Water Resour Manage (2015) 29:4379–4395

DOI 10.1007/s11269-015-1065-0

A Machine Learning Approach for the Mean Flow

Velocity Prediction in Alluvial Channels

Vasileios Kitsikoudis1 · Epaminondas Sidiropoulos2 ·

Lazaros Iliadis3 · Vlassios Hrissanthou1

Received: 8 October 2014 / Accepted: 6 July 2015 /

Published online: 29 July 2015
© Springer Science+Business Media Dordrecht 2015

Abstract In natural alluvial channels, the determination of the flow resistance constitutes
a problem with additional complexity compared to rigid bed channels, due to the bed
morphology transformations and the alterations of the flow properties caused by sediment
transport. While there have been steps towards understanding the processes that contribute
to flow resistance in an alluvial channel, a robust quantitative model with wide applica-
bility remains elusive. Machine learning offers the ability to exploit available data and
generate equations that accurately describe the problem by taking implicitly into account
the contributing mechanisms that are difficult to be modeled. In this paper, four machine
learning techniques are employed for the mean flow velocity prediction, separately for sand-
bed and gravel-bed rivers, namely artificial neural networks, adaptive-network-based fuzzy
inference system, symbolic regression based on genetic programming, and support vector

 Vasileios Kitsikoudis
vkitsiko@civil.duth.gr

Epaminondas Sidiropoulos
nontas@topo.auth.gr

Lazaros Iliadis
liliadis@fmenr.duth.gr

Vlassios Hrissanthou
vhrissan@civil.duth.gr

1 Department of Civil Engineering, Democritus University of Thrace, Vasil. Sophias 12,

Xanthi 67100, Greece
2 Department of Rural and Surveying Engineering, Aristotle University of Thessaloniki,
Thessaloniki 54124, Greece
3 Department of Forestry and Management of the Environment and Natural Resources,
Democritus University of Thrace, Pandazidou 193, N. Orestiada 68200, Greece
4380 V. Kitsikoudis et al.

regression. The derived models are robust and their results are superior to those of some
widely used flow resistance formulae, which compute the mean flow velocity from similar
independent hydraulic variables.

Keywords Data-driven modeling · Flow resistance · Gravel-bed rivers ·

Machine learning · Sand-bed rivers · Stage-discharge relation

1 Introduction

Mean flow velocity in an open channel is needed for numerous engineering applications,
such as sediment transport computation, risk analysis, numerical modeling, and flood esti-
mation. In the study of open channel hydraulics with rigid boundary, the resistance is
specified by a roughness coefficient, which can be treated as a constant, and a resistance
formula can be applied directly for the computation of velocity, slope, or depth. In alluvial
rivers, flow resistance is influenced by grain or skin friction, which is the resistance that is
exerted to the flow from the bed particles, as well as by form drag induced from the devel-
opment of bed forms such as ripples, dunes, and bars. Therefore, the hydraulic roughness in
alluvial channels is a dynamic parameter that depends strongly on flow conditions as well
as on the bed sediment properties (Garcia 2008). In this case, a resistance formula cannot
be applied directly without knowledge of how the resistance coefficient will change under
different flow and sediment conditions (Yang 2003). In addition, bed load forms a layer that
alters the roughness height of the bed particles, while suspended load may affect turbulence,
and hence flow resistance.
In nonlinear problems that cannot be easily modeled, machine learning can provide an
alternative solution and be used to build regression models by utilizing a good and represen-
tative dataset. Machine learning has been effectively used in stage-discharge quantification
problems during the last years and this fact is mainly attributed to the continuously increas-
ing computational power. Giustolisi (2004) used genetic programming (GP) to determine
Chézy resistance coefficient for full circular corrugated channels, with data from an exper-
imental apparatus, and obtained parsimonious formulae, which provided very good results.
Baptist et al. (2007) applied GP on synthetic data to obtain an expression for vegetation
related resistance, based on Chézy coefficient, with the final formulation being a combi-
nation of a computer induced expression that fits the data well, and theoretically based
modifications to fit the theory. Dogan et al. (2009) employed relevance vector machine
based probabilistic models in alluvial channels for the prediction of flow regime, depth,
and total sediment transport. Maier et al. (2010) summarized the studies, which imple-
mented artificial neural networks in time-series analysis for the prediction and forecasting
of water quantity and quality variables in river systems. Shrestha and Simonovic (2010)
employed a fuzzy set theory based methodology for the analysis of uncertainty in the stage-
discharge relationship and performed fuzzy nonlinear regression. Azamathulla et al. (2011)
used GP and gene-expression programming, which is a an extension of the former, as an
alternative approach to model the stage-discharge relationship for Pahang River in Malaysia,
based solely on the flow depth and generated results superior to those of artificial neu-
ral networks and conventional regression techniques. Gene-expression programming was
used again by Azamathulla and Jarrett (2013) to develop a predictive model for Manning’s
roughness coefficient for high gradient streams, based on hydraulic radius, slope, and a
characteristic grain diameter, which performed better than neural networks and conventional
models.
Machine Learning Approach for the Mean Flow Velocity 4381

The aim and novelty of the present paper is the demonstration of the suitability of
machine learning in the quantification of mean flow velocity, individually for sand-bed and
gravel-bed streams, by providing an extensive comparison among the employed techniques
as well as with some well-known formulae, from which can be inferred the superior-
ity of the former. This study examines the potential of four machine learning techniques,
because each one has its own perspective and sometimes may be inadequate for a phys-
ical system representation (Witten et al. 2011). The tested techniques are artificial neural
networks (ANNs), adaptive-network-based fuzzy inference system (ANFIS), symbolic
regression (SR) based on GP, and support vector regression (SVR). In addition, the present
paper can be considered complementary to sediment transport quantification studies, which
employed similar methodology, and offer together a complete framework for machine
learning utilization in river hydraulics. Specifically, Kitsikoudis et al. (2014a, 2014c) and
Kitsikoudis et al. (2014b, 2014c) used the aforementioned techniques for bed-material load
prediction in sand-bed rivers and bed load transport in gravel-bed rivers, respectively, while
Kitsikoudis and Hrissanthou (2013) employed a custom-made ANN for the fractional bed
load prediction in gravel-bed rivers.

2 Alluvial Roughness and Resistance to Flow

Rouse (1965) suggested that flow resistance comprises the following mechanisms: surface
or skin friction, form resistance or drag, wave resistance, and resistance associated with flow
unsteadiness. Morvan et al. (2008) argued that roughness is implicit because it is a function
of the flow it governs and it is not a function of the grain size only, and as a result, it should
be a calibration parameter in computational models, especially in those modeling natural
geometries where the insights that have been gained in an idealized laboratory channel, do
not always apply.
Three popular relationships linking velocity and flow resistance are the Manning, the
Chézy, and the Darcy-Weisbach equations:


1 2/3  8gRh S
V = Rh S 1/2 = C Rh S = (1)
n f

where V is the mean flow velocity, Rh is the hydraulic radius, S is the slope, g is the grav-
itational acceleration, and n, C, f are the Manning, Chézy, and Darcy-Weisbach resistance
coefficients, respectively. In order to adapt these formulae to alluvial channels, Wu and
Wang (1999) and Yu and Lim (2003) have proposed equations to modify Manning’s formula
and estimate n for the case of sand-bed rivers, while van Rijn (1984) proposed a compu-
tational procedure for the determination of Chézy resistance coefficient C for various flow
conditions.
As the flow intensity increases, bed forms may develop and the bed roughness will
be additionally influenced by form roughness. Several researchers, subsequently, divided
the flow resistance into two additive parts, expressed by different variables. Einstein and
Barbarossa (1952) were the first to develop a depth-discharge predictor explicitly express-
ing the resistance caused by the bed forms and expressed the hydraulic radius Rh as
the summation of the hydraulic radius due to skin friction and the hydraulic radius due
to form drag. Engelund and Hansen (1967) based on a similar concept expressed the
4382 V. Kitsikoudis et al.

