Estimation of crown radii and crown projection area from stem size and tree position

Annals of Forest Science, Jul 2018

Rüdiger Grote

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Estimation of crown radii and crown projection area from stem size and tree position

Ann. For. Sci. Estimation of crown radii and crown projection area from stem size and tree position Rüdiger GROTE 0 1 0 Chair for Forest Yield Science, Department of Ecology and Landscape Management, TU Munich , Am Hochanger 13, 85354 Freising , Germany 1 [49] Utschig H., Analyzing the development of regeneration under crown cover: Inventory methods and results from 10 years of observation , in: Skovsgaard J.P., Burkhart H.E. (Eds.) , IUFRO - Recent Advances in Forest Mensuration and Growth and Yield Research, Danish Forest and Landscape Research Institute/Ministry of Environment and Energy , Tampere, Finland, 1995, pp. 234-241 - This paper describes a method for crown radii estimation in different cardinal directions using tree diameter, height, crown length, and stem position within the stand as independent variables. The approach can serve for the initialisation of crown dimensions if measured crown radii are not available in order to address various research questions. Test calculations are carried out with 4 pure spruce (Picea abies L. Karst), 5 beech (Fagus sylvatica L.), and 6 mixed stands with both species. Simulated tree radii, crown projection area and canopy cover are compared with measurements and simple estimation procedures based on logarithmic and linear equations. In beech stands and dense spruce stands the estimates with the new approach are similar or superior to those obtained with the other methods. However, in sparse plots or in stands, which have experienced a recent thinning crown size of trees is overestimated. 1. INTRODUCTION Many ecological and economic problems in forestry today (e.g. continuous cover forestry, wood production and quality) are approached using crown dimensional measures. For example, individual tree competition indices are derived from crown area estimates [ 6, 38 ] because crown dimension is a result of past competition as well as an indicator of the current growth potential [27]. Thus, crown dimensional measures are also used in more sophisticated single-tree models – particularly when forest growth in uneven-aged or mixed species stands is addressed [ 40 ]. Furthermore, crown size and canopy cover determine the probability of successful natural regeneration by its influence on the pattern of shade, light, and rainfall on the ground [49]. In general, many approaches of modelling light distribution (e.g. [48]), water balance (e.g. [ 2, 37 ]), tree growth (e.g. [ 7, 41 ]), and tree physiology (e.g. [50]) depend on information about crown dimensions of individual trees. Possibly, considering a more realistic crown shape will become increasingly important also for stem quality simulation, because branch dimension is one of the most important determinants [ 30 ]. Despite its importance crown extension remains difficult to determine. It can only be measured by optical methods from below [44] or from above [ 1 ], which both are subjected to a likely underestimation of crown width due to a limited visibility of crowns. The crown projection area can be estimated from stem dimensions [ 15, 52 ], but has to be thoroughly parameterised for specific stand conditions [ 18 ], which in most cases involves again a large number of direct measurements. Finally, canopy cover can not be assumed to be the sum of tree crown projection areas, because overlapping is a common phenomenon particularly in dense, uneven-aged, and mixed stands. The difficult measurements and the sensitivity of crown dimension on management makes it desirable to develop estimation procedures based on variables that are easier to measure than crown extension itself. Thus, maximum crown radius, which can be derived from stem diameter, has been used to estimate crown projection area [ 19, 51 ]. Because increasing stand density results in increasing overestimates an adjustment factor has been introduced that is generally derived from overlap estimates [ 13 ]. More recently, average crown radius and canopy cover in several types of conifer forests were successfully estimated with regression equations that have been derived from stem diameter, height, and/or crown length [ 17 ]. All of these methods are developed to give reliable results on the stand level, which is suitable for many of the purposes mentioned above. It is not sufficient, however, for analyses that account explicitly for the asymmetry of crowns. Information about asymmetric crown