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Estimation of individual knot volumes by mixed effects modelling

Journal: Canadian Journal of Forest Research

Manuscript ID cjfr-2019-0038.R1

Manuscript Type: Article

Date Submitted by the Author: 18-Apr-2019

Complete List of Authors: Manso, Rubén; Forest ResearchMcLean, J. Paul; Forest ResearchAsh, Adam; Forest ResearchAchim, Alexis; Universite Laval

Keyword: Knot geometry, Seemingly Unrelated Regression, Sitka spruce, mixed models, BLUPs

Is the invited manuscript for consideration in a Special

Issue? :Not applicable (regular submission)

https://mc06.manuscriptcentral.com/cjfr-pubs

Canadian Journal of Forest Research

Draft

Estimation of individual knot volumes by1

mixed effects modelling2

Rubén Manso1,*, J. Paul McLean1, Adam Ash1, and Alexis Achim23

1Forest Research, Northern Research Station, Roslin, Midlothian,4

EH25 9SY, UK5

2Départament des sciences du bois et de la fôret, Université Laval,6

2425 rue de la Terrasse, Québec, QC G1V 0A6, Canada7

*e-mail: [email protected], [email protected];8

Tlf.: +44 (0)300 067 59799

November 11, 201910

1

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INDIVIDUAL KNOT VOLUME ESTIMATION 2

ABSTRACT11

We present a new method to estimate individual knot volumes based on a knot12

geometry model coupled with observations on branch characteristics. X-ray Com-13

puter Tomography and image analysis were used to measure the volume and geom-14

etry of 424 knots of Sitka spruce. Knot geometry can be described mathematically15

by deriving functions for relative vertical position, diameter and slope dependent16

on radial position in the stem. These functions were parameterised using “Seem-17

ingly Unrelated Regression” and mixed-modelling techniques. This provided a base18

model for typical knots. In order to estimate individual-knot volume, we used avail-19

able data for branch diameter and insertion angle to obtain conditional predictions.20

We imputed the most likely knot trajectory as relative vertical position cannot be21

measured on branches. The model explained up to 96% of the variability in knot22

volume by incorporating the branch measurements in contrast to the 43% using23

the typical-knot model. Additionally, knot volume assessment based only on con-24

ditional predictions of diameter and marginal predictions of vertical position also25

accounted for 96% of the variability. Therefore measurements of branch diameter26

alone would be enough to obtain highly precise individual-knot volume predictions.27

This estimator is a first step towards a knot model to be used for management pur-28

poses of Sitka spruce in Great Britain.29

Keywords: Knot geometry, Seemingly Unrelated Regression, Sitka spruce,30

mixed models, BLUPs31

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1 INTRODUCTION32

The occurrence of knots in the wood of commercial species is a long standing33

issue for the sawn-wood industries. Knot frequency and size, often jointly referred34

to as knot area ratio, are considered as major detrimental factors for the quality of35

sawn wood (Zink-Sharp 2003), which is the end product out of which the highest36

value of the harvested tree is generally obtained. As a result of their economic37

impact, learning where, when and how knots occur has been a topic of research for38

some time, mainly through studies on the branching habit of the species of interest39

(Achim et al. 2006; Auty et al. 2012; Colin and Houllier 1992; Mäkinen and Colin40

1998; Osborne and Maguire 2016, among many others). These studies typically41

require a lot of manual measurements of branches and the aim is to produce models42

that describe variables such as branch diameter and angle of insertion/inclination,43

that can be used to infer knot geometry.44

An alternative to the measurement of branches and making inferences about45

knots is the application of X-ray Computer Tomography to directly measure knots46

in situ (e.g. Moberg 2000, 2001). Knot volume can now be measured from X-ray47

images of logs (Pinto et al. 2003) and used to model a multitude of parameters48

describing the shape of knots (Moberg 2006), which have recently been modelled49

for various North-American conifer species (Duchateau et al. 2015, 2013). The50

models by Duchateau et al. (2013) included a model for the z coordinate of knot51

trajectory (hereafter referred to as relative vertical position) and a knot diameter52

one. Their models can be described as specific to individual knots, as they provide53

predictions of knot geometry at the knot level over the stem radius (i.e. from pith54

to bark, or bark to pith) with an expansion of model parameters as a function of55

several external tree- and branch-related predictors. Among the latter, the more56

relevant ones are branch angle, height and diameter at the point of insertion into the57

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stem. This is useful as knot geometry can consequently be better predicted from58

measurements of trees that can be obtained in the forest. The downsides are that59

quite a range of branch measurements are required and that the combined effects of60

these branch measurements on the model parameters have a composite effect, and61

cannot be easily discerned on either a mathematical or a biological basis. Therefore62

an alternative empirical approach that can more parsimoniously account for such63

variability is desirable.64

One possible alternative is to use a mixed modelling approach (see Lappi and65

Bailey 1988). In a mixed modelling approach the fixed effects can be used to pro-66

duce marginal predictions, which represent the mean of a given process if the model67

is linear, while the random effects can be used to represent the variability around68

the mean at a given grouping level. In our case we modelled knot geometry as a69

function of radial position (the fixed component) and we allowed this to vary by in-70

dividual knots (the random component and grouping level). Individual knots differ71

from each other for many reasons that we could not feasibly measure (e.g. light72

availability of individual branches), but using only a relatively small number of73

measurements the random components of each knot can be quantified through their74