energy slope as above, while Alam and Kennedy (1969) decomposed in a similar way
the Darcy-Weisbach friction factor. This linear division of the cross-section is a use-
ful conceptual tool; however, there is no such strict geometric division (Vanoni 2006).
Other researchers proposed resistance equations that directly compute the water depth
(Brownlie 1983; Karim 1995). Yen (2002) classified loosely the channel bed forms as plane
bed, ripples, dunes, and antidunes and described qualitatively their flow resistance mecha-
nisms. He indicated that a plane bed channel can be similar to the plane rigid bed channel,
due to the fact that the source of resistance is skin friction, but he also stated that the major
differences between the two are the water movement through the voids among the bed par-
ticles of the sediment bed and the fact that in a plane loose bed, there is an energy and
momentum demand for picking up, transporting, and depositing the bed sediment. In ripple
bed channels, form resistance constitutes an additional flow resistance mechanism, while
for channels with dunes and antidunes, there is wave resistance as well.
Studies focused into the effect of suspended sediment transport on flow hydrodynamics
have provided contradictory results. Vanoni and Nomicos (1960) showed that the addition
of suspended sediment to clear water damped turbulence intensity and thus flow resistance
was decreased, Wang and Qian (1989) observed that the turbulence intensity of a sediment
laden flow is weaker compared to that of a clear water flow, and Khullar et al. (2007) found
that the friction factor decreases in the presence of wash load, provided that the bed form
geometry does not change significantly with change in suspended load concentration. On
the contrary, Muller (1973) reported opposite results, with sediment transport increasing
streamwise turbulence intensity, Lyn (1991) found that the presence of suspended sediment
will generally result to an increase of flow resistance, while van Ingen (1981) observed
that the longitudinal turbulence intensity does not seem to be significantly affected by the
presence of suspended sediment.
The existence of feedback mechanisms between flow resistance and bed load has been
verified by numerous studies, and, subsequently, Recking et al. (2008) suggested that a
flow-dependent bed roughness should be used. At least for intense bed load transport, a
higher apparent roughness height emerges, which is related to the thickness of the bed
load layer and is much larger than the sediment particle sizes (Wilson 1987; Sumer et al.
1996; Camenen et al. 2006). Nonetheless, Whiting and Dietrich (1990) didn’t observe any
roughness variation with transport stage in naturally packed, poorly sorted gravel bed with
non-intense bed load transport and considered the bed roughness to be dictated by the
larger static and slowly moving particles. Pitlick (1992) observed that weak bed load trans-
port in North Fork Toutle River, Washington, over a plane gravel bed had little effect on
flow resistance. Nikora and Goring (2000) analysis showed that the mean longitudinal flow
velocity can actually be increased in the presence of weak gravel bed load. Carbonneau
and Bergeron (2000) performed a series of flume experiments, in which bed load trans-
port sometimes caused an increase of flow velocity, depending on the hydraulic variables,
and demonstrated that bed load affects flow velocity by modifying the rate of dissipa-
tion of turbulent kinetic energy. Campbell et al. (2005) ran a series of relative experiments
and observed that with the transport of fine grains at the higher feed rate in a flume with
glued coarse grains, longitudinal mean flow velocities increase compared to the clear water
condition, because the fine grains saturated the troughs and interstices in the bed topogra-
phy, effectively smoothing the irregular bed. Flow resistance may therefore be increased,
decreased, or unchanged by bed load transport, but it is clear that mean and turbulent
flow properties over a mobile bed may differ significantly from those over a fixed bed
(Campbell et al. 2005).
Machine Learning Approach for the Mean Flow Velocity 4383

3 Data Analysis

Parker and Anderson (1977) expressed the stage-discharge relationship as a function of five
dimensionless variables and derived the following relationship for equilibrium flow in an
alluvial channel with a bed comprising noncohesive sediment:
 
V ∗ = f X̂1 , X̂2 , Rep , R, p (2)

where V ∗ is a dimensionless mean flow velocity, X̂1 and X̂2 are dimensionless parameters
pertinent to the problem, p is a dimensionless bed surface sediment factor, Rep denotes
an explicit particle Reynolds number, and R denotes the submerged specific gravity of the
sediment. Herein, after testing several variables, X̂1 and X̂2 are relative depth Rh /d and
slope S, respectively. The former is expressed as the ratio between the hydraulic radius Rh
and a characteristic grain diameter d. d84 /d50 is used as a surrogate for p in gravel-bed
streams, which shows the nonuniformity of the bed sediment mixture, while for sand-bed
rivers is omitted because the bed sediment is considered uniform. d84 and d50 are grain sizes
such that 84 % and 50 %, by weight, of the bed sample are finer, respectively. The variables
of Eq. 2 are summarized in the following Eqs. 3 to 7
√
Rgdd
Rep = (3)
ν

ρs
R= −1 (4)
ρ

V
V∗ = √ (5)
gd

Rh
X̂1 = (6)
d

X̂2 = S (7)

where ν is the kinematic viscosity of water, ρs is the sediment density, and ρ is the water
density. R in natural streams has, usually, a value of 1.65 and as a result is omitted from
Eq. 2. The selection of the variables, which serve as X̂1 and X̂2 , has been done consider-
ing the fact that the physics of flows with low relative depth can be different from those
with high relative depth (Smart and Jaeggi 1983; Ferguson 2007; Recking et al. 2008),
while the bed slope affects the threshold for the incipience of motion of the bed particles
(Shvidchenko and Pender 2000; Lamb et al. 2008; Recking 2009), and thus, modifies the
bed load roughness layer as well. Because gravel-bed rivers are poorly sorted and the bigger
4384 V. Kitsikoudis et al.

