Article
Techniques for quantifying theaccuracy of gridded elevationmodels and for mappinguncertainty in digital terrainanalysis
N. Gonga-SaholiarilivaUniversite Paris Diderot, France; TTI Production, France
Y. GunnellUniversite Lumiere Lyon 2, France
C. PetitUniversite de Nice, France
C. MeringUniversite Paris Diderot, France
AbstractWe first provide a critical review of statistical procedures employed in the literature for testing uncertainty indigital terrain analysis, then focus on several aspects of spatial autocorrelation that have been neglected in theanalysis of gridded elevation data. When applied to first derivatives of elevation such as topographic slope,a spatial approach using Moran’s I and the LISA (Local Indicator of Spatial Association) allows:(1) georeferenced data patterns to be generated; (2) error hot- and coldspots to be located; and(3) error propagation during DEM manipulation to be evaluated. In a worked example focusing on theWasatch mountain front, Utah, we analyse the relative advantages of six DEMs resulting from differentacquisition modes (airborne, optical, radar, or composite): the LiDAR (2 m), CODEM (5 m), NED10(10 m), ASTER DEM (15 m) and GDEM (30 m), and SRTM (90 m). The example shows that (apart fromthe LiDAR) the NED10, which is generated from composite data sources, is the least error-ridden DEM forthat region. Knowing error magnitudes and where errors are located determines where corrections to ele-vation are required in order to minimize error accumulation or propagation, and clarifies how they mightaffect expert judgement in environmental decisions. Ground resolution issues can subsequently be addressedwith greater confidence by resampling the preferred grid to terrain resolutions suited to the landscapeattributes of interest. Source product testing is an essential yet often neglected part of DEM analysis,with many practical applications in hydrological modelling, for predictions of slope- to catchment-scale masssediment flux, or for the assessment of slope stability thresholds.
Corresponding author:N. Gonga-Saholiariliva, 33 impasse des Arbousiers, F-30250, Sommieres, FranceEmail: [email protected]
Progress in Physical Geography1–26
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Keywordsdigital elevation model, LISA, Moran’s I, spatial autocorrelation, topographic slope, uncertainty mapping
I Introduction
The advent of remote sensing technologies has
meant that larger areas can be mapped by fewer
people in less time and at diminishing cost
(Smith and Clark, 2005). The widespread use
of digital elevation models (DEMs) is a direct
consequence of those advantages. Traditional
methods such as field surveying and photogram-
metry can yield high accuracy terrain data, but
they are time consuming and labour intensive
(Liu, 2008). As a result, DEMs have become the
most common method for extracting topo-
graphic information, also allowing the flow of
water across topography to be analysed, with
many practical applications in hydrological
modelling and for predictions of slope- to
catchment-scale mass sediment flux (Desmet
and Govers, 1995; Dietrich et al., 1993; Kamp
et al., 2003; Kirkby, 1990). Topographic infor-
mation in a DEM can be represented and stored
as: (1) a triangulated irregular network (TIN);
(2) discrete landform elements based on the
intersection of flow- and contour lines; or (3) a
grid. Here we examine gridded DEMs, which
despite drawbacks arising from higher demands
on computer memory (Claessens et al., 2005:
462; Yue et al., 2007: 161) nevertheless remain
the most commonly used data source for digital
terrain analysis because of their simple structure
and compatibility with other digitally produced
data (Wise, 2000: 1910).
Slope gradient and aspect, which are exam-
ined in this review, are two of the most basic
parameters affecting landscape-scale natural
processes. The accuracy of a slope map deter-
mines many of the predictions that geomorpho-
logists, engineers, hydrologists, and ecologists,
farmers or foresters make about the behaviour
of environmental variables, about natural
hazards, land-use potential, and for planning.
Errors on slope propagate to more sophisticated
DEM-derived terrain attributes such as slope
curvature, drainage networks, slope channel
lengths, slope widths or topographic wetness
indices. Although seemingly obvious, the
consideration of error distribution and error
propagation in DEMs has nonetheless been
neglected in routine DEM-based terrain studies,
perhaps because the appeal and immediacy of
DEM implementation in commonly available
software packages overrides any concern for
accuracy and error. The imperfections of
gridded DEMs, i.e. the fidelity with which they
represent the ‘real world’, have been extensively
explored in the last decades. Fisher and Tate
(2006) have described errors on gridded data sets
as the difference between a model value and the
true value, i.e. as deviations from reality (called
incorrect elevation values, misplaced elevation
values, data gaps, or missing values). True or
real values are absolute elevation measurements
collected on the Earth’s surface with the instru-
ments of terrestrial geodesy, whether satellite-
assisted or not. Like all numerical models
(Dietrich et al., 2003: 109–110), elevation
models are approximations of reality, but this
does not mean that sources of error are
unknowable, intractable or unquantifiable.
Any step in DEM production could be a source
of error, and error should be detected early in
the batch processes in order to avoid its
insidious propagation to subsequent steps and
its effects on the quality of the final product
(Darnell et al., 2008; Hengl et al., 2010).
The risk of introducing or ignoring distortions
is to generate bias in DEM-based quantitative
analyses of the Earth’s surface and in the accu-
rate characterization of landforms and processes.
The use of terrain models for environmental and
engineering applications, or even for heuristic
purposes, thus calls for an assessment of error
relating to data type, data precision, the way
2 Progress in Physical Geography
the data acquisition tools operate, and to the
comparative advantages of high ground-
resolution (g.r.) data such as LiDAR, SpotLight
(g.r.: 1 m, TerraSarX), Cosmo-SkyMed 4 (g.r.:
2–4 m) or Alos Prism (g.r.: 2.5 m) (Hengl
et al., 2010: 769). At a time when the number
and diversity of satellite, airborne and radar
platforms is on the increase due to the rapidly
evolving technology, this requires at least some
knowledge of data processing techniques and
appropriate means for meaningfully evaluating
the reliability of results used in terrain analysis.
Here we review the variety and relative per-
formance of existing approaches while empha-
sizing from the start that few studies have
embraced the analysis of spatial autocorrelation
as a prerequisite for developing error models.
Only random patterns exhibit no spatial autocor-
relation (Hunter and Goodchild, 1997; Oksanen
and Sarjakoski, 2005). DEM error, however, is
in essence already spatially dependent because
of the interpolation methods used in DEM
generation (Carlisle, 2005: 524; Liu and Bian,
2008: 308). This means that redundancy occurs
in spatial data and can bias the common assump-
tions of independence between values in many
statistical operations (e.g. least-squares regres-
sion analysis). A review of the techniques and
applications of spatial autocorrelation to DEM
analysis will accordingly occupy the later
sections of this paper and will focus around a
worked example providing new results for
illustrative and didactic purposes. Some recent
studies have addressed spatial autocorrelation
as a means of calibrating error models, but have
exclusively dealt with global autocorrelation
rather than with examining patterns of
local-scale variation (Hebeler and Purves,
2009; Lindsay and Evans, 2008; Liu and Bian,
2008; Wechsler and Kroll, 2006), and have
tended to focus on global data (e.g. GTOPO30,
SRTM: Holmes et al., 2000, Wechsler, 1999;
Wechsler and Kroll, 2006) and on the interpola-
tion of high accuracy data (e.g. LiDAR: Arrell
et al., 2008). Limited attention has been paid
to: (1) the potential for analysing local and
global spatial autocorrelation conjointly; (2) the
generation of continuous spatial representations
of error arising from DEM analysis; and (3) the
comparative sensitivity to error of DEMs
obtained from different sensors (optical, radar,
airborne). We will show how local spatial auto-
correlation analysis helps to locate the weak-
nesses of DEM structure, how it highlights the
spatial heterogeneity of error, and how, from the
use of local indices, it is possible to obtain error
maps useful to a wide range of terrain applica-
tions relevant to physical geography.
