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RESEARCH ARTICLE
10.1002/2015WR018299
Error modeling of DEMs from topographic surveys of rivers
using fuzzy inference systems
Sara Bangen
1
, James Hensleigh
1
, Peter McHugh
1
, and Joseph Wheaton
1
1
Department of Watershed Sciences, Utah State University, Logan, Utah, USA
Abstract Digital elevation models (DEMs) have become common place in the earth sciences as a tool to
characterize surface topography and set modeling boundary conditions. All DEMs have a degree of inherent
error that is propagated to subsequent models and analyses. While previous research has shown that DEM
error is spatially variable it is often represented as spatially uniform for analytical simplicity. Fuzzy inference
systems (FIS) offer a tractable approach for modeling spatially variable DEM error, including flexibility in the
number of inputs and calibration of outputs based on survey technique and modeling environment. We com-
pare three FIS error models for DEMs derived from TS surveys of wadeable streams and test them at 34 sites
in the Columbia River basin. The models differ in complexity regarding the number/type of inputs and degree
of site-specific parameterization. A 2-input FIS uses inputs derived from the topographic point cloud (slope,
point density). A 4-input FIS adds interpolation error and 3-D point quality. The 5-input FIS adds bed-surface
roughness estimates. Both the 4 and 5-input FIS model output were parameterized to site-specific values. In
the wetted channel we found (i) the 5-input FIS resulted in lower mean dz due to including roughness, and (ii)
the 4 and 5-input FIS resulted in a higher standard deviation and maximum dz due to the inclusion of site-
specific bank heights. All three FIS gave plausible estimates of DEM error, with the two more complicated
models offering an improvement in the ability to detect spatially localized areas of DEM uncertainty.
1. Introduction
Digital elevation models (DEMs) have become pervasive in the earth sciences as representations of land
surfaces, topographic boundary conditions in hydraulic models [e.g., Horritt et al., 2006; Cook and Merwade,
2009], hydrologic models [e.g., Johnson and Miller, 1997; Wu et al., 2008], morphodynamic models [e.g.,
Williams et al., 2011; Wheaton et al., 2013], ecohydraulic models [e.g., Pasternack et al., 2008; Wheaton et al.,
2010b], and are regularly used to quantify geomorphic processes of erosion and deposition [e.g., Milan
et al., 2011; Croke et al., 2013]. In fluvial geomorphology, DEMs are the standard by which we represent
stream channel and floodplain topography. While airborne near-infrared (NIR) Light Detection and Ranging
(LiDAR), terrestrial laser scanning (TLS), and Structure from Motion (SfM) are increasingly used to collect
topographic data, their signal is either scattered or attenuated in deep water [Heritage et al., 2009; Notebaert
et al., 2009; Fonstad et al., 2013]. Data for submerged parts of the channel have to be collected using other
survey methods. As a result, total stations (TS) and real-time kinematic global positioning systems (rtkGPS)
are often the most reliable and cost effective tools for conducting reach-scale surveys of wadeable sub-
merged channels [S. G. Bangen et al., 2014].
Uncertainties in the topographic representation of the Earth’s surface in a DEM have implications for com-
monplace DEM applications (e.g., geomorphic change detection, hydraulic modelling). Such uncertainties
are often represented quantitatively with error estimation. The error magnitude of DEMs derived from topo-
graphic surveys are affected by several factors, including: survey instrument, survey point quality, sampling
strategy, surface complexity, surface roughness, grid resolution, and interpolation method [Lane et al., 1994;
Lane, 1998; Wise, 1998; Wechsler and Kroll, 2006; Heritage et al., 2009; Schwendel et al., 2012]. Characterizing
uncertainty in DEMs is critically important as error propagates into metrics and analyses that derived from
DEMs [Holmes et al., 2000]. For example, ignoring errors can result in over estimation of net erosion and
deposition in DEM-based sediment budget estimates [Wheaton et al., 2010a].
Various approaches have been taken to characterize DEM error. Previous efforts characterized error as being
uniform across the entire DEM surface using metrics such as root mean square error or standard deviation
Key Points:
Compare fuzzy inference system
DEM error models for total station
surveys of streams
Including roughness decreases
in-channel mean error estimate
Including site-specific bank heights
increases in-channel standard
deviation and max error estimate
Supporting Information:
Supporting Information S1
Supporting Information S2
Supporting Information S3
Supporting Information S4
Correspondence to:
S. Bangen,
sara.bangen@gmail.com
Citation:
Bangen, S., J. Hensleigh, P. McHugh,
and J. Wheaton (2016), Error modeling
of DEMs from topographic surveys of
rivers using fuzzy inference systems,
Water Resour. Res., 52, doi:10.1002/
2015WR018299.
Received 10 NOV 2015
Accepted 25 JAN 2016
Accepted article online 28 JAN 2016
V
C
2016. American Geophysical Union.
All Rights Reserved.
BANGEN ET AL. ERROR MODELING OF DEMS FROM TOPOGRAPHIC SURVEYS OF RIVERS 1
Water Resources Research
PUBLICATIONS
of error calculated using validation point data [e.g., Brasington et al., 2000; Brasington et al., 2003; Milan
et al., 2007]. This approach was improved upon by Lane et al. [2003] by applying separate uniform error esti-
mates to wet and dry surfaces. However, due to the spatially variable nature of DEM error, applying a uni-
form error estimate results in overestimating error on flatter surfaces and underestimating error on more
complex, sloping surfaces. This pattern has been shown by Heritage et al. [2009] and Milan et al. [2011] who
found strong relationships between topographic variability and DEM error. Recently, focus has shifted
toward spatially variable DEM error models using empirical relationships [e.g., Milan et al., 2011], morpho-
metric parameter distributions [e.g., Sofia et al., 2013], and fuzzy inference systems [FIS; e.g., Wheaton et al.,
2010a; Prosdocimi et al., 2015].