grains have dominant effect in the bed roughness, since they protrude more to the flow, d84
is used, in contrast to the bed form dominated bed roughness of the sand-bed rivers
where the median grain diameter d50 is utilized. Hence, V50 ∗ and V ∗ correspond to
84
V ∗ obtained from Eq. 5 based on d50 and d84 , respectively, while Rep50 and Rep84
along with Rh /d50 and Rh /d84 are formed in a similar way, based on Eqs. 3 and 6,
respectively.
The training procedure in data-driven modeling is efficient as long as there is an abun-
dance of quality data. In this study, the exploited field data for the flow resistance in
sand-bed rivers originate from the database compiled by Brownlie (1981), while those
for gravel-bed rivers emanate from the data compilation of Rickenmann and Recking
(2011). For details regarding the exploited data, due to space limitations, the interested
reader is referred to these papers, which put together pertinent data from different sources.
For the former case, the sand-bed data were scrutinized, and those that didn’t satisfy
Rh /d50 > 100, in order to eliminate shallow water effects, and the geometric standard
deviation of bed particles being σg ≤ 5, in order to eliminate bimodal distributions, were
removed. For the gravel-bed rivers case, Rickenmann and Recking (2011) put the data
through a screening process, which is employed in the present paper as well. Specifically,
Rickenmann and Recking (2011) set an upper and a lower limit for (8/f )0.5 based on the
Keulegan law and the values predicted by the friction law proposed by Recking et al. (2008)
for very steep slopes and high sediment transport, respectively. The generalization capabili-
ties of data-driven modeling can be considered as the effect of a good nonlinear interpolation
(Haykin 2009). However, the data scarcity at the extreme ranges of the available dataset
could lead to spurious conclusions due to the inadequate number of available data for val-
idation and testing. Especially the extreme high values of the single output variable could
significantly affect the training process due to the fact that the training error criterion was
a sum of squared errors. Thus, since the proposed models can be actually utilized in rivers
similar to those studied herein, and for data within the data range of the training set, after
visual inspection of data distribution plots, measurements including extreme values, which
were considered to be outliers, were removed. The data preparation was concluded by loga-
rithmizing all the input and output variables in order to obtain a distribution closer to normal,
which benefits data-driven modeling (Pyle 1999), and additionally, generates only positive
flow velocities.
The proper training procedure of a machine learning model dictates the division of the
available data into three datasets, namely the training, validation, and testing datasets. The
goal is to create a robust model with minimal complexity that generalizes well to unseen
data, which should lie in the range of those constituting the training set. The model is trained
by exploiting the training data, aiming to minimize a sum of errors between the model out-
puts and the actual measured data, and utilizes the validation data as a training stopping
criterion to avoid overfitting. The testing data serve as an independent evaluation dataset to
assess the generalization capabilities of the derived model. This process is slightly differ-
entiated for the SVR modeling, where the 5-fold cross-validation is used, as explained in
Section 4.4. Figure 1 shows the modeling process flow chart for each employed technique.
All datasets (training, validation, and testing sets) should have similar statistical distribu-
tions (Bowden et al. 2002), while the data constituting the validation and testing sets should
be within the data range of the respective variables of the training set. However, the exploited
data compilations consist of datasets from different streams and by different researchers. To
ensure the proper representation of all the streams, the remaining data are arranged in their
original order and three successive measurements are selected for the training set followed
Machine Learning Approach for the Mean Flow Velocity 4385

Fig. 1 Flow chart of the modeling process for each utilized machine learning technique. The testing set is
the same for all four cases. ANNs, ANFIS, and SR employ the same training and validations sets, while SVR,
due to different modeling process, considers all these data for 5-fold cross-validation

by one for the validation and one for the testing set. Consequently, there are 493 measure-
ments for training, 164 measurements for validation, and 164 measurements for testing in
the sand-bed rivers study, while for the gravel-bed rivers study there are 984 measurements
for training, 328 measurements for validation, and 328 measurements for testing. Tables 1
and 2 show some statistical indices of the exploited data for the sand-bed and gravel-bed
case, respectively.

4 Stage-Discharge Modeling with Machine Learning

4.1 ANN Modeling

The employed ANN is the common multilayer feedforward neural network (Haykin 2009),
which sends the information solely in the forward direction and the neurons of each
layer are fully connected to those of the following layer. The Levenberg-Marquardt back-
propagation training algorithm (Hagan and Menhaj 1994) was utilized, and the mean square
error between the modeled and measured values served as the cost function. Hornik et al.
(1989) showed that an ANN is able to approximate any measurable function as long as
it consists of a sufficient number of hidden neurons with a squashing activation function.
However, the optimal architecture is determined based on a trial-and-error approach, and
4386 V. Kitsikoudis et al.

