per hectare had five levels (200, 400,600 800 and 1000), the second factor, representing the diameter distribution, was described using eight sets of the three parameters α, β, ν required by the Weibull distribution, while the third factor, representing the degree of tree association or clumpiness, had three levels, one indicating no clumps, one indicating significant grouping (i.e., 4 clumps/ha) and one intermediate grouping (i.e., 8 clumps/ha). The method used to generate the 3D stands (i.e., describe the location (x and y) as well as the magnitude of an attribute of interest, in this case diameter at breast height (dbh)) follows a sequential path with three steps:
In eventuality that sampling is of interest then a fourth step is required: namely, the overlapping of the generated stand with the sampling scheme.
To ensure the generality of the results, each combination of the factorial design was generated for a 100 ha stand. The size of the stand was recommended as accommodating both the management perspective, for which a stand is commonly less than 60 ha, and the unbiased requirement (i.e., larger populations are more accurate described by the theoretical distributions). Additionally, for each case of the factorial design, 10 replications were performed to fulfill the randomness requirement need for further statistical analyzes. The sample size was selected as 49 regardless the sampling scheme, size that guaranties that the estimates have a sampling error less than 10% and a confidence level of 95%. Consequently, a set of 1200 stands (i.e., 5 stand densities x 8 Weibull distribution sets of parameters x 3 degree of association x 10 replications) were generated. The stands were created using ArcGIS 9.3 (ESRI, 2009) and the statistics were computed using SAS 9.1 (SAS Institute, 2008).
A variety of distributions are used to describe the dbh distribution, distribution ranging from exponential to normal or Johnson SB. However, the most popular distribution used to represent the dbh distribution is the Weibull distribution as, depending on the parameters, could interpolate from exponential (β=1), to Rayleigh (β=2) and relatively normal (β>3), (Bailey and Dell, 1973; Little, 1983; Maltamo et al., 1995; Nanang, 1998; Zhang, 2006). The Weibull distribution with three parameters was used to describe the stand dbh, as provides greater flexibility and adjustment to the real forests than the Weibull distribution with two parameters. The objective of the study was not to identify the distribution that best describe a stand, but to generate 3D stands that could be used for generalization of the proposed methodology; situation recommending the usage of the three -parameters Weibull distribution. The probability density function for Weibull distribution of the random variable x>0 is (Ross 2006):
where α is the scale parameter, β affects the shape of the curve, and impacts the location of distribution curve (i.e., position of the mean on the abscissa)
The first two moments of the Weibull distribution are:
where Γ is the gamma function determined as
Figures 1 and 2 show how the Weibull distribution changes with changes in α and β, while ν = 0.
Weibull distribution has a great flexibility in representing natural stands, but can not address irregular diameter frequency distributions such as multimodal structures or highly skewed shaped diameter distributions (Zhang and Liu, 2006). The value of α and β can be interpreted as follows:
The third parameter from the Weibull distribution shifts to the right the entire distribution and helps describe stands with larger dbh. The increases in ν practically translates the Weibull distribution described by specific α and β, and ν = 0, with ν units, as shown in the Figure 3.
The Weibull distributions showing the diameter ranges and frequencies are displayed in Figure 4.
The Cartesian coordinates of each tree were determined by randomly assigning values between 0 and 1000 to each tree within a square with side 1000 m (i.e., the virtual stand has 100 hectare), one value being the abscissa, x, and the other one the ordinate, y. The details of the procedure are presented in Figure 5.
x & y: x,y location of tree center (on a 1 meter grid)
Tract width: 1000 meters, height: 1000 meters
Total_number_trees: tree_density(trees per hectare) x number_hectares
Weibull parameters: α, β,ν
Generate random numbers between 1 and 1000 for x and y
If a tree does not already exist at this location
Assign the coordinate to the tree
Generate a random number between Minimum_Tree_Diameter and
Maximum_Tree_Diameter for Diameter
(minimum_Tree_Diameter can not be less than the θ value of the Weibull function)
Calculate the Weibull_value for this Diameter
Generate a random number (Rand) between 0 and 1
If the Weibull_value > Rand then
Assign the Diameter to the tree
Until a diameter is assigned
Until total_number_trees are located
The forest obtained using this procedure is a forest that presents no trees association, either bilateral or multilateral. Tree diameters were assigned values according to Weibull distribution. The set of three parameters defining the three-parameters Weibull distributions used to describe the dbh distribution were [.5, 4, 0] [3, 4, 0] [5, 10, 0] [3, 4, 10] [5, 10, 10][3, 4, 20] [3, 4, 30] [3, 4, 40], where the first value is α, the second is β and the third is ν.
