PE&RS May 2017 Full - page 344

fixed under a given criterion, or
dynamic (i.e., variable through-
out the study area). Sullivan
(2008) applied a fixed HBreak
= 2 m, emulating traditional
canopy density measurements
made visually in the field. Fla-
herty et al (2012) considered a
fixed HBreak of 1.3 m, under the
above-mentioned logical crite-
rion used by Jensen et al (2006).
Alternatively, HBreaks may also
be variable throughout the study
area. Andersen (2005) used a
dynamic HBreak, which was de-
fined by the crown base height.
Næsset (2011) also proposed a
dynamic definition of fixed in-
tervals relative to the maximum
return elevation. The HBreak
may also be defined as func-
tion by another height metrics,
such as the mean or mode of ALS
heights (McGaughey, 2013) or
percentiles (Næsset, 2002). Since
those height metrics are affected
by MinH, their nested influence
complicates and obscures the
effect of reference heights on the
final ALS metrics.
Very few studies have inves-
tigated the effects of different
reference heights. Næsset (2011)
compared the values of HBreak
= 0.5, 1.3, and 2 m, finding no
clear effect on the accuracy of biomass estimations. Likewise,
Montaghi (personal communication: Wulder et al., 2013)
tested reference heights increasing in 0.5 m intervals from 0
to 2.5 m also finding no clear effects when models included
a large number of predictors. These studies only observed
the influence of changing one single reference height ap-
plied across all metrics, and only assessed the final model
that resulted from the whole process, including predictor
selection. In contrast, the objective of this research is to test
several reference height thresholds and observe their effects
on each ALS metric one-by-one, so that a different threshold
can be selected for each individual metric on the basis of its
relationship to forest stand volume. We propose that tailoring
reference heights to each metric could potentially increase the
accuracy of the model, by optimizing the explanatory power
of each metric on the given target response desired.
Material and Methods
Metric Computation and Effects of Reference Height Thresholds
We analyzed the ALS metrics available in FUSION (USDA
Forest Service; McGaughey, 2013). After extracting the returns
within the georeferenced plots, the ALS metrics (Table 1) were
repeatedly computed from them using different reference
height thresholds. Depending on the type of metric, these
thresholds may affect either MinH or HBreak, which we con-
sider worthwhile reviewing. All height metrics were affected
by MinH, and hence, in their case, those thresholds repre-
sented the height below which ALS returns were eliminated.
Conversely, in canopy cover metrics with fixed HBreak,
those were the thresholds above which the relative density
of returns was computed (i.e., not eliminating any return). In
the case of canopy cover metrics with variable HBreak, the
threshold height for computing relative density originated
from a height metric, e.g., the mean or mode. MinH , in turn,
affects those height metrics (Table 1).
Optimization Method
For each metric (X) computed using a specific reference
threshold, we studied the strength of its relation with the
response (Y), by means of the maximal information coef-
ficient (MIC ; Reshef et al., 2011), which considers entropy
as a measure of uncertainty (Speed, 2011). We chose it as an
alternative to the more widespread coefficient of correlation,
as it allows to identify non-linear relations as well, which
can be important for non-parametric estimation methods
such as nearest neighbor imputation. The mutual information
between two variables MI(X,Y), reveals the amount of vari-
ance in Y which is explained by X (Linfoot, 1957). During MI
computation, the variables are discretized into bins of a size
B, which jointly for X,Y generates a grid with relative propor-
tions p(X,Y) of realizations contained within each cell. The
naïve mutual information between these variables MI(X,Y) is
calculated as the entropy among grid’s cells (Linfoot, 1957;
Clark, 2013):
MI X Y p X Y p X Y
p X p Y
X Y
, ( , ) log ( , )
( ) ( )
,
( ) = ∑ .
The MIC consists in optimizing bin size by selecting the B
which maximizes MI(X,Y) (Reshef et al., 2011):
MIC X Y MI X Y
X Y
X Y Btotal
( ; ) max ,
log min ,
,
= ( )
( ) { }




< .
Table 1. Summary of ALS Metrics; See McGaughey (2013) for Details
Group Sub-group All returns First returns Description
ALS height metrics
Metrics expressing
the central
tendency of
ALS heights
distribution
MEAN MEANf mean
MODE MODEf mode
SQRT SQRTf quadratic mean
CUB CUBf cubic mean
Metrics expressing
the dispersion
of ALS heights
distribution
SD SDf standard deviation
VAR VARf variance
AAD AADf absolute average deviation
MAD.MED MADMEDf median absolute deviation from median
MAD.MOD MADMODf median absolute deviation from mode
IQ IQf interquartile range
CV CVf coefficient of variation
Metrics expressing
the shape of
ALS heights
distribution
SKEW SKEWf skewness
KURT KURTf Kurtosis
CRR CRRf canopy relief ratio
Percentiles of
the ALS heights
distribution
P05, P10…
…P95, P99
P05f, P10f…
…P95f, P99f
5th , 10th , 20th, 25 th, 30th , 40th , 50th, 60 th,
70th, 75 th, 80th , 90th , 95th, 99 th percentiles
Density metrics(Canopy cover)
Fixed Hbreak (#)
PROP# PROP#f proportion of returns above HBreak
RATIO#- ratio between all returns above HBreak
and total of first returns
Variable Hbreak
PROP.MEAN PROP.MEANf proportion of returns above mean
PROP.MOD PROP.MODf proportion of returns above mode
RATIO.MEAN ratio between all returns above mean and
total of first returns
RATIO.MOD ratio between all returns above mode and
total of first returns
344 May 2017 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING