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Experimental study on the
possibility of detecting internal
decay in standing Picea abies by
blind impact response analysis
JOAKIM AXMON*, MARIA HANSSON AND LEIF SÖRNMO
Signal Processing Group, Department of Electroscience, Lund University, PO Box 118, SE-221 00 Lund, Sweden
* Corresponding author. E-mail: [email protected]
Summary
This paper considers detection of internal decay in standing trees of species Picea abies (L.) Karst.
The novel approach is based on two-dimensional spatiotemporal modal analysis of a cross-section
which is excited by the hand-made impact of a hammer. An array of accelerometers is distributed
around the cross-section, and the resulting impact response is analysed. The temporal frequency for
a special spatial mode-shape is used for comparisons on a tree-to-tree basis. The mechanical
properties of wood are inherently variable as they are for most materials of biological origin. This
leads to a scatter of the analysed parameters that hinders detection of decay based on the temporal
frequencies alone. Using regression analysis, we show that by incorporating the additional
information on a surface wave propagation velocity, the scatter of sound trees is significantly
reduced. The performance of a detector rule which incorporates the frequency and the surface wave
propagation velocity is investigated and found to be better than performance reported for visual tree
examination. The analyses are based on the impact responses from 94 standing trees, with 66 sound
and 28 in various stages of decay. The proposed technique is yet to be considered an experimental
tool. Further research, e.g. on how the mechanical properties are influenced by various
environmental factors, is needed before the technique can be applied operationally.
Introduction
Picea abies (L.) Karst. (Norway spruce) is like
many other conifers susceptible to fungi, whose
colonization of the root system and trunk leads
to rot and decay of the wood. Generally, there are
no definite external signs of the decay and the
presence of rot is noticed first upon felling the
tree or, more dramatically, when it is wind© Institute of Chartered Foresters, 2004
thrown. Even trained and highly skilled foresters
fail to identify infected trees from visual examination (Vollbrecht and Agestam, 1995). The
difficulty lies in that the external indicators of rot
and decay may be due to other causes, and also
that trunks of decayed trees may appear to be
sound since the sapwood is more resistant to
decay than is the heartwood. For instance, resin
exudation is sometimes observed on severely
Forestry, Vol. 77, No. 3, 2004
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decayed trees (Vollbrecht and Agestam, 1995),
but can as well be a symptom of dead spots of
inner bark, and cracks, caused by frost late in the
growth season, or of drought (Cherubini et al.,
1997; Jönsson et al., 2001). On many occasions,
the fungus enters the host tree through the root
system and then progresses upward through the
trunk, leaving a column of rot covering a substantial part of the heartwood’s cross-section. In
the process, the fungus converts constituents of
the cell walls into nutritious carbohydrates and,
at the same time, alters the mechanical and aesthetic qualities of the wood (Wagener and
Davidson, 1954; Toole, 1971; Kirk and Highley,
1973; Wilcox, 1978; Pratt, 1979).
A number of methods have been proposed as
alternatives to visual tree assessment (visual tree
assessment here refers to its literal meaning and
should not be confused with the VTA method).
One of the most widely practised methods is to
visually study core samples taken from the trunk
using an increment borer (Stenlid and Wästerlund, 1986). The major drawbacks of this method
are unreliability due to the risk of missing possibly
decayed areas when taking the sample (Stenlid
and Wästerlund, 1986), and increased susceptibility of the tree to attacks by fungal spores due
to the wound caused by the increment borer.
Besides visual inspection of the core sample,
analyses such as bending and rupture tests have
been proposed (Mattheck et al., 1995). Other
methods that, although not based on core
samples, cause similar injuries are based on either
mechanical resistance (Barrett et al., 1987; Bethge
et al.,. 1996) or measurements of the wood’s electrical resistance (Shigo and Berry, 1975; Shortle
and Smith, 1987) in which needle-like probes are
punched or drilled into the trunk. Furthermore,
immunological probes for detection of certain
species of decay fungi in wood samples have been
proposed (Clausen, 1997). All of these methods
are destructive, although not to an extent that
acutely threatens the existence of the tree. Completely non-destructive methods based on, for
example, gamma-rays and computer tomography,
have been proposed (Habermehl, 1982a, b), but
the complexity of the measurements and the
expensive equipment represent major obstacles
for widespread application.
A large class of methods that cause only negligible damage and therefore are accepted as non-
destructive is based on the propagation of stress
waves (acoustical waves), mainly within the
audible frequency range (Mattheck and Bethge,
1993). The use of ultrasonics is under investigation (Sandoz et al., 2000; Socco et al., 2000).
The destructiveness is limited to attachment of
vibration transducers or sensors to the trunk, e.g.
using short needles or screws. These methods are
based on the fact that the propagation velocity of
stress waves decreases when the mechanical
properties of the wood are altered by decay (cf.