extension has been used e.g. for detailed ecosystem characterisation [47] or the simulation of wood quality [ 28, 45 ], radiation distribution [ 10, 11 ], susceptibility of trees to windthrow [46], crown biomass [22], and individual tree physiology [ 23 ]. Therefore, a method, which estimates crown radii in various cardinal directions for every tree in a given stand would be of great value for these research areas. In this paper, such an approach is presented that is based on the size of a tree and the size and position of surrounding competitors. Also, the sum of crown projection area and canopy cover (the total area covered by canopy) are both calculated based on the estimated radii and results are compared with those obtained with other methods. 2. MATERIALS AND METHODS 2.1. Stand description In order to test the proposed method for crown radii estimation, a number of forest stands were selected that include the most important tree species and stand structure types in Germany. The stands consist of trees with a coniferous (Picea abies L. Karst.) and a broadleaved (Fagus sylvatica L.) tree species either in pure or mixed stands. All of them belong to the network of long-term investigation plots in Bavaria, South Germany and are maintained by the Chair for Forest Yield Science in Freising. Thus, tree position, stem diameter, height, height of crown base, and crown radii length had already been measured at many trees. The plots of pure spruce (Eurach, 4 plots) and beech (Starnberg, 5 plots) both represent different degrees of stand density. The mixed plots (Freising, 6 plots) represent different age classes. All plots of one site are located closely together to minimise differences in site conditions. For a more detailed description see Table I. The plots of pure spruce and the mixed plots are furthermore described in connection with other investigations [ 20, 42 ]. Diameter at breast height had been measured with a girth tape at all trees. Tree height and crown base height of each tree within one plot had been determined from height-diameter relations that are derived from a subset of approximately 40 measured heights at each plot. The visible crown extension in each of eight cardinal directions had been measured by vertically looking up as described by Röhle [43]. Calculations are carried out with all trees within the plots, but trees at the plot boundaries are omitted from the results. This is necessary because in these cases no competitors at the outward side are considered and crown radii would thus be overestimated. 2.2. Distance dependent approach The suggested approach is based on two assumptions. The first is that the potential horizontal crown extension is a function of stem diameter, and the second is that the distance between the tree and its competitors determines the actual crown dimension within the limit of the potential crown extension. Following Arney [ 3 ], competitors are defined as trees with an overlap in potential crown extensions (Fig. 1A). Crown radius length in a particular direction is limited by the position of competitors within a certain angle on both sides of the radius (Figs. 1A and 1B) and by their crown width at the height where the maximum crown extension of a centre tree is assumed (Fig. 1C). The method is further on referred to as ‘maximum radii estimation’ (MRE). Firstly, in order to determine the height where maximum crown width occurs, a crown shape function is required similar to those that have been suggested by several authors during the last decades (e.g. [ 8, 25, 26, 29, 35 ]). However, these equations require many parameters that are not directly measurable (e.g. [ 25 ]), assume a steady increase with canopy depth (e.g. [ 35 ]), or end with a zero-radius at crown base height (e.g. [ 29 ]). In the current context, these properties are considered as disadvantages. Thus, a new one-parameter equation is used that describes crown radius at every height h (rh) as a function of crown base height (hcr), crown length (lcr), and the maximum radius in a particular cardinal direction (rmax). The term relH refers to the relative height within the crown, which is 0 at crown base and 1 at the tip of the tree. 1 – relH f h rh = rmax -m-----a---x--------------------------------------------- 1 – relH f h relH = -h----–----------- hcr lcr f h = base 100-h----–----------- hcr lcr2 . Eq. (1a) Eq. (1b) Eq. (1c) The effect of the base-term in equation (1c) is demonstrated in Figure 2, with rmax = 1 and a crown length of 15 m. In a detailed analysis of 12 trees, values of base were found to be 1.23 0.074 for spruce and 2.02 0.71 for beech [ 21 ]. However, the standard deviation can be decreased