Bayesian Calibration and Validation of a Large‐Scale and Time‐Demanding Sediment Transport Model

Abstract

This study suggests a stochastic Bayesian approach for calibrating and validating morphodynamic sediment transport models and for quantifying parametric uncertainties in order to alleviate limitations of conventional (manual, deterministic) calibration procedures. The applicability of our method is shown for a large‐scale (11.0 km) and time‐demanding (9.14 h for the period 2002‐2013) 2D morphodynamic sediment transport model of the Lower River Salzach and for three most sensitive input parameters (critical Shields parameter, grain roughness, and grain size distribution). Since Bayesian methods require a significant number of simulation runs, this work proposes to construct a surrogate model, here with the arbitrary Polynomial Chaos technique. The surrogate model is constructed from a limited set of runs (n=20) of the full complex sediment transport model. Then, Monte‐Carlo based techniques for Bayesian calibration are used with the surrogate model (10^5 realizations in 4 h). The results demonstrate that following Bayesian principles and iterative Bayesian updating of the surrogate model (10 iterations) enables to identify the most probable ranges of the three calibration parameters. Model verification based on the maximum a posteriori parameter combination indicates that the surrogate model accurately replicates the morphodynamic behavior of the sediment transport model for both calibration (RMSE =0.31m) and validation (RMSE =0.42m). Furthermore, it is shown that the surrogate model is highly effective in lowering the total computational time for Bayesian calibration, validation and uncertainty analysis. As a whole, this provides more realistic calibration and validation of morphodynamic sediment transport models with quantified uncertainty in less time compared to conventional calibration procedures.

Full text

Bayesian Calibration and Validation of a Large‐Scale and
Time‐Demanding Sediment Transport Model
Felix Beckers
1
, Andrés Heredia
1,2,3
, Markus Noack
1,4
, Wolfgang Nowak
2
,
Silke Wieprecht
1
, and Sergey Oladyshkin
2
1
Institute for Modelling Hydraulic and Environmental Systems, Department of Hydraulic Engineering and Water
Resources Management, University of Stuttgart, Stuttgart, Germany,
2
Institute for Modelling Hydraulic and
Environmental Systems, Department of Stochastic Simulation and Safety Research for Hydrosystems, SC SimTech,
University of Stuttgart, Stuttgart, Germany,
3
Faculty of Civil Engineering, Universidad Politécnica Salesiana, Quito,
Ecuador,
4
Faculty of Architecture and Civil Engineering, Karlsruhe University of Applied Science, Karlsruhe, Germany
Abstract This study suggests a stochastic Bayesian approach for calibrating and validating
morphodynamic sediment transport models and for quantifying parametric uncertainties in order to
alleviate limitations of conventional (manual, deterministic) calibration procedures. The applicability of our
method is shown for a large‐scale (11.0 km) and time‐demanding (9.14 hr for the period 2002–2013) 2‐D
morphodynamic sediment transport model of the Lower River Salzach and for three most sensitive input
parameters (critical Shields parameter, grain roughness, and grain size distribution). Since Bayesian
methods require a significant number of simulation runs, this work proposes to construct a surrogate model,
here with the arbitrary polynomial chaos technique. The surrogate model is constructed from a limited set of
runs (n= 20) of the full complex sediment transport model. Then, Monte Carlo‐based techniques for
Bayesian calibration are used with the surrogate model (10
5
realizations in 4 hr). The results demonstrate
that following Bayesian principles and iterative Bayesian updating of the surrogate model (10 iterations)
enables to identify the most probable ranges of the three calibration parameters. Model verification based on
the maximum a posteriori parameter combination indicates that the surrogate model accurately replicates
the morphodynamic behavior of the sediment transport model for both calibration (RMSE = 0.31 m) and
validation (RMSE = 0.42 m). Furthermore, it is shown that the surrogate model is highly effective in
lowering the total computational time for Bayesian calibration, validation, and uncertainty analysis. As a
whole, this provides more realistic calibration and validation of morphodynamic sediment transport models
with quantified uncertainty in less time compared to conventional calibration procedures.