Best Linear Unbiased Predictors (BLUPs) (Goldberger 1962; Henderson 1975) and75

projected onto new individuals in the wider population. We ultimately want to pre-76

dict knot geometries and volumes from external measurements that can be made77

on trees: the diameter of branches (knot diameter) and the angle of insertion (knot78

tangent) can be measured, whereas the relative vertical position of knots cannot.79

This is because knot relative vertical position represents the difference in vertical80

height between any point in the knot trajectory and knot origin at tree pith, the latter81

being obviously invisible from outside the tree. These terms are depicted in Fig 1.82

However, in order to calculate volume we need diameter and relative vertical posi-83

tion for each given radial position, whereas the tangent plays no direct role in the84

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calculation of volume. Herein lies the problem. Our proposed solution is based on85

the fact that the knot relative vertical position, diameter and tangent deviations from86

model marginal predictions are not likely to be independent (e.g. a steeper knot may87

tend to reach higher and have a larger diameter as well). In other words, random88

effects from the processes that define knot geometry can —and ideally should—89

be allowed to correlate in the model fit, which implies a simultaneous parameter90

estimation of the models representing those processes. It would then be possible91

to exploit the random effect correlation to impute the value of the missing random92

effect(s). Here we attempt to use knot diameter and knot tangent for these purposes93

and the theoretical approach to attain this is developed in this paper.94

(Insert Fig 1 here)95

In this study, we will consider the example of planted forests of Sitka spruce96

(Picea sitchensis (Bong.) Carr.) in Great Britain. Sitka spruce is the most important97

coniferous species in Great Britain in terms of area and harvested volume (50% and98

60% of the conifer total, respectively; Forestry Commission (2016)) and plays a99

crucial role in the economy, particularly that of rural areas. Consequently, there is100

an applied interest to understand knot geometry and volume in the species, both for101

purposes of sawn wood production and for evaluation of the potential for emerging102

markets such as biorefineries, which use raw materials from forests to produce a103

wide range of chemicals (Clark et al. 2006). Some of the desirable chemicals can104

be found in considerable quantities in conifer knots as extractives (Holmbom et al.105

2003; Kebbi-Benkeder et al. 2017)106

The main aim is to test the approach for individual knot geometry and volume107

estimation through the prediction of random effects and produce a model that can108

be made readily available for management purposes of Sitka spruce stands in Great109

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Britain. The hypothesis is that volume predictions obtained through this approach110

will be more precise than marginal predictions where only the typical (i.e. the mean)111

knot geometry and volume over the stem radius are estimated. The main objective112

of the present study was to test this hypothesis. In order to do this we were required113

to: (i) develop a model for knot geometry where error terms of the different sub-114

models are allowed to correlate, (ii) implement the theoretical methods required to115

predict individual knot volume, (iii) produce the necessary outputs for comparison116

between typical knot and individual knot volume predictions, and (iv) test the rele-117

vance of each geometry model component (knot relative vertical position, diameter118

and tangent) in individual knot volume prediction.119

2 MATERIALS AND METHODS120

2.1 DATA121

Six Sitka spruce trees were sampled from two sites in Scotland: two mature trees122

were sampled from a planted stand on the grounds of the Forest Research’s Northern123

Research Station, Roslin, and nominally represented specimens of large and small124

diameter. Four additional trees of varying diameter at breast height (DBH) were125

sampled from Aberfoyle Forest. The trees from both sites were growing in pure126

even-age stands at an initial 2 m spacing between plants (2500 stems ha−1) , which127

are the typical conditions for Sitka spruce planted forests in Great Britain. All trees128

were cross-cut into logs nominally 1 m in length, resulting in 88 logs with diameters129

ranging from 50 to 200 mm. Their internal structure was captured using X-ray CT130

scanning using the equipment and services of Scotland’s Rural College (SRUC).131

A series of cross-sectional X-ray images were collected in 1.5 and 5 mm spirals132

along log length for all logs from Roslin and Aberfoyle sites, respectively. The133

Roslin samples were measured first and it was determined that 5 mm resolution was134

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considered enough for the purposes of this study. The Roslin data were simplified135

accordingly. Following the methods of Duchateau et al. (2013) the scans from each136

log were visualised as image series and descriptively tagged using the Gourmands137

plugin (Colin et al. 2010) within the open source ImageJ image analysis software138

(Rasband 1997). The software enabled us to manually mark the locations of the139

pith and knots within each image and subsequently turned this data into coordinates140

describing the paths of each knot and estimates of their diameters as they develop141

from pith to bark Fig 1. Visualization was also possible by means of Bil3D software142

(Colin et al. 2010) (Fig 2).143

Subsequently, data were analysed to detect potential measurement errors and144

epicormic shoots mistakenly considered as true knots. Both potential errors and145

epicormics shoots were manually examined and these records removed (111 false146

knots found). After this process, 424 knots remained to carry out the intended147

analysis.148


2.2 MODELLING APPROACH150

The modelling of knot volume consisted of three stages: (i) modelling knot151

geometry features, (ii) assessment of correlation between these features, and (iii)152

knot volume estimation.153

In the first step of our analysis, the data described before were used to model154

the three main features that define knot geometry: relative vertical position of knot155

pith, knot diameter and tangent of knot pith, all three over the stem radius. Pre-156

liminary models were fitted to examine and select suitable functional forms. These157

responses were modelled as a function of stem radius and different transformations158

of this variable, such as fractional, power or exponential transformations, allow-159