The stands created with the previous algorithm exhibit a random distribution of trees. Since natural stands often display a clumped characteristic due to competition, tree dependence, and harvesting patterns a second algorithm was developed to create clumped forests for each of the Weibull distributions and tree densities listed above.
Clump centers were created randomly across the 100 hectare virtual stand to serve as “drawing points”. When the x-y coordinate of each tree is created, it is moved a random amount toward the nearest clump center as described by (Wang et al., 2009). The algorithm used to create clumping is presented in Figure 6.
Generate randomly spaced clustering centers
Generate the x and y coordinates for a tree
Generate a diameter using the Weibull function
Calculate the distance of this x,y point to the nearest clumping center
Multiply this distance by a random number between 0 and 1 to establish a
New coordinate for the tree center relative to the clumping center
If the distance to the nearest existing tree > 1 meter
Assign this coordinate and diameter to the tree
Until all trees have been placed
One of the weaknesses of the algorithm developed by Wang et al.(2009) is that it supplies biased results; the bias increasing with the reduction in the surface of the generated stand. The bias resulted by moving the position of the tree toward the closest clump center. Therefore, trees that are at the edge of the generated surface will be most likely moved toward the center of the surface, reducing the number of trees that should represent the edge area. Assuming that the edge area has a width of 50 m, as determined by Redding et al. (2004), then the bias could lead to results larger with than the actual values (where A is the area in m2 of the generated stand), which for A=106m2 (i.e., 100 ha) is 19%. However, the bias should be formally determined, not only the upper limit (i.e., the previous formula).
Generation of the 1200 stands using ArcGIS 9.3 proved to be unsuccessful, as besides programming the memory allocation played a significant role. As a result, the stands with 1000 trees/ha were not generated with ArcGIS which constantly crashed the computers. Alternatively, the 1000 trees/ha stands were created using Visual Basic. For the generation of 1200 stands 72 million trees were created. The dbh mean and variance of the generated stands fit the expected values computed using the formulas 2 and 3 and presented in Table 1.
|Theoretical||Range of generated stand||Theoretical||Range of generated stand|
|0.7||8||0||10.13||10.11 - 10.16||219.32||217.1-220.81|
|5||10||0||9.18||9.17 – 1.19||4.72||4.70 – 4.74|
|5||10||10||19.18||19.16 – 19.2||4.72||4.71 – 4.74|
|3||4||0||3.57||3.55 – 3.59||1.68||1.66 – 1.70|
|3||4||10||13.57||13.55 – 14.0||1.68||1.66 – 1.70|
|3||4||20||23.57||23.56 – 23.58||1.68||1.67 – 1.69|
|3||4||30||33.57||33.56 - 33.58||1.68||1.67 – 1.69|
|3||4||40||43.57||43.56 - 43.58||1.68||1.67 – 1.69|
The results from Table 1 shows that generating stands using the proposed procedures supplies valid data for the subsequent investigations based on the generated data. Visually, the effect of the three clumping strategies (i.e., no clumps, 400 clumps/100 ha and 800 clumps/100 ha) reveals the conformity with real stands (Fig. 7).
A project of this size requires significant computer memory resources, as data files with millions of records are generated. The final data set used for the statistical analysis had 72 million records. We found that the GIS software requires too much of the computer\\\\\\\\\\\\\\\'s resources and crashed before completing the processes for forests of density over 800 trees per hectare. A significant amount of time was spent to rewrite the code to streamline the processes but we did not succeeded in generating the stands with 1000 trees/hectare. Consequently, the generation of stands with 100 trees/ha within ArcGIS was not performed and was replaced with a new code written in Visual Basic. The change from ArcGIS to Visual Basic solved not only the stand generation problem but also reduced the computing time required to generate the stands.
The proposed algorithms can be easily adjusted to generate stands with desired tree densities, diameter distributions or clumpiness. The algorithms presented in this paper allow the creation of virtual forests with spatial consistent properties, allowing analysis of the parameters in question without the impeded variability that is encountered in nature. An extension of the algorithm would be the incorporation of multiple species and the addition of tree-level species information that would allow the modeling of the interactions between the species as well.
School of Forestry,
Louisiana Tech University,
1201 Reese Dr., Ruston, LA 71270South Campus, Lomax Hall 5
Office Phone Number: (318)257-3714
Fax Number: (318)257-5061
School of Forestry,
Louisiana Tech University,
1201 Reese Dr., Ruston, LA 71270South Campus, Reese Hall 201
Office Phone Number: (318)257-2168
Fax Number: (318)257-2168