Accordingly, we will focus on three issues
that arise when different DEM sources are avail-
able for a given geographical area. (1) How can
the accuracy of elevation data be assessed in
such situations? (2) How can errors affecting pri-
mary topographic attributes such as topographic
slope be evaluated, and what confidence can be
attributed to such data for hydrological or
geomorphometric applications? (3) What error
mapping techniques are most advantageous for
detecting regions susceptible to error propaga-
tion and for visually displaying bias in DEM-
based scientific predictions? We first present a
review of current knowledge about DEM
accuracy and the tracking of error propagation
in grid processing. The Local Indicator of
Spatial Association (LISA: Anselin, 1995),
similar to calculating neighbourhood filters in
raster analysis, is then introduced and tested as
an illustration of how to select the most efficient
error model based on clustering results.
We test six DEM grids obtained from differ-
ent categories of sensors and focus on techniques
for mapping the spatial distribution of error
affecting topographic slope values. The chosen
study area is the Wasatch mountain front, Utah,
USA. The data sources are a LiDAR grid, i.e. an
airborne source; the composite CODEM,
NED10 and GDEM, each compiled from multi-
ple data sources, i.e. maps and/or optical and/or
radar; the ASTER DEM (AstD), an optical source;
and the SRTM, a radar source. Each possesses
Gonga-Saholiariliva et al. 3
initial statistical characteristics relating to the
elevation interpolation procedures, to the error
corrections relating to physical acquisition
parameters, and to altimetric configurations
imposed by the initial choice of geoid reference
frame. Although the ground resolutions of the
DEMs analysed also happen to range from
2 m to 90 m, what we test here is the quality
of the initial products. Ground resolution
remains a separate issue. Although the com-
plexity of any land surface is likely to decrease
with increasing cell size, with ground resolu-
tion potentially affecting error distribution as
a consequence, the first step in digital terrain
analysis still involves choosing among an
increasingly wide range of imperfect digital
products, whether these are readily available
on the market or purpose-generated by stereo-
photogrammetry. Whatever the original ground
resolution of the digital product, it becomes
advantageous to resample or degrade the DEM
to the ground resolution best suited to the
application required only after quality checks,
such as those proposed in this paper, have been
performed. After the most satisfactory digital
product available has been selected in this way,
work on terrain roughness indices, which mea-
sure the variability of elevation based on a slid-
ing window specific to each DEM (Grohmann
et al., 2009), is one way of addressing the
effects of scale on the analytical accuracy
affecting the landscape attributes of interest.
II Evaluation of accuracy and errorpropagation in gridded DEMprocessing
According to Hunter and Goodchild (1997: 36),
the notion of uncertainty covers undefined error
(i.e. when a true value is unknown) and struc-
tural error. Undefined error covers a range of
subcategories defined by Thapa and Bossler
(1992: 836), including blunders, gross errors
and systematic errors (see Appendix). Often
these cannot be removed, whether by repeated
observation or by recourse to a deterministic
search function, because they tend to evade any
clearly detectable pattern. After Burrough
(1986) and Wise (1998), Wechsler (1999: 3) has
identified three possible sources of structural
error: (1) errors due to an incomplete density
of observations, or to consequences of spatial
sampling; (2) measurement errors such as posi-
tional inaccuracy, which might arise from
improper orientation of the stereo-model or by
neglecting lens distortion in photogrammetric
procedures; and (3) random errors, which can
normally be detected after removal of gross and
systematic errors (see Appendix) and might be
due to data entry faults, observer bias, or
computer-generated numerical errors, interpola-
tion errors, or problems of classification and
generalization.
Finer differentiation among structural errors
distinguishes between horizontal (positional
accuracy) and vertical components (attributes
accuracy), respectively. These may be difficult
to separate (Ehlschlaeger and Shortridge,
1996) but an expression both in terms of the
metric accuracy of elevation and in terms of
planimetric deviation values is possible (Arrell
et al., 2008: 945). Planimetric and metric accu-
racy have long been evaluated and calculated
from routine statistical methods such as mean
error (ME), mean absolute error (MAE), error
in standard deviation units (S), or as the RMSE,
which is the square root of the variance of a dis-
tribution (e.g. Carlisle, 2005: 522; Erdogan,
2010; Fisher and Tate, 2006: 470; Wechsler and
Kroll, 2006: 1082; see Appendix). Within that
class of statistical test, RMSE-based comparison
of DEM elevation values with sample values
from another, more accurate source seems to
be the most appropriate way of providing error
estimation (e.g. Arrell et al., 2008: 945; Hirano
et al., 2003; Wise, 2007: 1353). However, a
major drawback of the RMSE is that results
merely provide one global value per DEM grid,
i.e. they ignore the finer-scale spatial variability
of the error (Wechsler, 2003: 58) that often
4 Progress in Physical Geography
contains a great deal of useful information for
terrain-based applications. Global indices are
thus insufficient (Li and Wong, 2010: 252) and
any useful study of DEM accuracy should also
investigate the spatial variation of error values
(Carlisle, 2005: 523; Wood, 1994).
Methods of error investigation in DEMs have
been widely explored (Fisher and Tate, 2006;
Giles and Franklin, 1996: 1165; Wechsler,
2003: 58; Wilson and Gallant, 2000: 15). The
simplest methods are based on criteria such as:
(1) differences in elevation between adjacent
points (e.g. Binh and Thuy, 2008; Hannah,
1981); (2) systematic detection by inspection
of anomalous values within a given moving win-
dow (e.g. Chaplot et al., 2006; Felicısimo, 1994;
Zhou and Liu, 2004b); (3) the minimization of
total error, which is the sum of errors originating
from the input data sources and operational
errors generated by data manipulation (e.g.
Walsh et al., 1987; Younan et al., 2002); or
(4) spectral (e.g. Tempfli, 1980) and elevation
histogram analyses (e.g. Carrara et al., 1997;
Holmes et al., 2000), which each focus on the
detection of structure in the statistical distribu-
tion of elevation values and on the identification
of unexpected variations interpreted as indicat-
ing systematic or isolated error. More complex
methods introduce remote sensing and/or
geostatistical processes. Goodchild (1982) and
Polidori (1991: 12–20) have investigated the
value of Brownian processes in a fractal terrain
simulation model for improving DEM accuracy.
Brown and Bara (1994) used semi-variogram
and fractal dimension to describe the pattern of
systematic errors. Lopez (1997, 2000) devel-
oped a method based on principal component
analysis (PCA) to locate random errors in DEMs
and to extract uncorrelated patterns. After
comparisons between Felicisimo’s method
(1994) and Lopez’s method (Lopez, 1997),
Lopez (2000) revised the PCA approach in order
to handle the significant spatial autocorrelation
observed in the data. By identifying gross and
random errors, systematic errors were avoided.