1. Introduction
Rivers have political, economic, and environmental relevance and have been of great importance for the
development of urban regions (Kostof & Castillo, 2005). Many rivers have undergone anthropogenic change
during this development (Kondolf & Pinto, 2017). Such change includes the reduction of flood plain areas
due to settlement, river course simplification achieved by straightening and narrowing, and the construction
of hydraulic structures for a variety of purposes (e.g., for bank/bed protection, improved navigability, hydro-
power production). As a result of these measures, the natural hydromorphodynamic behavior of river sys-
tems has been changed, and excessive erosion and deposition can be a medium to long‐term consequence
(e.g., Habersack & Pigay, 2007; Hinderer et al., 2013; Reisenbüchler et al., 2019; Stecca et al., 2019). To study
these consequences, physical‐deterministic numerical models are in use. They are capable to reproduce the
hydromorphodynamic system behavior to predict sediment transport processes and riverbed evolution
(James et al., 2010; Pinto et al., 2012).
By now, many morphodynamic sediment transport models exist and numerical methods are constantly
improving. Despite this, challenges remain since accurate computational description of sediment dynamics
requires to include several model‐and river‐specific parameters, which results in highly parameterized
models (e.g., Merritt et al., 2003). Many of the model‐specific parameters are based on empirical approaches
(e.g., critical Shields parameter), and most of the river‐specific parameters are extremely difficult to measure
in both space and time (e.g., grain size distribution in the surface and subsurface layer, sediment transport
rates, or transported grain sizes). Additionally, monitoring programs are rarely continuously available for
©2020. The Authors.
This is an open access article under the
terms of the Creative Commons
Attribution License, which permits use,
distribution and reproduction in any
medium, provided the original work is
properly cited.
RESEARCH ARTICLE
10.1029/2019WR026966
Key Points:
•We reduce a time‐demanding
sediment transport model with a
surrogate technique based on the
arbitrary polynomial chaos
expansion (aPC)
•Bayesian model calibration and
validation in a fraction of
computational time compared to
conventional (manual,
deterministic) methods
•We achieve a more realistic
calibration, a more successful
validation, and valuable information
in the form of uncertainty intervals
Supporting Information:
•Supporting Information S1
Correspondence to:
F. Beckers,
felix.beckers@iws.uni-stuttgart.de
Citation:
Beckers, F., Heredia, A., Noack, M.,
Nowak, W., Wieprecht, S., &
Oladyshkin, S. (2020). Bayesian
calibration and validation of a
large‐scale and time‐demanding
sediment transport model. Water
Resources Research,56,
e2019WR026966. https://doi.org/
10.1029/2019WR026966
Received 15 DEC 2019
Accepted 12 JUN 2020
Accepted article online 15 JUN 2020
BECKERS ET AL.1of23

long‐term periods (Hinderer et al., 2013), which explains why information on river‐specific parameters is
often confined to local surveys. These model‐and river‐specific parameters carry a large degree of uncer-
tainty that must be thoroughly considered during modeling (e.g., Schmelter et al., 2012; Villaret et al.,
2016). Consequently, sediment transport models require a reasonable calibration and validation process to
evaluate the reliability, accuracy, and quality of the model results (Merritt et al., 2003; Simons et al.,
2000). Model calibration is performed by adjusting independent and uncertain model‐and river‐specific
parameters within a physically meaningful range to reproduce as faithfully as possible the morphodynamic
changes observed in the field within a given period (Cunge et al., 1980; Maren & Wegen, 2016; Oreskes et al.,
1994; Simons et al., 2000). In the following validation process, the model response is tested on an indepen-
dent morphodynamic data set with the selected parameters found during calibration.
The conventional approach in morphodynamic model calibration is manual calibration through heuristic
exercise: The calibration parameters are manually adjusted and the simulation results are checked after each
model run. This iterative procedure results in a single, allegedly best solution. Unfortunately, the combina-
tion of required model runs and constant monitoring of results leads to a labor‐intensive and
time‐consuming workflow, especially for large‐scale and time‐demanding morphodynamic models (e.g.,
Klar et al., 2012, 2014; Mohammadi et al., 2018). Another challenge is related to the possible ill‐posedness
of the calibration problem, that is, the model can achieve similar calibration quality under different para-
meter combinations representing different river conditions (Chavarrías et al., 2018). This, in turn, means
that a unique deterministic solution does not exist and additionally increases the complexity of calibration.
Thus, manual calibration techniques may be criticized because of the total time requirement and potential
for subjective bias on the part of the modeler when selecting one deterministic solution among multiple
potential solutions (e.g., Muehleisen & Bergerson, 2016; Schmelter et al., 2012).