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ing for a different behaviour depending on whether the target knot belonged to a160

whorl or not. Based on visual evidence, we classified a given knot as a whorl161

knot when its vertical distance to the nearest knot taken at tree pith was less than162

10 mm. Tree DBH and absolute height up the tree were also tested as covariates,163

although no significant effect was found. These tests were carried out through a164

visual check to standardized model residuals. Residual variances were modelled165

to correct from heteroscedastic residual patterns when sensible. In the following166

formulae, the different observations within the knots and the knots themselves are167

assigned the indices i and j respectively. Next the indices m = 1,2,3 are assigned to168

the knot relative vertical position, diameter and tangent models, respectively. The169

preliminary models were as follows170

y1,i j = β1ri j

ri j +1+β2ri jWi j +β3ri j(1−Wi j)+ ε1,i j (1)

y2,i j = γ0(1− eγ1ri j)+ γ2ri j + ε2,i j (2)

y3,i j = θ0 +θ1(1−θ2)ri j + ε3,i j (3)

where ym,i j and εm,i j are the responses and the residual errors respectively for171

knot relative vertical position, diameter and tangent models. ri j stands for stem ra-172

dius, Wi j is a dummy variable that adopts the value 1 when the knot belongs to a173

whorl and 0 otherwise. β T. = βββ , γT

. = γγγ and θ T. = θθθ are vectors of estimable pa-174

rameters. εm,i j are assumed normally distributed such as εm,i jiid∼ N(0,σ2

m), where175

σ2m is the model residual variance. In the case of the relative vertical position model,176

the residual variance was modelled as a function of ri j to meet the homoscestatic-177

ity assumption, such that σ21 = σ ′2

1 · e2ϕri j , with ϕ an additional parameter to be178

estimated. Note that these functions are empirical and therefore the parameters179

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themselves hold little meaning.180

These models provide typical or mean knot features conditional on a given set of181

covariates through their parameters that are considered fixed effects. The descrip-182

tion of the variance around that mean can be best provided with the incorporation183

of random effects (Gregoire et al. 1995) in a mixed modelling approach. This is184

the second stage of our analysis. Our hypothesis is that these random effects would185

be correlated and therefore we can take this correlation into account through so-186

called “Seemingly Unrelated Regression” (SUR) (Gallant 1987). The principle of187

SUR is to couple various models through dummy variables and weight their resid-188

ual variances according to the response type. Once each of the preliminary models189

in Eqs. 1, 2 and 3 were determined to behave satisfactorily independently, SUR was190

used as a second analytical step to carry out a simultaneous fit which will then allow191

random effects to correlate between models. Different structures of random effects192

at the knot level were tested, the standardised residuals being further checked at193

each step. Expressing the covariates of the three models in their matrix form as194

Xm, the definitive model structure with random effects under the SUR formulation195

yields196

yyym = qqqTmmm

X1βββ

f (X2,γγγ)

X3θθθ

+qqqTmmm

ri j

ri j+1b1, j

(1+ eγ1ri j)b2, j

(1−θ2)ri jb3, j

+qqqTmmmεεε (4)

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yyymmm =

y1,i j

y2,i j

y3,i j

, qqqmmm =

q1,i j

q2,i j

q3,i j

,

εεε =

ε1,i j

ε2,i j

ε3,i j

iid∼ N3(000,R),

bbb jjj = [b1, j,b2, j,b3, j]T iid∼ N3(000,G)

where qqqmmm is a vector of dummy variables that alternatively adopt the value 1 for197

a given m and 0 otherwise; b1, j, b2, j and b3, j are the elements of the vector bbb jjj of the198

random effects at the knot level and R and G are the variance-covariance matrices of199

the distribution of the residuals and the random effects, respectively. In this case, R200

can be factorized as σ2mW2, where W is a diagonal matrix of the variance weights for201

the different models with elements wwwmmm (Pinheiro and Bates 2000). Constant weights202

for each model corresponding to the variance function g(wwwmmm) = δδδ mmm are compulsory203

in SUR, δδδ mmm being a vector of additional parameters to estimate. Elements of www111 are204

conventionally set to 1 (thus δδδ 111 as well) so that σ2m = σ2

1 δm. Additionally, we still205

set σ21 = σ ′2

1 · e2ϕri j in order to deal with heteroscesaticity in the first model. All206

parameters were estimated through the maximization of model likelihood.207

The third analytical step focused on the use of parameter estimates from the208

relative vertical position and diameter models in Eq. 4 to assess the total volume209

of a given knot (Vj). Each knot was considered as a series of consecutive and210

tapered, cylinders. The cross sectional area of the cylinder was considered as the211

cross sectional area at the mid point between two consecutive predicted sections.212

This cross sectional area was assumed elliptical. Although vertical and horizontal213

diameters of knots measured in a vertical section can be assumed identical in the214

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species (Pearson correlation coefficient 0.99), relative to the major knot axis, the215

vertical diameter is greater than the horizontal diameter (Achim et al. 2006). Hence216

the cross sections perpendicular to knot axis must be an ellipse. The minor axis of217

the ellipse Rmin is equal to the diameter that was measured (equation 5a) and the218

major axis Rmax must be calculated geometrically (equation 5b). The length of the219

section l is the length of the line between the centre of the two consecutive sections220