Wilcox, 1978; Graff, 1991). In many methods,
the estimated propagation velocity is compared
with a species-specific value for sound trees: a
velocity below a selected threshold is interpreted
as due to decay or other internal defects. A severe
drawback with this approach is that mechanical
parameters which have a strong impact on the
various wave propagation velocities may differ
among members of the same species. For
instance, the modulus of elasticity may differ
more within a species than the average value
differs from one species to another; the coeffcient
of variation (the ratio of the standard deviation
to the mean) for mechanical parameters of wood
from individuals of the same species is often as
large as 20 per cent (Schniewind, 1989). The
mechanical properties of an individual tree
depend on many factors, e.g. whether it is suppressed by, or dominates, neighbouring trees
(Brüchert et al., 2000). As a consequence, sound
trees may exhibit stress wave propagation velocities that deviate considerably from the speciesspecific average value. In tomographic
approaches, this issue is avoided since relative
propagation velocities within the trunk are
examined. Essentially, tomographic approaches
correspond to multiple stress wave propagation
velocity measurements along different stretches
through the trunk (Divos, 2000; Rust, 2000);
performed in an organized manner, the presence
and location of decay may be unveiled and visualized as a coarse image. Such systems are already
in commercial use (PICUS®, Argus Electronic
GMBH, Rostock, Germany; ARBOTOM®,
RinnTech, Heidelberg, Germany).
In recent years, methods accounting for the
dynamic impact response of the trunk have been
proposed (e.g. Axmon, 2000; Lawday and
Hodges, 2000; Axmon et al., 2002). In these
methods the frequency content of the measured
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signals is analysed. In Axmon and Hansson
(1999) and Axmon et al. (2002), we reported on
an estimation method and modal frequencies
sorted for different radial displacement modes
(i.e. spatial mode-shapes) of a cross-section for
trunks of standing Picea abies. Differences were
found between the modal frequencies of sound
and decayed trees; decayed trees tended to yield
lower frequencies. It was also found that the two
groups partially overlapped, and that the scatter
with respect to the modal frequencies of sound
trees with matching physical dimensions was substantial. This implied that the method suffered
from individual variations of the mechanical
properties. Moreover, it was observed that, in
addition to stress waves travelling through the
cross-section, surface waves were travelling bidirectionally from the point of excitation and
along the circumference. The propagation
velocity of the surface wave depends primarily on
the outer parts of the trunk, i.e. the sapwood,
whereas the modal frequencies depend on the
global properties of the trunk, i.e. both heartwood and sapwood. Hence, under the assumption that the individual variations in mechanical
properties are somewhat similar for sapwood and
heartwood, it may be possible to reduce the
scatter of the frequencies for sound trees with
matching physical dimensions, by accounting for
the surface wave propagation velocity, hence
using it as a proxy for the mechanical properties.
Provided that the sapwood is intact, it may even
be possible to reduce the influence of individual
variations for decayed trees such that the effect
due to the altered heartwood on the measured
frequencies is unveiled. Robust algorithms for
estimation of the propagation velocity of the
surface wave have been presented in Axmon and
Hansson (2000) and Axmon (2000).
In this paper, we examine the hypothesis that
the surface wave propagation velocity can be
used as a proxy for the mechanical properties and
hence, together with the circumference, serves as
an explanatory variable for the modal frequencies of sound trees. Furthermore, we demonstrate
how this relationship can be used to detect decay;
given the circumference and an estimate of the
surface wave propagation velocity, it is possible
to determine a tree’s expected modal frequency
under the presumption that it is sound. Should
the measured (estimated) frequency deviate
181
Table 1: Age of the trees examined
Stand
A
B
C
Stems
Age (years)
10*
26*
15*
11*
32*
58
30
37
23
42
* Not thinned.
significantly from this value, the tree is likely to
be afflicted by internal defects or decay. At
present, the proposed technique uses about the
same number of sensors that tomographic stress
wave methods do, and hence requires almost a
similar effort when a tree is examined. It is,
however, based on global properties of the trunk
and, with increasing knowledge on waveforms
and other characteristics of the signals, in the
future the number of sensors may be reduced to
two or three and the examination of a single tree
would only take a couple of minutes.
Materials and methods
Trees in the database
A total of 94 trees (Picea abies) was subjected to
the experiment during the summers of 1998 and
1999. The trees were selected in three geographically separated, managed forest stands in central
Scania, the southernmost part of Sweden. The
stands had a similar history in that they had been
established on former arable land, had been
thinned regularly, and the trees within each stand
or sectors of a stand were evenly aged. However,
the age of the stands differ; the study contains
trees of age spanning 23–58 years, with breast
height circumference ranging from 0.26 to
1.15 m. The number of trees at a particular age
is given in Table 1.
