The
MIC
normalizes the mutual information, dividing it by
its log{
MIN
(
X,Y
)}, so that its values are confined within the
interval [0,1], becoming comparable across scales and units.
The outcome is a non-parametric measure of the strength
(
viz.
absence of noise) in the relation between
X
and
Y
, which
provides with a similar value whether the relation is linear,
exponential, periodic, circular, parabola, or any other type of
shape (Reshef
et al
., 2011). For a detailed description of
MIC
,
please refer to Reshef
et al
(2011).
The package “minerva” (Filosi
et al
., 2014) was employed
for implementation in R statistical environment (R Develop-
ment Core Team, 2014).
MIC
is interpreted in a similar manner
as a correlation coefficient, and therefore the higher the
MIC
,
the stronger the relation between that given metric with stand
volume.
MIC
= 1 indicates a noiseless functional relationship,
whereas
MIC
= 0 denotes independence between predictor and
response. The optimization method consisted in iterating the
operation of metric computation for different reference heights,
and selecting the one which maximized
MIC
. Assuring a range
of reference heights reaching up to the base of tree crowns, we
considered reference height threshold at 0, 0.1, 0.3, 0.5, 0.7, 1,
1.3, 2, 3, 4, 5, 6, and 7 meters, which corresponded to either
MinH
or
HBreak
depending of each metric, as explained above.
Study Case
The approach was tested in five different sites within a com-
mercial
Eucalyptus
plantation managed to supply a pulp mill
company located in Brazil. The climate is subtropical, with
wet and hot summers, and dry and cold winters. Annual mean
temperature is 21°C and annual rainfall is 1,300 mm. The
plantations consist of hybrid
Eucalyptus
clones of
E. urophylla
(S.T. Blake) x
E. grandis
(W.Hill ex Maiden). The stand ages
range from 2 to 8 years old (Table 2), covering the full span of
ages usually considered in Brazil to define the rotation cycle
of commercial
Eucaliptus
plantations. The trees are systemati-
cally planted over a 3 × 2 m regular grid, which results in a
stand density of 1,666 trees∙ha
-1
. This density is therefore stable
throughout the area and roughly constant along the stand age,
since mortality rates are very low. We presumed that changes
in stand density could be a significant factor of covariability
affecting the relations between
ALS
metrics and stand volume
found for different reference heights. For this reason, we select-
ed this study area, since it allowed us to minimize such effect.
A total of 105 field plots were established and georefer-
enced. Using a caliper, all tree diameters at breast height were
measured within circular plots sizing 400 m². Tree heights were
determined by an electronic clinometer for 15 percent of trees
within a plot. Unmeasured heights were estimated based on
diameters using a locally adjusted allometric model. Next, re-
spective diameters at breast height and heights were employed
into individual tree volume equation. The company in charge
of forest operations supplied all of the field measurements,
including the allometric and the volume models. Finally, stand
volume was calculated at plot-level, and used as response vari-
able (
Y
) for this study. All the fieldwork was performed on the
same months around the flight campaign.
The
ALS
survey was conducted in April 2012, and pro-
duced point clouds with an average density of 5 points·m
-
².
The laser returns were filtered and classified as ground and
non-ground using the interpolation algorithm proposed by
Kraus and Pfeifer (1998), considering a moving window equal
to six meters and nine interactions. A digital terrain model
(
DTM
) with a 1 m cell size was created assigning to each cell
the average elevation of the corresponding ground returns.
The value of this
DTM
underneath each
ALS
return was sub-
tracted from its absolute elevation above the ellipsoid, obtain-
ing the height above ground for the entire
ALS
cloud.
Results
Changes in reference height thresholds had a significant
influence in the computation of
ALS
metrics, and thereby their
explanatory capacity for stand volume, as signified by our
MIC
results. Figures 1 and 2 show the evolution in the study
case of
MIC
(y-axes) for increasing reference heights (x-axes)
for each of the
ALS
metrics. The x-axis has been modified in
Figure 3, which instead shows the results obtained by dif-
ferent
MinH
for increasing percentiles (x-axis). The scale of
y-axis has been consistently fixed for all these plots, allowing
direct comparison.
ALS
height metrics (Figures 1 and 3) computed from all
returns (left) were more influenced by changes in reference
height thresholds than those computed from first returns only
(right). We shall affirm that increasing reference height thresh-
olds was generally positive for most of the height metrics, in
the sense of strengthening their relation with stand volume,
as
MIC
increased. Not excluding returns from the ground at
all (
MinH
= 0) was particularly negative for metrics describ-
ing the central tendency of
ALS
height distributions, such as
the mean and mode, as the strength of the relation increased
sharply when the returns near the ground where excluded
(Figure 1a and 1b.1). Similar tendencies were found for met-
rics describing the shape of the distributions (Figure 1a.3 and
1b.3), which showed analogous results whether computed
from all returns or first only.
The case was less clear for metrics describing the disper-
sion (variation) in the
ALS
height distribution (Figure 1a.2 and
1b.2). As
MinH
increased, the strength of the relation of stan-
dard deviation and variance with stand volume decreased.
When
SD
and
VAR
(Table 1) were computed using no reference
height at all (
MinH =
0), the resulting value of
MIC
= 0.813
was among the highest obtained in this study case. Metrics
expressing relative dispersion were more robust to changes in
MinH
, such as the coefficient of variation or the inter-quartile
range, however having a weaker relation with stand volume.
Table 2. Summary of Stands Characteristics; Figures are Averages, with Ranges in Parenthesis
Age (yr)
Number of plots
Basal Area (m².ha
–
¹)
Height (m)
Dominant Height (m)
Volume (m³.ha
–
¹)
2
14
7.9
(2.5 – 12.0)
12.1
(5.1 – 14.8)
13.4
46.3
(6.9 – 87.8)
3
19
20.8
(16.6 – 23.6)
18.6
(16.1 – 20.7)
20.1
173.8
(117.3 – 217.5)
5
17
27.2
(20.1 – 31.6)
21.5
(18.8 – 23.0)
24.2
266.5
(200 – 337.8)
6
14
27.3
(24.2 – 30.6)
23.7
(22.0 – 25.3)
26.5
292.1
(241.7 - 344)
7
23
33.8
(23.9 – 40.44)
24.6
(21.5 – 27.5)
32.0
410.6
(278.0 – 536.6)
8
18
30.2
(26 – 36.1)
25.2
(19.8 – 31.8)
30.2
362.3
(277.7 – 452.8)
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
May 2017
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