Table 1 Statistical measures for the machine learning utilized data for sand-bed rivers

Statistical measures Rh /d50 S Rep50 ∗

V50 V
(m/s)

Train. set Minimum value 127.32 0.96 × 10−5 3.885 5.063 0.325
Maximum value 88306 0.0032 123.10 39.192 2.384
Mean value 12935 7.39 × 10−4 28.124 16.858 0.916
Standard deviation 18287 6.97 × 10−4 26.142 7.884 0.387
Skewness coefficient 1.791 0.829 2.377 0.885 1.158
Val. set Minimum value 139.82 1.00 × 10−5 4.087 5.257 0.268
Maximum value 81407 0.0031 123.10 37.040 2.167
Mean value 13685 7.38 × 10−4 27.051 16.723 0.890
Standard deviation 19120 7.01 × 10−4 26.014 7.713 0.376
Skewness coefficient 1.618 0.830 2.551 0.705 1.218
Test. set Minimum value 130.65 1.08 × 10−5 4.353 5.486 0.320
Maximum value 77990 0.0031 123.10 39.046 2.423
Mean value 13782 7.31 × 10−4 26.499 16.793 0.900
Standard deviation 19704 7.00 × 10−4 24.524 7.992 0.385
Skewness coefficient 1.776 0.818 2.721 0.977 1.378

subsequently, several network architectures were tested, which comprise one and two hid-
den layers with up to 20 neurons. The training efficiency of an ANN relies heavily on the
initial values of the synaptic weights, due to their impact on the entrapment of the cost
function to a local minimum, which will serve as the solution of the optimization prob-
lem and determine the strength of the synaptic weights. Consequently, a supplementary

Table 2 Statistical measures for the machine learning utilized data for gravel-bed rivers

Statistical measures Rh /d84 S d84 /d50 Rep84 ∗

V84 V
(m/s)

Train. set Minimum value 0.191 0.0020 1.256 111602 0.036 0.050
Maximum value 4.808 0.0750 3.958 1518742 1.937 3.725
Mean value 1.319 0.0252 2.413 541579 0.608 0.861
Standard deviation 0.843 0.0163 0.655 419655 0.398 0.479
Skewness coefficient 1.170 1.406 0.721 1.087 0.626 0.508
Val. set Minimum value 0.257 0.0026 1.256 111602 0.049 0.100
Maximum value 4.049 0.0750 3.958 1518742 1.937 2.136
Mean value 1.324 0.0253 2.413 532479 0.609 0.859
Standard deviation 0.818 0.0167 0.661 418786 0.397 0.478
Skewness coefficient 0.923 1.399 0.717 1.128 0.511 0.247
Test. set Minimum value 0.191 0.0026 1.256 111602 0.043 0.060
Maximum value 4.387 0.0750 3.958 1518742 1.919 2.600
Mean value 1.320 0.0254 2.402 538029 0.609 0.863
Standard deviation 0.808 0.0162 0.650 418367 0.393 0.474
Skewness coefficient 1.039 1.443 0.720 1.104 0.598 0.377
Machine Learning Approach for the Mean Flow Velocity 4387

code to the MATLAB neural network toolbox was written, which involves 2000 differ-
ent training executions, with random initial synaptic weights for each repetition. The flow
chart of the modeling process is shown in Fig. 1. Finally, the best results were obtained
from ANNs with one hidden layer, comprising 10 neurons for the sand-bed rivers, and
15 neurons for the gravel-bed rivers case. The activation functions for the hidden neu-
rons and for the single output neuron are the hyperbolic tangent and the linear function,
respectively.

4.2 ANFIS Modeling

ANFIS is a powerful computational tool that generates a set of fuzzy if-then rules based
on fuzzy sets (Zadeh 1965) with data-driven tuning of the membership function parameters
and was presented by Jang (1993). For this study, it employs a hybrid learning algorithm
for training (Jang 1993), which consists of a combination of the least-squares and the
back-propagation gradient descent method (Werbos 1990), with the sum of squared errors
constituting the error function. A typical ANFIS model consists of five layers. In the first
one, the inputs are introduced to the membership functions and the product of their outputs,
which represents the firing strength of the rule, is calculated in the second layer. In the third
layer, the firing strengths are normalized with respect to the sum of all rules firing strengths
and subsequently they are multiplied by a constant or linear function of the ANFIS input
variables in the fourth layer. Finally, the fifth layer sums up all the incoming signals and
provides the output.
The ANFIS stage-discharge modeling was carried out with the aid of MATLAB fuzzy
logic toolbox. The utilized ANFIS is a Sugeno-type system with no rule sharing and unity
weight for each rule, while the maximum training epochs were set to 20000. The ANFIS
architecture, the membership functions associated with each variable, and the step size
setup are determined after a trial-and-error process, as shown in Fig. 1. The optimal ANFIS
configuration for the sand-bed rivers comprises two bell-shaped curves that serve as mem-
bership functions for each input and a linear output function, while for the gravel-bed rivers
it consists of three Gaussian curves as membership functions for each input and a constant
output function.

