Introduction

\cite{Davies_2017}
The main use of plantation-grown Eucalyptus species is the production of biomass for the pulp and paper, and bioenergy industries. These trees are fast-growing and can potentially produce high-quality timber for appearance, structural and engineered wood products. Unfortunately, this potential is hindered by the frequent presence of large growth-strains, which are associated with log splitting, warp, collapse and brittleheart, imposing substantial costs on processing.
A few technological mitigation strategies have been developed to reduce the incidence of wood defects caused by growth-strain, but they are costly and only partially effective. An alternative approach to the problem is to rely on the genetic control of growth-strain to select and grow individuals with low growth-strain. However, measuring growth strains in large numbers of trees (as needed for a successful breeding programme) has been difficult, time consuming and expensive until now. As an example, the largest reported studies to date assessed only 164 \cite{Murphy2005} and 216 \cite{Naranjo2012} trees.
The University of Canterbury has developed and implemented a rapid growth-strain testing procedure, based on the work by \citet{Chauhan2010} and \citet{Entwistle2014}. In order to minimise the time taken to measure growth-strain on each tree, the rapid testing procedure does not account for negative values, where the wood in the centre of the stem is under tension rather than compression, assigning instead a zero that results in left-censored datasets.
Left-censored data are common in research areas where detection limits are high compared to the measured values, such as testing for the presents of drugs in an animal. There are many approaches to deal with censoring \cite{Senn_2012} and in this article we use a Bayesian framework to impute the missing data from known data, reducing the error induced by zero inflation. A Bayesian approach makes it easier to include model complexity (e.g. censoring) while accounting for the hierarchical nature of the data. In addition, one can easily obtain complex distributions of functions of covariance components, like heritabilities, as a byproduct of the estimation process \cite{Cappa_2006}. There are several examples of Bayesian applications in forest genetics; for example: \cite{Soria_1998} (univariate analysis of growth traits), \cite{Cappa2006} (multivariate analysis of growth traits) and \cite{Apiolaza2011} (multivariate analysis of early wood properties).
We ran a pilot study consisting of two Eucalyptus bosistoana F.Muell. progeny tests from both seed and coppice grown stems, which included 623 individual stems from 40 half sibling families. Our estimates of narrow-sense heritability were obtained from left-censored growth-strain data and other wood properties, utilising a Bayesian approach. These results were used to design a much larger evaluation of the E. bosistoana breeding population currently underway.

Materials and method

An E. bosistoana open-pollinated progeny trial was established at an irrigated nursery site in Harewood, Christchurch, New Zealand. The trial represented 40 families from two provenances, for a total of 423 seedlings planted into 100 L bags, which were coppiced after the first harvest, giving a total of 623 tested samples. Two separate plantings (or trial sections) occurred in 2010 and 2012. The 2010 families originated from South East Australia, were harvested and coppiced in 2012, and harvested again in December 2014. The 2012 families originated from higher elevation in New South Wales and were harvested in 2013 (these data were not included in the analysis, due to the magnitude of errors induced by small, malformed stems) and again in October 2015 (at age two). All seedlings were established following a completely randomised design.
When harvested, each sample was processed for growth strain, volumetric shrinkage (displacement method, before and after drying), stem diameter (measured under bark using digital calipers), basic density (mass and displacement method) and dry acoustic velocity (resonance). Growth-strain was measured using a modified version of the \citet{Chauhan2010} and \citet{Entwistle2014} log-splitting methods. The newly developed “rapid splitting test” substantially reduces measurement time, enabling larger numbers of samples to be processed. The modified method involves stripping the bark and measuring the under-bark large-end diameter of a clear section of the stem, giving an over estimate of the average diameter used by \citet{Chauhan2010}. The sample is then cut lengthwise, with the slit length determined by the length of clear wood and diameter of the sample. Diameter and slit length are measured, recorded and the stem cut. The small-end of the sample is left intact with the large-end free to distort, which removes the need to clamp the two halves together. Finally, the opening is measured and recorded. The calculation of strain is unchanged (with the exception that average radius is now large-end radius) from \citet{Chauhan2010} and calculated using Equation \ref{eq:strain_LC}. It is important to note that the over estimate of radius slightly reduces the strain value, but does so linearly over all samples.