(equation 5c). Cylinder volumes can then be numerically integrated to produce knot221

volume Vj (equation 5d)222

Rmini,i+1, j =di+1, j +di, j

2(5a)

Rmaxi,i+1, j = Rmini,i+1, j cosηi,i+1, j (5b)

li,i+1, j =

√(hi+1, j −hi, j

)2+(ri+1, j − ri, j

)2 (5c)

Vj =K−1

∑i=1

πRmini,i+1, jRmaxi,i+1, j li,i+1, j (5d)

where di j, hi j and ri j are the ith observation on diameter, relative vertical position223

and location at the stem radius in knot j, respectively, ηi,i+1, j is the angle of the224

slope between sections i and i+ 1 of knot j, and K is the number of segments in225

which the knot is divided. hi j and di j were estimated using Monte Carlo simulation226

from the relative vertical position and diameter models. The theoretical distribution227

of the volume of a typical knot can be obtained through a sufficient number of Monte228

Carlo simulations (10, 000 in this case). The expected value for knot volume at a229

given stem radius would be the mean of such distribution. All sources of uncertainty230

were considered, i.e. fixed effects, random effects and residual variance. Fixed-231

effect and random-effect realizations were held constant within knots. Models in232

this section were fitted with package nlme (Pinheiro et al. 2013) in R (R Core Team233

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2015).234

2.3 VOLUME ESTIMATION OF INDIVIDUAL KNOTS THROUGH PRIOR REL-235

ATIVE VERTICAL POSITION SAMPLING236

In the previous subsection, methods were developed to predict knot volume for a237

given stem radius. At this stage we can consider that we have the ability to describe238

typical knots in a stem. However we also want to quantify how observations of239

branches will affect the volume of individual knots. Therefore the next step was to240

develop methods for individual knot volume that incorporate observed branching241

parameters, i.e. insertion diameter and insertion angle of branches. Those methods242

rely upon the prediction of the random effects in Eq. 4 for new observations.243

It is well known that the prediction of random effects in a linear context can be244

carried out through the BLUPs approach as245

bbb = GZTV−1(yyy−Xβββ ) (6)

where Z is the matrix of the covariates affected by the random effects, V is246

the estimated matrix of the variance of the model, which can be calculated as V =247

ZGZT + R, yyy is a vector with new observations and Xβββ are the marginal model248

predictions for the individuals that yyy belongs to. Ideally we could use the branch249

geometry at the point of branch insertion into the stem as new observations yyy. In250

our case, we considered knot geometry at the stem periphery from our own dataset251

as a proxy (see Table 2 for some basic statistics).252

Again it is not feasible to obtain knot relative vertical position from measuring253

dimensions on the tree. Therefore the relative vertical position was systematically254

sampled over its observed range, each simulation being used as an observation it-255

self. As a result, a set of predicted random effects was obtained for each combi-256

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nation of observed diameter and tangent, plus sampled relative vertical position.257

Each of these sets can be seen as a sample of the probability density distribution of258

the random effects. The set of maximum density corresponds to that of the most259

likely relative vertical position given the observations on knot diameter and knot260

tangent at the stem periphery. We thereby used these optimal BLUPs as a means to261

obtain conditional predictions on individual knot relative vertical position and knot262

diameter over the stem radius. Subsequently, knot volume was estimated follow-263

ing the Monte Carlo procedure introduced in the previous section. In order to test264

the robustness of the approach, we additionally computed individual knot volume265

by imputing the most likely relative vertical position with the diameter-related ran-266

dom effects only (i.e. ignoring the tangent), and simply combining conditional knot267

diameter predictions and marginal relative vertical position predictions.268

(Insert Table 2 here)269

2.4 EVALUATION OF TYPICAL AND INDIVIDUAL KNOT VOLUME ESTIMA-270

TOR271

The different versions of the SUR were compared through their AICs (Akaike272

Information Criterion). Knot volume estimates based on the marginal predictions273

from the SUR model were evaluated by examining the empirical coverage of their274

confidence intervals. Confidence intervals for the prediction of each knot were cal-275

culated through the percentile rank method out of 10, 000 Monte Carlo simulations.276

The empirical coverage was assessed as the proportion of the knots whose observed277

volume fell within those confidence intervals for different levels of significance278

1−α or “nominal coverage”. For example, if α = 0.05, 95% of their correspond-279

ing confidence intervals should contain the observed volume. If the predictions are280

unbiased, then nominal and empirical coverage should coincide. The observed vol-281

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ume was computed through Eq. 5d using the measurement data from the X-ray CT282

scans and it was assumed to be a good estimate of the true volume.283

Additionally, absolute and relative estimation bias (E, Er), root mean square284

error (RMSE) and estimation efficiency (EF , Soares et al. (1995)) were calculated285

by comparing the expected volume of each individual knot in our dataset (Vj) with286

their respective observed volume (V j). V j was computed out of 10 000 Monte Carlo287

simulations. In order to determine the relative merit of each of the introduced ap-288

proaches, these metrics were computed for typical knot volume predictions (i.e.289

using the procedure in subsection 2.2) and for the predictions from the different290

versions of the individual knot volume estimator (i.e. following the steps in subsec-291

tion 2.3).292

3 RESULTS293

The fitting process of the model for knot geometry under SUR (Eq. 4) success-294

fully converged. The resulting parameter estimates are shown in Table 1. In light of295

the correlations among the random effects, it is possible to conclude that knots with296

a steeper profile (i.e. higher relative vertical position and tangent than the average)297

tend to exhibit larger diameters. Also, deviations in relative vertical position vary298

positively with deviations in tangent as stem radius increases. These results mirror299

the empirical correlations found among geometry components at the stem periphery300

(Table 2).301

In order to illustrate the expected evolution of knot relative vertical position,302

diameter and tangent along with stem radius, simulations based on the model were303

run for a hypothetical knot that belongs to a whorl occurring at a stem radius of 150304

mm (Figs. 3, 4 and 5).305

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(Insert Fig 3, 4 and 5 here)306

The empirical coverage probability closely matched their nominal values (Fig. 6).307

This indicates that model predictions and their confidence intervals are reasonably308

unbiased.309



Concerning individual knots, volume estimations carried out by combining con-312

ditional knot diameter predictions and marginal knot relative vertical position led to313

a massive improvement with respect to the mean knot volume predictions (96%314

and 43% of the variability explained, respectively). Observed and predicted values315

from both approaches are shown in Figs. 7 and 8. In contrast, additionally using the316

BLUPs from the diameter model or the tangent model to impute the most likely tra-317

jectory did not lead to further improvements of individual knot volume predictions.318

This information is summarized in Table 3.319

(Insert Fig 7 and 8 here)320


4 DISCUSSION322

A novel approach that allows for individual knot volume estimation was intro-323

duced. First, it was possible to simultaneously fit mixed models for knot geome-324

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try that eventually served to estimate the volume of typical knots. The process of325

simultaneous fitting allowed the assessment of between-model random effect cor-326

relations, which helped describe how individual knot geometry and consequently327

volume deviate from the mean predicted values. This information was combined328

with pseudo external branch observations (i.e. derived from knot measurements at329

stem periphery) to predict model random effects and, in turn, individual knot vol-330

ume with a high degree of precision. All of these aspects contribute to the originality331

of this paper and to the state of the art in the field of knot modelling.332

The most complete empirical work on knot geometry based on in situ measure-333

ments so far (Duchateau et al. 2013) sought to describe relative vertical position and334

diameter of individual knots as functions of stem radius. The parameters of these335

functions were in turn made dependent on different tree and branch level covari-336

ates, which may not be always available. We circumvented this problem through337

the aforementioned mixed-modelling approach, which drastically reduced the num-338

ber of variables needed to model individual knot geometry and volume. We consider339

this reduction as a major advantage over previous empirical approaches in this field.340

The functional form used to model knot relative vertical position and diameter341

in Duchateau et al. (2013) was defined so that some boundary conditions were met342

(e.g. relative vertical position and diameter at the pith had to be zero and relative343

vertical position at the point of branch insertion into the stem should match that344

observed). In our case, preliminary tests with these equations and others aimed at345

imposing those conditions resulted in either unacceptable patterns in model residu-346

als or convergence failure of the fitting algorithm. Consequentially, a simpler and347

more flexible approach was trialled that still proved satisfactory in terms of the348

model assumptions. These unbounded functional forms may not be adequate for349

other species or growing conditions, therefore this is an aspect that requires further350

research.351

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As far as we are aware, simultaneous parameter estimation of the models defin-352

ing knot geometry had not previously been attained. The merit of this approach353

has already been studied in the context of branch geometry, although with no suc-354

cess (Mäkinen and Colin 1998). Simultaneous fitting allows between-model error355

correlations to be taken into account. In our case, these correlations allowed the356

prediction of random effects even when a critical external observation parameter357

(relative vertical position) was omitted to represent the fact that it is physically358

unobtainable. This ultimately led to more realistic predictions of knot character-359

istics. Neglecting between-model error correlations can be of further importance360

when it comes to error propagation to obtain knot volume. For example, here knots361

steeper than the average were observed to have the tendency to have also larger362

cross-sections than the average. Simulations carried out through uncorrelated ran-363

dom effects could therefore result in predictions where relative vertical position364

increases notably while diameter for the same knot remains at low levels. The es-365

timated volume and volume prediction intervals of such a knot would be definitely366

unrealistic.367

Unsurprisingly, incorporating branch observations to estimate individual knot368

volume largely outperformed the simple application of the relative vertical position369

and diameter models to obtain typical knot volume, in line with what has been ob-370

served in other areas of forest science (see, for example Calama and Montero 2004;371

Mehtätalo 2004). Interestingly, imputing the most likely relative vertical position372

by means of the correlation of the random effects yielded similar results to simply373

using the conditional knot diameter predictions plus the marginal knot relative ver-374

tical position. An evaluation of the predicted most likely relative vertical position375

shows a relative improvement in terms of explained variability when knot diameter376

observations were used (45%, no imputation; 55%, imputation) but not further im-377

provement when the knot tangent observations were additionally considered (57%).378