Finally, Arrell et al. (2008) identified and
removed artifacts associated with gross error
using Fast Fourier Transform analysis as a means
of discriminating between signal and noise.
Techniques of error location and visualization
have also been considered. Fisher and Tate
(2006) have reviewed research in this domain
and noted that DEM contour maps could be a
useful tool for error observation, with the
possibility of detecting gross errors and blun-
ders. The reliability of this method, however,
is not proven. The method of Hunter and Good-
child (1995) based on the use of the RMSE as a
threshold criterion for testing the accuracy of
elevation appears more useful. Other more
sophisticated methods have focused on error
visualization by using TIN or discrete data and
by displaying in 3-D the magnitude of local dif-
ferences and curvature errors from contoured
DEM data (Gousie, 2005). Perhaps a more work-
able method, proposed by Oksanen (2003:
2412), detects and visualizes gross errors in
gridded DEMs from relief shading because of its
ability to render a 3-D effect on elevation data.
Wise (2007: 1352) has argued that visualization
techniques used in error identification could also
be efficient at demonstrating strong links
between error patterns and interpolation-related
distortions. Wood (1996: 64) demonstrated this
by developing a spline-fitting algorithm on a
fractal surface that allowed errors and other
impacts of the algorithmic process to be detected
and visually displayed. More recent attention
has been paid to DEM interpolation algorithms
and their error-causing impacts (1) on elevation
in the case of high g.r. data (e.g. LiDAR: Arrell
et al., 2008: 944; Bater and Coops, 2009;
Lindsay and Evans, 2008; Liu, 2008), (2) on the
calculation of mathematical derivatives of ele-
vation (e.g. Florinsky, 1998; Jones, 1998; Liu
and Bian, 2008; Schmidt et al., 2003; Zhou and
Liu, 2004a), but also (3) on flow simulation and
its repercussions on drainage network extraction
(e.g. Li and Wong, 2010; Lindsay and Evans,
2008).
Gonga-Saholiariliva et al. 5
In terrain analysis, error propagation analyses
have usually focused on slope and aspect as test
variables. These mathematical derivatives of
elevation are recognized to provide a more use-
ful test of error location and propagation, and
thus for DEM quality assessment, than elevation
itself because of their greater sensitivity to error
(Wechsler, 2003; Wise, 2000: 1911). Liu and
Jezek (1999: 250) have listed three approaches
for treating the propagation of DEM error. These
include empirical approaches (e.g. Bolstad and
Stowe, 1994; Gatziolis and Fried, 2004;
Lindsay, 2006), analytical methods (e.g.
Florinsky, 1998; Heuvelink et al., 1989; Leick,
1990; Oksanen and Sarjakoski, 2005) and
computer simulations (e.g. Canters et al., 2002;
Ehlschlaeger, 1998; Hebeler and Purves, 2009;
Oksanen and Sarjakoski, 2005; Wechsler and
Kroll, 2006). (1) The empirical approaches rely
on a derivation of the spatial properties of error
based on a more accurate reference data set,
where results for a series of simulations are pre-
sented with error fields characterized by a range
of spatial properties (Lindsay and Evans, 2008:
1590). This compensates for the often unknown
spatial characteristics of error. (2) Analytical
methods can involve Taylor series expansion
(Heuvelink et al., 1989). Under the assumption
that error distribution follows a Gaussian distri-
bution, this approach constitutes the basis for
least-squares adjustment techniques (Hunter and
Goodchild, 1997: 36). (3) The simulation meth-
ods involve stochastic approaches such as Monte
Carlo techniques. This requires a number of
equally probable outcomes upon which selected
statistics are performed. Uncertainty is quanti-
fied by evaluating the statistics associated with
the range of outputs under the assumption of
spatial autocorrelation. A series of random equi-
probable error maps, one of which could be the
‘true’ error map, is provided. Such simulations,
however, are non-unique and do not ensure that
the true error map has been generated by the pro-
cedure (Wechsler, 2006: 2360; Wechsler and
Kroll, 2006).
The analytical treatment of terrain algorithms
is complicated and laborious (Lindsay and
Evans, 2008: 1590). Oksanen and Sarjakoski
(2005) confronted an analytical approach with
Monte Carlo simulations to evaluate the propa-
gation of error in slope and aspect simultane-
ously. They concluded that the two methods
produced similar results even though the data
had not been inspected for spatial autocorrela-
tion in the analytical approach. The work of
Oksanen and Sarjakoski (2005: 1016) and Liu
and Bian (2008: 308) has revived a debate about
the value of performing analytical manipulations
of gridded DEMs that disregard the spatial
autocorrelation of error — the most important
consideration being the choice of autocorrela-
tion parameter rather than the shape of the spa-
tial autocorrelation. These aspects are presently
examined.
III Spatial autocorrelation and theproduction of error maps
Spatial autocorrelation characterizes the correla-
tion between a variable and itself (univariate
autocorrelation), or between that same variable
and another variable’s values (bivariate
autocorrelation), in neighbouring cells. Spatial
autocorrelation functions are used to quantify
the spatial regularity of a phenomenon (a form
of spatial complexity) (Aubry and Piegay,
2001). It is based on the assumption that in all
data sets near things are more related than distant
things (Tobler’s first law of geography). Some
approaches avoid the condition of heteroscedas-
ticity, i.e. the necessity to consider variations
in error value as a function of variations in
another variable. This occurs in the case of
spatial autoregressive models (e.g. Hunter and
Goodchild, 1997) and of semi-variance analysis
(e.g. Carlisle, 2005; Giles and Franklin, 1996).
Other approaches, such as pixel swapping and
spatial moving averages using low-pass filters
(e.g. Fisher, 1998; Wechsler, 2006), introduce
spatial correlation without explicitly addressing
6 Progress in Physical Geography
error patterns or mapping error surfaces
(Hebeler and Purves, 2009: 6). Methods such
as sequential Gaussian simulation (e.g. Oksanen
and Sarjakoski, 2005) use global spatial autocor-
relation information without examining the finer
structure of the error. Given that the study of
spatial autocorrelation and of its potential conse-
quences in gridded DEMs has been unsyste-
matic, here we investigate the definitions of
spatial autocorrelation, its uses, how to deal with
it, and how error maps displaying local autocor-
relation clusters can reveal error propagation
phenomena.
Moran’s I (Moran, 1950) is historically the
first index to have been used for understanding
spatial autocorrelation, and still a standard.
In the context of gridded DEMs, Moran’s I is a
global indicator of spatial association and is used
to measure average deviations in slope or eleva-
tion for an entire grid. This global measure is
given by the following formula:
I ¼ NPi
Pj wij
�P
i
Pj wijzizjPi z2
i
ð1Þ
When there are N units, zi (zi ¼ xi � �x) is
the variable value at particular location, zj
(zj ¼ xj � �x) is the variable value at another
location, is the mean of the variable, and wij
expresses the level of interaction (connectivity)
between units i and j. This contiguity matrix
compares the sum of the cross-products of
values at different locations, two at a time,
weighted by the inverse of the distance between
the locations (if i is next to j, the interaction
receives a weight of 1). Moran’s I ranges
betweenþ1 (strongly positive) and –1 (strongly
negative spatial autocorrelation). A value of
0 indicates a random spatial pattern.