Rivers and streams exhibit a diversity of character, and as such, applying a universal DEM error model with
hard-set values across river types is often inappropriate. The purpose of this paper is to illustrate the trade-
offs associated with increasing the complexity (i.e., few versus many inputs) of FIS DEM error models (see
Supporting Information S1 for the .fis files described). A secondary objective of this study is to provide
researchers examples demonstrating how to parameterize a site-specific FIS to build more robust, spatially
variable DEM error models. As a general rule, FIS model input selection should consider what is relevant
given the survey technique, what is readily available/derivable from the raw topographic data, and the envi-
ronment being surveyed. We focus our attention on examples of topographic surveys from wadeable
streams. In this paper, we build on Wheaton et al. [2010a] by developing and comparing three separate spa-
tially variable FIS error models. The FIS error models are designed to estimate error associated with DEMs
derived from TS surveys of streams. Our models differ in the number of inputs considered, input source (i.e.,
derived from topography versus auxiliary data), and the extent of site-specific parameterization. An existing,
relatively simple, FIS is tested that uses two inputs derived from the topographic data set (point density, DEM
slope). A second FIS includes two additional inputs that are derived directly from the topographic data set
(interpolation error, 3-D point quality) and is calibrated using site-specific estimates of bank heights. The third
FIS builds on the second by considering auxiliary information on bed-surface roughness as a model input.
1.1. Fuzzy Inference System Background
Fuzzy inference systems are founded on fuzzy set theory and fuzzy logic [Zadeh, 1965]. In fuzzy set theory,
continuous variables are grouped into classes with overlap between classes allowing for partial membership
in multiple classes. This contrasts with classical set theory where classes are crisp and class membership is
discrete. An advantage of fuzzy sets is they account for real-world uncertainties and measurement
imprecision.
An inference system is simply a rule table that translates unique combinations of categorical inputs into an
output value. An FIS allows the inputs to be continuous variables, and for any given combination of inputs
instead of one rule applying, multiple rules may apply based on overlapping input membership. Continuous
variables (e.g., point density) are mapped to classes (e.g., ‘‘low’’) using membership functions (MFs). For exam-
ple, a ‘‘low’’ point density MF may be assigned a numerical range of 0–0.25 pts m
22
, and a ‘‘medium’’ point
density MF may be assigned a numerical range of 0.1–1 pts m
22
. Ambiguity for inputs in the 0.1–0.25 pts
m
22
would map into partial membership into both the ‘‘low’’ and ‘‘medium’’ classes. The consequence of this
is realized in the inference system, which combines the relevant rules based on the respective membership
values in the rule inputs. The total consequence combines the outputs into a fuzzy number output, which is
akin to a probability density function of the output explicitly representing the range of potential outputs. In
practice, FIS outputs must be ‘‘defuzzified’’ to a crisp single value. Advantages of FIS are that they are concep-
tually easy to understand, are rooted in transparent language, are flexible in the ease of adjusting model
inputs, can handle imprecise data, and can incorporate expert knowledge [Jang and Gulley, 2014].
When modeling topography, the ‘‘true’’ surface is unknown since all instruments incorporate some mea-
surement error. As such, our ability to estimate DEM error is limited because validation surfaces can only
approximate the ‘‘truth.’’ This is in stark contrast to purely empirical approaches wherein errors are known
(or are assumed to be) and modeled using relationships to covariates and/or spatial autocorrelation func-
tions [e.g., Leon et al., 2014]. An additional advantage of FIS is thus their ability to leverage-limited empirical
evidence (i.e., a locally relevant error domain) and expert knowledge to incorporate known sources of error
into a robust error model. For further reviews on FIS, we refer the reader to Bandemer and Gottwald [1995]
and Klir and Yuan [1995].
Water Resources Research 10.1002/2015WR018299
BANGEN ET AL. ERROR MODELING OF DEMS FROM TOPOGRAPHIC SURVEYS OF RIVERS 2
2. Methods
2.1. Study Region
The Columbia Habitat Monitoring Program (CHaMP; https://www.champmonitoring.org/) collects topo-
graphic and auxiliary habitat data in 15 subbasins of the Columbia River basin, encompassing several hun-
dred sites. We selected 34 sites to test the three error models described in this paper (Figure 1). To ensure a
range of sampling conditions was represented, we randomly selected sites from three reach-length strata
which correspond to streams of different size: small (<160 m), medium (160–320 m), and large (>320 m).
We report results for all 34 sites, but for illustrative purposes highlight results in further detail for two sites:
Mad River and Murderers Creek (Figure 1). The Mad River site has moderate gradient (2.1%), a straight plan-
form (sinuosity: 1.03), and a D
84
of 103 mm. The Murderers Creek site has low gradient (1.26%), a sinuous
planform (sinuosity: 1.6), and a D
84
of 149 mm. More information on these sites can be found at
champmonitoring.org.
2.2. Data Collection and Processing
Topographic data were collected using Nikon NIVO 5C TSs following the CHaMP protocol [CHaMP, 2013].
Crews used a topographically stratified sampling strategy where the surveyor selects points and breaklines
that best represent channel morphology with an emphasis on capturing significant breaks in slope [e.g.,
Vall
e and Pasternack, 2006; Milan et al., 2011]. Fuller et al. [2003] and Heritage et al. [2009] found that this
approach results in lower DEM errors in complex areas than other approaches (e.g., cross-sections). This
strategy translates to a higher density of points in areas with greater topographic complexity (e.g., pools)
and a lower density of points in areas with lower topographic complexity (e.g., tops of bars).