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The reason why the 10% improvement when knot diameter observations were used379

did not translate into a better volume prediction may be due to how and where knot380

volume is allocated, plus how the random effects affect knot trajectory. Relative ver-381

tical position increases linearly over the whole stem radius except in the very close382

proximity to the stem pith, where knot diameter is nearly negligible and therefore383

so is the share of volume. Further, here the single random effect included in the rel-384

ative vertical position model only affects the parameter linked to the term r/(r+1),385

which basically governs the height at which knot trajectory evolves in a linear fash-386

ion. In consequence the individual knot trajectories, defined by that random effect,387

do not have a major impact in terms of volume as they do not differ in the linear388

part, which accounts for most of the volume. This is in contrast to the conditional389

predictions of knot diameter. As it can be seen in Fig. 4, the uncertainty around the390

expected value for a typical knot dramatically increases already in the proximity391

to the tree pith. Most of this uncertainty is due to the variance of the associated392

random effect. This means that the between-knot variability in terms of knot diam-393

eter is considerable all along the knot trajectory. In consequence, the conditional394

predictions of knot diameter have a great impact on knot volume, as observed in395

the simulations. In summary, while knot diameter greatly increases the accuracy of396

our predictions of knot volume, we do not require the sampling approach to infer a397

more precise knot trajectory. This technique may be useful for other criteria where398

knot trajectories are relevant, like the proportion of the height of the stem affected399

by knots. The degree of improvement to be expected is limited though.400

Our findings also showed that diameter behaviour over stem radius of a given401

knot can accurately be obtained from a single diameter measurement if the model402

is unbiased and the random effects are properly specified. It is important to note403

that our model may not meet these requirements under a more active management,404

such as intensive thinning, which could temporarily change the rate of growth of405

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the branch. Otherwise, the possibility of using a single measurement under the con-406

ditions for which the model was fitted is most convenient as branch diameter at the407

point of branch insertion into the stem is well known to be an excellent proxy of408

knot diameter at the periphery of the stem (e.g. Duchateau et al. 2013). Contrast-409

ingly branch insertion angle and knot angle at the stem periphery may exhibit a poor410

correlation (Duchateau et al. 2013; Lemieux et al. 2001). In spite of this limitation,411

modelling insertion angle is still a major goal in many studies on knot features,412

some of them dealing with knot volume (e.g. Duchateau et al. 2013; Fredriksson413

2012; Mäkinen and Colin 1998; Osborne and Maguire 2016, among many others).414

Here we have demonstrated that insertion angle can safety be omitted when pre-415

dicting knot volume, at least in the type of planted Sitka spruce forests investigated,416

which represents a worthwhile advantage over other modelling alternatives. Thus417

investigating whether this behaviour occurs in other species and/or under other man-418

agement regimes may prove highly valuable.419

From a practical perspective, we believe our approach is highly applicable in420

practical forest management. Sitka spruce is a species of high economic interest in421

Great Britain, with a wide range of uses where knot occurrence is seen either as a422

disadvantage (e.g. structural timber) or as an opportunity (e.g. chemical extractive423

production, Moore 2011). In either case, a prior assessment of knot geometry and424

volume in standing timber may prove useful. As shown, both features can be ascer-425

tained if branch diameter at insertion is known. This metric could be obtained by426

a direct manual measurement at breast height, or there is potential that this may be427

derived for a larger portion of the tree bole using laser scanning technology (Dassot428

et al. 2012; Hackenberg et al. 2015; Liang et al. 2016). Consequently knot assess-429

ment is possible at the forest inventory stage and our model makes it possible to430

consider knots in future planning. Alternatively, branch diameter at insertion could431

be obtained from existing models that were developed to predict the number, distri-432

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bution and diameter of branches in Sitka spruce in Great Britain (Achim et al. 2006;433

Auty et al. 2012), in an attempt to integrate branches in dynamic growth models.434

Further, these models could be coupled with that proposed here to estimate the im-435

pact of different silvicultural choices on knot volume. This would help maximize436

the profit at the end of the rotation. In the case of structural timber, an optimal sil-437

vicultural scheme may lead to a reduction in the number and size of knots, which438

in turn would enhance timber stiffness. This mechanical property is well-known439

to be the limiting factor when grading Sitka spruce timber (Moore 2011). Conse-440

quently, addressing this issue may significantly increase the value of British timber.441

In the case of chemical extractive production, the goal would be to promote knots.442

Given the current development of the bioeconomy business, this could rise the ben-443

efits from poor or exposed sites where the grades for structural timber cannot be444

reached.445

While the parameterisation presented here could provisionally be used for the446

aforementioned purposes, a more robust model calibration based on a larger sample447

of knots from different locations would be desirable. Additionally, it is important448

to note that the sample trees used in this study did not originate from the British449

selective breeding programme for Sitka spruce, aimed at improving growth rates450

and stem form including branching. According to Lee (2001) a considerable reduc-451

tion in the number and size of branches has been observed in improved individuals.452

Given that improved material has been used in the vast majority of the new planta-453

tions of the species in Great Britain over the last 25 years (Rook 1992), an extended454

sample for model recalibration would need to include improved trees as well.455

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AKNOWLEDGEMENTS456

The authors wish to express their gratitude to Madeleine Murtagh for her contri-457

bution with image processing, the staff at SRUC for X-ray CT scanning and Frédéric458

Mothe (Unité Mixte de Recherche Silva) for use of his software. Special thanks to459

Tom Connolly, Tommaso Locatelli, Gustavo López and Helen McKay (Forest Re-460

search) for their useful comments on an earlier version of this paper.461

REFERENCES

Achim, A., Gardiner, B., Leban, J.-M., and Daquitaine, R. (2006). Predicting the

branching properties of sitka spruce grown in Great Britain. New Zealand Journal

of Forestry Science, 36(2-3):246–264.

Auty, D., Weiskittel, A. R., Achim, A., Moore, J. R., and Gardiner, B. A. (2012).

