A triangular form-based multiple flow algorithm to estimate overland flow
distribution and accumulation on a digital elevation model
Petter Pilesjö and Abdulghani Hasan
GIS Centre, Department of Physical Geography and Ecosystems Sciences, Lund University
Sölvegatan 12, SE-223 62 Lund, Sweden
Running head: Triangular form-based multiple flow algorithm
Corresponding Author:
Petter Pilesjö
Sölvegatan 12
223 62 LUND
Keywords: digital terrain modelling, digital terrain analysis, hydrological modelling, surface flow
estimation, flow routing algorithm.
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Abstract
In this study, we present a newly developed method for the estimation of surface flow paths on a digital
elevation model (DEM). The objective is to use a form-based algorithm, analysing flow over single cells
by dividing them into eight triangular facets and to estimate the surface flow paths on a raster DEM. For
each cell on a gridded DEM, the triangular form-based multiple flow algorithm (TFM) was used to
distribute flow to one or more of the eight neighbour cells, which determined the flow paths over the
DEM. Because each of the eight facets covering a cell has a constant slope and aspect, the estimations
of—for example—flow direction and divergence/convergence are more intuitive and less complicated
compared to many traditional raster-based solutions. Experiments were undertaken by estimating the
specific catchment area (SCA) over a number of mathematical surfaces, as well as on a real-world DEM.
Comparisons were made between the derived SCA by the TFM algorithm with eight other algorithms
reported in the literature. The results show that the TFM algorithm produced the closest outcomes to the
theoretical values of the SCA compared with other algorithms derived more consistent outcomes and
was less influenced by surface shapes. The real-world DEM test shows that the TFM was capable of
modelling flow distribution without noticeable ‘artefacts’, and its ability of tracking flow paths makes it
an appropriate platform for dynamic surface flow simulation.
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1. Introduction
Topography is critical for modelling distributed hydrological processes, and especially surface/overland
flow. The key parameter in catchment topography that has to be estimated is flow distribution, which
indicates how overland flow is distributed over the terrain. Slope direction controls the pathway of the
overland flow and also substantially influences the sub-surface flow pattern.
Many authors have discussed different flow routing algorithms and how they have been applied on
digital elevation models (DEMs). The following paragraphs are based mainly on discussions by Zhou et
al. (2011) and Pilesjö (2008). One can conclude that the use of a DEM has made it possible to estimate
flow at each location over a surface. Based on the flow distribution estimation, the drainage pattern over
a surface—as well as various hydrological parameters, such as catchment area and up-stream flow
accumulation—can be modelled (Wilson et al., 2008).
The most critical spatial data required for digital hydrological modelling are the surface elevations,
which are typically modelled as discrete samples over the land surface. Although there have been
numerous models developed for this purpose—such as the triangulated irregular network (TIN), and
digital contour and hybrid structures (i.e., a grid with intermeshed break-lines) (Ackerman and Krauss,
2004)— the gridded DEM is the most commonly used data source for terrain analysis because of its
simple data structure, ease of implementation, and rapidly growing applications of digital
photogrammetry and remote sensing technology (Li et al., 2005).
Grid DEM-based hydrological modelling algorithms have been developed since the early stages of
geographical information systems (GIS) (Beven and Moore, 1994; Wilson and Gallant, 2000; Zhou et
al., 2008). However, because a grid DEM itself is an approximation of the real-world continuous surface
using regularly spaced samples, the implementation of terrain analysis models is inevitably affected by
the DEM grid structure. In practice, numerous assumptions and optimisations about mass transportation
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and movement on a specific local surface have to be made to establish a workable mathematical model
(Holmgren, 1994), resulting in significantly different approaches and methodologies towards terrain
modelling (Zhou and Liu, 2002). It has been recognised that such modelling algorithms may produce
significantly inconsistent results related to the terrain complexity and DEM data property (Zhou and Liu,
2004; Zhou et al., 2006), which lead to the presence of some significant artificial patterns (known as
‘artefacts’) in the output.
Generally speaking, on a grid DEM, the surface flow and local catchment area are approximated by