One possibility for handling these challenges would be accounting for the uncertainty in the calibration
parameters and their resulting values by applying a stochastic calibration. Contrary to the deterministic
approach (one best solution), a stochastic calibration approach conceptualizes the values of calibration para-
meters and model results as random variables (e.g., Kim & Park, 2016). Random variables mean that a range
of values are admitted, where each possible value is equipped with an associated probability or probability
density. Thus, multiple possible solutions, occurring in proportion to how likely they are, will match data
and assumptions. During stochastic calibration, the main goal is not only to match the measurements as clo-
sely as possible with the simulation results, but additionally to assess the precalibration through postcalibra-
tion uncertainty and to reduce, in a manner consistent with the measurements, the uncertainty through
modification of the model inputs (Muehleisen & Bergerson, 2016). A well‐known stochastic calibration
approach is Bayesian inference, where the goal is to learn about the unknowns (i.e., calibration parameters)
given observed or measured data (Box & Tiao, 1973). The approach allows for the inclusion of a prior idea or
information about the unknowns via prior probability distributions and also information about how
observed data depend on the unknowns through a likelihood function. Thus, the Bayesian framework infers
a posterior distribution (multiple solutions) of the unknowns in light of known information. This procedure
ensures that linear and nonlinear model relations are accounted for and that corresponding uncertainties
are quantified, both of which can hardly be realized through a manual approach.
Although widely employed in a variety of related disciplines such as ecology, hydrology, and environmental
sciences (see Schmelter et al., 2011), little information on Bayesian approaches in sediment transport mod-
eling are available as they are relatively new in fluvial sediment transport (Schmelter & Stevens, 2013). Wu
and Chen (2009) attribute this to the little resemblance of Bayesian methods to conventional (deterministic)
approaches for parameter estimations and missing demonstration to hydraulic engineers. These authors
employ a Bayesian framework to update parameters of a sediment entrainment model and obtain more
accurate and realistic results after Bayesian updating (Wu & Chen, 2009). Schmelter et al. (2011) introduce
a Bayesian sediment transport model for unisize bed load and employ it to a pedagogical sediment transport
problem. Schmelter et al. (2012) extend this model and demonstrate the applicability to a large gravel river
for sediment budget predictions. In the study of Schmelter and Stevens (2013), the authors present potential
benefits of a Bayesian statistical approach over a traditional curve‐fitting approach to simulate critical stres-
ses observed in laboratory flume experiments. More recent studies are available from Mohammadi et al.
(2018), who present a benchmark study on Bayesian selection of hydromorphodynamic models under com-
putational time constraints and Shojaeezadeh et al. (2018) who use a Bayesian network to model
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stochastically the suspended sediment load given river discharge volumes. Independently of each other, all
these studies recommend the use of Bayesian methods in the field of sediment transport modeling due to
their various benefits including uncertainty quantification, parameter estimation, or model selection to
finally obtain more robust and confident predictions. For this reason, Schmelter et al. (2011) generally con-
clude that Bayesian modeling can provide a tool for innovation in sediment transport research, though blind
use of Bayesian principles could be misleading and requires quality control (Mohammadi et al., 2018). As it
turns out, a stochastic calibration of a morphodynamic sediment transport model seems promising.
However, the main drawback of Bayesian inference is that numerous calibration runs are necessary to
obtain posterior distributions of the investigated parameters. Relatedly, a large number of model executions
(i.e., independent realizations) would be necessary, which is unfeasible for a full complex large‐scale and
time‐demanding sediment transport model. Hence, the limiting factor of Bayesian calibration in morphody-
namic sediment transport modeling is the total time requirement, making model reduction techniques and
surrogate models attractive.
The main goal of a surrogate model (also known as response surface, emulator, metamodel, reduced model,
etc.) is to replicate the behavior of the full complex physical‐deterministic model from a limited set of runs
without sacrificing a lot of detail and accuracy. Therefore, intelligent input parameter combinations that
cover the range of the parametric space as good as possible are strategically selected (called collocation
points or training sets). Since surrogate models are an approximation of any full physical‐deterministic
model, they substantially reduce the computational effort, making them attractive to quantify the uncer-
tainty in the real system using Bayesian statistics. Recent developments based on the polynomial chaos
expansion technique (Wiener, 1938) express the stochastic solution of the surrogate model as orthogonal
polynomials of the input parameters to achieve better convergence (e.g., Oladyshkin & Nowak, 2012).
With this technique, a surrogate model can be constructed that is suitable even for very computationally
demanding problems (Köppel et al., 2019). For example, it can treat morphodynamic sediment transport
modeling including a profound uncertainty analysis followed by Bayesian model calibration.
This work proposes a Bayesian calibration of a large‐scale and time‐demanding morphodynamic sediment
transport model of the Lower Salzach River (Germany, Austria) (Beckers et al., 2016). In an initial step,
we will construct a surrogate model using the arbitrary polynomial chaos technique (aPC) (Oladyshkin &
Nowak, 2012) based on sensitive calibration parameters in order to enable a stochastic analysis. The calibra-
tion will be achieved by implementing an iterative Bayesian update of the aPC surrogate model via a
likelihood‐controlled strategy (Oladyshkin, Schröder, et al., 2013), to match a necessary number of random
Monte Carlo surrogate model realizations to available measurements of the riverbed. This Bayesian analysis
identifies regions in the prior ranges of analyzed calibration parameters where the surrogate model achieves
the best agreement with the measured data. Based on these findings, we aim to obtain an adequate posterior
distribution of the calibration parameters, resulting in multiple suitable parameter combinations. In a next
step, the calibration results will be validated against an additional set of measured riverbed data and we will
provide the statistical information for both calibration and validation, in order to assess the quality of the
stochastic analysis. Further, we will verify the approximations made during stochastic model inversion
and the construction of the aPC surrogate model against the morphodynamic sediment transport model
using the best deterministic solution from the posterior distribution. Finally, we will present the residuals
of the riverbed evolution obtained with the stochastically calibrated surrogate and full complex model.
Moreover, we will compare these findings with a manually calibrated morphodynamic sediment transport
model (Beckers et al., 2016). We expect that obtaining statistical information will help to improve the repro-
duction of the morphodynamic processes, increase the robustness of the physical‐deterministic sediment
transport model and eventually result in a more reliable and realistic calibration in less time.
2. Materials and Methods
2.1. The Lower River Salzach
The River Salzach is an alpine river that originates in the Kitzbuehel Alps and drains via the River Inn to the
River Danube. Many parts of the River Salzach have been subject to anthropogenic modifications that have
changed the entire hydraulic and morphodynamic system behavior. Thus, restoration measures are dis-
cussed in order to restore a sustainable morphodynamic equilibrium (Habersack & Piégay, 2007).
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Especially the German/Austrian section of the Lower River Salzach in the basin of Freilassing (47°51′32.1′′N
12°59′50.2′′E) has suffered from heavy river course modifications in the past. River course straightening,
bank stabilization, and hindered longitudinal sediment continuity due to lateral engineering structures
has led to restricted river dynamics, missing sediment supply, and an increased transport capacity. As a
result, the presence of historically existing alternating gravel bars has declined. Further, the increased
flow velocities and shear stresses have induced progressive riverbed erosion and have pushed the river
into a critical erosion state. The constant risk of riverbed erosion was particularly evident during flood
Figure 1. Overview of the study area in the Lower River Salzach (chainage 63.4 to 52.4 km). Riverbed erosion in the
upstream section (chainage 63.4 to 59.3 km) is prevented due to three lateral structures (ground sills). The investigated
section of interest covers the region downstream of the confluence with the River Saalach from river chainage 59.3 to
52.4 km.
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events in August 2002 and June 2013. The peak discharge at the gaging station Laufen (chainage 47.0 km)
during both flood events reached and exceeded the discharge of a 100‐year flood (Q
100
= 3,100 m
3
s
−1
,
Q
2002
= 3,000 m
3
s
−1
,Q
2013
= 3,530 m
3
s
−1
, for reference Q
mean
240 m
3
s
−1
). Significant riverbed deepening
was the consequence. The current situation of the Lower River Salzach is aggravated by the fact that only
a thin layer of protective gravel material covers a layer of fine‐grained and erosion‐sensitive material due
to altered sediment supply (Mangelsdorf et al., 2000). Once the protective material is eroded, rapid bed inci-
sion and deep scour formations are likely and have been already observed in upstream regions (Stephan
et al., 2003). Moreover, the morphodynamic behavior of the Lower River Salzach is dominated by the sedi-
ment supply from the River Saalach that is the largest tributary. Thus, the hydromorphological situation of
the Lower River Salzach requires measures that can mitigate the ongoing erosion process and protect the
river from further morphological and environmental degradation without decreasing the degree of flood
protection. To obtain an improved understanding of the morphodynamic system and to evaluate the
long‐term riverbed behavior, a 2‐D morphodynamic sediment transport model of the Lower River Salzach
using Hydro_FT−2Dwas applied by Beckers et al. (2016) on behalf of the water authorities of Traunstein
(WWA) and of Salzburg (ASL).
2.2. The Morphodynamic Sediment Transport Modeling Software Hydro_FT−2D
The morphodynamic sediment transport modeling software Hydro_FT−2D(Nujic et al., 2019) is coupled
with the hydraulic solver of the two‐dimensional flow modeling software Hydro_AS−2D(Nujic &
Hydrotec, 2017). Preprocessing and postprocessing of the 2‐D mesh including mesh generation, definition
of boundary conditions, and evaluation of simulation results is conducted with the software SMS
(Surface‐water Modeling System) (Aquaveo, 2013). Hydro_FT−2Dsolves the shallow water equations by
means of a finite volume approach for spatial discretization and the explicit Runge‐Kutta method for tem-
poral discretization (Nujic & Hydrotec, 2017). Bed roughness coefficients are taken into account by
Strickler values k
St
, which are the reciprocals of Manning's values n(k
St
=1/n). Total roughness (k
St
) and
grain roughness k′
St

can be specified separately. Bed load can be calculated with the equation of
Hunziker (1995), an extended form of the equation by Meyer‐Peter and Mueller (1948), to consider multi-
fraction transport (see also Hunziker & Jaeggi, 2002). Total load can be calculated with the equations of
Engelund and Hansen (1967) or Ackers and White (1973). Sediment continuity to calculate riverbed evolu-
tion is ensured by solving the Exner (1925) equation. The software uses a multiple layer approach to simulate
riverbed evolution and allows the use of multiple particle sizes d
i
(i=1,…,nwhere nmax ¼12) to reconstruct
the stratification of the riverbed. The composition of the grain size distribution in the mixing layer is calcu-
lated using the approach of Hirano (1971), and the vertical exchange over the layers is calculated according
to Hunziker (1995).
2.3. Model Setup of the Lower River Salzach and Data Availability
In comparison to the earlier model employed by Beckers et al. (2016), the model of the Lower River Salzach
used in this study is slightly reduced in length (11.0 km instead of 13.2 km) because the main focus is set on
the most relevant river section. This section is located downstream of the confluence with the River Saalach
between river chainage 59.3 and 52.4 km. Consequently, the presented results are limited to this section of
interest with a length of 6.9 km. Figure 1 gives an overview of the study area, the model domain, and the
section of interest.
The applied model setup uses eight grain size classes (0.5, 2.0, 8.0, 16.0, 31.5, 63.0, 90.0, and 140.0 mm) to cal-
culate bed load transport using the equation of Meyer‐Peter and Mueller (1948) in the extended form by
Hunziker (1995) (see also Hunziker & Jaeggi, 2002). Suspended load is not considered since the model aims
at simulating the long‐term riverbed behavior of the Lower River Salzach. The computational mesh consists
of 16,113 elements (14,040 nodes) of which 1,525 elements (1,756 nodes) represent the movable riverbed in
the entire Salzach and 950 elements (1,138 nodes) represent the movable riverbed in the section of interest
(riverbanks are nonerodible, average movable width is approximately 82 m). The average cell size along
the riverbed is 16 m in width and 36 m in length. The initial riverbed geometry is based on cross‐section mea-
surements of the year 2002, which are available at a regular distance of 200 m for the entire model domain.
The typical slope is 0.11% for the upstream model region and reduces to 0.09% in the section of interest. In
addition, cross‐section measurements at the same distance are available for the years 2005, 2010, and
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2013. Information on the grain size distribution is based on measurements that were conducted in different
years (1991, 1993, 1996, and 2001) at only a few places in the river (total number of available measurements:
eight for the surface layer and nine for the sublayer) and from previous studies (Beckers et al., 2016). The
mean grain size diameter of the active layer and the sublayer is d
m,al
= 49.6 mm and d
m,sl
= 45.8 mm, respec-
tively (values averaged over length and width). Three ground sills are located at a river chainage of 62.0, 60.6,
and 59.8 km to prevent ongoing riverbed erosion in the upstream River Salzach. At these locations, the riv-
erbed in the model is defined as nonerodible. Gaging stations at the rivers Salzach and Saalach provide the
hydraulic input data. Bed load input from the upstream river regions (Salzach and Saalach) is considered
using river‐specific rating curves that are derived from measurements at a hydroelectric power plant located
at Saalach chainage 2.4 km and available from previous studies (Beckers et al., 2016; Stephan et al., 2002). A
small tributary (Fischbach) flows into the River Salzach at a chainage of 59.9 km. It is only considered
hydraulically since bed load input from this tributary is negligible (Mangelsdorf et al., 2000; Stephan et al.,
2002). The downstream boundary condition is defined as a constant energy slope of 0.1%. Further model
parameterization includes a sediment density of 2,650 kg m
−3
, a sediment porosity of 37%, and an active layer
thickness that corresponds to the maximum grain size class dmax .
2.4. Previous Manual Calibration, Validation, and Uncertainty Analysis
Beckers et al. (2016) first calibrated and validated their model hydraulically against water level measure-
ments to identify local changes in the total roughness composition along the main channel (13 regions iden-
tified, of them 8 in the section of interest). Subsequently, Beckers et al. (2016) manually calibrated their
morphodynamic model for the period of 2002–2010 against the observed riverbed evolution (erosion and
deposition) and validated the model for the period of 2010–2013. For these periods, Table 1 provides the
minimum, mean, and maximum value of the hydraulic and morphological boundary conditions for the
three upstream inflows Salzach, Saalach, and Fischbach. The hydrographs were reduced by a bed load rele-
vant discharge threshold, which was found to be 350 m
3
s
−1
with regard to the total flow rate in the system
(Sadid et al., 2016). It is worth noting, that the Saalach did not exceed a sediment input rate of 2,500 kg s
−1
during these periods.
The manually calibrated and validated parameters were found by testing more than 200 parameter combi-
nations (Beckers et al., 2016) in the same number of model runs. In this study, the accepted manually cali-
brated parameter combination is applied to the shortened model of the Lower River Salzach (Figure 1) for
the equivalent periods (Table 1). The manually determined values of the calibration parameters are listed
in Table 2. The table also summarizes the grain size classes as well as the total bed roughness coefficients
found during their initial hydraulic calibration. For better comparison with the stochastically calibrated
results, the roughness parameters are presented as weighted average over the existing mesh elements for
the entire Salzach and for the section of interest.
The computation time for one run of the shortened model is about 7.14 hr for the calibration period (years
2002–2010) and 2.0 hr for the validation period (years 2010–2013) on a conventional computer with a max-
imum speed of 3.2 GHz (four cores). The model fit between simulation results and measured values results in
a root‐mean‐square error of 0.68 m for the calibration period (2002–2010), and a root‐mean‐square error of
0.91 m for the validation period (2010–2013). Details are provided in section 3.5.
Beckers et al. (2018) applied the first‐order second‐moment method to quantify the uncertainties of the
initial sediment transport model of the Lower River Salzach. The impact on model results have been evalu-
ated for eight parameters including four model‐specific parameters (critical Shields parameter, scaling factor
of bed load transport equation, active layer thickness, acceleration factor) and four river‐specific parameters
(total roughness of river channel, grain roughness, bed load input of River Salzach and Saalach). The results
indicate that the most sensitive parameters are the critical Shields parameter θ
crit
and the grain roughness
k′
St;jamong the model‐specific and river‐specific parameters, respectively.
2.5. Selection of Parameters for the Bayesian Calibration and Validation
In the current study we apply a Bayesian approach to stochastically calibrate and validate the morphody-
namic model of the Lower River Salzach according to the measured riverbed geometry from the years
2002, 2005, 2010, and 2013. The selection of calibration parameters is based on the previous studies of
Beckers et al. (2016, 2018). Included are the critical Shields parameter θ
crit
(global parameter) and the
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grain roughness k′
St

(local parameter). In the model of the Lower River Salzach, 13 different regions of total
roughness exist that stem from the hydraulic calibration (Table 2). This is why the varied grain roughness is
denoted as k′
St;j(j=1,…,13). In addition, the grain size distribution d
i
(i=1,…,8) is selected as third calibration
parameter to account for the uncertainty in the grain sizes given their year of sampling (1991, 1993, 1996,
and 2001), their low number of available measurements (surface layer: eight samples, sublayer: nine
samples), and the potentially underestimated surface armoring. Since the grain size distribution takes into
consideration the uncertainty in the measured data but is also used as a pure numeric calibration
parameter in some sediment transport models, it can be considered both a model‐and river‐specific
parameter.
