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\begin{document}
\title{Analysis of the precision of the Splitting and Rapid Splitting test for
growth strain in trees}
\author[1]{Nicholas Davies}%
\author[1]{Luis A. Apiolaza}%
\author[1]{Awaiting Activation}%
\affil[1]{University of Canterbury}%
\vspace{-1em}
\date{\today}
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\section*{Abstract}
{\label{632569}}
Growth strain is a recent addition to wood properties of interest to
breeders. The splitting tests are the fastest methods for testing for
growth strain, which is commonly believed to be the driver for
deformation during green sawing. Unfortunately they are destructive
tests, making directly testing their precision impossible . Here a
method using multiple testing scenarios and a minimization scheme was
used to estimate the precision of a single splitting test measurement to
be~\(\pm 1006\) micro-strain. It is suggested that the resulting
interval makes the splitting tests only suitable for removing very
poorly performing individuals to make future breeding trials more
efficient.~~
\par\null
\section*{Introduction}
\par\null
In recent years wood quality parameters have started to be included in
tree breeding programs~\cite{Davies2017,Chauhan_2013,Apiolaza_2011}. Particular wood properties
provide advantages for different applications of timber. For example,
high stiffness timber is sought after for structural uses, and a premium
is paid. Growth-strain has previously been identified as a wood quality
parameter which reduces the value of some Eucalypt species
\cite{Yamamoto2007,Chauhan2010}. Some timber species, particularly hardwood such as
Eucalypts have a tendency to produce high internal strains during
growth, when these strains are released during processing, the cut
boards deform resulting in an excessive value recovery loss as they must
be re-sawn to gain straight boards. For more details on growth-strain
refer to~\citet{Alm_ras_2016} and for an older but comprehensive review
see~\citet{Kubler1987}.
\par\null
Until recently the quickest growth strain test required approximately
half an hour and substantially sized trees rendering growth-strain too
time consuming and expensive to incorporate as part of a breeding trial
particularly within an early selection trial with small
trees.~\citet{jacobs1945} developed the paring test (splitting stems
down the pith and measuring the movement of each side)~ and reported
results from \emph{Eucalyptus gigantea} along with a number of other
species. However he did not take the final steps of calculus required to
convert his deformation measurements to strains.~~\citet{Chauhan2010}
developed the splitting test which (although the paper makes no mention
of Jacobs) uses the same method with a number of assumptions to complete
the calculus required to estimate surface strain from the paring test.
The updated pairing test (now refereed to as the splitting test)
substantially reduced the time and cost involved in measuring
growth-strain, this method was further refined for production breeding
trial assessment and a pilot study conducted by~\citet{Davies2017}, who
showed it had potential for suitably large scale trials on Eucalypts.
The splitting test essentially involves cutting a stem longitudinally
along the pith and measuring the opening, along with diameter and cut
length a numerical value related to the strain in the sample can be
obtained.
\par\null
\citet{Entwistle_2014} calculated the theoretical values of strain lost due
to kerf, these become negligible with stems of significant
size.~\cite{Chauhan_2010} took strain gauge measurements at the surface
of the stem and used them to predict the test opening, which showed a
reasonable correlation on large stems. The results of this analysis were
misleading for two reasons; one, the strain gauges were placed on on the
surface perpendicular to the cut, and therefore only indicate the
reliability of the splitting test to produce results when measured in
such a way as to maximize the correlation and two, were conducted on
large stems which was not the intended use for the test (very early
selection). Larger diameter stems (anecdotally) exhibit similar growth
strain variation by angle as smaller stems, as a result higher
correlations can likely be created when tested in this particular way on
larger stems, as there is a higher margin for error with gauge
placement.~\citet{Cramer2018} analysed~\emph{Eucalyptus nitens} stems
with a diameter ranging from 58 to 120mm under the same method
as~\citet{Chauhan_2010} and found correlation between strain gauge
surface strain and splitting test predicted surface strain of
approximately 0.27~compared to~\citet{Chauhan_2010} reporting of 0.92
on~\emph{Eucalyptus nitens} stems with diamters between 111 and 192mm.
