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NB: This is an author’s version of a manuscript to be published by Elsevier, Inc., in Remote Sensing 1

of Environment with a DOI: 10.1016/j.rse.2015.05.009 2

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Original Research Article 4

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Title 6

Geometrically explicit description of forest canopy based on 3D triangulations of airborne laser 7

scanning data 8

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Authors 10

Jari Vauhkonen1,2,*, Markus Holopainen2, Ville Kankare2, Mikko Vastaranta2, Risto Viitala3 11

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Affiliations 13

1 University of Eastern Finland, School of Forest Sciences, Yliopistokatu 7 (P.O. Box 111), FI-80101 14

Joensuu, Finland 15

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2 University of Helsinki, Department of Forest Sciences, Latokartanonkaari 7 (P.O. Box 27), FI-00014 17

Helsinki, Finland 18

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3 HAMK University of Applied Sciences, Saarelantie 1, FI-16970 Evo, Finland 20

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* Corresponding author. E-mail jari.vauhkonen@uef.fi, tel. +358 50 442 4432 22

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Abstract 24

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Geometrically explicit parameterizations of the locations, orientations and properties of trees and 26

canopy elements are useful for forest ecosystem modeling. Detailed reconstructions based on real 27

tree geometry are, however, hard to obtain based on conventional field measurements and 28

sampling. We describe and evaluate an alternative approach to reconstruct the forest canopy from 29

sparse, leaf-off airborne laser scanning (ALS) data with a wall-to-wall coverage. The approach 30

employs computational geometry and topological connectivity to generate filtrations, i.e., ordered 31

sets of simplices belonging to the three-dimensional (3D) triangulations of the point data, and 32

numerical optimization to select the set of simplices with a quasi-optimal relationship with field-33

measured forest biophysical attributes. The approach was evaluated by predicting the quantities and 34

spatial patterns of biomass-related forest attributes according to the characteristics of the filtration. 35

When the filtration parameters were optimized for 245 sample plots of 300 m2 located in southern 36

boreal forest in Finland, the coefficients of determination (R2) between total volumes of the 37

filtrations and basal area, stem volume, total above-ground biomass, and canopy biomass were 0.93, 38

0.87, 0.87, and 0.62, respectively. Considerably less accurate results (R2=0.44–0.64) were obtained 39

when the filtration parameters were predicted with a limited number of the calibration field plots. 40

However, these accuracies could be obtained with modest field training data of 20-30 plots. The 41

proposed approach is a compromise between the parameterization of the forest scene by artificial 42

tree/crown level turbid media and realistic 3D models. The results particularly suggest that obtaining 43

coarse wall-to-wall descriptions does not require separate data acquisitions, but may be based on 44

data existing due to previous practical inventories. 45

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Keywords: Remote sensing; Light Detection and Ranging (LiDAR); Tree allometry; Crown 47

architecture; Delaunay triangulation; Alpha shape 48

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1. Introduction 50

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Forest structure affects the amount and distribution of light in the canopy and forest floor, and its 52

estimation is required by many terrestrial ecosystem models. Also, studies of reflective properties of 53

specific land cover types require parameterization of the three-dimensional (3-D) forest scene, in 54

which locations, orientations, properties of the trees and, at the finest levels of geometric detail, 55

branches, leaves and shoots are explicitly defined (Brunner 1998). Obtaining these detailed 56

parameterizations (referred to in the following text as “geometrically explicit descriptions of forest 57

canopy”) is complicated based on conventional field measurements and sampling. Therefore, the 58

analyses typically reduce to the use of a dimensionless indicator such as the leaf area index (LAI), 59

canopy cover or closure to characterize forest canopies (e.g., Smith et al. 2008). Even the explicit 60

approaches (e.g., North 1996, Ni-Meister et al. 2001) typically simplify the structural representation 61

of individual tree crowns by using simple geometric primitives such as cones or ellipses to 62

approximate the distribution of canopy elements and the actual crown shape (see also Rautiainen et 63

al. 2008). 64

65

Remote sensing of forest structure provides an ecologically significant advance over the 66

conventional methods (Lefsky et al. 2002, Vierling et al. 2008, Bergen et al. 2009). Particularly 67

techniques based on Light Detection And Ranging (LiDAR) allow 3-D descriptions based on pulses 68

emitted to and reflected from forest canopy. LiDAR operated from an airborne platform, i.e., 69

airborne laser scanning (ALS), allows the collection of wall-to-wall data for vast geographical areas 70

with high spatial resolution and positional accuracy. A variety of studies demonstrates the 71

quantification of canopy height and the linking of extracted height metrics with field-observed forest 72

biophysical properties (Nilsson 1996, Naesset 1997a,b, 2002, Magnussen and Boudewyn 1998, 73

Means et al. 2000) and canopy properties such as the LAI, canopy cover and canopy gap fraction 74