Influence of early re-spacing on Sitka spruce branch structure. Annals of Forest

Science, 69(1):93–104.

Calama, R. and Montero, G. (2004). Interregional nonlinear height-diameter model

with random coefficients for stone pine in Spain. Canadian Journal of Forest

Research, 34:150–163.

Clark, J. H., Budarin, V., Deswarte, F. E. I., Hardy, J. J. E., Kerton, F. M., Hunt,

A. J., Luque, R., Macquarrie, D. J., Milkowski, K., Rodríguez, A., Samuel, O.,

Tavener, S. J., White, R. J., and Wilson, A. J. (2006). Green chemistry and the

biorefinery: a partnership for a sustainable future. Green Chemistry, 8(10):853.

Colin, F. and Houllier, F. (1992). Branchiness of Norway spruce in northeastern

France: predicting the main crown characteristics from usual tree measurements.

Annales des Sciences Forestières, 49(5):511–538.

Colin, F., Mothe, F., Freyburger, C., Morisset, J.-B., Leban, J.-M., and Fontaine, F.

Page 21 of 34



Draft


(2010). Tracking rameal traces in sessile oak trunks with X-ray computer tomog-

raphy: biological bases, preliminary results and perspectives. Trees, 24(5):953–

967.

Dassot, M., Colin, A., Santenoise, P., Fournier, M., and Constant, T. (2012). Ter-

restrial laser scanning for measuring the solid wood volume, including branches,

of adult standing trees in the forest environment. Computers and Electronics in

Agriculture, 89:86–93.

Duchateau, E., Auty, D., Mothe, F., Longuetaud, F., Ung, C. H., and Achim, A.

(2015). Models of knot and stem development in black spruce trees indicate a

shift in allocation priority to branches when growth is limited. PeerJ, 3:e873.

Duchateau, E., Ung, C., Auty, D., and Achim, A. (2013). Modelling knot morphol-

ogy as a function of external tree and branch attributes. Canadian Journal of

Forest Research, 43(3):266–277.

Forestry Commission (2016). Forestry Statistics 2016-A compendium of statistics

about woodland, forestry and primary wood processing in the United Kingdom.

Technical report, Forestry Commission, Edinburgh.

Fredriksson, M. (2012). Reconstruction of Pinus sylvestris knots using measur-

able log features in the swedish pine stem bank. Scandinavian Journal of Forest

Research, 27(5):481–492.

Gallant, R. (1987). Nonlinear equation estimation. John Wiley & Sons, New York.

Goldberger, A. (1962). Best linear unbiased prediction in the generalized linear

regression model. Journal of the American Statistical Association, 57:369–375.

Gregoire, T., Schabenberger, O., and Barrett, J. (1995). Linear modelling of irregu-

larly spaced, unbalanced, longitudinal data from permanent-plot measurements.

Canadian Journal of Forest Research, 25:137–156.

Hackenberg, J., Spiecker, H., Calders, K., Disney, M., and Raumonen, P. (2015).

Page 22 of 34



Draft


Simpletree — an efficient ppen source tool to build tree models from tls clouds.

Forests, 6:4245–4294.

Henderson, C. (1975). Best linear unbiased estimation and prediction under a se-

lection model. Biometrics, 31(2):423–447.

Holmbom, B., Eckerman, C., Eklund, P., Hemming, J., Nisula, L., Reunanen, M.,

Sjöholm, R., Sundberg, A., Sundberg, K., and Willför, S. (2003). Knots in trees

– A new rich source of lignans. Phytochemistry Reviews, 2(3):331–340.

Kebbi-Benkeder, Z., Manso, R., Gérardin, P., Dumarçay, S., Chopard, B., and Colin,

F. (2017). Knot extractives: a model for analysing the eco-physiological factors

that control the within and between-tree variability. Trees, 31(5):1619–1633.

Lappi, J. and Bailey, R. (1988). A height prediction model with random stand and

tree parameters: an alternative to traditional Site Index methods. Forest Science,

34(4):907–927.

Lee, S. (2001). Revised predicted gains from Sitka spruce seed orchards. Forestry

and British Timber, March.

Lemieux, H., Beaudoin, M., and Zhang, S. (2001). Characterization and mod-

eling of knots in black spruce (Picea mariana) logs. Wood and fiber science,

33(3):465–475.

Liang, X., Kankare, V., Hyyppä, J., Wang, Y., Kukko, A., Haggrén, H., Yu, X.,

Kaartinen, H., Jaakkola, A., Guan, F., Holopainen, M., and Vastaranta, M. (2016).

Terrestrial laser scanning in forest inventories. ISPRS Journal of Photogrammetry

and Remote Sensing, 89:63–77.

Mäkinen, H. and Colin, F. (1998). Predicting branch angle and branch diameter

of Scots pine from usual tree measurements and ‘stand structural information.

Canadian Journal of Forest Research, 28:1686–1696.

Mehtätalo, L. (2004). A longitudinal height–diameter model for Norway spruce in

Page 23 of 34



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Finland. Canadian Journal of Forest Research, 34:131–140.

Moberg, L. (2000). Models of internal knot diameter for Pinus sylvestris. Scandi-

navian Journal of Forest Research, 15(2):177–187.

Moberg, L. (2001). Models of internal knot properties for Picea abies. Forest

Ecology and Management, 147(2-3):123–138.