Topographic points and breaklines were interpolated into a triangulated irregular network (TIN). TINs are an
efficient tool for representing the complex morphology (e.g., breaks in slope) of streams and rivers [Moore
et al., 1991]. Each crew edited the TIN derived from their survey to remove obvious busts that may have
resulted from survey errors (e.g., incorrect rod height) using the CHaMP Topo Toolbar (http://champtools.
northarrowresearch.com/) ArcGIS add-in. After editing, each TIN was interpolated into a 10 cm DEM using a
natural neighbors interpolation algorithm.
2.3. Fuzzy Inference System DEM Error Modeling
We compared three FIS error models with varying degrees of complexity for TS surveys of wadeable
streams and rivers. The first step in constructing a FIS model is determining the model inputs and outputs.
The three FIS described here (Table 1) have the same output (DEM elevation uncertainty, dz) but include a
different assemblage of inputs (DEM slope, point density, interpolation error, 3-D point quality, roughness).
The model inputs are described in greater detail in section 2.3.1.
Once model inputs and outputs are determined, building the FIS entails:
1. Defining the input categories and fuzzy MFs;
2. Defining the output categories and calibrating fuzzy MFs to independent estimates of DEM error;
3. Defining the rules that relate the inputs to the output; and
4. Specifying the FIS type, fuzzy operation methods, rule implication method, aggregation method, and
defuzzification method.
Figure 2 illustrates a 5-input FIS with input and output MFs. The process of defining MFs includes three
steps. The first step is deciding the number of classes or linguistic adjectives for each input and output. Sec-
ond, the MFs are defined using a range of numerical values to represent each adjective. The shape of the
membership function is determined in this second step (e.g., a trapezoid-shaped MF will have four sets of
values).
The three FIS models defined here differ slightly in linguistic adjectives, associated MF values, and fuzzy
rules. The FIS are similar in that they shared the same default values for FIS type (Mamandi), rule implication
method (minimum), aggregation method (maximum), defuzzification method (centroid), and MF shape
(trapezoidal). For more detail on these default values, see Jang and Gulley [2014].
2.3.1. FIS Inputs and MFs
Three of the FIS inputs were derived from the raw survey point data and DEM (DEM slope, point density,
and interpolation error) while one (3-D point quality) required additional information on the occupation
Water Resources Research 10.1002/2015WR018299
BANGEN ET AL. ERROR MODELING OF DEMS FROM TOPOGRAPHIC SURVEYS OF RIVERS 3
Figure 1. (top) Columbia River basin location map and sample site map. Site photographs and water depth maps for the two highlighted sites: (bottom) Mad River and Murderers Creek.
Water Resources Research 10.1002/2015WR018299
BANGEN ET AL. ERROR MODELING OF DEMS FROM TOPOGRAPHIC SURVEYS OF RIVERS 4
point error and instrument accuracy. The fifth FIS input required a grain size estimate (i.e., bed-surface
roughness) of the surveyed site. Each input was represented as a 10 cm raster.
2.3.1.1. DEM Slope
DEM slope is a common input in FIS DEM error models because it can be derived from the point cloud and
is a reasonable proxy for topographic complexity [Wheaton et al., 2010a]. When surveying streams and riv-
ers, small horizontal errors can lead to large vertical errors, and this increases where slope increases. In the
context of TS and rtkGPS surveys, these errors generally occur along the channel margin as a result of (i) the
surveyor holding the rod at a slight angle over the bank edge resulting in an offset point, (ii) inadequate
representation of complex bank edges, and/or (iii) lack of proper breakline representation [S. Bangen et al.,
2014].
We calculated local DEM slope in degrees as the maximum rate of change in a 3 33 cell neighborhood. It is
preferable to use degrees slope as opposed to percent slope, because degrees has a consistent upper end
in its range (i.e., 908). The 2-input FIS has three slope MFs (‘‘low,’’ ‘‘medium,’’ ‘‘high’’), while the 4-input and 5-
input FIS include an additional ‘‘extreme’’ MF. The ‘‘extreme’’ MF is used to isolate bank features where
higher vertical errors are likely to occur.
2.3.1.2. Point Density
Point density is a proxy for survey sampling effort and is calculated from the point cloud. Point density
(pts m
22
) was calculated using a 3 m radius circular neighborhood. The three point density MFs (‘‘low,’’
‘‘medium,’’ ‘‘high’’) are from Wheaton et al . [2010a] and were based on empirical studies by Wheaton [2008].
Point density was included in all three FIS.
2.3.1.3. Interpolation Error
Interpolation error represents error introduced when translating survey points and breaklines that are
exactly represented in a TIN, but approximated by interpolation when converted into a continuous raster
DEM. Interpolation error is typically quantified using the DZ between a raw surveyed point and the DEM
[e.g., Desmet, 1997; Chaplot et al., 2006]. In the context of this study, the Z
DEM
–Z
Survey Point
calculation is not
appropriate since crews directly edited the TINs, and not the survey points themselves, to remove survey
errors during data postprocessing before interpolating the TIN into a DEM. In cases such as this it is more
appropriate to use the TIN nodes to calculate interpolation error. Here, we converted the TIN nodes to
points and calculated interpolation error as Abs(Z
DEM
–Z
TIN Node
).
While interpolation error using TINs is generally low [Schwendel et al., 2012] (Figure 3), we wanted to ensure
rare and localized instances of high interpolation error had a consequent of high DEM elevation uncertainty
in the models. The two MFs (‘‘low,’’ ‘‘extreme’’) were used as a threshold to capture instances of high inter-
polation error. Interpolation error was included in the 4-input and 5-input FIS.