(Lovell et al. 2003, Morsdorf et al. 2006, Korhonen et al. 2011, Korhonen and Morsdorf 2014). ALS-75

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based analyses have matured and are routinely operated in various forest inventories, of which a 76

recent handbook of best practices (White et al. 2013) and a recent textbook focused specifically on 77

forestry applications (Maltamo et al. 2014) are concrete examples. 78

79

ALS data collected for terrain elevation modeling at regional to national scales by the land survey 80

authorities of several countries (e.g., Nord-Larsen and Riis-Nielsen 2010, Dalponte et al. 2011, 81

Gomez-Gutierrez et al. 2011, Bohlin et al. 2012, Villikka et al. 2012) are currently the most plentiful 82

ALS data sets for environmental mapping, monitoring and modeling. However, due to the sparse 83

densities (< 1 pulses m-2), these data are mainly analyzed for their distributions of height values 84

(White et al. 2013), while geometrically explicit modeling would, in particular, require preserving the 85

full 3-D geometry of the point data. Although geometric modeling techniques based on ALS data 86

exist (e.g., Koch et al. 2014), the use of these techniques typically demands a considerably higher 87

data density. 88

89

Studies carried out by Vauhkonen (2010a,b), Vauhkonen et al. (2010, 2012, 2014) and Korhonen et 90

al. (2013) propose triangulations of the 3-D point data as potential structures for multiple scales and 91

varying densities. Figure 1 gives an example of triangulating point data, i.e., subdividing the 92

underlying space of the points into simplices, which results in weighted simplicial complexes (e.g., 93

Edelsbrunner 2011) with weights quantifying the (empty) space delimited by the points. To 94

reconstruct the canopy volume populated by biomass and to exclude that volume from canopy voids 95

(see also Figure 1), Vauhkonen et al. (2014) proposed a filtering approach in which the simplices 96

were classified into either canopy biomass or voids based on numerical optimization with field-97

measured biomass. The main interest of the present study is to further test this filtering approach 98

for constructing geometrically explicit, wall-to-wall descriptions of forest canopy structure, which are 99

assumed to be highly useful for structural ecosystem modeling. 100

101

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Figure 1. Triangulation and filtration of an example set of 3D points. Subplot (A) shows the extreme 102

values of the filtration: the initial set of points and the convex hull of the point data denoted by 103

dashed lines. The red lines in subplots B, C, and D show three example simplicial complexes 104

belonging to a filtration in which the subcomplexes were ordered according to parameter 105

increasing between B and D. 106

107

108

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Earlier, Nord-Larsen and Schumacher (2012) and Villikka et al. (2012) verified the suitability of the 110

data acquired for ground elevation modeling also for forest inventories, while Maltamo et al. (2010) 111

and Vauhkonen et al. (2012, 2014) have developed triangulation-based approaches for sparse point 112

densities. Particularly, our study is a continuation of Vauhkonen et al. (2014), who introduced the 113

concepts of triangulation and filtration for modeling volumetric canopy surfaces from sparse ALS 114

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data and assessed canopy volumes derived from 40 sample plots. Our study extends the setup of 115

Vauhkonen et al. (2014) in two particular ways: We focused the evaluation on (i) the geometrically 116

explicit models produced and (ii) the potential to produce these as a part of a practical forest 117

inventory based on a limited set of field plots for model fitting/calibration. Regarding (i), we 118

assessed the correspondence of the spatial arrangement of the obtained filtrations and the trees 119

mapped in the field. Regarding (ii), we fitted the models based on leaf-off ALS data acquired for 120

ground elevation modeling and simpler forest variables measured in typical forest inventories rather 121

than total or canopy biomasses (Vauhkonen et al. 2014). Finally, the data of 245 plots altogether 122

allowed the separation of training and testing data, an evaluation of whether the workflow was 123

operationally feasible, and a more robust validation of the previous results (Vauhkonen et al. 2014). 124

125

The purpose of this study is thus to evaluate the workflow for the 3-D reconstruction of forest 126

canopy from the leaf-off, sparse (0.8 points m-2) ALS data employing triangulations of the 3-D point 127

data. In particular, we focus on testing a filtering and optimization procedure for deriving “biomass-128

optimal” filtrations of the triangulations of ALS point clouds examined at a resolution of 300 m2. We 129

evaluate the filtrations based on their correspondence with forest biophysical properties observed 130

from the field sample plots considered. 131

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2. Material and methods 133

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2.1. Study area and data 135

136

The study area is located in a southern boreal forest zone in Finland (61.19°N, 25.11°E). The area 137

comprises a broad mixture of forest stands, varying from natural to intensively managed. Scots pine 138

(Pinus sylvestris L.), Norway spruce (Picea abies [L.] H. Karst.) and birches (Betula spp. L.) are the 139

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dominant tree species. The conifer species pine and spruce constitute approximately 80% of the 140

total volume. 141

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2.2. Field measurements 143

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The study area was surveyed using circular plots with radii of 9.77 m between May 2011 and 145

