# Optimizing Queensland’s Timber Industry with Mathematics

The southern pine is a commercially-important species of tree that grows along Australia’s Fraser coast in southeast Queensland and parts of northern New South Wales. The trees are the chief source of timber in these regions and are harvested primarily for structural purposes, such as the creation of trusses and housing frames. Yet timber properties are highly variable, which can impact the lumber’s processing and profitability from the industrial perspective. For example, structural timber from the southern pine is worth $350 per cubic meter, while non-structural timber is valued at$80 per cubic meter. As the forest sector becomes increasingly dependent on end-product value, the associated industry is attempting to discern trees’ projected uses earlier in the harvest process.

During a scientific session at the 9th International Congress on Industrial and Applied Mathematics, currently taking place in Valencia, Spain, Steven Psaltis of the (QUT) discussed his involvement with a collaborative project between QUT and the forestry industry to understand the variability of southern pine properties. The large, \$2.6 million project is industry-driven, though Psaltis’s mathematical analysis—focused on a single board—comprises only one small part. “Our overall goal is to improve profitability across the value chain through precision forestry,” he said.

Much of the project involves data collection to understand the current state of pine trees. The team collected data from 68 sacrificial southern pines. They sawed the trees into four discs and took diametrical cores—in 20-millimeter segments—from each disc. Of these discs, the middle part of the log— called the sawlog—is most important for Psaltis’s purposes because boards are sawed and measured from the sawlog. Two peeler billets peel the log into veneers that are valuable for three-dimensional reconstruction.

Psaltis's overall goal is to improve profitability across the value chain through precision forestry in Australia.
To address the mathematics of wood variability, Psaltis considered two related approaches: directly using measured data, and modeling data using a logistic function. The southern pine is a stiffness-limited resource characterized by the modulus of elasticity (MOE). “We want to try to get an estimate of the MOE throughout the sawlog, and be able to take a board from that sawlog and estimate the properties of that board,” he said. High-quality timber is associated with a higher MOE and usually found towards the outside of a log.

Data is available at four distinct heights within a tree’s structure, and Psaltis used data from these four heights to complete the rest of the log. To fill in the unknown regions, he utilized penalized radial basis functions with thin-plate splines and assumed that data for 20-millimeter segments is given by the surface at a point (plus some error). Psaltis then solved a minimization problem to obtain the smoothing parameter. This yielded a nice representation of the MOE in terms of a tree segment’s radius and height.

The southern pine’s value is linked to the value of its products, and the value of the products is linked to its properties (in this case, the MOE). Thus, Psaltis aimed to estimate the MOE of the board. However, doing so requires knowledge of where those boards sit within the log (based on the location of the studs). So he devised a method to verify stud orientation and determine which ones are aligned. This yielded the coordinates of each individual board’s location. Overlaying this information with the surface describing the MOE produced the value of the MOE at each point within the boards. A quick comparison between the measured and predicted MOE for individual timber boards indicated reasonable agreement.

Next, Psaltis wished to examine groups of pines comprised of trees with varying diameters and heights. To do so, he reevaluated his approach based on cambial and apical tree ages. He then assumed that one can model the MOE via a five-parameter logistic function. “That was something that the industry was comfortable with as well,” Psaltis said. To obtain parameters for the logistic function, he employed orthogonal distance regression, which allowed for error in both the dependent and independent variables. This approach also works for individual trees, where researchers use fitted curves (rather than data) to fit the radial basis function. These curves act as “fingerprints” on the resulting plots, and plot-level growth generates an estimate of the quality of “good” timber. Although the predicted MOE is not quite as good as when based directly on the data, it is still fairly reasonable. The advantage of this approach is that one can effectively examine many trees at once. “For the whole plot of trees, we can compare the individual segment data,” Psaltis said.

To summarize, Psaltis developed a mathematical method to reconstruct timber properties based on the diametrical cores of sawlogs. This ultimately allowed him to estimate the individual board MOEs. “Industry can use this data to guide their rotations,” he said, including the planting and harvest of future southern pines.

 Lina Sorg is the associate editor of SIAM News.