if base is derived from crown length according to equation (1d) (1.23 0.071 for spruce and 2.08 0.345 for beech) with ps equal to 0.018 (R2 = 0.65) and 0.0756 (R2 = 0.54) for spruce and beech respectively. base = 1 + ps lcr Eq. (1d) ps: shape parameter. To determine the maximum crown radii of one tree, the maximum crown extension for the competitor trees j are needed but generally not known. Thus, for a given competitor, rmax,j is calculated from the distance to tree i (dij) and from diameter at breast height (dbh) of both trees according to equation 2. The distance between a tree and its competitor dij can easily be calculated from stem positions. rmax j = dij -d---b---h--d-j---b---h---j---------. Eq. (2) + dbhi However, rmax of any tree is limited to its potential radius (rpot), which describes the physical maximum is hardly affected by site conditions [ 24 ]. Since no open grown trees were available, rpot is estimated from the 5% relative largest crown radii found at the trial plots. To determine these radii, firstly all radii (with 8 radii measured per tree) are exponentially fitted to the stem diameter at ground height do (MS Excel software package). The 5% selected radii are the ones with the largest positive deviation from this relation. Another exponential fit through these radii according to equation 3 derives the parameter pr1 and pr2. The diameter at ground height is derived from dbh by assuming a certain diameter decrease of the bole with increasing height (0.3 cm m–1). It is used as independent variable instead of dbh because otherwise equation 3 would imply that small trees (< 1.3 m height) have no crowns at all, which would restrict the generality of the approach. Parameters are determined separately for each tree species and for pure and mixed stands although the differences between the relations for spruces in different stand structure types were not significant (Fig. 3). Values for pr1 and pr2 together with the number of radii that have been used to derive the functions are given in Table II. rpot = pr1 dopr2 Eq. (3) (rpot and do in m). From rmax,j the potential crown extension of all competitor trees is calculated for every height according to equation 1 in height steps of 0.5 m. In this calculation, rh of competing trees below the height of maximum crown diameter is set to rmax to better account for the influence of light competition in deeper canopy layers. In the next step, the angle ij between the tree (i) and its competitor (j) is calculated from tree positions (Fig. 1A). Assuming that a branch will grow until it reaches the crown circumference of a competitor tree, the length of each crown radius is calculated as follows (Fig. 1C): rh i = min TiS rpot h i TiS = cos ij dij – l l = rpot h j T T j j Eq. (4a) Eq. (4b) Eq. (4c) rh,i: actual radius of centre tree i at height h; rpoth,i: potential radius of centre tree i in height h; rpoth,j: potential radius of competitor tree j in height h; TiS: distance between stem position and the point of intersection; l: help variable; TjT’j: distance between competitor tree j and a point, mirrored at the radius prolongation (T’j is described as a ‘virtual mirror tree’ in Fig. 1C). The actual radius of rh,i is calculated as the minimum radius determined by considering every competitor within angle s on both sides of the radius (see illustration in Fig. 1A). Based on former investigation results [ 33, 44 ] and test calculations with different angles, s is set to 45o (8 radii). Since first calculations showed that the largest crown radii in one direction was too often equal to the potential radii, a further restriction was introduced to get more realistic results for rmax. As illustrated in Figure 1B, the assumption is made that a radius can not grow beyond the stem position of a competitor tree. Despite these limitations, it should be recognised that an overlap between crowns can result from the elliptical connection between two adjacent radii (see further down). 2.3. Other calculations Currently, the most common estimation procedures of crown projection area are based on linear [ 17 ] or logarithmic [ 31, 52 ] relationships between stem and crown diameter. Thus, simple calculations are carried out using linear correlations between dbh and radius length (rmax in dm = alin1 + blin1 dbh), and dbh and crown projection area (A in m2= alin2 + blin2 dbh in cm) of individual trees. Crown projection area is also calculated with a logarithmic relation to stem cross-sectional area (lnA = alog + blog ln(dbh2 0.25)). The parameter alog and blog are derived analytically from the same data set as pr1 and pr2 and are also presented in Table II for each tree species and for pure and mixed plots (not for each plot!). In order to derive crown projection area from measured and simulated crown radii, the area between the radii is considered as a fraction of an ellipse [44]. Canopy cover is calculated with a computer program that draws the crown projection area of every tree on a grid and counts the number of coloured pixels. 