applying one out of two common approaches, namely, the use of single flow direction (SFD) and/or
multiple flow direction (MFD) algorithms, according to whether the algorithms consider flow
divergences. The SFD algorithms (O’Callaghan and Mark, 1984; Fairfield and Leymarie, 1991; Lea,
1992; Orlandini et al. 2003) do not allow flow divergence and restrict the mass movement to only one
downhill direction at a time, even though the flow may be proportionally distributed into two adjacent
downstream cells (e.g., Tarboton, 1997). The MFD algorithms (Freeman, 1991; Quinn et al., 1991;
Costal-Cabral and Burges, 1994; Qin et al., 2007; Seibert and McGlynn, 2007; Gallant et al., 2011)
consider flow divergence and assume that mass (or flow) on a grid DEM can be transported to more than
one flow direction. In many cases, SFD algorithms can produce satisfactory results over horizontally
concave surfaces where convergence of flow is assumed. On plane or horizontally convex slopes, where
parallel or divergent flows are more likely, it is often more appropriate to divide the flow into two or
more directions. Combinations of the two types of algorithms are often preferred when modelling water
flow over natural surfaces (Pilesjö et al., 1998).
Studies show that a large part of the uncertainty in derived hydrological parameters can be explained by
the grid data structure of the DEM (Zhou and Liu, 2004). When a natural continuous surface is
represented by regularly distributed spot heights, it is inevitably difficult to determine the way that
surface flow distributes over adjacent cells (Olsson and Pilesjö, 2002). On the other hand, flow
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estimation over a surface with a constant slope and aspect, such as a facet in a TIN, would be
considerably more consistent, and an SFD algorithm would be adequate without the complication of
flow divergence.
In this study, we are developing a new flow routing method, taking advantage of the SFD algorithms
concerning consistency, the MFD algorithm concerning the flexibility/flow diversity, the TIN modelling
regarding the continuity, and applying it to the grid DEM. The grid cells in the DEM (each of them
treated as a ‘centre cell’ when estimating flow from that cell) are further partitioned into triangular
facets, which have constant slopes and aspects. Based on these conclusions, the surface flow path from
each facet can be uniquely tracked and distributed to other facets covering the centre cell, and/or to one
or more of the eight cells surrounding the centre cell. A comparison between the proposed algorithm and
other algorithms reported in the literature is made using a data-independent test method (Zhou and Liu,
2002), as well as a test on a real-world DEM.
To summarise, the aim of this study is to create and evaluate a flow-algorithm that can simulate overland
flow in a realistic way on a digital elevation model, and overcome the previously mentioned problems.
How flat areas and man-made barriers are taken care of, discussed by Hasan et al. (2012a), is beyond the
scope of this study.
2. Methodology
A key challenge when estimating flow over grid-based DEMs is how to model the flow movement over
each grid cell (Zhou et al., 2006). A typical assumption for this has been to assign a ‘flow package’ (or a
package of water) to the centre of each grid cell. Based on this, the grid cell is treated as a point on a
continuous surface, where subsequently, a unit of flow package is generated and flows to the next
downhill point(s). This assumption makes it difficult to incorporate the form of the surface overlapping
the cell into the flow routing.
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In this study, we attempt to create a computational model that is capable of simulating the surface flow,
with less impact than the grid data structure of a DEM. The realistic simulation of the flow pattern is
considered valuable not only as a means to estimate drainage areas but also in estimating the amount of
water in time and space (Pilesjö, 2008). The approach is to sub-partition the grid cell according to the
surface form so that the flow direction over a local surface can be uniquely defined (Tarboton, 1997). A
grid cell is further partitioned by creating eight local, triangular facets between the cell centre and the
eight surrounding cell centres of the neighbouring cells.
2.1 Computation of slope and aspect for each triangular facet
When a facet is defined, the slope and aspect values are constant for this surface. The coordinates of the
three vertices of a triangle (compare to Figure 1) are known as C1(x1, y1, z1), C2(x2, y2, z2) and M(x3, y3,
z3). The facet is formed as a plane as follows:
z  f ( x, y )  ax  by  c
(1)
where a, b and c can be derived as:
( y1  y3 )( z1  z 2 )  ( y1  y2 )( z1  z3 ) 

( x1  x2 )( y1  y3 )  ( x1  x3 )( y1  y2 ) 
( x  x )( z  z )  ( x1  x3 )( z1  z 2 ) 
b 1 2 1 3

( x1  x2 )( y1  y3 )  ( x1  x3 )( y1  y2 ) 

c  z1  ax1  by1


a
(2)
Let p and q denote the gradients in the W-E and N-S directions, respectively. According to Equation 1,
for a triangular facet with known vertices, we have
p  fx 
f
a
x
, and
q  fy 
f
b
y
The slope () and aspect () of the facet can therefore be derived as:
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(3)


q
p
b
a
  180  arctan  90  180  arctan  90 
p
p
a
a
  arctan p 2  q 2  arctan a 2  b 2
(4)
The aspect angle is calculated clockwise from North.
2.2 Flow routing within a cell
The proposed triangular form-based multiple flow algorithm (TFM) combines the advantages of
different flow distribution algorithms in a very simple but innovative way. The work is partly based on,
and can be seen as a further development of, the TFN algorithm presented by Zhou et al. (2011). The
major improvement considers form. The TFM algorithm is based on multiple flow distribution allowing
overland flow to all lower cells surrounding a centre cell while TFN is a ‘single flow algorithm’. BY
dividing cells into sub-cells the proposed TFM algorithm topographic form is treated in more detail,
allowing flow to all neighbouring cells. It is developed to be consistent for all terrain types: convex,
concave, plane terrain, as well as their combinations.
Around the midpoint (M, see Figure 1) of the cell in question (the centre cell from where the flow is
estimated), eight planar triangular facets are constructed with midpoints of two adjacent cells (C1 and
C2). With the aid of these eight triangular facets, our current grid cell (centre cell) is divided into eight
triangular facets (1-8 in Figure 1). The slope and slope direction (aspect) of each of these triangular
facets can be calculated. The form of the current grid cell is represented by the combined surface of the
eight triangular facets. The area of each facet is equal to 1/8 of the cell size, and represents the flow
portion contributed by that facet. The overland flow of each triangular facet is to be routed out of the
facet (towards other facet(s) or neighbouring cell(s)), or stays in the same triangular facet depending on
slope direction.
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In the following sections, we take triangular facet number one in Figure 1 as an example to explain the
possible estimations of flow routing within the eight facets. Naturally, the same approach is applied
when estimating the flow from another of the eight facets (2-8) to its neighbouring facets.
The first step is to calculate the slope direction/aspect value for the facet to be analysed. As illustrated in
Figure 2, depending on the aspect value, three flow routing alternatives are possible; water can be
directly routed from a facet to a neighbouring cell. This alternative results in NO routing to other facets
and is denoted as stay; ALL water on a facet can be routed to ONE other neighbouring facet and is
denoted as move; water on a facet can be routed to a neighbouring facet AND a neighbouring cell, or to
two neighbouring facets and is denoted as split. The three alternatives are described in further detail
below:
1. If the aspect value of facet 1 is between 0 and 45 degrees (see Case 1, Figure 2a, where 0 ≤ ASP ≤
45), then the amount of water (directly proportionate to the area of the facet) to be transported from
this facet to a neighbouring cell will stay as it is (1/8 of a cell).
2. If the aspect value is between 90 and 180 degrees (see Case 3, Figure 2d, where 90 ≤ ASP ≤ 180), or
between 225 and 270 degrees (see Case 3, Figure 2e, where 225 ≤ ASP ≤ 270), then all the water on
the facet will be moved to one adjacent facet; that is, to facet number 2 and/or facet number 8. This
means that the amount of water to be routed from facet 1 to a neighbouring cell in these cases will be
0, and this water (1/8) will instead be added to facet number 2 and 8, respectively.
3. If the aspect value is between 45 and 90 degrees (see Case 2, Figure 2b, where 45 < ASP < 90), or
between 270 and 360 degrees (see Case 2, Figure 2c, where 270 < ASP < 360), then the water on
facet 1 will be split, and partly routed into one neighbouring facets according to a vector split (e.g.,
Pilesjö, 2008). The amount of water to be later routed to a neighbouring cell will initially stay on the
facet, while the other portion will be moved to a neighbouring facet. In the example given in Figure
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2b, some water will stay (for further routing to the cell above the facet; see below), while some water
will be moved to facet number 2. In the example given in Figure 2c, some of the water will be
moved to facet number 8. If the aspect value is between 180 and 225 degrees (see Case 4, Figure 2f,
where 180 < ASP < 225), then the water on facet 1 will be split between two adjacent facets; one
portion will be added to the amount of water (proportional to the area) of facet number 2, while
another portion will be added to facet number 8. The resulting amount of water of facet number 1
will be 0.
Note, in the case of two facets sloping towards each other, i.e., contradictive slopes, no water is routed
between the two facets, but the alternative stay is applied.
Because water can be transported between more than two facets (from facet 1 to facet 2; to facet 3 to
facet 4; and then to neighbouring cells), flow routing between facets continues until all water has
reached the outflow facet(s) of the cell. Water is then transported to neighbouring cells. The result of
this first step is the redistribution of water within a cell. It also estimates where (to which neighbour
cells) water will route.
2.3 Flow distribution to neighbouring cells
After doing the flow routing for each cell—which consists of eight individual facets—the routing of
water to adjacent cell(s) takes place. In one or more of the eight facets, water is ‘waiting’ to be routed to
neighbouring cell(s). If the cell represents a concave landform, we can have only one facet holding all
the water (directly proportional to the area, 1, of one cell); if the cell represents a convex surface
(compare to a pyramid), there might be water in all eight facets.
Each facet has two neighbouring cells (e.g., see cells C1 and C2 for facet number one in Figure 1), and
the next step is to distribute the accumulated water between these cells. Two different cases can then be
found:
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1. If one neighbouring cell is lower in elevation than the centre cell (where the facet is located) and the
other cell is higher or equal to the centre cell, then the water accumulated in the facet will all be
distributed to the lower cell.
2. If both neighbouring cells are lower in elevation than the centre cell, then the accumulated water in
the facet will be proportionally distributed to both lower cells to slope (an x value of 1, Equation 5):
fi 
tan i x
 tan  
2
j 1
1. , for all ß > 0
(5)
x
j
where i,j = flow directions (1…2) to lower cells, fi = flow proportion (0…1) in direction i, tan ßi =
slope gradient between the centre cell and the cell in direction i, and x = variable exponent.
3. If both neighboring cells are higher or equal in elevation than the central cell and there are other
lower elevation cells then the accumulated water in the facet will be proportionally distributed to
all lower cells.
2.4 Accumulated flow estimation and specific catchment area
Flow distribution to neighbouring cells is estimated for all cells except for the border cells in the DEM.
Starting from cells with no incoming water—that is, peaks or cells at the edge of the DEM—flow
accumulations (in the unit of cells) are estimated as going downhill in the catchments. The results are
estimated as values of flow accumulation for every cell, with the exception of the border cells in the
DEM.
When comparing results between different flow routing algorithms, the specific catchment area (SCA)
was used. The SCA can be estimated using the definition and method reported by Costa-Cabral and
Burges (1994) as:
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SCA 
TCA


TCA
g ( sin A  cos A )
(6)
where SCA is the specific catchment area defined as the upstream catchment area per unit contour line
(m2/m); TCA is the total upstream catchment area at a given point (m2);  is the flow width of the
catchment outlet (m); A is the slope aspect at the centre of the grid cell (º), estimated by approximation
of a second order trend surface and representing the general flow direction of that cell; and g is the grid
size (m).
3. Experiment
Two different methods have been used to test the proposed flow estimation algorithm (TFM). The first
one is a data-independent test based on the method proposed by Zhou and Liu (2002), testing the results
of flow estimation on a number of mathematical surfaces. This makes it possible to derive and compare
quantitative measurements of the accuracy for different methods. We also apply the proposed algorithm
to a real-world DEM. The spatial patterns of estimated flow and accumulated flow are then visually
examined, to detect if there are significant artefacts.
3.1 Data-independent assessment
Repeating the experiment carried out by Zhou et al. (2011), the following surfaces were used: ellipsoid
(representing a convex slope), inverse ellipsoid (representing a concave slope), plane (representing a
straight slope) and saddle (representing a convex/concave slope association). The four different
mathematical surfaces are visualised below, in Figure 3. The generation of the surfaces, including the
formula defining the elevation at each and every cell, can be found in Zhou et al. (2011). According to
Zhou et al. (2011), the cell size for all surfaces is 5 metres, and the number of cells varies between 15
000 and 40 000. For the plane surface, the slope is set to 66º, and the aspect to 243º.
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At each cell of the mathematical surfaces, the theoretical ‘true’ specific catchment area (SCA) was
calculated according to the method described by Zhou and Liu (2002). When the ‘true’ SCA values, as
well as the estimated flow algorithm dependant ones are known, the error at each cell can be computed
as:
Ei  SCAi  SCAt
(7)
where Ei denotes the error (or residual) at the ith cell using a selected algorithm, SCAt and SCAi denote
the theoretical and estimated value (using the flow algorithm to be tested) of the SCA at the ith cell,
respectively. The Root-Mean-Standard Error (RMSE), Mean Error (ME, denoted as E ) and Standard
Deviation of the residuals (SD, denoted as ) were then computed according to Zhou et al. (2011) for
the assessment and comparison of the selected algorithms:
n
E
RMSE 
i 1
2
i
(8)
n
n
E
E
i 1
i
(9)
n
 E  E 
n

i 1
2
i
(10)
n 1
where n denotes the total number of grid cells for each mathematical surface DEM.
The proposed TFM algorithm was applied on the four mathematical surfaces, and the estimated flow
accumulation values for each cell were converted to SCA in the unit of grid cells according to Equation
6. Then, the estimated SCA values were compared with the ‘true’ values computed. Statistical analyses
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of the differences (RMSE, E , and ) were carried out to estimate overall accuracy as well as spatial
distribution of errors.
3.2 Selection of algorithms for comparison
The proposed TFM algorithm was compared with eight commonly used, tested, and well documented
flow algorithms. Three of these can be classified as grid based single flow algorithms (D8, D8-LTD, and
D), three as grid based multiple flow algorithms (MD, FMFD, and QMFD), and two as (partly)
vector-based algorithms (DEMON and TFN). The selected algorithms are as follows:

Deterministic Eight-Node (D8) (O’Callaghan and Mark, 1984)

Deterministic Eight-Node Least Transversal Deviation (D8-LTD) (Orlandini et al., 2003; Orlandini
and Moretti, 2009)

Deterministic infinite number of possible single flow directions (D) (Tarboton, 1997)

Triangular Multiple Flow Direction (MD) (Seibert and McGlynn, 2007)

Freeman Multiple Flow Direction (FMFD) (Freeman, 1991)

Quinn Multiple Flow Direction (QMFD) (Quinn et al., 1991)

Digital Elevation Model Networks (DEMON) (Costa-Cabral and Burges, 1994)

Triangular Facet Network (TFN) (Zhou et al., 2011)
Which algorithm to select for comparisons can always be discussed. However, the selected eight
algorithms were judged to represent a wide variety of solutions, involving raster as well as vector-based
algorithms, all widely available to the research community. We used the System for Automated GeoScientific Analysis, SAGA (2012) for testing D8, D, MD, FMFD and DEMON algorithms, the
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original FORTRAN code available from the author’s website1 for the D8-LTD algorithm, and the
original code for the TFN algorithm available directly from the authors. Software implementation for the
QMFD algorithm was conducted using the C++ programming language from the Zhou et al. study
(2011). ArcGIS2 GIS software was used for comparison and visualisation of the results.
3.3 Real-world testing
To visually examine the result of the proposed TFM algorithm, it was applied to a real-world DEM. This
1 metre resolution DEM was interpolated form LiDAR data measured at the Stordalen mire and its
catchment area. Stordalen is a peat land area in the Arctic region 10 km west of Abisko (68º 20' N, 19º
03' E), north of Sweden. The LiDAR data, as well as the interpolation procedure, are described in detail
in Hasan et al. (2012b).
Because the differences between the different flow algorithms are relatively small (see below), the main
purpose of real-world testing is to identify possible artefacts created by the proposed TFM algorithm.
However, to highlight differences, we have also included accumulated flow estimations based on the D8
algorithm.
4. Results
The tests of the different flow algorithm can be divided into two parts: first, the quantitative test of the
significant differences between estimated values and ‘true values’ of SCA over the mathematical
surfaces, and second, the qualitative test examining spatial distribution of errors over the mathematical
surfaces, as well as visual interpretation of the real-world modelling.
4.1 Quantitative accuracy assessment
1
2
http://www.idrologia.unimore.it/orlandini/download.html.
ArcGIS 9.3, © Environmental Systems Research Institute, Inc., http://www.esri.com.
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The significant differences between the nine different flow algorithms and the ‘true values’ of SCA for
the four mathematical surfaces are presented in Table 1. For three out of four of the mathematical
surfaces, the results show that the proposed TFM algorithm outperforms the other eight algorithms in
terms of RMSE. It has the lowest RMSE values out of 5.88 m, 8.28 m, and 7.66 m, for the ellipsoid, the
inverse ellipsoid, and the saddle, respectively. For the plane, the TFM algorithm has a higher (6.13 m)
RMSE value than the TFN algorithm (1.86 m) and is also just higher than the DEMON algorithm (5.18
m).
Regarding the systematic bias, represented by the Mean Error ( E ) value, the TFM algorithm shows
better results than the all other algorithm for two of the four mathematical surfaces: the ellipsoid ( E = 0.99 m) and the saddle ( E = 0.61 m). For the inverse ellipsoid the differences between the TFM result (
E = 4.00 m) and the slightly better TFN ( E = -3.95 m) and DEMON ( E = 3.88 m) are marginal. For
the plane, only the TFN algorithm shows a better result ( E = 1.85 m) than the TFM algorithm ( E =
3.12 m).
If we assume normal distribution, then the Standard Deviation of the Residuals () indicates the
representability of the Mean Error; the lower the Standard Deviation, the more narrow distribution of
errors around the Mean Error. For the ellipsoid surface, the TFM algorithm has a Standard Deviation of
5.80 m, which is the third lowest of the eight algorithms. However, because the FMFD ( = 2.32 m) and
the QMFD ( = 4.61 m) algorithm both have considerably higher mean errors (6.87 m and 6.64 m
compared to -0.99 m) the deviation from zero (no error) is actually smaller for the TFM algorithm. This
is also the case for the saddle, where only the FMFD algorithm has a lower Standard Deviation than
TFM (6.95 m compared to 7.63 m), but a much higher mean error (13.19 m compared to 0.61 m). For
the plane, the TFM standard deviation (5.27 nm) is higher than the deviations of DEMON (3.91 m) and
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TFN (0.11 m) algorithms, and for the inverse ellipsoid the TFM algorithm shows the lowest standard
deviation out of all the tested algorithms.
The Mean Error and Standard Deviation also indicate possible skewness in the estimation of flow
accumulation. If the standard deviation is less than the absolute value of the mean error, this indicates
that at least 83% of the errors are either all positive or all negative. Such a systematic error is not
desirable. Referring to Table 1, the absolute mean error never exceeded the standard deviation for three
algorithms (DEMON, TFN, and TFM); exceeded it at one surface, for three algorithms (D8, D8-LTD,
and D); exceeded it at two surfaces, for one algorithm (MD); and exceeded it at three out of four
surfaces, for two algorithms (FMFD and QMFD). When comparing the frequency distribution of