October 2012. Selection of the field plot locations was based on pre-stratification of existing stand 146

inventory data with an aim to distribute the plots over various site types, tree species and stand 147

development classes (Kankare et al. 2013). The field plots were located with a GEOXM 2005 global 148

positioning system (GPS) device (Trimble Navigation Ltd., Sunnyvale, CA, USA). The locations were 149

post-processed with local base station data resulting in an accuracy of approximately 0.6 m. All trees 150

with a diameter at breast height (DBH) more than 5 cm were measured for species, crown position 151

in vertical canopy layers (in multicohort stands), DBH, height and location relative to plot center. A 152

Vertex hypsometer (Haglöfs, Sweden) was used to measure the tree heights and the distance and 153

angle from the tree stems to the plot center. 154

155

The forest attributes considered in this study were basal area (G), total volume (V), above-ground 156

biomass (AGB; consisting of both stems and foliage) and foliage biomass (FB) separately, determined 157

at a resolution of 300 m2. G was computed based on summing from the diameter measurements. 158

Individual stem volumes and single-tree biomass fractions were estimated by models of Laasasenaho 159

(1982) and Repola (2008, 2009), respectively, both employing the DBH, height and tree species as 160

predictors. The models for birch were used for all deciduous trees. The tree-level values were 161

summed to V, AGB and FB. General descriptive statistics of the field data employed in this study are 162

shown in Table 1. 163

164

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Table 1. Descriptive statistics of the 245 field plots. Min – minimum, Max – maximum, Sd – standard 165

deviation. 166

Attribute Min Mean Max Sd

Basal-area weighted mean diameter, cm 7.5 23 56.2 8.8

Basal-area weighted mean height, m 4.4 20.2 37 6.2

Number of stems, 1/ha 133 1020 3001 616.2

Basal area, m2/ha 1.2 22.2 53.8 9.6

Stem volume, m3/ha 4.7 211.5 653.6 119.6

Above-ground biomass, t/ha 2.8 107.7 302.8 56.2

Canopy biomassa, t/ha 0.8 24.6 87.2 13.8

a Comprising foliage and branches. 167

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2.3. ALS data 169

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The ALS data used in the study were acquired by the National Land Survey (NLS) of Finland as a part 171

of its data acquisition campaign for creating a nationwide digital elevation model. The data were 172

downloaded from a file service (NLS 2014), from which they are available for free and with extensive 173

permissions of use. The data were acquired on May 13, 2012, with Leica ALS50 scanner operated in a 174

multipulse mode at a wavelength of 1064 nm. The flying altitude was 2200 m, yielding a nominal 175

pulse density of approximately 0.8 m-2. The data provider had detected and classified the ground 176

points and normalized the vegetation height values. As the data are meant specifically for ground 177

elevation modeling, we assumed the accuracy of this classification to be appropriate for our 178

purposes. The analyses were focused only on the first echoes (i.e., “only” and “first of many” of up 179

to 4 echoes recorded per pulse) to extract the main information in the data, but still to be able to 180

generalize the obtained results for sensors recording a different number of echo categories and 181

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therefore different distributions of the echoes (see also Morsdorf et al. 2006 and Næsset 2014, for 182

example). 183

184

2.4. Outlier removal and stratified subsampling of plots to training and testing data 185

186

Altogether, 263 of the field plots originally covering a larger area were located beneath the ALS data. 187

Because the systematic field sampling design allowed plots to be located on stand boundaries, for 188

example, we assumed that some of the plots could depict the prevailing forest structure incorrectly 189

between the data sets and therefore become coarse outliers in our analyses. As the relationships 190

between the ALS and field measured height (e.g., Næsset et al. 2004) and the proportion of echoes 191

reflected above a vegetation threshold and canopy cover (Korhonen et al. 2011) are well 192

documented and strong, we assumed that the outliers (e.g., plots located on stand boundaries or 193

plots harvested between the data acquisitions) could be identified and the data stratified by 194

examining these relationships. 195

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The outlier removal was solely based on the correspondence of the field and ALS-based heights in 197

the areas of the sample plots, quantified as a linear regression model, , where 198

was the ALS-based 60th height percentile, calculated as in Section 2.6., was the field-199

measured dominant height, calculated as the mean height of the 100 thickest trees per hectare (or 3 200

thickest trees of each plot) and was the model parameter. We excluded the plots not within the 201

prediction interval of the model (residual > 2 × standard error; see also Breidenbach et al. 2010). 202

Altogether, 245 plots were retained for the analyses. 203

204

Our aim in the further analyses was to mimic a practical ALS-based forest inventory (e.g., Næsset 205

2002), in which (i) models to predict the forest attributes of interest are fitted based on a limited set 206

of training plots and (ii) the resulting models are applied to a prediction grid covering the entire 207