3. RESULTS The relation between simulated and measured radii is shown in Figures 4A–4D. The coefficient of determination ranges from 0.2 for pure spruce to 0.45 for beech in mixed stands. A small bias is obvious in every figure, which indicates an overestimation of small radii and an underestimation of large radii. This is at least partly due to radii that had been measured with zero length, which can not be represented with the MRE method due to the assumption made in equation 2. Slope values with the regression line forced through the origin are presented in Table III separately for species and sites together with the respective R2 values. The table shows that despite the bias positive correlation coefficients had been Location Species obtained with the MRE method in all cases, but not with the estimation based on the linear approach. Figure 5 and Table IV show that MRE does not decrease the accuracy of crown projection area estimates compared with the fitted logarithmic (LOG) and the linear method (LIN). The slope values obtained with every method are similar (in average over all plots separated by species: MRE = 1.00, LOG = 0.91, LIN = 0.96) although the standard deviation of MRE is highest (MRE = 0.25, LOG = 0.12, LIN = 0.15). The R2 values of MRE are similar to those obtained with the LOG approach and are higher than R2 values obtained with the linear approach (MRE = 0.64 0.15, LOG = 0.61 0.16, LIN = 0.50 0.23, with all negative values excluded from the average). However, crown projection area for spruce is somewhat overestimated, particularly in the mixed plots (+4 and +28% mean deviation from measurements for pure and mixed plots respectively), whereas for beech it is generally underestimated (–15 and –17%). The goodness of fit apparently depends on the density of the plot and of the thinning intensity that the stand has been treated with (see Tab. I). In the plots Eurach 1 and Starnberg 2, which are the most dense for each species, the deviation from the 1:1 line is only marginal (spruce –4%, beech +1%) and the simulated values are closely correlated with measured crown projection area (R2 = 0.7 and 0.8 for spruce and beech respectively). In spruce, overestimation increases in sparser plots (up to 23% in the sparsest plot Eurach 2), whereas for beech crown projection area is underestimated in thinned plots but no particular trend with the intensity of thinning is obvious. The sum of crown projection areas within one plot is similar to that calculated from the measurements although an overestimation for spruce (+9%) and an underestimation for beech (–19%) is obtained (Tab. V). Again, the simulation of the densest plots for both species are closest to the measurements (Eurach 1: –6%, Starnberg 2: –4%). Table V shows canopy cover values derived from either measured or simulated crown radii. Additionally, crown overlap is calculated from the difference between the sum of single tree crown projection areas and canopy cover. This demonstrates that the overlap derived with the MRE method is generally too small. In spruce stands, however, this underestimation is only slight (–3%), whereas it is in average –14% for beech stands. Mixed stands are in between (in average –6%). 4. DISCUSSION Results indicate that the MRE method can be used to estimate crown radii for beech and spruce in dense stands but has to be applied cautiously. Although some of the variance may be due to the high inaccuracy of crown measurements [43], crown radii of trees from sparse plots or in recently thinned stands are generally overestimated. This is consistent with the underlying assumption of a balanced crown extension, which can not be expected in heavily thinned stands and which is more likely with morphological flexible tree species like beech than with spruce [ 4, 16 ]. Future tests and improvements of the MRE approach will focus on crown shape estimation, which is based on a quite small sample size of trees yet. Only a larger sample provides the possibility to establish dependencies of crown shape on spacing and competition that have been already found in other investigations [ 5, 12, 14, 32, 34 ]. Further improvements could be based on the finding that in mixed stands spruce radii are generally over and beech radii are underestimated. This would be mitigated if a species-specific