2.3.1.4. 3-D Point Quality
Three-dimensional point quality (3-DPQ) is a proxy for survey point quality. Most rtkGPS devices report
an estimate of 3-DPQ based on satellite geometry at the time of point acquisition. Here, we devel-
oped a TS-specific 3-DPQ calculation. The TS point 3-DPQ is calculated from each raw point observa-
tion and the station occupation error. The basic premise is the occupation error should propagate
into each measured point and grow as a function of slope distance and station occupation error. The
TS point (TSP) horizontal distance error (HDE) and vertical distance error (VDE) is calculated for each
point as:
HDE
TSP
5Slope Distance
TSP
HDE
OCP
=Slope Distance
OCP
ðÞ
VDE
TSP
5Slope Distance
TSP
VDE
OCP
=Slope Distance
OCP
ðÞ
Table 1. Fuzzy Inference System (FIS) Model Input and Output
FIS Model
FIS Inputs
FIS Output
DEM Slope Point Density Interpolation Error 3-D Point Quality Roughness Bank Height Calibration
2-Input x x
4-Input x x x x x
5-Input x x x x x x
Water Resources Research 10.1002/2015WR018299
BANGEN ET AL. ERROR MODELING OF DEMS FROM TOPOGRAPHIC SURVEYS OF RIVERS 5
where Slope Distance
TSP
is the straight line distance between the survey point and the station set up, Slope
Distance
OCP
is the straight line distance between the backsight and the station set up, and VDE and HDE
are the vertical and horizontal occupation errors for the station set up the point was surveyed from. The
Figure 2. Input and output fuzzy MFs used in the 5-input FIS. Inputs: point density, slope, interpolation error, 3-D point quality, roughness. Output: elevation uncertainty. Note, with the
exception of the 2-input FIS which lacks an ‘‘extreme’’ slope MF, input MFs for the 2-input and 4-input FIS are same as shown here.
Water Resources Research 10.1002/2015WR018299
BANGEN ET AL. ERROR MODELING OF DEMS FROM TOPOGRAPHIC SURVEYS OF RIVERS 6
occupation errors are calculated during the instrument setup by comparing the measured slope distance,
horizontal angle, and vertical angle against what the established values should be based on the geometry
between the occupied control point and the backsight point. The VDE and HDE are then combined into an
estimate of 3-DPQ using a simple error propagation calculation:
3DPQ
TSP
5sqrt HDE
2
TSP
1VDE
2
TSP
As in the case of interpolation error, 3-DPQ errors are generally quite low and not limiting (Figure 3). How-
ever, we wanted to capture exceptions where error is high. These typically relate to poor station setups.
Since the 3-DPQ is a function of slope distance, shots collected with longer baselines, will have higher mag-
nitudes. The two MFs (‘‘low,’’ ‘‘extreme’’) are used as a threshold to capture instances of high 3-DPQ. Three-
dimensional point quality is included in the 4-input and 5-input FIS.
2.3.1.5. Roughness
Topographic complexity can be represented using an estimate of roughness height. In many cases rough-
ness, whether grain size or vegetative, can limit accurate topographic representation. A measure of rough-
ness height in the wetted channel (hereafter referred to as in-channel) is grain size. Here, spatially variable
in-channel roughness was represented using D
50
calculated from channel unit substrate estimates made by
the survey crew [CHaMP, 2013]. At each channel, unit crews visually estimated the percent fines, sand, fine
gravel, coarse gravel, cobble, boulders, and bedrock. CHaMP employs this method so that the time spent at
a site is allocated optimally across several different habitat assessments, with the main emphasis being
placed on topographic point collection. Past work has shown that although more time-consuming methods
(e.g., pebble counts) offer greater precision, ocular approaches are adequate for capturing coarse
Figure 3. Total station point interpolation error and 3-DPQ distributions across the 34 sites.
Water Resources Research 10.1002/2015WR018299
BANGEN ET AL. ERROR MODELING OF DEMS FROM TOPOGRAPHIC SURVEYS OF RIVERS 7
differences in substrate conditions across space and time [e.g., Faustini and Kaufmann, 2007]. The D
50
value
was attributed to the surveyed channel unit shapefile and then converted to a 10 cm raster concurrent with
the other input rasters.
In this study, we lacked a direct measure of out-of-channel roughness. We approximated out-of-channel
roughness using a resampling approach. At each site, we selected a random subset of 80% of the sur-
vey points, interpolated a TIN using the survey point subset and all breaklines, and interpolated a DEM.
The mean DZ (Abs[Z
DEM
–Z
Subset DEM
]) value across all 34 sample sites was assigned to all out-of-
channel areas.
The in-channel and out-of-channel roughness rasters were merged into a single roughness raster. Four MFs
(‘‘Fine Gravel,’’ ‘‘Coarse Gravel,’’ ‘‘Cobble,’’ ‘‘Boulder’’) were modeled after size classes used in the ocular sub-
strate estimates. We included fines and sand with fine gravel since roughness on the scale of fines and sand
(i.e., particles < 2 mm) will have nominal influence on DEM error. Roughness was only included in the 5-
input FIS.
2.3.2. FIS Output and MFs
The four output linguistic MFs and associated values differ among the three FIS models. The minimum, or
‘‘low’’ MF, error values were calibrated using TS point precision from an empirical study [Wheaton, 2008].