4.3 SR Modeling

GP is inspired by the Darwinian principle of reproduction and survival of the fittest and
seeks the computer program, with a tree-like structure, that performs best in a given prob-
lem in terms of fitness measure, which is a sum of prediction errors (Koza 1992). After a
random population generation, the genetic operators of sexual recombination (crossover)
and mutation alter the individual programs and facilitate their conversion to more potent
ones, which are nonlinear functions able to fit the training data. The new population is again
subjected to this procedure for a predetermined number of generations or until satisfactory
accuracy is achieved.
GPTIPS (Searson 2009) is an open source MATLAB toolbox that performs multigene
SR, which is a linear combination of several genes, each of which has a tree-like structure
and is evolved according to GP. Table 3 shows the parameters and genetic operators config-
urations that were tested for training, while Fig. 1 depicts the modeling process flow chart.
Searson (2009) suggested that relatively compact models, which are linear combinations of
low order nonlinear transformations, may be obtained by keeping the tree depth relatively
short.
4388 V. Kitsikoudis et al.

Table 3 Parameters and genetic

operators tested in GPTIPS for Parameters and genetic operators Adjustments
SR implementation
Population size 7500-10000
Number of generations 250-2000
Max genes 2-3
Max gene tree depth 4-6
Functions + −, ×, ÷, x 2 , x 3
Probability of GP mutation event 0.10-0.35
Probability of GP crossover event 0.85-0.60
Probability of GP direct copy event 0.05
Error function Root mean square error

4.4 SVR Modeling

Support vector machines (SVMs) were initially developed for classification and pattern
recognition (Vapnik 1995), and their modeling capabilities expanded to function approxi-
mation (Smola and Schölkopf 2004). In SVR, the aim is to determine a function that has
at most ε deviation from the actual measurements, and at the same time is as flat as possi-
ble (Smola and Schölkopf 2004). The determination of such a function involves solving a
minimization problem (Iliadis et al. 2011) with an objective function heavily influenced by
the parameter CSV R , which determines the trade-off between the flatness of the generated
function and the tolerated deviations (Smola and Schölkopf 2004). Another parameter that
defines the performance of a SVR model is the kernel parameter γ , which nonlinearly maps
samples into a higher dimensional space.
The computational tool for the regression is provided by LIBSVM 3.17
(Chang and Lin 2011), which is encoded in C++ and offers a MATLAB interface. The
selection of the CSV R , ε, and γ values has been done after a trial-and-error process. The
radial basis function kernel is employed, while the data are scaled to [-1 1], and the 5-fold
cross-validation method is applied (Geisser 1993), in which the data for training are split
into five groups (folds) of equal size and the ε-regression algorithm is executed five times.
Each time, a different fold is chosen as the validation set, while the remaining folds serve
as training data. The values [CSV R , γ , ε] that provide the minimum error between the
measured and the computed values in the validation sets, are subsequently used to train the
whole training data population (all the five folds). Figure 1 shows the SVR modeling pro-
cess. The optimal models were considered those with [CSV R , γ , ε] = [0.5625, 7, 0.0488]
for sand-bed rivers, and [CSV R , γ , ε] = [5.25, 5, 0.0859] for gravel-bed rivers.

5 Results and Discussion

The modeled outputs Pi are compared to the observed values Oi and to the mean observed
value Ō on the basis of the root mean square error (RMSE),

 N

 (O − Pi )2
 i=1 i
RMSE = (8)
N
Machine Learning Approach for the Mean Flow Velocity 4389

mean absolute error (MAE),

N
|Oi − Pi |
i=1
MAE = (9)
N
mean normalized error (MN E),

N
100 Oi − Pi
MN E = (10)
N Oi
i=1

Nash and Sutcliffe (1970) coefficient of efficiency (CE),

N
(Oi − Pi )2
i=1
CE = 1 − (11)
N 2
Oi − Ō
i=1

and discrepancy ratio (DR). The latter denotes the percentage of the model outputs that lie
within some predetermined limits of the respective measured values. Moreover, the principle
of parsimony was followed, which suggests the preference for models with simpler structure
against complicated ones with similar or even slightly better performance.
Tables 4 and 5 show the machine learning generated mean flow velocities in terms of
m/s, as can be derived from Eq. 5, for the sand-bed and gravel-bed rivers, respectively.
The derived test set results are compared to two formulae for the sand-bed case and nine
formulae for the gravel-bed one. The machine learning models exhibit superior results, even

Table 4 Performance evaluation of machine learning for mean flow velocity prediction in sand-bed rivers,
in terms of m/s