The trees in the study were chosen from groups
that were to be commercially harvested, except
for the 11 youngest trees (see stand B; Table 1)
which were selected from a group of trees that
had to be felled when changing the stretch of a
road. Many of the trees had injuries: in forest
stand A, the main cause was winter-time grazing
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6
7
8
TD
5
9
4
PW
3
10
SW
(a)
Time
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2
12
SW
0
1
(c)
1
2
3
4
5
6 7 8
Sensor
9 10 11 12
1
2
3
4
5
6 7 8
Sensor
9 10 11 12
TC
(b)
Time
ce
Pr
es
su
re
wa
v e pa
th
wa
ve
pa
th
Sensor
r fa
Su
Excitation
0
(d)
Figure 1. Initial wave propagation: (a) distribution of sensors and a screw around the cross-section and
initial propagation of a stress wave (PW) and a surface wave (SW); (b) initial propagation paths. Time-ofarrival for (c) stress waves propagating through the cross-section at a velocity corresponding to covering a
distance of one diameter in TD seconds, and (d) surface waves propagating along the circumference at a
velocity corresponding to covering a distance of one circumference in TC seconds.
by fallow deer and red deer, and in stands B and
C it was a combination of grazing game and the
use of heavy machinery in the forest management. In the region, the incidence of butt- and
heart-rot in spruce is most often due to the fungus
Heterobasidion annosum (Fr.) Bref.
Measurement equipment
Impact responses of cross-sections of trunks at
breast height (~1.2 m above ground) were collected using a uniform linear array consisting of
12 accelerometers (PCB Piezotronics M353B03)
positioned along the circumference. Each
accelerometer had a sensitivity of 10 mVg–1 with
1 g = 9.804 ms–2, a frequency range of
0.7–11 000 Hz at 10 per cent tolerance, and a
weight of 10 g. An aluminium plate with a needle
was used to attach each of the accelerometers
firmly to the sapwood. The outputs of the
accelerometers were sampled by a PC equipped
with an I/O board (National Instruments PCIMIO-16E-1 enhanced multifunction I/O board)
at a rate of 100 kHz per sensor. The data collection was monitored using software developed in
National Instruments’ LabView environment. To
generate an exciting force, the impacts by handforce of a standard hammer with a weight of
~0.2 kg were used. To avoid damage to the bark
and attenuation of the exciting force, the impacts
were applied onto a screw positioned between
two of the accelerometers in the array. The screw
penetrated the trunk to a depth of about
15–20 mm, such that it was firmly attached to the
sapwood. The exciting force was not measured,
and hence the method is referred to as blind; the
output of the system is known, but the input that
caused the output is unknown. The distribution
of sensors and the screw around a cross-section
is shown in Figure 1a, along with sketches of
waveforms that will be discussed below.
Measurement procedure
Before collecting the impact responses, each tree
was given a unique identifier for the database. The
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183
Table 2: Classification based on visual inspection of the fell cut at ground level for the 94 subjects of Picea abies
in the database
Class
Subclass
Description
Stems
Sound
I
43*
Sound
II
Decay
I†
Decay
II‡
Decay
III‡
Subjects with a fell cut without signs of decay and without comments
on injuries, resin exudation and other remarkable signs.
Subjects with a fell cut without signs of decay, but with comments
on one or more of the issues above.
Subjects in incipient decay, noticed by light-shaded discoloration
of the still hard wood.
Subjects in advanced decay, noticed by dark-shaded discoloration of
the still hard wood, and by minor changes of the wood texture.
Subjects in the final stage of decay, noticed by soft and brownish
wood.
23*
7*
7*
14*
* Only 42 trees are used in the subsequent analysis due to one tree lacking a frequency estimate.
† The mechanical properties of the wood are somewhat affected (Habermehl, 1982b).
‡ The mechanical properties of the wood are impaired (Habermehl, 1982b).
circumference and two perpendicularly taken
measurements of the trunk’s radius at breast
height were registered. Observations of
abnormalities of the tree, e.g. resin exudation,
dead branches and injuries, were added to the
database. When positioning the array of sensors,
we avoided as far as possible putting accelerometers on knots or injuries. A total of 20 responses
was recorded from each tree, to allow a study on
possible variations due to the hand-made hammer
blows, and to rule out outliers, e.g. where the
hammer touches the bark after hitting the screw.
Post-measurement assessment procedure
To obtain the true status of each tree, which is
required for evaluation of the method, the trees
were felled. The status of each tree was determined visually from the fell cut at ~0.2 m above
ground level. Since wounds at the surface of the
trunk increase the risk of rot not necessarily
developed at the butt, two subclasses of sound
trees were used (see Table 2). The decayed trees
were arranged into three subclasses, each describing how developed the decay was.
Sensor array signal processing
The measured signals were subjected to two
analyses: (1) modal analysis and (2) surface wave
propagation velocity estimation, whose results
are the basis for the work presented in this paper.
In the modal analysis, temporal and spatial frequencies were estimated. While the former are
measured in hertz (Hz), the latter are associated
with different shapes that the cross-section of the
trunk attains when it vibrates. For the modes
examined in the analysis, each temporal frequency is associated to a spatial mode-shape.
When using impact excitation, one has little or no
control over which modes will be excited, and
furthermore, several modes are excited simultaneously, hence the measured responses contain
the combined vibrations of many modes.
However, the spatial mode-shape can be used as
an identifier, such that corresponding modes can
be compared on a tree-to-tree basis although the
temporal frequencies differ. The algorithm used
for the modal analysis and the results are
described in Axmon et al. (2002). Since it was
found that the mode for which the cross-section
attains an oval shape was found in recordings
from all but one tree, i.e. 93 out of 94, our attention is on that mode.
The surface waves, which travel from the point
of excitation along the circumference in both
directions, generate a pattern in the measured
signals that is clearly separable from the pattern
created by the stress waves. In an isotropic,
homogeneous and linearly elastic medium, a
stress wave travels along the straight line from
the point of excitation to the respective sensor,
whereas a surface wave follows the contour of
the boundary, i.e. the circumference, see
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F O R E S T RY
Acc. [m/s2 ]
Acc. [m/s2 ]
1000
1000
0
0
−1000
−1000
2.5
2
1.5
1
0.5
(a) Time×10−3 [s]
0 1
2
3
4
5
6
7
8
9
12
11
10
0.6
0.4
0.2
Sensor
(b) Time×10−3 [s]
0 1
2
3
4
5
6
7
8
9
12
11
10
Sensor
Figure 2. Measured accelerations for a sound tree with a circumference of 0.81 m subjected to an impact
by a hammer: (a) the initial 2.5 ms and (b) 0.7 ms. For the purpose of visualization, the spatiotemporal
samples have been connected by a mesh grid at every fifth temporal sample and shaded according to the
amplitude of each sensor output. In (b) a half period of a sinusoid (white line) is superimposed on the
spatiotemporal impact response.
Figure 1a and b. The arrival times to the respective sensors are for stress waves, governed by the
law of cosine, which for uniformly distributed
sensors yields a profile which is half a period of
a sinusoid (Figure 1c). The surface waves, travelling along the circumference with constant
velocity, generate a linear time-of-arrival for each
direction (see Figure 1d). These clearly distinguishable kinds of time-of-arrival profiles are
found in all 93 recordings used in the present
work, and one representative example for a
sound tree with a circumference of 0.81 m is
given in Figure 2. The acceleration for the initial
2.5 ms of the sensor outputs is shown in
Figure 2a. There is obviously an agreement
between the time-of-arrivals of the different
wavefronts visible in Figure 2a and the slopes in
Figure 1c and d, even though wood is not
isotropic, but constitutes a structured anisotropy
(Bodig and Jayne, 1982; Côté, 1989). However,
the anisotropy in the radial–tangential plane of
the trunk (the cross-section) is much smaller than
in, for example, the radial–longitudinal plane.
The half period of a sinusoid assumption for the
initial stress wave is supported by the zoomed
plot in Figure 2b showing the initial 0.7 ms. The
algorithm used for estimating the surface wave
propagation velocity is detailed in Axmon and
Hansson (2000), and further developed in
Axmon (2000) where results for the trees in the
database are reported.
Signal processing results
The sensor array signal processing results are
summarized in Figure 3, where the frequency of
the ovaling mode in Figure 3a, and the propagation velocity in Figure 3b, are plotted versus
the circumference of each tree. The results are
taken from Axmon et al. (2002) and Axmon
(2000), respectively. The markers are coded
according to the status of each tree (cf. Table 2).
Evidently, there is a considerable scatter for
sound trees with similar dimensions. This scatter
may to some extent depend on individual variations of the mechanical properties.
Regression analysis and hypothesis testing
For trees in subclass Sound I, a nearly reciprocal
relationship between the frequency of the ovaling
mode and the circumference is observed; see
Figure 3a. Hence, the circumference C is obviously an explanatory variable in the sense that it
explains the general functional dependency
between measured frequency and dimension of
the cross-section. The hypothesis is that the
surface wave propagation velocity υ, too, is an
explanatory variable, such that the scatter in
Figure 3b explains some of the scatter in Figure
3a. The hypothesis is based on the theory of
isotropic, homogeneous, linearly elastic cylinders, for which the functional relationship
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2500
440
2250
420
Sound I
Sound II
Decay I
Decay II
Decay III
1750
1500
2,4
1,3
1250
1000
3
750
500
0.3
400
1,4
380
360
340
3
320
3
300
1,4
(a)
1,3
2,4
Propagation velocity [m/s]
Frequency [Hz]
2000
185
280
3
0.4
0.5
0.6 0.7 0.8 0.9
Circumference [m]
1.0
1.1
1.2
260
0.3
(b)
0.4
0.5
0.6 0.7 0.8 0.9
Circumference [m]
1.0
1.1
1.2
Figure 3. Signal analysis results coded according to the status determined visually from the fell cuts shown
in Table 2. (a) Temporal frequencies for the ovaling mode and (b) surface wave propagation velocities. Trees
with remarks had (1) heavy resin exudation, (2) remarkably many dead branches, (3) strange geometry of
the cross-section or (4) two stems from the same root with one surrounded by the other (of 93 trees, 65
were sound and 28 decayed).