The maximum error value was set at 1.5 m in the 2-input FIS model and to site-specific bank height values
in the 4 and 5-input FIS models. The 2-input FIS output used default MFs (‘‘low,’’ ‘‘medium,’’ ‘‘high,’’
‘‘extreme’’) and values from Wheaton et al. [2010a]. The 4-input FIS used the same default ‘‘low,’’ ‘‘medium,’’
‘‘high’’ MFs and values as the 2-input FIS, but the ‘‘extreme’’ MF was calibrated to represent site-specific
bank heights (Table 3). The 5-input FIS differed from the two other FIS in the ‘‘high’’ and ‘‘extreme’’ MFs; it
has two ‘‘extreme’’ MFs, one capturing areas with high roughness (‘‘extreme – roughness’’) and one captur-
ing banks (‘‘extreme – banks’’).
The ‘‘extreme’’ MF values in the 4-input and the ‘‘extreme – bank’’ MF values in the 5-input FIS were cali-
brated to sites using bank-height distributions. The maximum bank heights represent the maximum
expected vertical error which could manifest itself from a horizontal positional error between neighbor-
ing DEM grid cells along a steep bank. Bank height (Z
Top of Bank
2 Z
Toe of Bank
) was calculated at 20 ran-
dom cross sections at each site. The second inflection point in the ‘‘extreme – banks’’ output MF was
set to the median bank height while the third and fourth inflection points were set to the upper hinge
(i.e., Q
75
1 1.5 * [Q
75
2 Q
25
]). If the maximum bank height was less than the upper hinge, it was used
instead.
For further information on calibrating FIS error model membership function values, please see Hensleigh
et al. [2016].
2.3.3. FIS Rules
The FIS rules for the 2-input, 4-input, and 5-input FIS models are shown in Tables 2–4. Most rules used ‘‘and’’
statements. There were four exceptions where ‘‘or’’ statements were used to isolate instances where the
error associated with a single input is likely high enough to trump all other inputs. Instances where ‘‘or’’
rules are invoked include:
1. if interpolation error is ‘‘extreme,’’ then elevation uncertainty is ‘‘extreme’’ (4-input); ‘‘extreme – rough-
ness’’ (5-input FIS)
2. if 3-DPQ is ‘‘extreme,’’ then elevation uncertainty is ‘‘extreme’’ (4-input); ‘‘extreme – roughness’’
(5-input FIS)
3. if DEM slope is ‘‘extreme,’’ then elevation uncertainty is ‘‘extreme -banks’’ (4 and 5-input FIS)
4. if roughness is ‘‘boulders,’’ then elevation uncertainty is ‘‘extreme-roughness’’ (5-input FIS)
The ‘‘extreme-roughness’’ output MF was also used as a consequent in the 5-input FIS under the antece-
dents of (i) ‘‘high’’ DEM slope, ‘‘low’’ point density, ‘‘cobbles’’ roughness, and (ii) ‘‘high’’ DEM slope, ‘‘medium’’
point density, ‘‘cobbles’’ roughness. Note that while in a regular inference system, any combination of inputs
can only have one rule that applies, an FIS input falling into an overlapping MF region results in multiple
rules applying.
2.4. FIS Implementation
The FIS models were implemented using a batch p rocessed version of the Geomorphic Change
Detection s oftware (http://gcd.joewheaton.org). The file format of the FIS models (*.fis) is consistent
Water Resources Research 10.1002/2015WR018299
BANGEN ET AL. ERROR MODELING OF DEMS FROM TOPOGRAPHIC SURVEYS OF RIVERS 8
with the file format of the Matlab Fuzzy
Logic Tool box [ Jang and Gulley, 2014],
but is implemented with o ur own core
C11 open source library. These FIS
models are available in the online
supplement and at our online FIS D EM
Error Repository (https://bitbucket.org/
pipbailey/fis-dem-error-repository). Within
CHaMP, input derivation and FIS DEM
error modeling are automated for all
surveys via cloud computing (EC2
instancesonAmazonWebServices)
and administered and disseminat ed
through CHaMP’s data repository http ://
champmonitoring.org. However, the FIS described here can also b e run manually wit hin the GCD
software.
3. Results
3.1. FIS Inputs: Mad River and Murderers Creek
FIS inputs varied considerably across space, both within and between the two focal streams, Mad River and
Murderers Creek (Figure 4). Site-averaged point density (pts m
22
) for the Mad River site was lower (in-chan-
nel mean [sd]: 0.15 [0.05]; out-of-channel mean [sd]: 0.07 [0.04]) than Murderers Creek (in-channel mean
[sd]: 0.39 [0.09]; out-of-channel mean [sd]: 0.18 [0.14]). Site-averaged DEM slope was similar in-channel (Mad
mean [sd]: 8.28 [7.68]; Murderers mean [sd] 5 7.78 [6.98 ]), but slightly higher out-of-channel at the Mad River
site (Mad mean [sd]: 21.18 [13.28]; Murderers mean [sd]: 16.38 [11.88]). In-channel roughness (i.e., D
50
) at the
Mad River site was dominated by cobbles. One unit had D
50
in the coarse gravel class and one unit had D
50
in the boulders class. In-channel roughness was more variable at the Murderers Creek site, but was still
dominated by cobbles, with several units in the coarse gravel D
50
class and one unit in the fine gravel D
50
class. There were areas of high in-channel 3-DPQ at the Mad River site, whereas in-channel 3-DPQ was rela-
tively low at Murderers Creek, with the exception of an area along the river-right floodplain. Interpolation
errors were generally well distributed across both sites. The distribution of heights at the Mad River site
were higher (median 5 1.18 m; max 5 3.26 m) than Murderers Creek (median 5 0.26 m; max 5 3.42 m).