positive and negative errors for these eight algorithms, only the TFM algorithm did not exceed skewness
of the proportion 80-20 (%) for any of the four surfaces. D8-LTD exceeded 80-20 for one surface, D8,
D, MD, and TFN did it for two surfaces, DEMON for three, and FMFD and QMFD exceeded the 8020 proportion for all four surfaces. It is also worth noting that the skewness was one-sided (either over
or underestimating SCA for all mathematical surfaces) for all tested algorithm but TFM.
In Figure 4 we illustrate the results of one single flow algorithm (D8), one multiple flow algorithm
(QMFD), and the proposed form-based algorithm (TFM) applied on the four mathematical surfaces.
Differences in flow routing are obvious, especially for the saddle, where only the TFM algorithm
logically routes 50% of the water in NW and SE directions.
Regarding the quantitative comparison between the proposed triangular form-based multiple flow
algorithm (TFM) and the eight alternative algorithms it clearly shows that the TFM algorithm produces
the best over-all results. Even if it did not produce the best result in every test over every mathematical
surface, it shows by far the best and most consistent results. Referring to Table 1 one can conclude that
out of the 12 measurements of accuracy (RMSE, E , and  for the four mathematical surfaces) the TFM
16
algorithm outperforms all other algorithms in six of these. The algorithms producing the second best
result, the TFN algorithm, outperforms all other algorithms in three out of these 12 cases. It should also
be noted that these three cases are all linked to estimation on the plane surface.
Generally it can be concluded that the raster-based single flow algorithms (D8 and D8-LTD) produce
the poorest results, with RMSE values between 50 and 224 m (average = 130.0 m). The raster-based
multiple flow algorithms (D, MD, FMFD, and QMFD) produce better results, with RMSE between 7
and 66 m (average = 32.7 m). The (partly) vector-based algorithms (DEMON and TFN) produce even
more reliable results, with RMSE between 2 and 56 (average = 14.7 m). The reason for this can be
explained by the obvious over-simplification adopted by the single flow algorithms, in combination with
the advantages in geometric precision connected to the vector solutions. The average RMSE value for
the TFM algorithm is 10.0 m.
4.2 Spatial distribution of errors on the artificial surfaces
The ‘true values’ of SCA, as well as the estimated values applying the proposed TFM algorithm for the
four mathematical surfaces, are presented in Figure 5. Even if differences can be observed, it is clear that
the modelled SCA values, to a high degree, coincide with the true values.
The spatial distribution of residuals of derived SCA—with respect to the theoretical ‘true’ value of SCA
on the four mathematical surfaces, for all of the nine tested algorithms—is shown in Figure 6. Logically,
and based on the errors reported above (see Table 1, for example), the TFM, TFN and DEMON
algorithms show relatively less errors compared to the other algorithms tested. A general difference
between the raster-based algorithms (D8, D8-LTD, D, MD, FMFD, and QMFD) and the vectorbased algorithms (DEMON and TFN) can also be identified: the vector-based algorithms show a more
random pattern of error distribution, while the errors in the raster-based algorithms seem to be more
additive. However, for the TFM algorithm, this systematic bias seems to be relatively low.
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The less-biased pattern of the TFN algorithm is explained by the way SCA values are calculated. By
counting the number of flow lines passing each cell, in vector mode, the derived SCA values of the
upper stream cells do not affect that of the down-stream cell (see Zhou et al., 2011). However, as will be
further discussed, the TFN algorithm does not produce better overall results and is much more CPU
demanding.
4.3 Real-world testing
Even if visual differences are often difficult to detect when modelling flow over real-world surfaces, the
derived SCA values between the TFM and D8 algorithm were estimated for a peat land area in northern
Sweden. Figure 7 presents the visual comparison between the results of the two algorithms. The results
show that the TFM has produced a more balanced simulation on the flow distribution on this terrain
surface, with a more realistic spatial pattern on the different terrain types (straight slopes, converging
slopes and diverging slopes). In comparison, the D8 algorithm shows a possible advantage in
representing the drainage network, but it also demonstrates its mostly criticised weakness: an
unacceptable 45-degree bias that results in significant artefacts, such as unrealistic parallel drainage
lines.
5. Discussion
Even if the proposed TFM algorithm takes a raster approach when estimating the overland flow
pattern—although this has been recognised as one of the major sources of error for flow estimation
(Zhou and Liu, 2004)—it seems superior to all other tested algorithms, including the (partly) vector
based DEMON and TFN algorithms.
5.1 Accuracy comparison
18
The accuracy estimations have been carried out using both quantitative comparisons between estimated