From this index Anselin (1995) later
elaborated the LISA, which can also be used
on continuous DEM grid surfaces. Here we use
Moran’s I and LISA as techniques for
visualizing error propagation, and for adding
information about spatial autocorrelation not
provided by Moran’s I. LISA is used for estimat-
ing the local contribution of each pixel value to
the global index. Local indicators are based on a
gamma statistic (Getis, 1991) given by the fol-
lowing formula (Oliveau, 2005):
G ¼X
i
XjwijWij ð2Þ
where Wij ¼ ðzi�zÞðzj�zÞPiðzi�zÞ �
nm
and Gi ¼P
j wijWij
In these formulae, variables are as in equation
(1), n is the total number of subjects in the cross
section (P
i), m is the neighbourhood pair num-
ber (P
i
Pjwij), and Wij is a measure of relation-
ship in attribute space between data sites i and j.
LISA measurements take into account spatial
and numerical similarities between contiguity
matrices. The LISA of each observation gives
an indication of the extent of statistically signif-
icant spatial clustering of similar values around
the observation point. The LISA is obtained by
local spatial clustering procedures using the
following equation (Anselin, 1995):
Ii ¼ zi
Xj
wijzj ð3Þ
with the sum of local indicators:Xi
Ii ¼X
i
zi
Xj
wijzj ð4Þ
where zi and zj are expressed in deviations from
the mean, the summation of is such that only
neighbouring values j 2 Ji are included, and is
the contiguity matrix of weights. The sum of
local indices is proportional to Moran’s I from
g¼P
i
Pj
wijm2; with m2 ¼P
i
z2i =n. LISA
results can be standardized by dividing the
LISA by. The LISA helps to identify local out-
liers (where negative local spatial autocorrela-
tion shows up on the cluster map as high–low
or low–high occurrences, meaning that the
Gonga-Saholiariliva et al. 7
variable of interest exhibits dissimilar values
among its neighbours); and local clusters
(where positive local spatial autocorrelation
shows up on the cluster map as high–high or
low–low occurrences, meaning that the variable
of interest shows values that are similar to its
neighbours). Local deviations from the global
pattern of spatial autocorrelation can also be
delineated.
Bivariate LISA-type cluster analysis explores
the spatial distribution of DEM errors affecting
elevation and measures its transmission to errors
affecting slope gradient. It follows a scheme of
pattern recognition (Jacquez, 2008: 400) made
up of five components: (1) a null hypothesis,
H0, which describes the spatial pattern expected
when the alternative hypothesis is false (i.e. an
absence of significant clustering of errors in
slope at sites where errors in elevation occur);
(2) an alternative hypothesis, HA, which states
that the clustering of errors in slope closely
matches the spatial distribution of errors in
elevation; (3) a statistical significance test that
quantifies the spatial pattern of the data; (4) a
neutral spatial model, which is a numerical
device for generating the reference distribution
based on a Monte Carlo simulation (achieves
randomly operated permutations of values);
(5) a reference distribution, which is the
distribution of the statistical test scores when
the null hypothesis is true, and is set by the
p-values, which in statistics measure strength
of evidence. For the LISA tests, pseudo
p-values are computed as (mþ1) / (rþ1),
where r is the number of replications and m
is the number of instances where a statistic
computed from the permutations is either
equal to or greater than the observed value
(in the case of positive LISA scores), or equal
to or smaller than the observed value (for
negative LISA scores) (Anselin, 1995). Results
are filtered with the choice of an arbitrary
significance threshold, a (0.05, 0.01, 0.001, or
0.0001), which determines confidence intervals
used for assessing results.
IV Patterns of error in griddedDEMs: a worked example fromthe Wasatch Mountains, Utah,USA
Among the few existing studies to have explored
the local structure of spatial autocorrelation in
raster analysis, Goovaerts et al. (2005) detected
anomalies on high-resolution hyperspectral
imagery while testing the significance of LISA
values through randomization. Erdogan (2010)
investigated spatial variation as well as spatial
associations of error by comparing a range of
interpolation methods. Monckton (1994) used
Moran’s I to simulate the spatial structure of
error in elevation data. Results, however, were
inconclusive, probably because of the spatial
lags (which exceeded 250 m) that were used in
dealing with spot heights on Ordnance Survey
maps (Carlisle, 2005: 524).
In the following worked example designed to
illustrate the power of autocorrelation analysis
as a tool in DEM error evaluation, we performed
tests on a *29 km2 study area encompassing
part of the Weber Segment, one of the longest
(61 km) and seismically most active segments
of the Wasatch normal fault in Utah (Figure 1).
On this segment, faceted spurs and their triangu-
lar mountain-front hillslopes are well preserved
(McCalpin and Berry, 1996; Nelson and Perso-
nius, 1993; Petit et al., 2009; Zuchiewicz and
McCalpin, 2000). The geomorphology of fault-
generated range fronts is reflected in the drai-
nage basin morphology as well as in the faceted
spur geometry. In order to extract precise terrain
parameters with minimal bias, DEM evaluation
was examined through slope analysis. Using
both LISA and Moran’s I, the methodology aims
to single out the DEM best suited to geomorpho-
metric or hydrological applications. The best
DEM should provide the model with the
smallest error, i.e. with the highest accuracy and
the lowest level of error clustering.
The grid characteristics of the six DEMs
available for the study area (see Introduction),
8 Progress in Physical Geography
with differing column and row grid-cell sizes,
are as follows (Figure 2) – LiDAR (2 m): 2979
� 2370; CODEM (5 m): 1260 � 857; NED10
(10 m): 630 � 474; AstD (15 m): 420 � 316;
GDEM (30 m): 210 � 857; SRTM (90 m):
70 � 53. General statistics are usually available
for each DEM from supporting documentation,
in which major errors and correction processes
are mentioned. Out of the six in this study,
only the AstD was generated directly from raw
data by the authors using digital stereo-
photogrammetric techniques summarized in
Figure 3. The operational error we use is the dif-
ference between the z values provided by each of
the five tested DEM grids on the one hand, and
the z values provided by the LiDAR data on the
other, the latter being closest to any ‘true’ eleva-
tion measurement. Any observed difference in
z is interpreted as being due to one or several
of the error categories defined in the Appendix.
Those refer to the various defects originally
present in the gridded elevation data. Their
distribution pattern across the grid remains
unknown until we proceed to the mapping tech-
niques proposed later in this worked example.
They depend on the type of sensor from which
the data was collected, and how this data was
preprocessed before making it available for use
by geographers.
1 Vertical accuracy
The first approach to error estimation relied on
the RMSE, which integrates errors relating to
interpolation methods and to geographical posi-
tion in x, y and z. Vertical accuracy assessment at
pixel scale was first calculated from the LiDAR
model (RMSE & 0.086 m) with a regular
sample extraction interval of 100 m, each point
extracted (3000 for each DEM) being a substitute
Figure 1. The Weber Segment in the Wasatch Mountains, Utah, USA. Elevations range from 257 m to 3618 m.The 2-D and 2.5-D views highlight the active normal boundary fault with its triangular faceted spurs.