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inventory area. In the absence of a prediction grid or separate testing data, we applied repeated 208

stratified subsampling to divide the set of 245 plots into training and testing data. 209

210

Following Maltamo et al. (2011), the plot data were stratified according to ALS-based median height 211

(H) and canopy cover (CC) estimates, calculated as the median value of echo heights and the 212

proportion of the echoes, respectively, above a 2 m height threshold. A subsample of n×10 plots was 213

obtained by dividing the range of the H × CC estimates into 10 bins of equal width and selecting n 214

random plots from each bin. This subsample was assigned as the training data, while the rest of the 215

plots were preserved for evaluating accuracies. The subsampling was repeated 500 times with n = 1, 216

2 and 3, resulting in, altogether, 1,500 different series of data with the number of training plots 217

being 10, 20 or 30. 218

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2.5 Deriving triangulations and optimized filtrations 220

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Our main aim was to depict the forest canopy according to the filtered triangulation of the ALS point 222

data (Vauhkonen 2010b, Vauhkonen et al. 2014). As the entire triangulation would have included 223

simplices representing canopy gaps and thus empty space (Figure 1), the fundamental problem was 224

to determine a proper filtration such that the simplices considered in the computation represented 225

the areas populated by the forest biomass as closely as possible and excluded those representing 226

empty space, based on the training data (section i in Figure 2). A further problem was related to 227

obtaining wall-to-wall estimates for the entire study area (section ii in Figure 2), i.e., predicting the 228

filtration parameters for the areas of interest in which the true forest biomass was unknown. To 229

solve these problems, we considered the workflow illustrated in Figure 2, explained at a general 230

level below and in detail in an earlier publication (Vauhkonen et al. 2014). 231

232

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Figure 2. The workflow proposed for optimizing the filtration parameter with sample plots (i) and 233

predicting it for the entire study area (ii). 234

235

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To filter the triangulations of the ALS point data (step 1 of Figure 2), we employed the computational 237

geometric or topological concept of the family of 3-D alpha shapes (Edelsbrunner and Mücke 1994). 238

Alpha ( ) is a threshold value for the squared radius of the circumscribing sphere of each simplex 239

and therefore a criterion determining which simplices of the triangulation belong to an -complex. 240

Although an -shape is defined for every non-negative real , there are only finitely many different 241

-complexes (see Delfinado and Edelsbrunner 1995, p. 782).Ordering the obtained complexes 242

according to the values of produces a filtration of simplicial complexes (Delfinado and 243

Edelsbrunner 1995) such as that illustrated in Figure 1. The triangulations and filtrations were 244

derived for each individual plot separately using C++ and an open-source Computational Geometry 245

Algorithms Library (Pion and Teillaud 2013, Da et al. 2013). 246

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247

To identify the -complex that optimally depicted forest canopy (step 2 of Figure 2), we changed the 248

values of iteratively and examined the relationship between the volume of the underlying space of 249

the -complex and the field-measured plot basal area (G). Although the ALS point data could be 250

expected to represent canopy biomass (branches and foliage) rather than that originating from the 251

stems, G is the most typical forest biophysical property measured in forest inventories with strong 252

allometric links to variables such as AGB or FB. Therefore, we wanted to optimize the relationship 253

based on this variable (see further notes in Discussion). 254

255

The optimization considered each set of plots obtained by the subsampling (Section 2.4.) 256

separately. For convenience, let in the following text denote the volume of the underlying 257

space of the i:th -complex of the ALS point data of plot , in a filtration in which the 258

subcomplexes were ordered according to an increasing . The objective was to minimize the 259

residual errors of a linear regression model 260

, (1) 261

where was the reference basal area, and the model parameters, and the residual error 262

of plot p. The initial values of and were obtained by fitting the model to with i selected 263

randomly for each plot. In the optimization, of plot q which produced the largest residual error 264

was altered to in successive rounds of 500 iterations, until either the error, determined as 1 265

minus the obtained coefficient of determination (R2), decreased below 0.0009 or did not change 266

during the last round of iterations. The optimization was run for each subsample of the plots, thus 267

providing altogether 1,500 sets of filtration parameters (i.e. 1,500 different combinations of plot-268

specific -values) that were in a quasi-optimal linear relationship with G. 269

270

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2.6. Evaluation and performance measures 271

272

To obtain a wall-to-wall representation of the modeled canopy (step 3 of Figure 2), the optimal 273

filtration parameter needed to be predicted for each cell of a grid covering the entire inventory area. 274

In the absence of data to validate the predictions for the grid, we made the prediction and 275

evaluation on the plots included in the training data, following the two-stage procedure proposed by 276

Næsset (2002). First, the optimized values of the filtration parameter were correlated against the 277

most common ALS-based predictor variables. These variables were the mean and standard deviation 278

of the height values, the proportion of echoes above 2 m, and the 5th, 10th, 20th, …, 90th and 95th 279

percentiles and the corresponding proportional densities of the ALS-based canopy height 280

distribution. The variables were calculated according to Korhonen et al. (2008, pp. 502–503). 281