flow accumulation/SCA and ‘true values’ over mathematical surfaces, and visual interpretation of a
natural surface in northern Sweden. The main purpose of the latter is to reveal possible artefacts, caused
by over-simplifications and ‘illogical flow routing’.
Common problems connected to the multiple flow algorithms include the limited possibilities in
adjusting the overland flow to the terrain form. Logically, we assume the flow to convert over concave
surfaces and divert over convex surfaces. This can be regulated by the x exponent in Equation 5, but this
is rarely performed (see Pilesjö et al., 1998). However, this is taken care of in the proposed form-based
algorithm. The logic in the flow routing is exemplified in Figure 4, where it can be observed how water
is and should be divided over a ridge (the saddle).
The results of visual analysis over both artificial and real-world surfaces show no obvious systematic
bias or artificial spatial pattern for the TFM algorithm. The single flow algorithms have the capability of
modelling concave landforms relatively well, but show illogical results for plane and convex landforms.
The opposite is valid for most of the multiple flow algorithms; because flow is normally distributed to
all neighbour cells with lower elevation this creates unrealistic flow accumulation values for concave
surfaces.
5.2 Potential implementation
The source code of the proposed TFM algorithm is available from the authors. In this code, also
solutions how to tackle e.g., sinks, flat areas, and man-made barriers are implemented (Hasan et al.,
2012a). Compared to vector based methods (e.g., the TFN algorithm), the computational time (CPU) of
the proposed TFM algorithm is also fast, making it more useful. Most likely, the major hurdle for the
implementation of the TFM algorithm, as well as other more advanced multiple flow algorithms, is the
slow response from the research and industry communities. This, in combination with the question if
19
‘better’ flow algorithms are really needed (and they not always are), will probable slow down the
potential implementation.
5.3 Assessment methods
The proposed TFM method has been evaluated and compared with eight other commonly used
algorithms reported in the literature. Both quantitative and qualitative evaluations of flow
accumulation/SCA, on mathematical surfaces (ellipsoid, inverse ellipsoid, plane, and saddle) as well as
real-world data, have been carried out. Root mean standard errors, mean errors, and standard deviations
of the residuals have been calculated and compared. Visual interpretations have been made. However,
more thorough and rigorous evaluation methods can always be performed, not only on the quantitative
errors of the estimated catchment area (such as the area estimation error) but also on the uncertainty of
the flow path or their derivatives’ morphology (see Orlandini and Moretti, 2009).
6. Conclusion
In this study we have tested a newly developed triangular form-based multiple flow algorithm (TFM).
The algorithm is form-based in the sense that it divides the cell from where the flow routing takes place
into eight triangular facets to estimate flow over the cell. The flow is then distributed to one or more of
the eight neighbour cells, determining the flow paths over the DEM. Because each of the eight facets
covering a cell has a constant slope and aspect, the estimations of, for example, flow direction and
divergence/convergence are more intuitive and less complicated compared to many traditional rasterbased solutions. Experiments undertaken by estimating the specific catchment area (SCA) over a
number of mathematical surfaces, as well as on a real-world DEM, clearly indicate an all-round
advantage of estimating flow and flow accumulation in all forms of slopes when using the TFM
algorithm compared to other algorithms. The TFM algorithm outperforms eight commonly used
compared algorithms (D8, D8-LTD, D, MD, FMFD, QMFD, DEMON, and TFN), proven by both
20
quantitative measurements including RMSE, ME and SD, and qualitative evaluation including the visual
analysis of spatial patterns of residuals on the artificial surfaces and SCA over a real-world DEM.
It is also concluded that the fact that the TFM algorithm is raster-based in terms of flow distribution
between different cells in the DEM, making it considerably faster than vector based alternatives (e.g.,
TFN), is highly desirable when processing larger DEM.
Overall, the proposed TFM algorithm is considered both adequate in representing overland flow over a
DEM, resulting in accurate estimations of flow accumulation and SCA, and efficient enough to have a
high potential for broad implementation.
7. Acknowledgements
The European Union funding programme Erasmus Mundus ‘External Cooperation Window’ (EMECW
lot8) financed the second author of this study. The authors are grateful to Qiming Zhou, Hong Kong
Baptist University, and Yumin Chen, Wuhan University, for fruitful cooperation and data exchange.
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24
Table 1. Accuracy comparison between the TFM algorithm and eight other selected algorithms on the
four mathematical surfaces (unit = metres). Partly from Zhou et al. (2011).
RMSE
D8
D8-LTD
D
MD
FMFD
QMFD
DEMON
TFN
TFM
Ellipsoid
63.53
Inverse ellipsoid
224.31
Plane
154.53
Saddle
103.68
E