between frequency F of the ovaling mode, circumference C and shear wave propagation
velocity υs is
F ≈ xC–1υs
(1)
where x is a dimensionless factor which is dependent upon the Poisson ratio, and for the case of
hollow cylinders, on the relative thickness of the
walls (Axmon, 2000). The propagation velocity
of the shear wave is dependent upon the modulus
of shear G and the mass density ρ through
υs = √G/ρ. A similar relationship can be established for the stress wave. Shear and stress waves
are the only kinds that can propagate within this
simplified medium. However, at the boundaries,
Rayleigh surface waves which are neither pure
stress waves, nor pure shear waves, propagate
(Graff, 1991). Although the precise characteristics of the surface wave observed in trees is
unknown, the wave is most certainly dependent
on the mechanical properties of the outer parts of
the trunk: modulus of shear, mass density and
Poisson’s ratio. For the simplified medium, the
latter has been found to have a smaller impact than
the former two on equation (1) (Axmon, 2000).
To test the hypothesis, the following two
regression models are examined:
θ1
F1(C; θ1) = ——
,
C
(2)
θ1υ
F2(C, υ; θ1) = ––––
C
(3)
where in equation (2) it is assumed that the frequency is independent of the surface wave propagation velocity, and in equation (3), that the
frequency depends on the propagation velocity in
a manner similar to the idealized case in
equation (1). It is assumed that C and υ are independent. Although it is understood that both C
and υ are estimates, it will be ignored in the
analysis; all other estimates are marked by a circumflex. The regression analysis of the models F1
and F2 is performed under a least-squares error
criterion using the data for the trees in Sound I.
In the selection between competing models, bias
and standard deviation of the regression residuals are studied, where the standard deviations are
adjusted for the degrees of freedom in each model
(Rawlings, 1988).
To verify the assumption that the propagation
velocity may be considered independent of the
circumference, the following competing regression functions are examined:
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υ1(θ1) = θ1,
(4)
υ2(C; θ1, θ2) = θ1(1 + θ2(C – 1))
(5)
where in equation (4), independence, and in
equation (5), a linear relationship is assumed. In
both functions, the average propagation velocity
for a sound tree with 1 m in circumference is
given by the estimate θˆ1.
Non-destructive detection of decay
To be able to detect internal decay from the
measured frequency F̂, which results from modal
analysis of the impact response, one has to know
the frequency the tree would have if it were
sound. Hence, it is desirable to predict the frequency for a sound tree, given its circumference
and surface wave propagation velocity. A clear
discrepancy between the measured and the predicted frequency would indicate that the tree
under test is likely to suffer from decay or other
internal defects.
The regression models F1 and F2 in the
previous section are both based on the functional
relationship equation (1) which is valid for an
idealized material. However, wood is neither
homogeneous nor isotropic. While lacking a
precise functional relationship between the
measured frequency and the explanatory variables for sound trees, a regression function that
produces unbiased residuals is desired for the
detector. Therefore a relaxed (and heuristic)
regression function, which is based on F2 with the
inclusion of a first-degree monic polynomial in
each explanatory variable, is used;
FP(C, υ; θ1, θ2, θ3) =
θ1υ
− ))
–––– (1 + θ2(C – 1))(1 + θ3(υ – υ
C
(6)
The role of the additional factors, which are
expressed around a circumference of 1 m and an
−, respectively, is to
average propagation velocity υ
capture some of the deviations from the idealized
case in equation (1). After tuning the parameters
θi, i = 1, 2, 3, of FP by performing a regression
analysis on a set of sound trees, one may use
FP(C, υ; θˆ1, θˆ2, θˆ3) in a predictive manner.
In a testing situation, one measures the circumference Ĉ and then records an impulse response.
From the recorded impulse response, the surface
wave propagation velocity υ̂ and the frequency F̂
of the ovaling mode are determined, upon which
the residual
ε = F̂ – FP(Ĉ, υ̂ ; θˆ1, θˆ2, θˆ3)
(7)
is formed. The residual corresponds to the difference between the measured frequency and the
predicted frequency for a presumed sound tree. A
sound tree is expected to yield a small residual,
whereas a decayed tree is expected to yield a large
negative residual since decay reduces the propagation velocity of the waves within the trunk.