Second, the absolute value of and the corresponding percentile of the distribution of potential 282

plot-specific -values were predicted for the plots based on these correlations by means of 283

regression analysis. The fits of the prediction models were analyzed with 1–3 of the most correlated 284

predictors, but no exhaustive search for the best combination was attempted. Finally, the forest 285

scene was constructed from the triangulations filtered according to the predicted . 286

287

The evaluation of our approach was based on assessing the correspondence of the total tetrahedral 288

volume (Tvol) of the optimized and predicted filtrations with various forest attributes observed in the 289

full set of plots . This relationship was quantified by fitting an allometric power equation (White 290

and Gould 1965) 291

, (2) 292

where was the considered forest attribute (either AGB, FB, G, or V), the volume of the 293

underlying space of the selected subcomplex , b and k the model parameters, and the residual 294

error of plot . The model fitting was carried out with the nls function of R (R Core Team 295

2012). 296

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297

The spatial pattern of simplices of the filtrations was assessed by means of the Clark-Evans index 298

(CEI; Clark and Evans 1954) of the aggregation of a point pattern. The positions compared in the 299

analyses were those of the tree stems and the centroids of the simplices. The computations 300

regarding the CEI were carried out using the spatstat package of R (Baddeley and Turner 2005). 301

302

The correspondence of the estimates with the reference data was evaluated by graphical 303

assessments and in terms of R2, root mean squared error (RMSE) and mean difference (bias), the 304

latter two calculated according to 305

306

(3) 307

308

(4) 309

310

where n is the number of observations, and and are the predicted and reference values, 311

respectively. The relative RMSE and bias were calculated by dividing the absolute RMSE and bias 312

values by the mean value of the reference attribute. A paired t-test was used to test the null 313

hypothesis that the mean difference between the predicted and reference values was 0. In the 314

following text, the term “significant” refers to the statistical significance of this test statistic at the 315

95% confidence level. 316

317

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3. Results 318

319

3.1. Correspondence between the optimized filtrations and the forest attributes 320

321

Approximately 80% of the optimized 1,500 solutions (Eq. 1) yielded R2 ≥ 0.99, and the average R2 322

was 0.98. Unless otherwise noted, the following assessments are based on using the plot-specific 323

mean values derived from the pooled optimizations. Based on a visual assessment, the -324

complexes based on the mean values showed a fairly good geometrical correspondence with the 325

tree locations measured in the field (Figure 3). 326

327

The CEI values between the locations of the centroids of the simplices and the trees measured in the 328

field were only moderately correlated (R2 ≤ 0.1). The maximum CEI obtained for the simplices was 329

about 1, i.e., the simplices failed to show regular patterns. Although some regular tree patterns 330

could be identified in terms of low proportion of excluded simplices relative to the total amount, the 331

spatial pattern of tree stems could not generally be predicted by examining that of the filtered 332

triangulations. It is noted that the low correspondence in terms of CEI may be related to the 333

different (multiple) amount of the simplices compared to the trees, i.e., a better correspondence 334

could potentially have been obtained by aggregating the simplices within the plots to simulate the 335

allocation of the foliage in individual trees. 336

337

The correspondence between the selected forest biophysical attributes and the Tvol of the filtrations 338

extracted with the mean value resulting from the pooled optimizations is shown in Figure 4. As 339

expected, the Tvol had the highest correlation with G that was used in the optimization. However, 340

due to the allometric relationships, the Tvol showed good correspondence also with other tree (stem) 341

dimensions. The correspondence was weakest with respect to FB (Figure 4, Table 2). 342

343

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Figure 3. Examples of filtered triangulations for sample plots with clustered (CEI=0.6, above), 344

random (CEI=1.0, middle) and regular (CEI=1.5, below) pattern of trees. The selected simplices are 345

shown on top of the full triangulation and the initial point cloud (left column), and cylinders 346

representing the locations and dimensions of the tree stems measured in the field (right). 347

348

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Figure 4. The relationships between volume of the filtered triangulations and selected forest 349

biophysical properties. The coefficients shown are based on fitting Eq. 2 to the data. 350

351

352

353

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3.2. Prediction of the filtration parameters 354

355

The mean values of the filtration parameter and particularly the percentile values of the 356

distributions of potential values per plot were found to be moderately correlated (R2 up to 0.41) 357

with several ALS-based height and density metrics. A model, 901.8 + 32.6 × CC2 − 881.4 × D95, where 358

CC was the ALS-based canopy cover estimate (proportion of echoes above 2 m) and D95 the 359

proportional density at the 95th percentile height, predicted the value of the percentile of the 360

distributions of with R2=0.49. The local variation in the point density between the plots was 361

insignificantly correlated with the difference of the predictions (R2 < 0.02). 362