3.2. FIS Output: Mad River and Murderers Creek
There was substantial variation in the FIS output ( dz in meters) among the three models at both the Mad
River and Murderers Creek sites (Figure 5). The 2 and 4-input FIS dz rasters were similar in-channel, with the
exception of areas along the wetted chan-
nel edge. These are areas where the 4-
input dz estimates were either higher or
lower due to the inclusion of the bank
slope threshold. Maximum dz was also
higher in the 4-input FIS due to the inclu-
sion of site-specific bank heights in the
‘‘extreme’’ output MF, which were higher
than the hard-set (i.e., nonsite-specific)
values used in the 2-input FIS. These same
trends hold for the 5-input FIS dz esti-
mates. Additionally, including roughness
in the 5-input FIS resulted in higher dzin
out-of-channel areas and greater spatial
variability in dz estimates for in-channel
areas.
To understand the spatial distribution of
dz estimates and FIS inputs, we analyzed
Table 2. Two-Input Fuzzy Inference System Ruleset for DEM Elevation Uncer-
tainty (d
z
)
a
Rule
Inputs
Output
Slope (%) Point Density (Pts m
22
) d
z
(m)
1 Low Low High
2 Low Medium Medium
3 Low High Low
4 Medium Low High
5 Medium Medium Medium
6 Medium High Low
7 High Low Extreme
8 High Medium High
9 High High Medium
a
The two inputs are DEM slope and point density.
Table 3. Four-Input Fuzzy Inference System Ruleset for DEM Elevation
Uncertainty (d
z
)
a
Rule
Input Output
Slope
(%)
Point
Density
(Pts m
22
)
3-D
Point
Quality
Interpolation
Error (m) d
z
(m)
1 Low Low NA NA High
2 Low Medium NA NA Medium
3 Low High NA NA Low
4 Medium Low NA NA High
5 Medium Medium NA NA Medium
6 Medium High NA NA Low
7 High Low NA NA High
8 High Medium NA NA High
9 High High NA NA Medium
10 Extreme NA NA NA Extreme
11
b
NA NA High High Extreme
a
The four inputs are DEM slope, point density, 3-D point quality, and interpo-
lation error.
b
‘‘OR’’ rule.
Water Resources Research 10.1002/2015WR018299
BANGEN ET AL. ERROR MODELING OF DEMS FROM TOPOGRAPHIC SURVEYS OF RIVERS 9
patterns in relative to the channel unit types assigned by survey crews using a classification system adapted
from Hawkins et al. [1993] and Bisson et al. [2007]. The Mad River site channel units included: rapids (n 5 2;
D
50
5 236 mm), riffles (n 5 3, D
50
5 121 mm), and scour pools (n 5 6, D
50
5 177 m). The Murderers Creek
site had a greater number and more diverse assemblage of channel units, which included: fast nonturbu-
lent/glides (n 5 6, D
50
5 82 mm), riffles (n 5 7, D
50
5 88 mm), scour pools (n 5 3, D
50
5 59 mm), and beaver
pools (n 5 2, D
50
5 16 mm).
Assessing dz rasters by channel unit type (Figures 6 and 7) revealed trends similar to those observed at the
site-level. There was a general shift in the tail of the distributions for each FIS. Isolating higher slope banks
in the 4 and 5-input FIS resulted in fewer large dz estimates in-channel that skew summary statistics, such
as the mean and sd. This was most apparent at the Murderers Creek site where there were negligible differ-
ences in the shape and median of dz distributions, but the mean and sd were higher. The greatest differen-
ces among in-channel unit median dz estimates at Murderers Creek were for beaver pools, where grain size
was also the smallest. At the Mad River site, we observed lower median and mean dz estimates in riffles and
scour pools using the 5-input FIS compared to the 2 and 4-input FIS due to their lower roughness. In con-
trast, dz estimates were higher in rapids due to their high roughness.
The two sites have two channel unit types in common that allow across-site comparisons: riffle and scour
pools. The Mad River site has lower point density and higher bank heights and in-channel roughness.
Accordingly, the 4 and 5-input dz estimates for these units were higher at the Mad River site than at Mur-
derers Creek.
3.3. FIS Output: All Sites
Across the 34 sites, including additional inputs and different MF values in each of the FIS error models
resulted in distinct dz distributions both in and outside the wetted channel (Figure 8).
Table 4. Five-Input Fuzzy Inference System Ruleset for DEM Elevation Uncertainty ( d
z
)
a
Rule
Input Output
Slope (%) Point Density (Pts m
22
) Roughness (mm) 3-D Point Quality Interpolation Error (m) d
z
(m)
1 Low Low Fine Gravel NA NA Average
2 Low Low Coarse Gravel NA NA Average
3 Low Low Cobble NA NA High
4 Low Medium Fine Gravel NA NA Low
5 Low Medium Coarse Gravel NA NA Average
6 Low Medium Cobble NA NA Average
7 Low High Fine Gravel NA NA Low
8 Low High Coarse Gravel NA NA Low
9 Low High Cobble NA NA Average
10 Medium Low Fine Gravel NA NA Average
11 Medium Low Coarse Gravel NA NA Average
12 Medium Low Cobble NA NA High
13 Medium Medium Fine Gravel NA NA Low
14 Medium Medium Coarse Gravel NA NA Average
15 Medium Medium Cobble NA NA High
16 Medium High Fine Gravel NA NA Low
17 Medium High Coarse Gravel NA NA Low
18 Medium High Cobble NA NA Average
19 High Low Fine Gravel NA NA High
20 High Low Coarse Gravel NA NA High
21 High Low Cobble NA NA Rough
22 High Medium Fine Gravel NA NA Average
23 High Medium Coarse Gravel NA NA High
24 High Medium Cobble NA NA Rough
25 High High Fine Gravel NA NA Average
26 High High Coarse Gravel NA NA High
27 High High Cobble NA NA High
28 Extreme NA NA NA NA Bank
29
b
NA NA NA High High Rough
30 NA NA Boulder NA NA Rough
a
The five inputs are DEM slope, point density, roughness, 3D point quality, and interpolation error.
b
‘‘OR’’ rule.