It is suspected that one of the reasons for this marked difference in
correlation between~\citet{Cramer2018} and~\citet{Chauhan2010} is due
to the substantially smaller openings observed by~\citet{Cramer2018}
due to the smaller tree size, which are similar to the tree size used in
this study. It has been well reported that surface growth strain can
vary markedly over small areas of stem~\cite{Okuyama_1994,Saurat_1976}. The de-facto
standard procedure for measuring surface strain on logs with CIRAD or
strain gauge methods is to take 8 measurements around the
perimeter~\cite{cirad} this is to account for the radial
variation which can occur over small distances on the log
surface~\cite{Fournier_1994}. The major trade-off with the speed of the
splitting test is the resolution which can be achieved, surface strains
near the cut are suppressed due to the geometry of the test, while the
strains perpendicular to the cut are primarily responsible for the
opening. Hence placing one strain gauge on each side, perpendicular to
the plane of cut of the splitting test produces a misleading assessment
of how accurately the splitting test predicts the average growth strain
of the sample.~
\par\null
It should be noted that the opening in the test is not solely caused by
the longitudinal strain, tangential and radial strains also likely play
a role. The way longitudinal, tangential and radial strains develop and
interact is still unknown, with no work (to our knowledge) having been
conducted to understand the underlying mechanical relationship, past
work such as \cite{Chauhan2010} have reported correlations obtained
from strain gauges.~ Grain angle with respect to the longitudinal cut
direction likely influences the results of both the splitting and strain
gauge tests, however no known research has investigated this.~
\par\null
The splitting tests are used within breeding
programs~\cite{Davies2017} in order to gain a quantitative measure of
how much the stem would deform during green sawing. The primary
objective of this paper is to present a method (and results from) for
estimating how reliable splitting test measurements are when predicting
this trait. A cut directly through the pith is a procedure which while
not a typically used cutting strategy for eucalypt timber production,
allows for (in normal circumstances) the largest manifestation of
movement during cutting. Having knowledge of how much trust to place in
splitting test measurements of individuals is important for tree
breeders when they are deciding on selection weights for various traits.
174 samples were measured using the original splitting
test~\cite{Chauhan2010}, the rapid splitting test~\cite{Davies2017}~
and the newly proposed ``quartering test''. The results from each were
compared, and used to estimate the reliability of the testing procedures
when considered at an arbitrary angle.~
\par\null\par\null\par\null
\section*{Method}
The trees used in this study were thinnings of five year
old~\emph{Eucalytus argophloia} grown on a moderately steep east facing
slope with a rainfall of approximately 700 mm per year in Marlborough
New Zealand. The 176 samples from 115 individual trees were labeled and
cut when they provided at least 400 mm of suitably straight stem with an
estimated under bark diameter of greater than 20 mm. Under bark
diameters ranged from 21 mm to 71 mm with a mean of 40 mm. The samples
were cut from the stems in autumn using a chainsaw and were packed into
air tight containers with excess water and transported to a cool store
where they were stored at 5C until processing, which took place over
four weeks, there where no signs (visual or statistical) of sample
quality degradation.
\par\null
Testing procedures: Samples were removed individually from their
containers. Bark was removed using knives to hand peel the samples, with
care taken not to damage the underlying wood. The longest sufficiently
straight section of sample which could be obtained from the big end was
marked (and the length from the big end recorded), with a maximum length
being 50mm short of the small end. The diameter of the big end was
recorded as was the small end diameter at the marked point.
Rapid splitting test: The Rapid Splitting test outlined
in~\citet{Davies_2017} involves splitting with a band saw through the
pith from the big end to the marked point. The resulting opening is
measured and recorded; From the diameter~\emph{d}, slit length~\emph{L}
and opening~\emph{o}, the strain within the sample can be estimated
using Equation {\ref{eq:split_eq}}.
Splitting Test: The original splitting test, presented
by~\citet{Chauhan_2010} involves splitting the sample the whole way down
the pith (in this case it was achieved by docking the remaining intact
end from the above procedure to gain two half rounds), clamping the two
halves in the centre and measuring the opening at each end. Here the
method is slightly modified due to the (compared to the original paper)
low openings to reduce measurement error. In this case, the small end
was clamped and the opening was measured at the big end. Note that when
the curvature is sufficiently low, these two methods are approximately
equivalent. The opening, big end diameter, small end diameter and split
length are measured and processed through Equation --- to obtain strain.