RMSE
-12.07
-86.22
-109.87
-2.06
62.37
97.21
207.08
174.98
108.67
50.08
103.66
172.01
E

RMSE
-14.04
-26.21
-42.54
11.52
96.19
21.51
-10.65
173.01
66.03
-21.43
26.43
62.75
-49.92
171.62
27.67
-5.71
18.69
62.45
38.02
27.07
14.52
-11.81
56.37
-18.55
62.75
-49.92
17.32
-4.41
8.46
53.23
38.02
16.74
7.25
6.87
34.94
14.11
29.62
25.26
14.91
13.19
2.32
31.97
15.47
6.95
8.08
6.64
41.24
17.09
42.26
31.61
15.76
13.11
4.61
37.53
28.06
8.75
17.21
4.47
11.21
3.88
5.18
3.41
55.64
6.67
16.62
10.52
3.91
55.23
8.42
1.97
10.25
-3.95
1.86
1.85
8.11
-0.92
8.18
9.46
0.11
8.06
5.88
-0.99
8.28
4.00
6.13
3.12
7.66
0.61
5.80
8.23
5.27
7.63
E

RMSE
E

RMSE
E

RMSE
E

RMSE
E

RMSE
E

RMSE
E

25
C1
C2
8 1
7
2
M
3
6
5 4
Figure 1. In a 3 by 3 cells window, the centre cell is divided into eight triangular facets (1-8). Each facet
is formed from three points; one is the centre cell (M), and the two other are two adjacent cells (e.g., C1
and C2).
26
1
2
2
2.
3.
4.
3
3
4
5.
6.
7.
Figure 2. An illustration of how water can be routed from one triangular facet to other facets. Different
aspect values lead to different actions (stay, move and split).
27
Ellipsoid
Inverse Ellipsoid
Plane
Saddle
Figure 3. Illustrations of the four different mathematical surfaces used in the data-independent
assessment of flow algorithms.
28
Ellipsoid
Inverse ellipsoid
Plane
Saddle
Elevation
99.3 100 99.3
100.4 107.2 114.2
99
101 103
99.6 100 100.7
98.6 100 98.6
93.1 100 107.2
98
100 102
100 100 99.8
97.3 98
97.3
85.9 93.1 100.4
97
99
100.4 99.8 98.8
100
100
100
101
D8
100
QMFD
4
4
12
12
29
21
42
21
26
12
19
34
29
37
13
17
13
55
TFM
4.4
4.4
4.9
21.1
21.1
31
18.1 12.8 18.1
38
46.1
31
38.4 10.6
50
7.3
7.3 35.4
Figure 4. Distribution of flow from a centre cell to the eight neighbouring cells using different flow
algorithms over sub-surfaces (3 x 3 cells) of four mathematical surfaces (inverse ellipsoid, plane,
ellipsoid, and saddle).
29
A1
C1
A2
B1
C2
D1
B2
D2
Figure 5. A comparison between the ‘true’ SCA values (A1, B1, C1, D1) and the ones modelled by
applying the proposed TFM algorithm (A2, B2, C2, D2) for four different mathematical surfaces
(ellipsoid, inverse ellipsoid, plane and saddle). Low SCA values (dark brown) to high SCA values (dark
blue).
30
Ellipsoid
Inverse Ellipsoid
Plane
Saddle
D8
D8-LTD
D
> 1,000
100 - 100
10 - 100
1 - 10
-1 - 1
MD
-10 - -1
-100 - -10
-1,000 - -100
< -1,000
FMFD
QMFD
DEMON
DEMON
31
TFN
TFM
Figure 6. The spatial distribution of residuals of the derived SCA compared to the theoretical ‘true’
values over four mathematical surfaces.
32
D8
TFM
Figure 7. A comparison between the D8 and TFM algorithms on estimated SCA over a real-world
DEM.
33
Figure captions
Figure 1. In a 3 by 3 cells window, the centre cell is divided into eight triangular facets (1-8). Each facet
is formed from three points; one is the centre cell (M), and the two other are two adjacent cells (e.g., C1
and C2).
Figure 2. An illustration of how water can be routed from one triangular facet to other facets. Different
aspect values lead to different actions (stay, move and split).
Figure 3. Illustrations of the four different mathematical surfaces used in the data-independent
assessment of flow algorithms.
Figure 4. Distribution of flow from a centre cell to the eight neighbouring cells using different flow
algorithms over sub-surfaces (3 x 3 cells) of four mathematical surfaces (inverse ellipsoid, plane,
ellipsoid, and saddle).
Figure 5. A comparison between the ‘true’ SCA values (A1, B1, C1, D1) and the ones modelled by
applying the proposed TFM algorithm (A2, B2, C2, D2) for four different mathematical surfaces
(ellipsoid, inverse ellipsoid, plane and saddle).
Figure 6. The spatial distribution of residuals of the derived SCA compared to the theoretical ‘true’
values over four mathematical surfaces.
Figure 7. A comparison between the D8 and TFM algorithms on estimated SCA over a real-world
DEM.
34