Gonga-Saholiariliva et al. 9
for ground GPS information (see Appendix for
details). Given that, for comparison purposes,
the grids could not be used with their original
characteristics, DEM resampling was also nec-
essary. Accordingly, for the RMSE calculation
resampling procedures were performed on each
DEM by transforming cell size to the reference
size, i.e. 2 m. To avoid stair-stepping and
jagged artifacts in the resampling process, a
bicubic spline resampling method was used
instead (see Appendix) (Keys, 1981). The
resampled grids were used for a comparative
assessment of vertical accuracy between
DEMs. In order to obtain percent values of
transformation from the resampled DEM, a
new RMSE value, noted RMSEc, was
calculated. The formulae given in the Appendix
were applied to the entire DEM grid and assess-
ment of the vertical accuracy of surface objects
was carried out using the RMSEc criterion.
Figure 4 and Table 1 illustrate the distribution
of error values in elevation among the six DEM
grids. Quartile distributions (first and third)
describe a band of +8 m for minimum and
+22 m for maximum deviations from the mean
elevation (Table 2). Overall, DEM height distri-
butions exhibit a similar pattern despite the
fact that the elevation sources are different
(i.e. airborne, radar, optical or compilation data).
The more outstanding exception is the AstD,
Figure 2. Extract of each studied DEM representing the different grid-cell sizes used in the worked example.(A) LiDAR data (2 m, available at http://gis.utah.gov/elevation-terrain-data/2-meter-bare-earth-lidar). (B)CODEM (5 m, http://gis.utah.gov/elevation-terrain-data/5-meter-auto-correlated-elevation-model-dem).(C) NED10 (10 m, http://gis.utah.gov/elevation-terrain-data/national-elevation-dataset-ned). (D) ASTERDEM (15 m, http://www.echo.nasa.gov/reference/astergdem_tutorial.htm). (E) GDEM (30 m http://aster-web.jpl.nasa.gov/gdem.asp). (F) Shuttle Radar Topography Mission, or SRTM (1 arc second, i.e.*90 m, http://www.cgiar-csi.org/data/elevation/item/45-srtm-90m-digital-elevation-database-v41).
10 Progress in Physical Geography
which presents significant offsets with respect to
the other DEMs. These offsets attain a difference
in values of about +15 m (roughly equivalent to
one AstD pixel) for height and RMSE measure-
ments as well as for statistical moments. This
exception is ascribable to the initial AstD extrac-
tion process, in which the precision of the DEM
tools used was greater than half a pixel (see Fig-
ure 3). The initial RMSE for the AstD (11.8 m)
was accordingly too large. The global RMSE
should be about half a pixel (7.5 m). Additional
ground control points were checked in order to
newly interpolate the AstD, finally providing it
with elevation ranges similar to the other DEMs,
i.e. [1355–1298 m]. As a result, the new value
obtained for the AstD RMSEc (Table 2) fell into
line with the overall RMSEc trend for the other
DEMs, i.e. about 12.9 m. Thus RMSEc results
(incorporating new values given in brackets for
the AstD) yield results similar to those obtained
for the other DEMs: the larger the grid cell size,
the higher the RMSEc [þ5.7;þ18.4] and RMSE
[þ4; þ16 m]. The key conclusion at this stage
is that whatever the DEM’s data source,
Figure 3. Flow chart explaining the ASTER DEM extraction process from the stereoscopic VNIR 3N andVNIR 3B bands. VNIR: visible near-infrared bands; SWIR: short-wave infrared bands; TIR: thermal infraredbands. VNIR, SWIR and TIR are ortho-rectified bands (geometric corrections). GCP are ground controlpoints extracted from the ortho-rectified VNIR bands. The RMSE is calculated for the (x,y) plane coordinatesand the vertical plane z.
Gonga-Saholiariliva et al. 11
DEM generation does not produce large errors in
z. Values for the SRTM and GDEM also fall
close to the reference LiDAR value and thus
perform almost as well as the most accurate
reference grids despite their comparatively
lower initial resolution.
As a further test, linear regression analysis
was carried out between the elevation data of
each DEM and the LiDAR reference grid,
respectively (Figure 5). Overall, the DEMs pro-
duced a goodness-of-fit, measured by the deter-
mination coefficient r2, close to þ1, with
intervals ranging betweenþ0.9979 for the worst
fit and þ0.9999 for the best. The CODEM and
NED10 show the strongest correlation with the
LiDAR, followed by the AstD and the GDEM.
Model fits reveal outliers at different locations
in the AstD, GDEM and SRTM scatter plots.
The outliers reflect some of the error present
within the DEM, which appears largest in the
case of the AstD. However, neither data scatter
in the linear regression models nor vertical accu-
racy constitute sufficient criteria to achieve a full
definition of DEM error because the error is
expressed globally (lacks any indication of spa-
tial distribution), and because when untested
spatial autocorrelation occurs in data, linear
regression models tend to overestimate both
strength of correlation and precision.
2 Residual maps and computation ofprimary topographic attributes
One approach to locating error distribution
patterns visually can be achieved through the
creation of so-called deterministic error
surfaces. These can be generated for a DEM
by using reference data assumed to possess
higher accuracy than the DEM (Hebeler and
Purves, 2009), in this case the LiDAR, but
equally ground control points or other surface
model data (Holmes et al., 2000). In this way,
maps of differences between the reference grid
and other grids, termed here residual maps of
either elevation or slope gradient, can provide
an understanding of error sources and form a
basis for further analysis (Van Niel et al.,
2004). Residual maps were generated by apply-
ing an algebraic operation of pixel by pixel
subtraction between each DEM and the LiDAR.
In order for these tests to be meaningful, degra-
dation of the LiDAR to the relevant ground
resolutions, i.e. from 5 m to 90 m, was necessary.
This operation resulted in a series of numerical
images representing Re, the pixel-wise residual
value (Figure 6).
Figure 6 shows that (1) the range of Re stan-
dard deviations is similar (0–11 m) for the three
most accurate DEMs, namely CODEM, NED10
and AstD. (2) The SRTM and GDEM show the
greatest deviations (up to 41 m) from the refer-
ence surface, which is in keeping with the lower
ground resolution of these two data sets. Further-
more, the bicubic spline resampling procedure
resulted in artificial smoothing of the surfaces,
an aspect clearly apparent in the SRTM residual
map (Figure 6E). (3) High values of standard
deviation tend to cluster around the same sites
in the landscape whatever the grid-cell size, with
detectable variations in the sharpness of the dis-
continuities. (4) The highest values, i.e. greater
Figure 4. Statistical results for vertical accuracyassessment. Cumulative elevation (in %) versus ele-vation values for the studied DEM grids. The offsetin the ASTER DEM curve reveals a shift (systematicerror) requiring systematic correction to achieverealignment with the other grids (see text).