However, other abnormalities (e.g. effects of
drought) may lead to large positive residuals,
since the mechanical properties of wood are
dependent on, for example, moisture content
(Bodig and Jayne, 1982; Booker et al., 1996).
Furthermore, in a severely decayed tree, the
mechanical properties of the outer wood of the
trunk may have been significantly altered. In that
case, it may lead to a lower propagation velocity
of the surface wave. This would then result in a
small residual, since a low measured frequency
then is explained by a low surface wave propagation velocity used in the prediction by FP. To
avoid such situations, a threshold on the surface
wave propagation velocity is used. Hence, the
following detection rule may be defined, where
εTH and υ TH are thresholds acting on the residual
and the propagation velocity, respectively:
Consider the tree sound if both
ε ≤ εTH and υ̂ > υ TH
(8)
are fulfilled.
To evaluate the performance of the detector
rule for a set of thresholds εTH and υ TH, the
fraction of correct detections within each major
class Sound and Decay are studied. Since it is
observed in Figure 3b that none of the sound
trees exhibit a propagation velocity below
310 ms–1, the threshold on the velocity is chosen
to υ TH = 310 ms–1. The threshold on the prediction residual εTH is examined for increasing
values beginning at εTH = 1 Hz. The parameters
θˆi, i = 1, 2, 3 are estimated using the trees in the
subclass Sound I.
Results
The results for the regression analysis of the competing models F1 and F2 are shown in Table 3.
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187
Table 3: Estimated parameters and residuals from regression analysis performed on the subclass Sound I
Regression model
Estimates
Regression residuals
———————————————– ———————————
θˆ1
θˆ2
θˆ3
Mean
SD
υ1(h) = θ1
υ2(C; h) = θ1(1 + θ2(C – 1))
353.2 ms–1
353.8 ms–1
θ1
F1(C; h) = —–
C
774.4 ms–1
–5.5 Hz
80.3 Hz
θ1υ
F2(C, υ ; h) = —–
C
2.193 ms–1
–4.1 Hz
44.2 Hz
0.3 Hz
40.1 Hz
5.9E-3 m–1
θ1υ
FP(C, υ ; h) = —–
(1 + θ2(C – 1))(1 + θ3(υ – υ–)) 2.256 ms–1 9.10E-2 m–1 –3.65E-4 sm–1
C
–3E-14 ms–1
–5E-15 ms–1
23.8 ms–1
24.1 ms–1
The standard deviations are adjusted for the number of parameters used in each regression model.
Solved around υ– = 353.2 ms–1.
Both models result in biased regression residuals,
but the adjusted standard deviations differ significantly. When including the propagation velocity,
F2, the standard deviation is reduced to 55 per
cent of the one obtained for F1 which only uses
the circumference as explanatory variable. This
means that much of the scatter in the measured
frequency is explained by the scatter of the
propagation velocity, and hence, the propagation
velocity is an explanatory variable that should
not be neglected. The results from the regression
analysis of υ1 and υ2 are shown in Table 3. The
bias is negligible for both models, but the
adjusted standard deviation of the regression
residuals is somewhat lower for υ1 than for υ2.
Thus υ1 is selected, i.e. the propagation velocity
of the surface wave is considered independent of
the circumference of the trunk at breast height,
− = 353.2 ms–1 for
yielding an average value of υ
the sound trees in Sound I. Conclusively, the
hypothesis that the surface wave can be used as
a proxy for the mechanical properties is supported, and the assumption that its propagation
velocity is independent on the circumference is
justified.
The coefficients for the frequency predictive
function FP which is used in the detector rule are
shown in Table 3. They were determined for the
− = 353.2 ms–1 which was
average velocity of υ
obtained from υ1. By inclusion of the two first-
degree monic polynomials, the bias is significantly reduced, and the adjusted standard deviation somewhat reduced, compared with the F2
model. A scatter plot of residuals from equation
(7) versus propagation velocity is shown in
Figure 4. It is found that many of the decayed
trees separate from the group of sound trees
either by a low propagation velocity or by a large
positive or negative residual. The standard deviation for the subclass Sound I was 40.1 Hz, and
for the major class Sound (i.e. Sound I plus Sound
II) is 46.9 Hz. Most of the decayed trees yield
residuals that fall outside the plus one to minus
one standard deviation of the major class Sound.
The performance of the detector rule is shown in
Figure 5 for υTH = 310 ms–1 and 1 ≤ εTH ≤ 120 Hz.
The curves describing the success of the classification intersect at εTH = 53 Hz, where 74 per
cent of the sound and the decayed trees, respectively, are correctly classified. For εTH < 53 Hz,
an increasing number of decayed trees are correctly detected as decayed at the expense of an
increasing number of sound trees falsely detected
as decayed, and for εTH > 53 Hz vice versa.