363

The Tvol extracted with the predicted percentile of had a considerably lower correspondence with 364

the field reference attributes than those based on the optimization (Table 2). Notably, the accuracy 365

mainly decreased due to the inaccuracy of the prediction model, which was observed already in the 366

cross validation of the model predictions. On average, making a corresponding prediction with a 367

limited number of training plots did not further reduce the prediction accuracy (Table 2). 368

369

Although the variation of the Tvol increased by predicting the filtration parameter, the averaged Tvol 370

were rather close to each other regardless of the amount of training data (Figure 5). Further, except 371

for the training data consisting of 10 plots, the averaged Tvol did not show statistically significant 372

deviations from the relationship with the forest attributes that was observed in the optimization 373

(Figure 5, Table 2). 374

375

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Table 2. The correspondence between the total volumes of the filtrations and the forest attributes 376

considered. All figures are based on applying Eq. 2 with model parameters shown in Figure 3. 377

Attribute Strategya R2 RMSE biasb

G, m2/ha o 0.95 2.2 (9.9%) -0.2 (-0.7%)

p, cv 0.57 7.6 (34.4%) 0.9 (4.2%) *

p, 30 0.58 7.3 (32.9%) -0.7 (-3.3%)

p, 20 0.59 7.5 (33.9%) -0.4 (-1.8%)

p, 10 0.58 8.5 (38.2%) 1.7 (7.5%) ***

V, m3/ha o 0.88 42.1 (19.9%) -1.4 (-0.7%)

p, cv 0.57 94.2 (44.5%) 14.4 (6.8%) **

p, 30 0.57 90.2 (42.7%) -6.0 (-2.8%)

p, 20 0.58 93.7 (44.3%) -1.1 (-0.5%)

p, 10 0.58 109.0 (51.5%) 26.0 (12.3%) ***

AGB, t/ha o 0.88 19.1 (17.7%) -0.6 (-0.5%)

p, cv 0.63 40.5 (37.6%) 6.3 (5.9%) **

p, 30 0.63 38.8 (36.0%) -3.2 (-2.9%)

p, 20 0.64 40.1 (37.2%) -1.0 (-1.0%)

p, 10 0.63 46.3 (43.0%) 11.2 (10.4%) ***

FB, t/ha o 0.64 8.3 (33.8%) -0.1 (-0.6%)

p, cv 0.45 11.2 (45.3%) 1.2 (4.9%) *

p, 30 0.45 10.9 (44.4%) -0.8 (-3.1%)

p, 20 0.45 11.2 (45.5%) -0.4 (-1.5%)

p, 10 0.44 12.1 (49.1%) 2.1 (8.5%) ** a Indicates the method and reference data to derive the filtration parameters. ‘o’ – optimization 378

based on all 245 plots (Figure 3); ‘p, cv’ – optimization and prediction based on all 245 plots (cross-379

validation); ‘p, N’ optimization with N plots and prediction to the other 245-N plots, when the 380

reported values are averages of the 500 iterations (see Section 2.4.). 381

b The asterisks refer to the statistical significance at confidence levels 90% (*), 95% (**), and 99% 382

(***). 383

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384

Figure 5. The total canopy volumes extracted with the optimized (red circles) and predicted filtration 385

parameters vs. G, when the (sub-)samples of all (A), 30 (B), 20 (C), or 10 (D) plots were used for the 386

prediction. The black dots indicate the averaged volume, while the dashed lines in B–D show the 387

ranges between the minimum and maximum values based on the 500 subsamples. 388

389

390

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4. Discussion 391

392

As reviewed in Section 1, detailed reconstructions of 3-D forest canopy, particularly based on real 393

tree geometry, are hard to obtain. The methodology presented here allows a description that is 394

based on real canopy elements, as extracted from the triangulations based on ALS data, but not 395

explicit prescription of leaf or shoot locations and orientations, for example. Therefore, regarding 396

the degree of detail in describing the distribution of canopy elements, the proposed approach is a 397

compromise between artificial tree/crown level turbid media parameterized by a list of tree 398

locations and dimensions (e.g., Ni-Meister et al. 2001) and more realistic 3-D models of vegetation, 399

such as those constructed by terrestrial laser scanning (e.g., Côté et al. 2011). In turn, the approach 400

can be operated based on ALS and field data, which already cover broad areas and are readily 401

available for practical inventories. 402

403

To the best of our knowledge, to date, the only example of geometrically explicit modeling based on 404

ALS data is given by Schneider et al. (2014), who parameterized a physically based radiative transfer 405

model with two information extraction approaches. First, they concluded that the tree canopy could 406

not be sufficiently described using simple geometric primitives with position and properties derived 407

by means of individual tree detection. Similar assumptions on ellipsoidal or conical crown shapes 408

would not be met in the tree species considered here (Rautiainen et al. 2008), which could result in 409

significant errors in retrieved crown parameters (Calders et al. 2013, Widlowski et al. 2014). Rather, 410