12 Progress in Physical Geography
than 9 m, follow narrow valley floors in the
mountainous relief. Value distributions differ
in the piedmont, where they are located along
wider valleys. (5) High deviations mostly affect
slopes and valley bottoms. Where hillslopes are
affected by anomalous deviations, errors are
likely to propagate to higher order topographic
attributes (e.g. slope curvature), and hence
adversely affect predictions made from the
DEM for applications such as hydrology or nat-
ural hazards. (6) Flat areas, which predominate
in the southern part of the piedmont zone,
display values that are close to the minimal
deviation, i.e. 0 m. This is most visible with the
higher grid-cell sizes in Figure 6, B, C and D.
Flat areas, which are often problematic in hydro-
logical network extraction (e.g. Tarboton, 1997),
thus seem to be less affected by high deviations.
However, this finding does not exclude the
possibility that erratic features, such as pits and
holes, could be present in this kind of area.
Figure 6E (SRTM) shows that regions affected
by large error are larger, form continuous
patches, and are concentrated in valleys and on
elevated ridges whereas they are absent or more
discontinuous in the other DEMs.
The following step in the procedure involved
the computation of slope gradient values in order
to assess error patterns in slope gradient. Slope
maps depend on the choice of algorithm and
on grid-cell size. We used the Zevenbergen and
Thorne (1987) algorithm due to existing
consensus over its relative robustness and con-
sistent behaviour in various terrain geometries
(e.g. Florinsky, 1998; Jones, 1998; Liu and Bian,
2008; Zhou and Liu, 2004b). Calculations were
Table 1. Distributions of elevation in the six DEMs.Min and Max are the minimum and maximum values; 1stand 3rd Qu. are the first and third quartile. For the ASTER DEM, corrected values (see text) are in brackets.For other DEMs, tabulated values are initial values.
Elevation value (m)
Data source SRTM GDEM ASTER NED10 CODEM LiDAR
Min 1355 1355 1340 (1355) 1357 1360 13511st Qu. 1775 1769 1754 (1769) 1769 1776 1771Median 2076 2073 2057 (2072) 2070 2078 2071Mean 2073 2069 2051 (2065) 2067 2073 20693rd Qu. 2366 2358 2338 (2352) 2358 2363 2359Max 2897 2884 2871 (2886) 2892 2900 2900
Table 2. RMSE results for the six DEMs.STD is the standard deviation and ME is the mean error for eachDEM grid. RMSE is the initial RMSE with respect to the reference geoids used for the production of eachDEM.RMSEc is the new RMSE calculated with respect to the LiDAR reference grid.
Error values (m)
DEM Grid cell size STD ME RMSE RMSEc RMSEc–RMSEr
LiDAR (reference) 2 387.3 Ø þ0.86 Ø ØCODEM 5 408.3 4 þ4 þ5.6 þ1.6NED10 10 406.5 1.9 &7 þ6.7 þ0.3ASTER DEM 15 403.9 21.8 (3.1) þ11.8 (þ9.4) þ25.9 (þ12.9) þ14.1 (þ3.5)GDEM 30 405.1 0.5 þ9.35 þ13.6 þ4.25SRTM 90 407.0 3.7 [þ9;þ16] þ18.4 [þ2.4;þ9.4]
Gonga-Saholiariliva et al. 13
Figure 5. Scatter plots of elevation values (z) produced for various permutations of the studied DEM grids
Figure 6. Maps of Re standard deviation for elevation in the Weber Segment (colour version available online).The residual values in maps A to E express the difference between z values in each DEM and the z valuesprovided by the LiDAR reference. (A) CODEM, cell size¼ 5 m. (B) NED10, cell size¼ 10 m. (C) ASTER DEM,cell size ¼ 15 m. (D) GDEM, cell size ¼ 30 m. (E) SRTM, cell size ¼ 90 m.
14 Progress in Physical Geography
carried out on each of the six grids (Figure 7).
Slope values range from 0� to 89.9� (Table 3).
Lower values (0–1�) indicate flat interfluves or
valley bottoms, whereas the highest slope angles
represent unusual terrain values, typically
greater than 35� and located on faceted spurs.
Standard deviation and mean slope decrease
with grid-cell size, in line with previous research
on this topic (e.g. Kienzle, 2004; Warren et al.,
2004) but unlike results obtained for DEMs
extracted from contour lines (e.g. Ziadat, 2007).
The distribution of slope values is representa-
tive of the general slope trend for all the DEMs
(Figure 8). Values of 0–30� represent 98% of the
slope frequency range in the SRTM, whereas
other DEMs exhibit a greater proportion of
Figure 7. Slope maps of the Weber Segment as generated from each of the DEMs (colour version availableonline). (A) LiDAR, cell size ¼ 2 m. (B) CODEM, cell size ¼ 5 m. (C) NED10, cell size ¼ 10 m. (D) ASTERDEM, cell size ¼ 15 m. (E) GDEM, cell size ¼ 30 m. (F) SRTM, cell size ¼ 90 m.
Table 3. Slope values for the six DEMs.Slope-angle range is the interval between minimum and maximumslope; mean slope angle is the arithmetic mean of the slope value distribution; Std slope is the standard devia-tion of that distribution.
DEM Slope-angle range(degrees) Mean slope angle(degrees) Std(degrees)
LiDAR 0–89.9 26.33 1.52CODEM 0–75.8 26.08 1.37NED10 0–89.7 26.26 1.34ASTER 0–89.6 23.43 1.29GDEM 0–89.4 23.14 1.30SRTM 0.3–42.2 22.37 1.06
Gonga-Saholiariliva et al. 15
extreme values (for example, slope angles
greater than 40� account for 10% of the slope
frequency distribution). Field data have revealed
that the Weber Segment mountain front is char-
acterized by slope angles never exceeding 43�
(Eblen, 1995; Stock et al., 2009), with mean val-
ues close to 25–26�. The highest values, between
45� and 90�, are thus anomalous and represent
errors due to DEM uncertainty, which were propa-
gated to the slope map during transformation of the
elevation data. The geographical location of large
errors in slope coincides for each DEM with that of
the largest errors in elevation (Figure 7). For
angles up to 45�, slope values also increase with
cell size. However, anomalous values are fewer
for the larger grid-cell sizes than for the smaller
ones (CODEM, NED10, AstD). This also means
that the larger the grid-cell size, the greater propor-
tion of values <45� and the fewer outliers present.
Although results are so far directly ascribable
to misfits originally affecting z, results differ
between the mean and lower slope values. The
lower 50% of the distribution frequency is
marked by several offsets between the highest
and smallest grid-cell sizes (Figure 8): for exam-
ple, contrary to the expectation that error is
consistently proportional to grid-cell size,
between 0� and 25� the slope distribution values
of the CODEM increase faster than for the
NED10. Also, in the low range (i.e. 0� to less than
15�) the AstD curve locally crosses the SRTM
and GDEM curves. Consequently, lower and
mid-range slope values appear to reflect isolated
discrepancies in the DEM elevation structure.
The standard deviation of residuals (difference
between reference and calculated slope) was also
calculated (Figure 9), illustrating for each DEM
the deviation in mean distribution values for each
grid. In the six DEMs, the thalwegs of larger
valleys are well marked. Note also that the loca-
tions of slope errors are almost identical to the
locations of the errors in elevation except on
the main central facets of the Weber Segment
(Figure 9). The higher the terrain resolution, the
more sensitive the gridded data to errors in slope,
which are thus also linked to terrain roughness
characteristics. Accordingly, slope deviations
range between 0� and 14� for the larger grid-
cell sizes (SRTM, GDEM) and between 0� and
25� for the smaller (LiDAR, CODEM, NED10,
AstD; Figure 9).