In order to define the allowable range of variation in the selected calibration parameters as prior probability
distributions, an additional preliminary sensitivity analysis based on 16 model runs was conducted. This was
done to account for any changes regarding sensitivity in the shortened model and to test the additional para-
meter grain size distribution. The range of variation for the Bayesian calibration is generated by deviating
each selected parameter θ
crit
,k′
St;j, and d
i
between a minimum and maximum value considering the physical
meaning of each parameter and the results of the preliminary sensitivity analysis. Thus, the prior assump-
tion for each of the three calibration parameters can be represented in their limits via uniform distributions
as ωθcrit ∼Uð0:033;0:047Þ,ωk′
St ∼Uð−4;þ4Þand ω
d
∼U(−0.2, + 4), and the uncertain parameters form the
following vector ω¼fωθcrit ;ωk′
St ;ωdg. Uniform distributions are used to avoid any subjective elicitation on
the distribution shape of the parameters. It is worth noting that k′
St;jis varied relatively to the initial grain
roughness values (Table 2). Likewise, we relatively vary the globally defined grain size classes d
i
and not
the volumetric composition (Table 2). Consequently, the related physical parameters are ωθcrit ,k′
St;jþωk′
St
Table 1
Minimum (Q
min
,Q
S,min
), Mean (Q
mean
,Q
S,mean
), and Maximum Values (Q
max
,Q
S,max
) of Discharges (Q) and Sediment Input Rates (Q
S
) for the Calibration
(2002–2010) and Validation Period (2010–2013)
Period Inflow Qmin [m
3
s
−1
]Q
mean
[m
3
s
−1
]Qmax [m
3
s
−1
]QS;min [kg s
−1
]Q
S,mean
[kg s
−1
]QS;max [kg s
−1
]
Calibration Salzach 162 362 1,569 0 3 73
(2002‐2010) Saalach 24 91 810 0 13 2,500
Fischbach 1 8 72 —— —
Validation Salzach 171 380 2,281 0 6 259
(2010–2013) Saalach 35 94 1,072 0 19 2,500
Fischbach 1 8 99 —— —
Table 2
Total Roughness, Grain Size Classes and Manually Determined Calibration Values of Grain Roughness and Critical
Shields Parameter
Comment or additional
Model parameters Values applied information
Total k′
St;jm1=3s−1
 29.6
∗
/28.3
∗∗
Weighted average over mesh
roughness elements (Full Salzach
∗
/
section of interest
∗∗
)
Grain sizes d
i
[mm] 0.5, 2.0, 8.0, 16.0, 31.5, Eight grain size classes in
63.0, 90.0, 140.0 use (i=1,…,8)
Grain k′
St;jm1=3s−1
 29.6
∗
/25.0
∗∗
Weighted average over mesh
roughness elements (Full Salzach
∗
/
section of interest
∗∗
)
Critical θ
crit
[‐] 0.04 Global parameter
Shields
parameter
Elements used for weighted average calculation:
∗
1,525 and
∗∗
950
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and d
i
+ω
d
for the critical Shields parameter, grain roughness and grain size distribution. Table 3 contains
the three selected parameters, their investigated range of variation, and the prior assumptions for Bayesian
model calibration.
2.6. Stochastic Model Inversion
We incorporate fully Bayesian principles and perform parameter inference using the measured riverbed geo-
metry from the years 2002, 2005, 2010, and 2013 denoted as Z
meas
. As our prior knowledge on the uncertain
calibration parameters (θ
crit
,k′
St;j,d
i
), we consider the prior probability density functions (PDF) according to
Table 3. According to the Bayesian approach, given the prior knowledge and the measured data Z
meas
,we
calculate the posterior probability distribution, which is typically narrower than the prior distributions
(Box & Tiao, 1973). Following the Bayesian theorem, the matching of the outputs of the sediment transport
model Z(these are dependent on a set of selected calibration parameters ω) to the available measurement
data Z
meas
will be done in the following way:
pωZmeas
jÞ

¼
pZ
meas ω
jÞ

pðωÞ
pZ
meas
ðÞ (1)
where ω¼fωθcrit ;ωk′
St ;ωdgdenotes the vector of the calibration parameters, p(ω) is their prior PDF, p
Zmeas ωjðÞrepresents the likelihood function of Z
meas
given ω,pωZmeas
jðÞis the posterior PDF of
the unknown parameters and pZ
meas
ðÞis the so‐called Bayesian model evidence.