Discussion
The hypothesis that the scatter of the measured
frequency of the ovaling mode is explained by a
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F O R E S T RY
250
200
Sound I
Sound II
Decay I
Decay II
Decay III
150
Residual [Hz]
100
3
50
1,3
0
2,4
1,4
-50
3
-100
-150
-200
-400
-450
260
300
280
320
340
360
380
400
420
440
Surface wave propagation velocity [m/s]
Figure 4. Scatter plot of residuals ε versus surface wave propagation velocity υ for the 93 trees included
in the study. The horizontal lines indicate plus one and minus one standard deviation (46.9 Hz) for the
residuals in the major class Sound. For a legend on the remarks, see Figure 3.
100%
Success
80%
60%
40%
Sound
Decayed
20%
0%
0
20
40
60
80
Residual threshold [Hz]
100
120
Figure 5. Performance evaluation of the detector rule for a threshold υTH = 310 ms–1. Successful classifications within the major classes Sound and Decay, respectively, versus double-sided residual threshold εTH.
scatter of the surface wave propagation velocity
is supported by the results from the regression
analysis. Validation using an independent data
set, i.e. a data set that was not included in the
estimation of the parameters of the regression
function, is not only considered good practice,
but is important especially when the regression
function is used for a predictive rather than a
descriptive purpose (Rawlings, 1988). Such validation has not been performed, and the reason is
that there are too few trees in Sound I to split the
data set into one for regression analysis and one
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D E T E C T I O N O F I N T E R N A L D E C AY I N P I C E A A B I E S
for validation. All estimates in Sound I are
required to ensure a satisfactory excitation in the
analysis. Validation using Sound II resulted in
larger bias and standard deviation of the residuals, but did not affect the selections of the regression function F2 over F1 and υ1 over υ2. There is,
however, a reason for splitting the major class
Sound into the subclasses Sound I and Sound II,
cf. Table 2, and therefore validation using Sound
II has little credibility. The uncertainty on
whether the trees in Sound II suffer from decay at
breast height could have been avoided by cutting
the trunks at breast height, upon which the status
of the tree in the vicinity of where the measurements were performed could have been determined. The reason for not doing so was purely
economical: generally the lower part of the trunk
is the most valuable one. In the case of followups to this study, we recommend that the trunks
are cut at the height where the measurements are
performed; there would then be no need for two
subclasses of sound trees.
The detector rule is evident from studying a
scatter plot of prediction residual ε versus propagation velocity υ̂ for the 93 trees included in the
analysis (Figure 4). It is evident that many of the
decayed trees fall outside the plus one to minus
one standard deviation (± 46.9 Hz) for the
major class Sound indicated by the dashed, horizontal lines. Many of the trees in Decay III which
exhibit low propagation velocities had hollow
trunks filled with moist and spongy leftovers of
what once was wood; those trees are in the
ultimate stage of decay. Although the measurements are performed at breast height, it is reasonable to assume that a large area of the examined
cross-section for trees in Decay III is severely
decayed (Wagener and Davidson, 1954; Habermehl, 1982b). It is also expected for Decay II that
the decay extends to the examined cross-section,
but for trees in Decay I, it is likely that some trees
are perfectly sound in the vicinity of the crosssection. The reason for this is the coarse, qualitative separation of the continuum of stages of
decay into three subgroups. The least developed
decay in Decay I corresponds to a spot of discoloured wood, and since most heart-rotting
fungi have a vertical extent of 0.3–0.6 m beyond
the lowest visible sign of decay (Wagener and
Davidson, 1954), the rot column might not reach
to breast height. Similarly, some of the trees in
189
Sound II may be decayed at breast height,
although they appear to be sound at the fell cut,
since injuries of the trunk may serve as entrances
for fungal spores.
Studying the fraction of correct detections
within each class for a range of thresholds on the
residuals (Figure 5) we see that the curve describing the success in identifying decayed trees intersects with the curve describing the success in
identifying sound trees at 74 per cent for a
threshold εTH = 53 Hz. If there were no overlap,
a simultaneous success of 100 per cent would be
achievable. Hence, trade-offs have to be made.
Although the results may appear to be displeasing, a comparison is made with the study by Vollbrecht and Agestam (1995) where five trained
foresters individually assessed 372 trees of Picea
abies by visual examination of the standing trees.
In the study, it was found that the assessments by
individual foresters were only slightly better than
random. To be able to compare the results in the
study with the performance of the detector,
figures for the majority’s decision, i.e. where at
least three of the five foresters had the same
opinion on whether an individual tree was sound
or decayed, were extracted from the reported bar
graphs, and converted into appropriate success
figures for each class. It was found that ~53 per
cent of the sound trees and 67 per cent of the
decayed trees were correctly classified. A comparison with the present study shows that the
success of the detector rule in detecting decayed
trees is 86 per cent, given a success of 53 per cent
in detecting sound trees. The success in detecting
sound trees is also 86 per cent, given a success of
67 per cent in detecting decayed trees. Should
these results be representative for a larger study,
it means that there is a prospect of detecting
decay more accurately than possible from visual
tree assessment.
One may have objections concerning the predictive function FP used in the detector, and the
performance evaluation. The parameters θˆi, i = 1,
2, 3, were determined by regression analysis
using the presumed perfectly sound trees in
Sound I. This means that the parameters are
adapted to sound trees from the examined forest
stands. Season, climate and other environmental
conditions, e.g. soil, terrain and water-table, may
influence the mechanical properties of the trunks.
Hence, the parameters may be different for other
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forest areas, and may even differ for the
examined forest stands during the year. Should
one use such a detector, one would first have to
understand the impact of the influential factors,
or to tune the parameters on a subset of sound
trees representative for the forest area under the
scope. The objection concerning the performance
evaluation is that the predictive function was
tuned using the trees in Sound I, constituting
65 per cent of the trees in the major class Sound.
The standard deviation of the regression residuals for FP was 40.1 Hz. Thus at the residual
threshold εTH = 40.1 Hz and assuming a normal
distribution, 68 per cent of the trees in Sound I,
corresponding to 44 per cent of the major class
Sound, are successfully identified. As evident
from Figure 5, the relative success figure for the
class Sound is 56 per cent, meaning that 34 per
cent of the trees in Sound II are within this threshold. The success per subclass differs by 10 percentage units, merely corresponding to two trees
in Sound II. Thus FP seems appropriate in predicting the frequency for the trees in Sound II,
too. Of course, performance evaluation using
independent data would be more reliable but, as
discussed above, this has not been an option in
the present study.
The main achievement of this paper was to
show that the surface waves capture the mechanical properties and thus can be used in reducing
the scatter due to individual variations among
members of Picea abies. It ought to be examined
whether stress wave time-of-flight methods (e.g.
Mattheck and Bethge, 1993) and combined
single-sensor dynamical analyses (e.g. Lawday
and Hodges, 2000) would benefit from information on the surface wave propagation velocity.
Finally, and obviously, a characterization of the
surface waves arising due to impact excitation of
the trunks ought to be conducted. Knowledge on
the type of surface wave, e.g. whether it resembles a Rayleigh wave, may lead to a better understanding of the mechanics of standing trunks,
from which all impact-based methods would
benefit.
The measurement technique used for the
present work requires about the same effort as
tomographic stress waves methods, since an
array of sensors has to be distributed around the
trunk; it is time consuming and may require as
much as 5 min per tree. Therefore, at present the
method is more experimental than operational. A
stationary PC and a portable power supply were
used for the data acquisition; this increased the
total time spent on each tree. Today, the performance of budget laptop computers is more than
sufficient for managing the data acquisition and
the analyses, thus cumbrous equipment is no
longer required. However, a large reduction of
time consumed at each tree would be achieved if
the number of sensors were reduced. At present,
the main obstacle is the fashion in which the
surface wave propagation velocity is estimated.
Signal processing algorithms which are based on
information gathered from further analyses on
the characteristics of surface waves may probably
lead to a minimum requirement of two or three
sensors, from which a considerable reduction in
time spent per tree follows.
Conclusions
It has been established that the propagation
velocity of a surface wave arising from impact
excitation is correlated to the modal frequency
of the ovaling mode for cross-sections of trunks
of Picea abies. The surface wave captures the
mechanical properties of the wood, and thus can
be used for reduction of the effects of the
natural, individual variations of the mechanical
properties among members of Picea abies.
Furthermore, it has been demonstrated that the
joint information on the modal frequency of the
ovaling mode and the propagation velocity of the
surface wave can be used in the detection of
decayed trees. Comparison with other studies
(Vollbrecht and Agestam, 1995) shows that the
detector performs better than visual tree examination. Although the proposed detector suffers
from drawbacks, e.g. that it likely has to be
tuned using sound trees from similar forest
stands, and the necessary selection of a residual
threshold, it shows that there is a prospect of
detecting decay using modal frequencies.
Research on, and characterization of, the
surface wave is recommended, since it is likely
that all impact-based non-destructive testing
methods for detection of decay in standing trees
would benefit from a reduction of the effects of
the individual variations of the mechanical
properties.
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Acknowledgements
This work is financed by the Swedish Research Council
(Vetenskapsrådet), project 621-2001-3119. Additional
support has kindly been provided by SÖDRA and the
Forestry Research Institute of Sweden (SkogForsk). Mr
B. Bengtsson and Trolleholms Gods AB made access to
forest areas possible. Various kinds of technical assistance were provided by Mr B. Bengtsson, Mr L.
Karlsson and Mr L. Hedenstjerna, who are all with the
Department of Electroscience, Lund University,
Sweden.
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Received 1 July 2002