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BANGEN ET AL. ERROR MODELING OF DEMS FROM TOPOGRAPHIC SURVEYS OF RIVERS 10
In-channel, the 5-input FIS had a lower mean and median dz than the 2 and 4-input FIS. This is due to the
inclusion of the roughness input, which lowers dz in areas with lower grain size. The standard deviation and
maximum dz were higher for the 4 and 5-input FIS. Although some portion of increased error is contributed
by interpolation error and 3-DPQ, the increase in the maximum dz can be attributed to the inclusion of site-
specific bank heights in output MFs. This is suggested by the relatively narrow distribution of the 2-input
FIS where the ‘‘extreme’’ output MF used hard-set values.
Out-of-channel results further illustrate the effect of including bank height and roughness in the error
model. The dz summary statistic distributions were lowest for the 2-input FIS which relies solely on point
density and DEM slope. The 4-input FIS distributions demonstrate that adding interpolation error, 3-DPQ,
DEM Slope
Point Density
3D Point Quality
0 1020304050Meters01020304050Meters
3D Point Quality
High : 0.07
Low : 0.00
Point Density
pts/m
2
0.0 - 0.1
0.1 - 0.2
0.2 - 0.3
0.3 - 0.4
0.4 - 0.5
0.5 - 0.6
0.6 - 0.7
DEM Slope
Degrees
0 - 2
2 - 5
5 - 10
10 - 15
15 - 25
25 - 35
35 - 45
45 - 60
60 - 80
80 - 90
Mad River Murderers Creek
Interpolation Error
Elev. (m)
High : 0.40
Low : 0.00
Roughness
Roughness
Size Category
Fines, Sand
Fine Gravel
Coarse Gravel
Cobbles
Boulders
Figure 4. Fuzzy inference system input rasters (point density, DEM slope, interpolation error, 3-D point quality, roughness) for the Mad River and Murderers Creek.
Water Resources Research 10.1002/2015WR018299
BANGEN ET AL. ERROR MODELING OF DEMS FROM TOPOGRAPHIC SURVEYS OF RIVERS 11
and bank heights to the model generally increases dz, whereas the addition of roughness slightly lowers
the dz distribution.
4. Discussion
In this study, we developed and compared three DEM error models using TS survey data from 34 sites in
the Columbia River Basin. The 2-input model estimated error using point density and DEM slope, and cap-
tured higher errors near steeper features, such as banks. A major limitation of the 2-input FIS was the use of
a fixed, site-invariant error ceiling that can over or underestimate error along the channel margin due to
bank-height diversity. The 4-input error model expanded on the 2-input model by including an additional
class in the slope MFs to isolate bank features, point quality information from the raw survey data, interpola-
tion error estimates, and site-specific bank heights used to calibrate the error model output values. The final
model, a 5-input error model, also included an input for spatially variable roughness to avoid overestima-
tion of error in areas with fine-grained substrates, and underestimation of error in areas with coarse-grained
substrates. The 5-input model is likely the most robust of the three given that it is (1) calibrated to site con-
ditions (i.e., bank height) and (2) considers auxiliary data (i.e., roughness) that can influence DEM error in
space [Heritage et al., 2009; Milan et al., 2011]. Our study is novel in that we demonstrate the combined
effects of tailoring FIS DEM error models to survey technique, locally relevant error conditions and input
availability. Further, our work underscores the flexibility of FIS error modeling approaches.
Given similarities (e.g., approximate point densities) between TS and rtkGPS, the error models described
here can be extended to rtkGPS-based surveys. However, the typical positional accuracy of rtkGPS points
(vertical: 10–30 mm; horizontal: 10–20 mm) is slightly lower than TS (vertical: 7–20 mm; horizontal: 5–
15 mm). Thus, the ‘‘low,’’ ‘‘medium,’’ and ‘‘high’’ elevation uncertainty output MF values should be calibrated
appropriately for rtkGPS applications.
In fluvial geomorphology applications, DEM error models can be used for error reporting [e.g., Butler et al.,
1998; Reusser and Bierman, 2007; Notebaert et al., 2009; Javernick et al., 2014] or propagated into sediment
budgets and geomorphic change detection (GCD) estimates when repeat topographic surveys are available
Figure 5. Fuzzy inference system output rasters (elevation uncertainty) for the Mad River and Murderers Creek.
Water Resources Research 10.1002/2015WR018299
BANGEN ET AL. ERROR MODELING OF DEMS FROM TOPOGRAPHIC SURVEYS OF RIVERS 12
[e.g., Brasington et al., 2003; Lane et al., 2003; Erwin et al., 2012; Milan, 2012; Wheaton et al., 2013]. When
propagating error in such analyses, higher errors generally result in more conservative (i.e., lower) estimates
of erosion and deposition volumes, whereas lower errors result in the opposite [Wheaton et al., 2010a]. In TS
and rtkGPS surveys, bank margins are notoriously difficult features to accurately represent which can trans-
late to gross overestimation of erosion when little-to-no lateral erosion has occurred [S. G. Bangen et al.,
2014]. Additionally, there are time constraint versus coverage trade-offs in surveys that often limit a
Figure 6. Fuzzy inference system output (elevation uncertainty) distributions and summary statistics for the Murderers Creek, segregated
by channel unit type.