--- inc Figure~{\ref{171010}}
\par\null\par\null
\par\null
\begin{equation} \label{eq:split_eq}
strain\ =\ \frac{od}{1.74L^2}
\end{equation}
\par\null
Quartering Test: Due to the destructive nature of both splitting test
procedures, and inhomogeneity both within and between individuals,
repeatability is not directly testable, in the sense; if the cut plane
exists though the (longitudinal) pith at an arbitrary radial alignment,
how close is the result to the (hypothetical) result of a cut through
the pith at some other arbitrary alignment. In order to test
repeatability, after the splitting test had been conducted, each of the
two halves were halved again (into quarter rounds), and reassembled with
the small end in a self aligning jig set in a vice. Four measurements
were taken, providing the spatial relationship between every quarter
round. Openings between adjacent quarters were taken to be the average
distance between faces, as there were two measurements for each half
round (Figure~{\ref{751922}}).~ In the splitting tests
using half rounds the constant, 1.74, is derived from the distance to
the neutral plane of bending from the outer surface, following the same
logic as was presented in~\citet{Chauhan2010} to calculate 1.74, a new
constant (2.079) can be calculated for the quarter round tests. Assuming
the sample possesses a round cross section, the cord of the circle
perpendicular to the plane of the cut through the centre of mass for
each quarter can be calculated using
Equation~{\ref{eq:qt_cont_1}} . Half the cord can
then be calculated as a function of the radius
(Equation~{\ref{eq:qt_cont_2}})~ and the distance
from the outer edge of the circle to the centre of mass calculated as a
function of radius (Equation~{\ref{eq:qt_cont_3}}).~
Rearranging this constant (as per~\citet{Chauhan2010} Equations 12 - 14
and updating to use diameter rather than radius) gives a constant for
quarter round testing of 2.079 (Equation
{\ref{eq:qt_cont_4}}).
\par\null\par\null
\par\null
\begin{equation}\label{eq:qt_cont_1}
cord\ =\ 2\sqrt{r^2-\left(\frac{4r}{3\pi}\right)^2}
\end{equation}
\begin{equation} \label{eq:qt_cont_2}
\frac{cord}{2}=0.9055r
\end{equation}
\begin{equation} \label{eq:qt_cont_3}
\left(0.9055\ -\ \frac{4}{3\pi}\right)r\ =\ 0.4811r
\end{equation}
\begin{equation} \label{eq:qt_cont_4}
\frac{1}{0.4811}=2.079
\end{equation}
\textbf{\(\)}
When the rotated test is included the differences between the four tests
on each sample can be described by six equations, with four error terms
as in Equations {\ref{eq:prec_min_1}}
to~{\ref{eq:prec_min_6}}. Being an over determined
ill-conditioned system of linear equations, they can be solved
simultaneously as a minimisation problem using
python~\cite{Jones2001--}. Equations
{\ref{eq:prec_min_1}}
to~{\ref{eq:prec_min_6}} were solved for means giving
estimates on the existing bias of each test (relative to the others).
The system was also solved for variances, the resulting calculated
errors of the variances provide estimates (after conversion to standard
deviations) of the 95\% confidence intervals on an arbitrary arbitrary
arbitrary measurement.
\par\null\par\null
\par\null
\begin{equation} \label{eq:prec_min_1}
RS\ -\ OS\ =\ e_{rs\ }+\ e_{os}
\end{equation}
\begin{equation}\label{eq:prec_min_2}
RS\ -\ QS1\ =\ e_{rs}\ +\ e_{qs}
\end{equation}
\begin{equation} \label{eq:prec_min_3}
RS\ -\ QS2\ =\ e_{rs\ }+e_{qs}\ +\ e_{rot}
\end{equation}
\begin{equation} \label{eq:prec_min_4}
OS\ -\ QS1\ =\ e_{os}\ +\ e_{qs}
\end{equation}
\begin{equation} \label{eq:prec_min_5}
OS\ -\ QS2\ =\ e_{os}\ +\ e_{qs}\ +\ e_{rot}
\end{equation}
\begin{equation} \label{eq:prec_min_6}
QS1\ -\ QS2\ =\ 2e_{qs}\ +\ e_{rot}
\end{equation}
\par\null
\par\null
Where RS is the result from the rapid splitting test, OS is the result
of the original splitting test,~ QS1 is the result from the quartering
test along the same plane as the rapid splitting test and QS2 is the
quartering test along the plane perpendicular to QS1.~\(e_{rs}\)
and~\(e_{os}\) are the measurement error associated with the
rapid and original splitting tests,~\(e_{qs}\) is the
measurement error associated with the quartering test (note the
measurement error is the same regardless of splitting plane)
and~and~and~\(e_{rot}\) is the difference resulting from from
from which plane is cut.
