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A new approach for predicting board MOE from increment cores


Key message

Increment cores can provide improved predictive capabilities of the modulus of elasticity (MOE) of sawn boards. Multiple increment cores collected at different heights in a tree provide marginally increased accuracy over a single breast-height core, with higher labour costs. Approximately 50% of the variability of the static bending MOE of individual boards is explained by the predicted MOE obtained from a single increment core taken at breast height.


Prediction of individual board MOE can lead to accurate optimisation of the value extracted from forest resources, and enhanced decision-making on the management and allocation of the resource to different processors, and improve the processors ability to optimise grade allocation.


The objective of this study is to predict the MOE of individual sawn boards from the MOE measured from cores collected from standing trees.


A five-parameter logistic (5PL) function and radial basis function interpolants are used to obtain a continuous distribution of MOE throughout a log. By developing a “virtual sawing” methodology, we predict the individual board MOE for sixty-eight trees consisting of locally developed F1 and F2 hybrid pines (Pinus caribaea var. hondurensis × Pinuselliottii var. elliottii).


Moderate correlations for individual board predictions are observed, with R2 values ranging from 0.47 to 0.53. Good correlations between average predicted board MOE and average measured MOE are also observed, with R2 ≈ 0.83. A pseudo-three-dimensional approach, accounting for variation in height in the tree, affords marginally greater accuracy and predictive capability at the cost of increased data collection and processing. By using a single breast-height core, we can obtain a similar level of prediction of individual board MOE.


We have presented a novel non-destructive evaluation approach to predict the MOE of individual boards sawn from trees. This approach can be adapted to other wood properties, and other wood products obtained from trees.

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The authors also wish to thank the following for their valuable contributions to this work: Bruce Hogg, Tony Burridge and John Oostenbrink (Department of Agriculture and Fisheries, Gympie) for plantation sampling, log selection and harvesting, and Dan Field, Eric Littee, Rhianna Robinson, and Rica Minett (Department of Agriculture and Fisheries, Salisbury Research Facility) for sawing and testing.


This work was supported by Forest and Wood Products Australia (FWPA) through project PNC361-1415, and partners University of the Sunshine Coast, Queensland Department of Agriculture and Fisheries, HQPlantations, Forestry Corporation of NSW, Hyne and Son Pty Ltd, Queensland University of Technology and Hancock Victoria Plantations. SP wishes to thank the Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers (project number CE140100049) for postdoctoral research funding.

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Contribution of the co-authors Conceptualisation: Henri Bailleres, David J. Lee. Methodology: Steven Psaltis, Chandan Kumar, Ian Turner, Elliot J. Carr, Troy Farrell, Loic Brancheriau, Henri Bailleres, David J. Lee. Formal analysis and investigation: Steven Psaltis, Chandan Kumar, Ian Turner, Elliot J. Carr, Troy Farrell, Loic Brancheriau, Henri Bailleres, David J. Lee. Writing— original draft preparation: Steven Psaltis, Chandan Kumar. Writing — review and editing: Ian Turner, Elliot J. Carr, Troy Farrell, Loic Brancheriau, Henri Bailleres, David J. Lee; Funding acquisition: David J. Lee. Supervision: Ian Turner, Troy Farrell, Henri Bailleres, David J. Lee.



Our aim is to build a function that describes the property variation, given by

$$ s(\textbf{x}) = \sum\limits_{j = 1}^{3} d_{j} p_{j}(\textbf{x}) + \sum\limits_{i = 1}^{N} c_{i} \phi(\|\textbf{x} - \textbf{x}_{i}\|), $$

where p1(x) = 1, p2(x) = r, p3(x) = z, ϕ is the radial basis function, and c = (c1,c2,…,cN) and d = (d1,d2,d3) are the unknown coefficients (to be determined). There are numerous forms of radial basis functions available, such as linear, Gaussian, and multiquadric (Buhmann 2004). Here, we use a thin-plate spline radial basis function, given by Wahba (1990)