Schneider et al. (2014) found a voxel grid approach to produce more realistic descriptions. Also in 411

this approach, however, selecting an appropriate voxel resolution has a critical effect on the 412

obtained result (Widlowski et al. 2014). 413

414

In addition to the two approaches listed in the previous paragraph, geometric modeling techniques 415

that could be considered for solving the presented problem are an iterative surface wrapping 416

22

technique based on delineated tree crowns (Kato et al. 2009), 3-D clustering followed by convex 417

polytope reconstruction (Gupta et al. 2010) and stacking horizontal slices of the height values (Tang 418

et al., 2013). All these methods involve certain parameters to be set by the operator, and the 419

obtained results depend on these. Further, the methods are so far tested with data densities varying 420

from 4–5 m-2 (Gupta et al. 2010) to 10–20 m-2 (Kato et al. 2009, Tang et al. 2013, Schneider et al. 421

2014) as opposed to 0.8 m-2 of this study. The data studied here prevented reliable tree detection 422

(Kankare et al., unpublished results) and would have resulted in an extremely coarse voxel grid 423

resolution. On the other hand, requiring data with a higher density would mean distinct data 424

acquisitions and therefore considerably higher inventory costs. 425

426

Deriving the geometrically explicit models from the 3-D triangulations of the point data differs from 427

all the approaches listed earlier and has certain benefits over them. The triangulations are entirely 428

based on the properties of the point data, which can be extracted from practically available data 429

with sparse point densities (see also Vauhkonen et al. 2012, 2014, Maltamo et al. 2010). Applying 430

filtrations, one can adjust the level of detail in the triangulated point cloud and thus account for 431

canopy gaps and detailed properties existing in the data, which is controlled by filtration parameter 432

. Regarding the latter, this study presents two types of results: those obtainable in the presence of 433

both field reference and ALS data (i.e., for training plots only) and those obtainable for a prediction 434

grid (wall-to-wall estimates). We consider both of the results important, for simulations and 435

predictions, respectively. 436

437

The approaches to determine the filtration parameter were (i) to optimize the degree of filtration 438

with respect to field measured G and (ii) to predict this degree by means of a simple linear 439

regression model. The value selected for the parameter was highly sensitive toward the obtained 440

results. When the degree of the filtration was determined in an optimization, the filtrations and their 441

total tetrahedral volumes (Tvol) were more closely related to the biomass attributes than in the case 442

23

of prediction. An alternative would have been to determine the most feasible population-specific 443

fixed value like Vauhkonen et al. (2010, 2012), who evaluated a range of -values, but rather than 444

selecting one particular value, they used metrics quantifying the difference of the shape obtained 445

with the range of -values to the convex hull of the point data, which corresponds to -shapes with 446

. An exploratory analysis carried out prior to this study indicated that up to half of the 447

degrees of determination could be obtained by a fixed value. 448

449

The idea to predict plot-specific filtration parameters was introduced by Vauhkonen et al. (2014), 450

who obtained considerably more accurate (R2=0.83 vs. 0.49 of this study) parameter estimates 451

applying a prediction model fundamentally similar to this study. The properties of the applied data 452

sets pose differences in the properties of the ALS data, plot sizes and the applied optimization 453

scheme, for example. The leaf-off data acquisition could be expected to cause shifts of the vertical 454

point profiles affecting especially plots dominated by deciduous trees (cf. Villikka et al. 2012). 455

However, these did not seem to affect the reconstruction of the triangulations. For example, each 456

plot in Figure 3 is a mixture of all the species (pine, spruce and deciduous) occurring in the study 457

area, and except for the underestimation of the dominant height in the deciduous canopy as in the 458

lowermost subplot of Figure 3, the filtrations are realistically allocated and do not differ from pure 459

coniferous plots, for example, based on a visual assessment. Also, the density metrics used in the 460

model to predict the filtration parameter are considered more neutral to the phenological effects 461

than those based on height. Finally, the results were examined by the dominant species of the area, 462

and it was confirmed that none of the species contributed significantly to the variation of the 463

predicted parameter . 464

465

Because the initial field plot sampling design allowed the plots to be located on stand boundaries, an 466

outlier removal based on examining the relationships between the ALS and field data (Section 2.4.) 467

was carried out. This step may cause the results to be over-optimistic due to the absence of the 468

24

problematic plots. However, the so-called stand boundary or edge effects, for instance, have been a 469

well-known problem in forest inventories (e.g. Kangas 2006) and are often readily eliminated by 470

applying restrictions to the measurement protocol or using appropriate plot sizes (see, e.g., Section 471