\par\null
\par\null
Again if no measurement error exists in any of the three testing
procedures,
Equations~{\ref{eq:prec_min_1}},~{\ref{eq:prec_min_2}}
and {\ref{eq:prec_min_4}} and
hence~\(e_{rs}\),~\(e_{os}\)and~\(e_{qs}\) would
be 0. Equations 3, 5 and 6 (the equations containing~\(e_{rot}\))
would only also be zero if all of the stems contain a completely
homogeneous strain field - in reality this is not the case, and
therefore, if~\(e_{rs}\),~\(e_{os}\)
and~and~and~\(e_{qs}\) are zero, any differences between QS2
and other tests would be the result of the change in strain by cutting
the sample at a different angle. If this assumption were to hold, the
difference between the results from the first cut and the results from
the second would provide a distribution with a mean of zero and a
non-zero variance (assuming the samples are arbitrarily aligned). The
distribution would be the expected difference between the cuts and it
would become possible to estimate 95\% confidence bounds on the value of
another hypothetical cut on the same sample.~
\par\null
Unfortunately none of these tests have zero measurement error, however,
because there are three tests which~\emph{should} all produce the same
result (as they are measuring the same thing) we can estimate the error
associated with each test.~test.~test.~Note that this is not necessarily
the error from the ``true'' value, but error in the sense of what
confidence we can hold that the result of one sample from one test can
estimate the hypothetical mean of the distribution formed by the same
test run on the same sample in the same orientation an infinite number
of times, i.e. the precision of the testing procedure. times, i.e. the
precision of the testing procedure. times, i.e. the precision of the
testing procedure, the true value. The QS2 test encompasses two types of
error,~\(e_{rot}\), discussed above, and ~\(e_{qs}\),
the error associated with the quartering test with respect to the RS and
OS tests. Fortunately because this test is used twice on each sample,
the two errors are separable.
\section*{Results and Discussion}
{\label{638343}}
Population wide means and standard deviations for growth strain measured
by the three different tests are shown in Table
{\ref{929591}}.~
\par\null\selectlanguage{english}
\begin{table}[h!]
\centering
\normalsize\begin{tabulary}{1.0\textwidth}{CCC}
Test & mean & standard deviation \\
Original splitting test & 1556 & 610 \\
Rapid splitting test & 1807 & 662 \\
Quartering test (same plane) & 1369 & 534 \\
Quartering test (Perpendicular plane) & 1341 & 520 \\
\end{tabulary}
\caption{{Mean and standard devation for the testing population for each test.
{\label{929591}}%
}}
\end{table}Person correlations between each measurement were calculated and are
presented in Table {\ref{884031}}.
\par\null\selectlanguage{english}
\begin{table}[h!]
\centering
\normalsize\begin{tabulary}{1.0\textwidth}{CCCCC}
& Origonal splitting test & Rapid splitting test & Quartering test (same plane) & Quartering test (Perpondicular plane) \\
Origonal splitting test & 1 & 0.89 & 0.88 & 0.73 \\
Rapid splitting test & & 1 & 0.9 & 0.78 \\
Quartering test (same plane) & & & 1 & 0.89 \\
Quartering test (Perpondicular plane) & & & & 1 \\
\end{tabulary}
\caption{{Pearson correlations between test types on the same individuals
{\label{884031}}%
}}
\end{table}\par\null
The splitting tests are used for estimating a trees stability when sawn
(wet), using an estimated strain as a numerical indicator. Due to the
inhomogeneity of material properties properties around the stem, the
orientation of the cut measures a result which will differ from a
measurement taken from a different orientation.~ However the value of
the splitting tests is the speed at which they can be conducted, so the
test can only be conducted once per sample (the destructive nature of
the testing procedures also precludes repeated measurements), and this
number is used as a proxy for the value within the whole stem, and
consequently the trait value of the individuals genetics. It is
important to know how well this estimate performs, and the proportion of
error associated with the orientation as oppose to random measurement
error. error. The four tests used here all measure the `same' property,
in that they are all numerical proxies of how much a sample is likely to
deform when it is green sawn and consequently should all give the same
result. The first three tests (original, rapid, quartering (same plane))
should provide the same numerical result for each sample. Differences
between them can only be due to the testing procedure as they measure
the same plane. The final test (quartering (perpendicular plane))
compounds measurement error (which will be the same as the quartering
(same plane) test) with error resulting from measuring a different
orientation of the sample. By comparing these test results over multiple
samples an estimate of how much error is associated with each
measurement was produced. Note that although we call the numerical
result `strain', it is not a measure of the same sample property as when
a strain gauge procedure is used. The `strain' measurements from
splitting tests and strain gauges or the CIRAD method do not measure the
exact same thing, and are not directly comparable. Strain guages measure
the surface strain over over a given small area, while the splitting
test measures a non-uniform consolidation of the three dimensional
strain field.~Strain, in the sense it is used here is the name given to
the numerical proxy for how much deformation can be expected during
green sawing of a sample obtained from the splitting tests.
\par\null
Equations~{\ref{eq:prec_min_1}} to
{\ref{eq:prec_min_6}} were solved simultaneously to
estimate approximate measurement error on each test, and the error
resulting from the rotation of the sample.