The strength of the relationship between
errors in slope and errors in elevation (Figures
6 and 9) was tested by performing a linear
regression between elevation (Re on elevation)
and slope residuals (Re on slope). A non-linear
relationship between those two variables was
found (Figure 10). However, the form and inten-
sity of the relationship is biased by several out-
liers that stretch the scatter plot in several
directions. Determination coefficients with
values <0.3 for the SRTM and CODEM and
<0.1 for the AstD, GDEM and NED10, indicate
weak to no dependency between the errors in
slope and in elevation. In other words, only
10–30% of anomalous slope values (in particular
among the lower and mid-range values) can be
explained by anomalous elevation values.
Figure 8. Comparative distributions of slope values for each DEM grid
16 Progress in Physical Geography
Such a conclusion, however, is potentially
spurious because r2 is not a robust criterion for
explaining either the numerical or the spatial
behaviour of errors affecting slope. Accordingly,
as a further tool for evaluating whether higher
error affecting elevation was positively correlated
with higher error affecting slope, we inspected the
data grids for spatial autocorrelation.
3 Spatial autocorrelation analysis
Calculation and 2.5-D mapping of Moran’s I and
LISA were used to identify areas with a high risk
of error caused by pixels being surrounded by
similar areas or by spatial outliers, and as tools
for visualizing the location of uncertainty.
A spatial weighting matrix was used for each
variable, and was defined at varying distance
lags corresponding to the respective DEM reso-
lutions. The weighting process was carried out
using Rook’s case–contiguity kernel, with sym-
metrical spatial weights wij ¼ 1 when locations
i and j are adjacent, and wij ¼ 0 in other cases.
Rook’s case considers the neighbourhood of
four locations adjacent to each cell (Elobaid
et al., 2009). A Monte Carlo randomization pro-
cedure allowed us to assess significance levels
by performing 999 permutations (i.e. tests) for
each grid (Table 4).
The randomization test avoids relying on
assumptions about data distribution and devel-
oping statistical moments under a null hypoth-
esis, and avoids testing all types of statistic
analytically (Aubry and Piegay, 2001). Signifi-
cance levels are given by the p-values (arbitrary
threshold), are calculated for each cluster, and
Figure 9. Maps of Re standard deviation for slope angle in the Weber Segment (colour version available online).Key as in Figure 6.
Gonga-Saholiariliva et al. 17
allow significant autocorrelation values to be
identified. After trial comparisons with other
thresholds (0.0001, 0.001, and 0.01) for each
DEM, p-value significance levels were set to
0.05 because this configuration yielded the high-
est and most sharply differentiated levels of data
clustering. Table 4 summarizes global clustering
levels for bivariate and univariate variables,
each time for a run of 999 permutations.
The CODEM grid, with a Moran’s univariate
I of about –0.06, illustrates a situation of low
negative spatial autocorrelation, implying that
neighbours are dissimilar and that error values
affecting elevation do not have a unique source
or cause. In contrast, error values in the other
DEMs obtain a Moran’s index close to þ0.9,
indicating a strong spatial link between errors
in elevation. Error in slope systematically attains
values close to þ1, with a high level of spatial
dependence showing strong positive spatial
autocorrelation values. Where high spatial auto-
correlation scores were observed for elevation,
values likewise increased for slope, thus con-
firming a mechanism that drives the dispropor-
tionate (i.e. non-linear) propagation of errors in
elevation to errors in slope gradient. The bivari-
ate Moran’s index, which describes the magni-
tude and relationship of global error between
elevation and slope, is characterized by positive
values but none of these are close to þ1.
The strongest positive relations are observed for
the GDEM (þ0.4870) and SRTM (þ0.3048)
grids, whereas the lowest occur in the AstD
(þ0.1125). Results are thus similar to the uni-
variate values except for the AstD. The SRTM,
GDEM and NED10 present a high level of
clustering, with the CODEM ranking last.
The LISA helps to bring out local magnitudes
of error clustering. Each observation provides a
measure of the extent of statistically significant
Figure 10. Scatter plots highlighting links between errors in elevation and in slope for various permutationsof the studied DEM grids
18 Progress in Physical Geography
spatial clustering of similar error values around a
data point. LISA results reveal a high level of
differentiation within each DEM (Figure 11).
High–high and low–low clusters define a high
error value surrounded by high values and a low
value surrounded by low values, respectively.
Such situations are present in the SRTM, GDEM
and to a lesser extent the NED10 grids. A higher
risk of error propagation occurs where high–
high values are found. Relative stability, i.e.
a low risk of error propagation to higher order
topographic derivatives, prevails in low–low
situations. High–high clusters are found in the
same locations as in the univariate elevation and
slope error maps. In those cases, therefore, the
LISA maps merely corroborate previous results.
Uncertainty in slope is thus highly correlated
with initial elevation, and thus with error in
elevation, which in turn conditions the spatial
distribution of errors.
Table 4. Autocorrelation results for errors on elevation and on slope.Moran’s I elevation and Moran’s Islope are the univariate Moran’s I calculations for the dependent (elevation) and independent (slope) vari-ables, respectively. Bivariate Moran’s I refers to the errors in elevation and in slope and is the result of themultivariate LISA test.
DEM r2 Moran’s I elevation error Moran’s I slope error Bivariate Moran’s I
CODEM 0.24392 –0.0573 0.9149 0.1403NED10 0.07332 0.7523 0.9226 0.2772ASTER 0.01236 0.6729 0.9218 0.1125GDEM 0.09374 0.7702 0.9261 0.4870SRTM 0.20950 0.8869 0.9149 0.3048
Figure 11. Uncertainty cluster maps based on the LISA (colour version available online)
Gonga-Saholiariliva et al. 19
For grids other than the SRTM, GDEM and
NED10, LISA maps show somewhat different
results. AstD maps show all the peak occur-
rences already found in the SRTM and GDEM
grids, but low–high and high–low clusters are
more prominent between high–high clusters.
High–high levels are typical of major thalwegs
and low-lying areas. In the case of the NED10,
error distribution patterns bring out geomorpho-
logical features (particularly ridges and escarp-
ments) more sharply than the SRTM and the
GDEM. This arises whatever the level of p-value
significance (tested from 0.0001 to 0.05). The
CODEM stands as the exception: uncertainty
already detected in previous maps (Figures 7 and
9) is well represented in the CODEM LISA map
but fails to show the high or low clustering levels
encountered in the other DEM grids. Clustering
is less intense (note the predominance of half-
tones), illustrating uncertainty in the data, i.e.
uncertainty not linked to terrain characteristics
and presumably caused by other unknown fac-
tors which would remain to be investigated.