RMSE MAE MN E CE DR0.8−1.25 DR2/3−1.50 DR0.5−2

(m/s) (m/s) (%) (%) (%) (%)

Train. set ANN 0.177 0.126 14.38 0.792 79.31 95.13 99.80
ANFIS 0.195 0.136 15.43 0.745 76.67 93.10 99.80
SR 0.208 0.146 16.05 0.712 74.04 93.10 99.59
SVR 0.175 0.117 13.48 0.795 80.32 94.32 99.19
Val. set ANN 0.184 0.128 16.13 0.758 81.10 92.68 98.78
ANFIS 0.208 0.136 16.95 0.691 78.05 92.68 98.17
SR 0.214 0.149 17.96 0.673 77.44 90.24 99.39
SVR 0.181 0.113 14.25 0.766 84.15 93.90 98.17
Testing set ANN 0.187 0.127 15.00 0.762 83.54 93.90 100.00
ANFIS 0.211 0.142 16.33 0.696 75.61 91.46 99.39
SR 0.213 0.148 16.38 0.693 75.00 93.90 100.00
SVR 0.201 0.129 15.11 0.726 81.71 93.29 98.78
Brownlie (1983) 0.252 0.169 18.26 0.569 68.29 90.24 98.78
Karim (1995) 0.339 0.229 26.30 0.221 62.80 82.93 96.34
4390 V. Kitsikoudis et al.

Table 5 Performance evaluation of machine learning for mean flow velocity prediction in gravel-bed rivers,
in terms of m/s

RMSE MAE MN E CE DR0.8−1.25 DR2/3−1.50 DR0.5−2

(m/s) (m/s) (%) (%) (%) (%)

Train. set ANN 0.143 0.105 18.02 0.912 76.02 89.63 97.97
ANFIS 0.149 0.113 19.27 0.904 73.07 88.11 97.26
SR 0.180 0.134 22.38 0.859 68.70 85.57 97.05
SVR 0.133 0.104 17.86 0.923 82.22 90.96 97.36
Val. set ANN 0.147 0.103 17.69 0.905 75.61 88.41 97.87
ANFIS 0.163 0.115 19.50 0.884 69.82 86.59 98.17
SR 0.176 0.128 21.85 0.865 69.51 84.45 96.34
SVR 0.131 0.098 16.12 0.925 80.18 92.68 98.48
Testing set ANN 0.159 0.118 20.34 0.886 72.56 87.80 96.65
ANFIS 0.179 0.127 20.97 0.856 72.26 89.02 95.12
SR 0.193 0.136 22.55 0.834 68.60 88.11 96.04
SVR 0.170 0.118 20.45 0.871 75.00 89.02 95.43
Bathurst (1985) 0.376 0.274 54.87 0.367 48.17 66.16 85.98
Ferguson (2007) 0.334 0.265 36.40 0.502 34.15 63.41 89.94
Hey (1979) 0.323 0.250 39.51 0.533 41.46 70.73 85.67
Jarrett (1984) 0.405 0.328 48.65 0.267 23.78 45.73 88.11
Keulegan (1938) 1.387 1.223 238.08 −7.599 3.66 8.54 38.11
Manning-Strickler 1.257 1.111 231.29 −6.068 4.88 15.55 39.94
(Strickler 1923)
Recking et al. (2008) 0.316 0.238 38.91 0.554 46.95 71.65 85.98
Rickenmann and 0.331 0.246 36.44 0.509 42.99 66.46 84.15
Recking (2011)
Smart and Jaeggi 0.327 0.256 33.71 0.521 32.32 62.20 85.37
(1983)

though some of these formulae have the significant advantage that they have been partially
calibrated on data from the testing dataset. Figures 2 and 3 show the scatter plots between the
observed mean flow velocities and the respective predicted ones from the machine learning
models for the test set of the sand-bed and gravel-bed rivers, respectively. The models seem
to be robust since their performance in the test set is satisfactory especially for the higher
velocities. The better results, for both the sand-bed and gravel-bed studies, were obtained
from ANNs followed by SVR and ANFIS, while SR performed a little worse. While the
SVR results for the training and validation sets constitute, essentially, a single training set,
due to space limitations it was preferred to be shown in Tables 4 and 5 in accordance to the
other implemented techniques.
The generated models that are considered to be the optimal ones exhibit medium com-
plexity. Machine learning models with increased degree of complexity can emulate the
training data to a great degree of accuracy; however, this will lead to overfitting. Despite the
utilization of cross-validation, which can prevent the overtraining to some degree, a model
with many redundant parameters will be forced to learn from the measurements noise as
Machine Learning Approach for the Mean Flow Velocity 4391
Computed mean flow velocity (m/s)