weighing factor for the calculation of potential spruce and beech radii is introduced in equation 2. However, it is not clear from the limited set of test sites to which degree the effect is due to the stand structure rather than species-specific properties. Although they are older, most beeches of the mixed plots are smaller than the spruces. Thus, the assumption that crowns of small trees are restricted by the largest extension of competitor crowns rather than their actual extension may affect beeches more than the spruces at these particular plots. In this case, separate crown radii estimations for different crown layers may produce more favourable results but simulations of differently structured mixed stands are required to test this assumption. Improvements in crown radii estimates will generally positively affect crown projection area and canopy cover estimates. Nevertheless, the good agreement of simulated and measured canopy cover despite the underestimation of crown projection area in beech stands shows that the estimation of crown overlap is also subjected to errors. Again, species-specific differences have to be considered since the predicted overlap for spruce trees is quite close to the measurementbased calculations. This finding strengthen the assumption that a separate calculation of different crown layers may be necessary. The MRE method aims not preliminary on a precise estimate of crown projection area or canopy cover. Over all, the logarithmic approach, which is used here as an example for similar and sometimes more sophisticated procedures (e.g. [ 13, 17, 51 ]), produced slightly better results and would perform even better if parameters would have been fitted for each plot separately. Furthermore, the estimates produced with the MRE method seem to be more sensitive to stand density effects than established estimation methods [9] – at least for trees with inflexible crowns. However, the author has found no other approach that estimates crown radii for different cardinal directions. Thus, the demand for crown asymmetry-information that has been formulated in various fields of research (scaling, light modelling, estimation of windthrow susceptibility, wood quality, and crown biomass) can currently only be fulfilled with actual measurements. Despite the scatter, the immanent bias, and the dependency of accuracy on species and stand density, the MRE method may thus be used as a substitute for measured crown radii in cases where these are not available but information about crown asymmetry is needed. While stem diameter, tree height, and crown length are often directly measured or can be estimated with suitable equations (e.g. [ 24, 36 ]), the acquisition of tree position data in the field may be more difficult and expensive. However, also tree positions can be gener% 26 25 33 26 28 26 29 17 11 11 20 26 25 24 28 31 27 26 Measured (m2) 879 727 771 754 783 746 639 514 494 597 598 1759 2266 1112 1125 992 1215 1411 sd 58 91 454 MRE (m2) 956 881 866 899 901 700 668 499 602 491 592 1818 2597 1213 1189 960 1310 1515 % 109 121 112 119 115 94 105 97 122 82 99 103 115 109 106 97 108 107 sd 34 86 549 Measured (m2) 484 228 324 380 354 473 340 370 345 43 314 910 481 315 416 815 1027 661 % 36 24 30 34 31 39 35 42 41 7 34 34 18 22 27 45 46 32 ated based on stand inventory data (e.g. [ 39 ]), which may be sufficient for many of the purposes mentioned above. Acknowledgements: This research has been conducted within the framework of the joint-research project ‘Growth and Parasite Defence’, funded by the German Research Agency (DFG). The Chair of Forest Yield Science, lead by Hans Pretzsch, supported the research generously with the supply of basic data, collected and processed by Martin Nickel, Leonhard Steinacker, and Martin Bachmann. Furthermore, I’d like to thank Hans Pretzsch, Greg Biging (Berkeley University, California), and the two anonymous reviewers, who made valuable comments to the manuscript, as well as Thomas Seifert, who provided yet unpublished data for the parameter estimation of crown shape for spruce trees. [48] Stadt K.J., Lieffers V.J., MIXLIGHT: a flexible light transmission model for mixed-species forest stands, Agric. For. Meteorol. 102 (2000) 235–252. [50] Wang Y.P., Jarvis P.G., Description and validation of an array model - MAESTRO, Agric. For. Meteorol. 51 (1990) 257–280. [1] Akça A. , Aerophotogrammetrische Messung der Baumkronen , AFZ/Der Wald 30 ( 1983 ) 772 - 773 . 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Rüdiger Grote. Estimation of crown radii and crown projection area from stem size and tree position, Annals of Forest Science, 393-402, DOI: doi:10.1051/forest:2003031