11.4.3.3 in Næsset 2014). To account for the plot size, we ran exploratory analyses employing a 472

circular 10 m wide buffer around the plots and a canopy mask formed by thresholding the height 473

data above 2 m. The purpose was to select the simplices originating from wider area and within the 474

canopy, respectively, and from combinations of these. Overall, these experiments had a decimal-475

level effect on the R2 values obtained. It is concluded that the proposed filtering technique itself is 476

capable of excluding the simplices populated by empty space without any pre-selection, but better 477

correspondence with respect to forest attributes could potentially be obtained for larger areas due 478

to averaging, which is similar to other ALS analyses (see Section 11.4.3.3 in Næsset 2014). The main 479

bottleneck on the way to better accuracy is nevertheless the prediction of filtration parameter . On 480

the other hand, the results indicated that the best possible prediction could, on average, be 481

currently obtained with modest field data of 20–30 plots, which can be assumed typically available 482

based on practical forest inventories. 483

484

Beyond simple linear regression based on the height and density metrics, improvements to the 485

prediction of the filtration parameters could potentially be obtained by further examining the 486

inherent algebraic properties of the triangulations such as simplicial homomorphism and topological 487

persistence (Zomorodian 2005). For example, Martynov (2008) de-noised geometric models of 488

power line towers reconstructed from the ALS point data by focusing only on important features and 489

structures that persisted over a wide range of -values. Although such an analysis is acknowledged 490

to be considerably simpler with manmade objects than vegetation, which has natural structural 491

variations, computing persistence homology from vegetation point clouds is identified as an 492

interesting topic for future studies. 493

494

25

Finally, the present study also differed from Vauhkonen et al. (2014) in the way the optimization (Eq. 495

2) was carried out. The idea of Eq. 2 is to determine the filtering degree such that a quasi-optimal 496

relationship is formed between the considered forest attribute and an attribute characterizing the 497

filtrations. Vauhkonen et al. (2014) used the modeled canopy- and above-ground biomasses for this 498

purpose but found the resulting canopy volumes to also have a strong allometric relationship with all 499

other forest attributes considered (cf. Figure 4). In that case, however, the accuracy of the response 500

biomass depended on the ability of the biomass model to account for the local conditions. As a 501

variate of the tree diameter, G, on the other hand, is directly measured from the trees and is also 502

the most typical forest attribute obtained in aggregate-level forest inventories. The use of G was 503

thus justified not only by the practical applicability of the proposed approach, but also by the 504

unknown form of the “real” allometric relationship between the canopy volume and forest 505

attributes of interest (see Discussion by Vauhkonen et al. 2014). Provided that the form of this 506

relationship was known, Eq. 2 could have been modified accordingly. 507

508

The earlier studies have evaluated the geometrically explicit models, for example, in terms of the 509

prediction error of LAI (Calders et al. 2013), correlations between the simulated and measured 510

remote sensing data (Schneider et al. 2014) or differences in bidirectional reflectance factors derived 511

from simulated remote sensing data (Widlowski et al. 2014). Our study differs from the previous 512

ones in that the present evaluation was carried out with respect to the relationships between the 513

plant parts. However, based on a visual comparison between Figure 3 of this paper and Figure 5 of 514

Schneider et al. (2014), it is reasonable to propose our approach to produce 3-D canopy 515

reconstructions potentially suitable for similar studies of light interaction. To obtain the final truth, 516

our approach should be further evaluated in such applications and compared against detailed 517

canopy data acquired by terrestrial laser scanning, for example. 518

519

26

5. Conclusions 520

521

The filtering and optimization approach introduced in this paper for 3-D reconstruction of forest 522

canopy represents a compromise between the parameterization of the forest scene by artificial 523

tree/crown level turbid media and realistic 3-D models. Particularly, the results suggest that 524

obtaining coarse wall-to-wall descriptions does not require separate data acquisitions, but may be 525

based on ALS data acquired for digital elevation modeling and field data existing from earlier 526

practical inventories. When the filtration parameters were optimized for 245 sample plots of 300 m2 527

located in southern boreal forest in Finland, the coefficients of determination (R2) between total 528

tetrahedral volumes of the filtrations and basal area, stem volume, total above-ground biomass, and 529

canopy biomass were 0.93, 0.87, 0.87 and 0.62, respectively. Considerably less accurate results were 530

obtained when the filtration parameters were predicted mimicking an ALS-based wall-to-wall 531

inventory with a limited number of the calibration field plots. However, the prediction accuracy did 532

not further decrease in the separate testing data set, and this accuracy was obtained with modest 533

field training data of 20–30 plots. The main bottleneck on the way to better accuracy is nevertheless 534

the prediction of the filtration parameter, which could potentially be improved by further analyses 535

of topological persistence. 536

537

Acknowledgements 538

539

Our study was financially supported by the Research Funds of the University of Helsinki. It is also a 540

contribution to the Finnish Academy project Centre of Excellence in Laser Scanning Research (CoE-541

LaSR, decision number 272195). 542

543

27

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