Table~{\ref{618809}} shows the mean and 2nd standard
deviation (95\% confidence interval) of the theoretical error
distributions. In these cases, the mean indicates a systematic error,
while the 95\% confidence intervals are random error associated with
that measurement type (i.e. measurement error).~
\par\null\selectlanguage{english}
\begin{table}[h!]
\centering
\normalsize\begin{tabulary}{1.0\textwidth}{CCC}
& mean & 95\% confidence interval \\
eRS & 125.7 & 417 \\
eOS & 125.7 & 366.3 \\
eQS & 11.2 & 401.6 \\
eROT & 5.6 & 588.6 \\
\end{tabulary}
\caption{{Precision of each test. The mean column represents the systematic error
each test contributes to the difference between testing results (across
different testing types), when considered with the ordering of Equations
{\ref{eq:prec_min_1}} to
{\ref{eq:prec_min_6}}, the order is important as it
distinguishes the sign. The 95\% confidence intervals are calculated
from the varance of the difference distributions giving the bounds on a
given measurment, i.e. how repetable the measurement is.~
{\label{618809}}%
}}
\end{table}\par\null
From Table {\ref{618809}} it can be seen that if a
measurement is taken using the rapid splitting test (which is currently
being used in early selection breeding programs) a hypothetical repeated
measurement on the same sample with the same cut orientation would
provide a result within~\(\pm417u\epsilon\) 95\% of the time. Further, if
the test was conducted at a random cut orientation, any given result
would (95\% of the time) be within~\(\pm1006u\epsilon\)~\(\left(417+589\right)\)
of mean of the hypothetical distribution created by testing the same
sample at random orientations an infinite number of times.~
\par\null
These results suggest that using splitting tests on a breeding
population such as the one presented in
Table~{\ref{929591}} have the potential to remove poor
performing individuals as part of a non-intensive breeding selection,
for example taking the top 25\% of the population would likely remove
the bottom 20\% going forward. However the limited resolving power of
the splitting tests precludes the ability to select the ``best''
individual or even the top few percent individuals. Current early
selection programs utilizing these tests aim to remove the poorest
performing individuals in order to reduce the expense of further more
extensive breeding programs. These tests are suitable for this purpose,
however without further development accurately ranking individuals for
selection is problematic. Note that these estimates are based on the
population presented in Table {\ref{929591}}.
Populations with significantly different means or variances would yield
the testing more or less worth while.~
\par\null
Figure~{\ref{273819}} shows a simplistic visual
representation of selecting the top 25\% of individuals (in red),
generated by ordering the rapid splitting test data from lowest to
highest. However using the results in Table
{\ref{618809}} a representation of the `true' values
can be seen in Figure~{\ref{920901}}, (note these are
simply created by adding a value randomly sampled from a normal
distribution characterised by~ a mean of 0 and a standard deviation of
513 from the 95\% confidence interval~\(\pm1006u\epsilon\), the real value
is unknown). When the same `true' values are reordered from lowest to
highest, with the same point color as in
Figure~{\ref{109992}} it can be seen some poorer
individuals are included and some better individuals are missed because
the inaccuracies within the test.~
\par\null\par\null
\section*{Conclusion}
{\label{147437}}
A method was presented which enabled the estimation of the precision of
the destructive growth strain measurements obtained by the unrepeatable
splitting tests. The results showed that within a population with a mean
of~\(1807u\epsilon\) and a standard deviation of~\(662u\epsilon\) a
rapid splitting test result has a 95\% confidence interval
of~\(\pm1006u\epsilon\). Splitting tests show marginal suitabllity for use
in early selection breeding programs where identifying the best
individual is not of high concern, rather removing the worst individuals
to make future programs more cost effective is. However the splitting
tests were shown not to provide high enough precision to be used in
intensive selections due to their measurement errors.~
\par\null\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/split-test/split-test}
\caption{{Splitting test
{\label{171010}}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/IMG-20160602-135636905-01/IMG-20160602-135636905-01}
\caption{{Quartering test
{\label{751922}}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/plot1/plot1}
\caption{{Ordering of the the rapid splitting test results from lowest to highest,
with the lowest 25\% in red.
{\label{273819}}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/plot2/plot2}
\caption{{The same top individuals in red as in
Figure~{\ref{273819}} but reordered using the `true'
values seen in Figure~{\ref{109992}}
{\label{920901}}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/plot3/plot3}
\caption{{Same order and colouring as in Figure {\ref{273819}}
but after the testing error is added to the result
{\label{109992}}%
}}
\end{center}
\end{figure}
\selectlanguage{english}
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