Water Resources Research 10.1002/2015WR018299
BANGEN ET AL. ERROR MODELING OF DEMS FROM TOPOGRAPHIC SURVEYS OF RIVERS 13
surveyor’s ability to accurately represent each boulder or large roughness element. If these features are not
consistently represented across surveys, they can result in overestimation of erosion or deposition. Con-
versely, if an overly conservative error estimate is used, erosion and deposition in finer-grained channel
units (e.g., pools) may go undetected. In short, while not explicitly tested here, we believe that isolating
bank features and including spatially variable roughness parameters in DEM error models can improve the
accuracy of sediment budgets and geomorphic change detection methods.
While FIS-based error models are one of several approaches to modeling DEM error, they offer advantages
over competing methods. For example, factors known to affect error can be added as model inputs with rel-
ative ease. This was illustrated here through the addition of inputs and by adjusting the MF parameters to
isolate bank features and account for site-specific bank heights. This flexibility has been noted previously
elsewhere. Blasone et al. [2014] developed a 3-input FIS derived from TLS. Inputs included slope, point
Figure 7. Fuzzy inference system output (elevation uncertainty) distributions and summary statistics for the Mad River, segregated by
channel unit type.
Water Resources Research 10.1002/2015WR018299
BANGEN ET AL. ERROR MODELING OF DEMS FROM TOPOGRAPHIC SURVEYS OF RIVERS 14
density, and vegetation noise with the latter input included because of its demonstrated effect on DEM
quality [Alba and Scaioni, 2010; Coveney et al., 2010]. In a study of morphological changes in a braided river
where vegetation was nearly absent, Picco et al. [2013] developed a 3-input FIS for DEMs derived from TLS
data that included the inputs slope, point density, and roughness.
Ideally, all FIS model inputs would be derived from the original survey point cloud. Yet, not all factors affect-
ing DEM quality can be derived from TS and rtkGPS topographic data alone. Point clouds from TS and
rtkGPS are generally not dense enough to derive statistical roughness models [Brasington et al., 2012; Rych-
kov et al., 2012], whereas roughness can be easily estimated from ocular estimates or facies mapping
Figure 8. Elevation uncertainty summary statistic (mean, median, standard deviation, maximum) distributions for all 34 sites. Results are shown for each of the 3 FIS (2, 4, 5-input) and
spatially segregated by in-wetted channel versus out of the wetted channel.
Water Resources Research 10.1002/2015WR018299
BANGEN ET AL. ERROR MODELING OF DEMS FROM TOPOGRAPHIC SURVEYS OF RIVERS 15
[Buffington and Montgomery, 1999]. We showed that adding spatially variable roughness estimates as an
input (5-input FIS) resulted in a more conservative error estimates in areas dominated by boulders and cob-
bles (e.g., rapids) and a more liberal error estimate in areas dominated by finer grained material (e.g., beaver
pools). Stated differently, simply using inputs derived from the point cloud, as in our 2 and 4-input FIS, can
result in both over and underestimation of error which directly translate to over or underestimations of vol-
umetric changes in DEM-derived sediment budget and geomorphic change detection analyses.
While we demonstrated the utility of including spatially variable surface roughness as an input in the
DEM error model, future work should explore the best method for representing this model parameter.
The method of characterizing grain size used in thi s paper is semiquantitative and does not capture
within-channel unit variability in roughness. Potential errors arising from observer subjectivity and/or vari-
ability in the ocular estimate approach could be eliminated by adopting more rigorous, quantitative
approaches such as Wolman pebble counts. As well, finer scale grain size roughness maps could be cre-
ated by sampling substrate within the channel subunit or at individual x, y locations. Elsewhere, spatially
variable estimates of roughness have been calculated as the standard deviation of sample points using
TLS data [Brasington et al ., 2012] and as the standard deviation of the residual topography (Z
DEM
–
Z
Smoothed DEM
)usingLiDAR[Cavalli et al ., 2008; Cavalli and Marchi, 2008] and SfM [Picco et al., 2013] data.
In the context studies using high point density sampling techniques, the standard deviation of elevation
approach can be suitable for characterizing surface or grain roughness. However, in the case of lower
poi nt density TS and rtkGPS surveys, such approaches are likely insufficient, and may at the very most
capture coarser form roughness.
5. Conclusions
This paper describes three different approaches to estimating DEM error for TS surveys using FIS models.
The approach is flexible that its ruleset can, and arguably should, be calibrated to a specific stream data set,
and/or survey technique. The FIS presented here differ in whether additional field data are collected and
the degree of processing needed to create error model inputs. The 5-input FIS model is the most robust of
the three, and represents the most comprehensive TS error model to date for topographic surveys of
streams. This FIS model is preferred, particularly in the context of morphological sediment budgeting, as it
provides more realistic error estimates along bank edges and incorporates the effect of grain size. Extend-
ing this approach to studies using different survey techniques (e.g., airborne LiDAR) or surveying other
types of systems (e.g., large, mainstem rivers) will require selecting the appropriate model inputs and
adjusting the input/output domains. Last, future work should aim to develop empirical methods of calibrat-
ing elevation uncertainty estimates for use in fuzzy models and further refine approaches of characterizing
spatially variable roughness, both within the wetted channel and on out-of-channel surfaces (e.g., active
floodplain).
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Acknowledgments
The data used in this paper are
publically available online at www.
champmonitoring.org. This work was
funded by the Bonneville Power
Administration through the Columbia
Habitat Monitoring Program (Project#:
2011-06) and subcontracts from Eco
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