Moberg, L. (2006). Predicting knot properties of Picea abies and Pinus sylvestris

from generic tree descriptors. Scandinavian Journal of Forest Research,

21(S7):49–62.

Moore, J. (2011). Wood properties and uses of Sitka spruce in Britain. Forestry

Commission Research Report i-iv, Forestry Commission, Edinburgh.

Osborne, N. L. and Maguire, D. A. (2016). Modeling knot geometry from branch

angles in Douglas-fir (Pseudotsuga menziesii). Canadian Journal of Forest Re-

search, 46(2):215–224.

Pinheiro, J. and Bates, D. (2000). Mixed effects models in S and S-PLUS. Springer,

New York.

Pinheiro, J., Bates, D., DebRoy, S., Sarkar, D., and Team, R. D. C. (2013). nlme:

Linear and Nonlinear Mixed Effects Models. R package version 3.1-109.

Pinto, I., Pereira, H., and Usenius, A. (2003). Analysis of log shape and internal

knots in twenty Maritime pine (Pinus pinaster Ait.) stems based on visual scan-

ning and computer aided reconstruction. Annals of Forest Science, 60:137–144.

R Core Team (2015). R: A language and environment for statistical computing.

Vienna, Austria.

Rasband, W. (1997). ImageJ. U. S. National Institutes of Health, Bethesda, Mary-

land, USA.

Rook, D. (1992). Super Sitka for the 1990s. Forestry Commission Bulletin, Forestry

Commission, Edinburgh.

Page 24 of 34



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Soares, P., Tomé, M., Skovsgaar, J., and Vanclay, J. (1995). Evaluating a growth

model for forest management using continuous forest inventory data. Forest Ecol-

ogy and Management, 71:251–265.

Zink-Sharp, A. (2003). Mechanical properties of wood. In Wood quality and its

biological basis, pages 197–209. Blackwell Publishing, Oxfordshire, UK.

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TABLES

Table 1. Maximum likelihood parameters estimates and standard errors of the SURmodel. For explanation of parameters refer to equations 1-4 in the text.

Fixed effects Variance componentsparameter estimate standard error parameter estimateβ1 17.7659 0.8878 σ2

1 1.3424β2 0.5041 0.0035 ϕ 0.0250β3 0.4602 0.0053 δ2 1.2617γ0 31.3331 1.0341 δ3 0.5147γ1 -0.0213 0.0005 σ2

b1330.3888

γ2 -0.0957 0.0069 σ2b2

48.9116θ0 0.4740 0.0129 σ2

b34.7067

θ1 2.1417 0.1100 σb1b2 33.9414θ2 0.0765 0.0015 σb1b3 37.146849

σb2b3 4.1118

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Table 2. Basic statistics of knot relative vertical position (y1, mm), diameter (y2, mm)and tangent (y3, mm/mm) data at the stem periphery.

CorrelationResponse mean standard dev. y1 y2 y3y1 55.8873 27.7044 1 0.5310 0.2921y2 16.5543 6.2427 1 0.3225y3 0.4820 0.2237 1

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Table 3. Evaluation metrics for knot volume predictions. E (mm3) and Er are the abso-lute and relative estimation bias, respectively, RMSE (mm3) is the root meansquare error and EF stands for estimation efficiency

Predictions used E Er RMSE EFMean knot 887 0.0627 9754 0.4395

Diameter (conditional),

Relative vertical position (marginal)643 0.0455 2644 0.9588


Relative vertical position

(imputed from diameter model)

651 0.0460 2632 0.9592


Relative vertical position

(imputed from diameter and tangent models)

708 0.0500 2656 0.9584

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FIGURE CAPTIONS

Fig. 1. The terms used to describe knot geometry. The longitudinal section of

the tree is shown in the xz plane, while the cross section of the tree is shown in the

xy plane. This cross section of the tree is taken at the line ab. The diameter of the

knot denoted by cd is measured in the xy plane and we assume that it is equal to the

diameter in the xz plane. The tangent (line ef) of the knot trajectory is located at the

intersection of the knot midpoint (the centre of cd) with the cross section (line ab),

this intersection of cd and ab on the xz plane is the vertical position.

Fig. 2. Left: knot display by means of Bil3D software, with whorls, inter-

whorls and epicormics (the latter are the two thin flat lines at about one third of the

image; eventually removed in the analysis); Right: X-ray image analysis through

Gourmand software.

Fig. 3. Predictions of knot relative vertical position over stem radius and 95%

percentile rank confidence intervals for a 150-mm long whorl knot.

Fig. 4. Predictions of knot diameter over stem radius and 95% percentile rank

confidence intervals for a 150-mm long whorl knot.

Fig. 5. Predictions of knot tangent over stem radius and 95% percentile rank

confidence intervals for a 150-mm long whorl knot.

Fig. 6. Nominal coverage probability over empirical coverage probability. The

straight line represents a perfect agreement between observed and expected volume

distribution.

Fig. 7. Observed vs predicted mean knot volume. The solid line represents

a perfect agreement between observations and predictions. The dashed one is the

regression line between them.

Fig. 8. Observed vs predicted individual knot volume. The solid line represents

a perfect agreement between observations and predictions. The dashed one is the

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regression line between them.

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FIGURES

Figure 1

Figure 2

Figure 3

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Figure 4

Figure 5

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Figure 6

Figure 7

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Figure 8

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