Computed mean flow velocity (m/s)

1 1

ANN ANFIS
y=x y=x
y=1.25x y=1.25x
y=0.8x y=0.8x
y=1.5x y=1.5x
y=2x/3 y=2x/3
0.2 1 0.2 1
Measured mean flow velocity (m/s) Measured mean flow velocity (m/s)
(a) ANN (b) ANFIS
Computed mean flow velocity (m/s)

Computed mean flow velocity (m/s)

1 1

SR SVR
y=x y=x
y=1.25x y=1.25x
y=0.8x y=0.8x
y=1.5x y=1.5x
y=2x/3 y=2x/3
0.2 1 0.2 1
Measured mean flow velocity (m/s) Measured mean flow velocity (m/s)
(c) SR (d) SVR
Fig. 2 Scatter plots of measured and computed mean flow velocities for the test set of the sand-bed rivers
study. In the legend, x denotes the measured mean flow velocity, while y denotes the computed mean flow
velocity

well. Subsequently, the models with the minimal complexity that produced equally good
results, were preferred to the more complicated ones. After testing several architectures
and parameters, it was inferred that additional complexity doesn’t actually contribute to the
model generalization improvement and as a result, more complicated models weren’t exam-
ined. Especially for the time consuming SR, it was observed that the model that performed
best in the validation set was obtained several generations before the predefined maximum
number. SR training has a stochastic component in the random initial population genera-
tion, as well as in the crossover and mutation operators. Subsequently, numerous training
executions would be required for exploring an adequate parameter space; however, the slow
training procedure hinders such an effort. This could be the reason that SR provided the
4392 V. Kitsikoudis et al.
Computed mean flow velocity (m/s)

Computed mean flow velocity (m/s)

1 1

ANN ANFIS
y=x y=x
y=1.25x y=1.25x
y=0.8x y=0.8x
0.1 0.1
y=1.5x y=1.5x
y=2x/3 y=2x/3
0.1 1 0.1 1
Measured mean flow velocity (m/s) Measured mean flow velocity (m/s)
(a) ANN (b) ANFIS
Computed mean flow velocity (m/s)

Computed mean flow velocity (m/s)

1 1

SR SVR
y=x y=x
y=1.25x y=1.25x
y=0.8x y=0.8x
0.1 0.1
y=1.5x y=1.5x
y=2x/3 y=2x/3
0.1 1 0.1 1
Measured mean flow velocity (m/s) Measured mean flow velocity (m/s)
(c) SR (d) SVR
Fig. 3 Scatter plots of measured and computed mean flow velocities for the test set of the gravel-bed rivers
study. In the legend, x denotes the measured mean flow velocity, while y denotes the computed mean flow
velocity

least good results, since all the modeling outputs were obtained after approximately equal
total computational times.
A machine learning approach for the quantification of the mean flow velocity and
sediment transport in alluvial streams could provide a useful tool for water resources man-
agement. It can potentially accommodate the peculiarities of the investigated river system
without spending too much time and many resources on data acquisition and simulation,
which would otherwise be required for computationally expensive models. The explicit for-
mulae provided by machine learning can be incorporated in optimization schemes to satisfy
water demands and minimize environmental impacts.
Machine Learning Approach for the Mean Flow Velocity 4393

6 Conclusions

Although there have been huge steps towards understanding the physics of flow resistance
and the effects of bed roughness to the flow, a physics-based quantitative model based
on deterministic foundations that performs consistently well in alluvial channels remains
elusive. Machine learning provides a solution to quantifying the stage-discharge relation-
ship by exploiting large datasets, interpreting the knowledge embodied in these data and
subsequently transforming it into equations for engineering purposes.
Four machine learning techniques were employed in this paper for the mean flow veloc-
ity prediction separately in sand-bed and gravel-bed rivers, namely ANNs, ANFIS, SR, and
SVR. All of them generated robust models with good generalization capabilities and per-
formed better than some commonly used stage-discharge relations. ANNs derived the best
results in almost all the statistical indices used, and SR generated the least good results
compared to the other machine learning techniques employed.
The derived models can be utilized for predicting the mean flow velocity in rivers similar
to the ones the training data emanate from. Moreover, the proper utilization of a data-driven
model dictates that the predictions should be based on independent variables within the
range of those constituting the training dataset, which are shown in Tables 1 and 2. Addi-
tional high quality data from different flow ranges or from rivers with different peculiarities
could be embodied to the existing ones, or be exploited separately, for generating machine
learning models with a wider applicability.

Acknowledgments The authors are grateful to an Associate Editor and two anonymous Reviewers for their
constructive and insightful comments and suggestions, which improved the presentation of this paper.

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