IV Summary and conclusion
The aims of this article were: (1) to review suit-
able tools for assessing error in remotely sensed
data and for calibrating DEM studies that focus
on primary topographic attributes; (2) to investi-
gate a statistical tool more often used in ecology
and human geography than in digital terrain
analysis, namely LISA clustering, which helps
to produce and evaluate error maps in order to
track DEM error propagation; and (3) to assist
in decision-making when selecting among a
range of DEM products that are readily available
for a chosen study area. In a worked example
involving terrain analysis in the Wasatch Moun-
tains of Utah, we were able to use LiDAR data as
a surrogate for field control points. We tested
five other, freely available, DEM grids against
this very high-resolution reference grid.
Despite the existing catalogue of methods for
removing various categories of error in gridded
data, the overriding conclusion at present is that
errors can never be totally erased from DEMs.
Empirical approaches work best when
higher-quality reference data are available for
comparison and allow a range of statistical rank-
ing procedures to be carried out. Initially, basic
quantitative tools such as the RMSEc were of
little use in determining DEM accuracy. It was
shown that the larger the grid-cell size, the lower
the accuracy, with nevertheless a fair degree of
reliability in z whatever the sensor type and
whatever the original data ingredients entered
into the DEM. The CODEM, NED10, AstD,
GDEM and SRTM could therefore be used
almost indiscriminately for basic displays of
topography, depending simply on choice of
resolution for graphic output.
As an improvement on RMSE procedures we
generated a series of residual elevation and
residual slope maps (Re), which provided an out-
line of the spatial distribution of errors based on
calculations of the standard deviation of Re.
Two main conclusions arose from these find-
ings: (1) errors in slope and in elevation are
linked, but with a variable strength of depen-
dency as shown by linear regression results; and
(2) the SRTM, GDEM and NED10 stand out for
their lower error scores. Irrespective of their
ground resolutions, the CODEM and the AstD
presented either more extreme error values or
more confused results. This finding argues
against preferring these data sources for
morphometric or hydrological applications. Re
results emulated a further approach in error
analysis using Moran’s I and LISA maps.
This approach has revealed that the intensity of
spatial autocorrelation is not linked to grid-cell
size. It is more closely related to the intrinsic
characteristics of DEM products and to proce-
dures dealing with slope gradient calculation.
The LISA maps also help to define error hot- and
coldspots, i.e. high–high and low–low clusters.
Among the five DEMs analysed in this study, the
autocorrelation tests finally allowed the NED10
to stand out as the DEM least affected by the
20 Progress in Physical Geography
presence of error and its propagation. Like the
CODEM, the NED10 is a hybrid DEM compiled
from a range of digital and non-digital sources
rather than from unique radar or optical sources,
and also happens to be of a lower native resolution.
The trends and topics reviewed here have
emphasized the value of exploring gridded data
structures for detecting spatial error patterns
because results can be used for ranking DEM
quality. Such procedures seem critical for most
practical terrain applications in natural catch-
ments, but also for using the DEM grids as input
substitutes for ‘real topography’, i.e. as initial
conditions for running predictive surface pro-
cess models (e.g. Codilean et al., 2006) and
observing how the landscape evolves under
given conditions (heuristic terrain modelling).
Despite its advantages, the LISA still remains
a semi-quantitative tool (values express varia-
tions in clustering as ‘high’ or ‘low’). A move
towards more fully quantitative characteriza-
tions of error in DEMs, where, in addition to its
location, the intensity of error is provided
(deviation of elevation in metres, and/or slope
in degrees), could perhaps be achieved by
coupling the LISA with other error modelling
methods such as geostatistical simulations based
on kriging, or geographically weighted regres-
sion (Fotheringham et al., 2002).
Appendix
(1) Definitions of error
(a) Artifacts, blunders and gross errors
Reuter et al. (2009) defined artifacts as erratic
features that appear on a DEM as holes, ghost
lines, stripes and so on. Artifacts and associated
errors could even have been invisible in the
input DEM but can be inferred from unrealistic
values occurring on elevation derivative maps.
Blunders are vertical errors associated with data
collection. They can be mistakes caused by
misreading contours, by transposing numerical
values, by erroneous correlations, or by careless
observations (Wechsler, 2003). Gross errors
occur when a measurement process is subject
to occasionally large inaccuracies and generates
data outliers. It includes topographical misinter-
pretations, and long linear features along some
boundaries of the calculation window used in
DEM interpolation (Oksanen, 2003). Some
techniques such as ‘data snooping’ are efficient
at detecting them (Karras and Petsa, 1993), and
inspection of a shaded relief map can also help
(Gousie, 2005).
(b) Systematic errors
Sytematic errors result from deficiencies in
measurement or processing and affect the accu-
racy of the final data, i.e. the degree to which the
final data agree with reality. Systematic errors
are also identified as residual systematic error
surfaces caused by slightly inaccurately esti-
mated lens distortion parameters (Wise, 2000),
by translation effects, or false features such as
phantom tops, ridges, benches or striations
(Raaflaub and Collins, 2006). In general, they
reflect bias inherent in the data collection
method (Reuter et al., 2009).
(c) Random errors
Random errors result from mistakes such as
inaccurate surveying or improper recording of
elevation information. Often associated with
signal noise, random errors are extracted after
known blunders and systematic errors have
been removed (Podobnikar, 2009; Reuter
et al., 2009; Wechsler, 1999, 2006; Zhang and
Cen, 2008).
(d) Uncertainty
Uncertainty is a term employed when a mod-
elled value deviates from its true value, with a
confidence interval (Hebeler and Purves,
2009). Lack of knowledge about measurement
reliability and about its representation of true
values is referred to as its uncertainty and is a
measure of what we do not know (Wechsler,
1999, 2006). DEM uncertainty thus covers lack
of knowledge about parameters, model assump-
tions, and scenarios, i.e. spatial evolution mod-
els (Liu and Bian, 2008).
Gonga-Saholiariliva et al. 21
(2) Equations and formulas
(a) Root mean square error and mean error
RMSEðzÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
n
Xn
i¼1
zdem � zref
� �?
s
ME ¼ 1
n
Xn
i¼1
zdem � zref
� �
where zdem is DEM elevation, zref is reference
elevation, and n is the number of test points.
(b) Bicubic spline formula
pðx; yÞ ¼X3
i¼0
X3
j¼0
aijxiyi
where p is the interpolated surface, x and y are
rectangular coordinates of point (i,j), and a is the
polynomial coefficient. This interpolation
maintains the continuity of the function and its
first derivatives (both normal and tangential)
across cell boundaries.
(c) Residual maps and standard deviation
algorithmsPixel by pixel algebraic subtraction
of elevation values provides a series of residual
values (Re) that can be mapped:
Reði;jÞ ¼ zði;jÞdem � zði;jÞref
where i and j are lines and columns of the grid,
and and are the elevations of the DEM and the
reference grid, respectively. The standard
deviation of Re is calculated as follows:
Std ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni¼1 zdem � zref �ME� �2
n� 1
s
where ME is the mean error of the DEM. Stan-
dard deviations were calculated each time for
the entire image, cell by cell, yielding a map
of Re values in which deviations in elevation
values from those of the reference model can
be located.
Acknowledgements
We are grateful to Editor-in-Chief Nicholas Clifford
for steering our manuscript through a high-quality
reviewing process, with fine reports from several
anonymous referees who greatly improved the scope
and structure of this contribution.
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