$$ \phi(\rho) = \frac{1}{8 \pi} \rho^{2} \log(\rho), $$

where ρ = ∥xxi∥ is the distance between a data point, xi, (a centre of the RBF) and a point on the surface. Nonlocal bases, i.e. those where \(\phi (\rho ) \rightarrow \infty \) as \(\rho \rightarrow \infty \), may perform better than local bases. Furthermore, the thin-plate spline is not dependent on a user-set shape parameter (Holmes and Mallick 1998), and is invariant under translation and rotation transformations (Franke 1982).

To obtain the unknown coefficients, c and d, we begin by assuming the data, yi, can be modelled as

$$ y_{i} = s(\textbf{x}_{i}) + {\epsilon}_{i}, \quad i = 1, \ldots, N, $$

where \({\epsilon }_{i} \sim {} N(0, \sigma ^{2})\) is the error. We must solve the minimisation problem,

$$ \min_{\textbf{c}, \textbf{d}}\frac{1}{N} {\sum}_{i = 1}^{N} \left( y_{i} - s(\textbf{x}_{i})\right)^{2} + \lambda J(s), $$

where λ is the smoothing parameter and J(s) is the penalty functional, given by

$$ J(s) = {\int}_{-\infty}^{\infty} {\int}_{-\infty}^{\infty} \left( s_{x_{1} x_{1}}^{2} + 2 s_{x_{1} x_{2}}^{2} + s_{x_{2} x_{2}}^{2} \right) dA. $$

For λ = 0, s(x) becomes a surface that interpolates the data, and as \(\lambda \rightarrow \infty \) we obtain the linear least squares solution (Wahba 1990).

Wahba (1990) shows that the solution to the minimisation problem (Eq. (A.4)) can be found by solving the linear system,

$$ \begin{array}{@{}rcl@{}} (K + N\lambda I)\textbf{c} + P\textbf{d} &= \textbf{y}, \end{array} $$
$$ \begin{array}{@{}rcl@{}} P^{T} \textbf{c} &= \textbf{0}, \end{array} $$

where K is the N × N matrix with ij th entry ϕ(∥xixj∥), P is the N × 3 matrix with (i,k) entry given by pk(xi), I is the N × N identity matrix, \({\square }^{T}\) is the transpose operator, c = (c1,…,cN)T, d = (d1,d2,d3)T, y = (y1,…,yN)T, and 0 is the 3 × 1 zero vector.

We see from Eq. (A.6) that to account for the penalty term we simply adjust the diagonal elements of K. To compute the value of λ, we utilise generalised cross validation (GCV) (Wahba 1990). This involves minimising the GCV function, V (λ), where

$$ V(\lambda) = N \|(I - A(\lambda))\textbf{y}\|^{2} / \left[ Tr(I - A(\lambda))\right]^{2}. $$

Tr() is the trace operator, and A(λ) is known as the influence matrix which can be calculated from (Wahba 1990)

$$ I - A(\lambda) = N \lambda Q_{2} ({Q_{2}^{T}} (K + N \lambda I) Q_{2})^{-1} {Q_{2}^{T}}. $$

Here, Q2 is computed from the QR decomposition of P, namely

$$ P = \left[\begin{array}{c|c} Q_{1} & Q_{2} \end{array}\right] \left[\begin{array}{cc} R\\ 0 \end{array}\right], $$

where \(Q_{1} \in \mathbb {R}^{N \times 3}\), \(Q_{2} \in \mathbb {R}^{N \times (N-3)}\) and R is an upper triangular matrix.

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Psaltis, S., Kumar, C., Turner, I. et al. A new approach for predicting board MOE from increment cores. Annals of Forest Science 78, 78 (2021).

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  • Non-destructive evaluation (NDE)
  • Wood quality
  • Modulus of elasticity
  • Queensland southern pines
  • Sawn boards