Bayesian model evidence quantifies the likelihood of a model producing the observed data averaged over the
complete prior parameter space Ωwith PDF p(ω) (Kass & Raftery, 1995). For this reason, Bayesian model
evidence is often used as a rigorous selection or ranking criterion among competing physical models
(Mohammadi et al., 2018; Wöhling et al., 2015). The present study is based on one formulation of the phy-
sical model, and hence, Bayesian model evidence will merely be considered as a normalizing constant that
can be computed using the following equation:
pZ
meas
ðÞ¼∫ΩpZ
meas ω
j
ðÞpðωÞdω(2)
However, as it is merely a fixed constant for our case with a single model and a given data set, it will be
irrelevant in the following (more details about model selection can be found in, e.g., Mohammadi et al.,
2018). Assuming that the measurement errors ϵbetween Z
meas
and the true river evaluations are indepen-
dent and Gaussian distributed, the likelihood function pZ
measjωðÞis given as
pZ
measjωðÞ¼ð2πÞ−n=2jRj−1
2exp −
1
2Zmeas−Zx;y;t;ωðÞðÞ
TR−1Zmeas −Zx;y;t;ωðÞðÞ

(3)
where Ris the diagonal (co)variance matrix of size n×n. It represents the uncertainty in data as measured
and preprocessed (i.e., interpolation of measured data on 2‐D mesh). The vector Z
meas
contains all the
information available in the measured data in the calibration nodes (i.e., the nodes at the cross sections
with measured data) and n= 204 refers to the length of the observation data set Z
meas
.
Table 3
Calibration Parameters and Their Range of Investigation
Calibration parameters Investigated range Prior assumption
Critical θ
crit
[‐] 0.033 0.047 U(0.033,0.047)
Shields
parameter
Grain k′
St;jm1=3s−1
k′
St;j−4k′
St;jþU−4þ4ðÞ
roughness
Grain size d
i
[mm] d
i
−0.2 d
i
+4 d
i
+U(−0.2, + 4)
distribution
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2.7. Surrogate Model via Arbitrary Polynomial Chaos Expansion
The current study focuses on a very computationally demanding, large‐scale morphodynamic sediment
transport model of the Lower River Salzach. Therefore, a direct implementation of Bayes theorem in
Equation 1 is not feasible using any version of Monte Carlo (MC) simulation and could not even be realized
via a Markov Chain Monte Carlo approach. Therefore, we alleviate this strong computational limitation by
constructing a surrogate model to replicate the behavior of the original, complex morphodynamic model for
the three selected calibration parameters. Referring to a recent benchmark comparison study by Köppel et al.
(2019), we construct the surrogate model using the arbitrary polynomial chaos expansion technique (aPC)
introduced in Oladyshkin and Nowak (2012).
The data‐driven aPC approach can be seen as a machine learning approach which approximates the model
output by its dependence on model parameters via multivariate polynomials. It can be interpreted as a
high‐order extension of first‐order second‐moments approaches, where higher‐order statistical moments
beyond mean and variance are considered (Oladyshkin & Nowak, 2018). Compared to the original polyno-
mial chaos expansion introduced by Wiener (1938) and later extensions by Xiu and Karniadakis (2002), the
aPC offers complete flexibility in the choice and representation of probability distributions. Generally, it can
be seen as a mathematically optimal method for constructing a polynomial surrogate model in the range of
the prior parameter distribution. By reducing the morphodynamic sediment transport model into a surro-
gate model based on selected calibration parameters, the method makes it possible to perform a fully blown
stochastic analysis at a much faster speed.
We assume that the model parameters are distributed according to Table 3. For the purpose of
surrogate modeling, the model Zcan be expressed as a function Zx;y;t;ωðÞof the modeling parameters
ω¼fωθcrit ;ωk′
St ;ωdg, and physical space x,y,t. The influence of all modeling parameters on the model
output Zcan be expressed as the following multivariate polynomial expansion:
Zx;y;t;ωðÞ≈e
Zx;y;t;ωðÞ¼∑
P
i¼1
ciðx;y;tÞϕiðωÞ(4)
The full model Zx;y;t;ωðÞis approximated by a surrogate model e
Zx;y;t;ωðÞin Equation 4 using the
expansion coefficients c
i
(x,y,t). The polynomials ϕ
i
(ω) follow, according to polynomial chaos expansion
theory, directly from the prior distributions of the selected parameters on an orthonormal basis
(Oladyshkin & Nowak, 2012). The surrogate model in Equation 4 is truncated at a finite number P. This
number Pdepends on the total number of input parameters Nand on the largest considered degree of the
polynomials d
p
as P¼Nþdp

!=ðN!dpÞ.
The expansion coefficients c
i
(x,y,t) quantify the dependence of the model output on the set of modeling para-
meters and have to be obtained for every desired point in space and time. There are several approaches for
obtaining the expansion coefficients. For practical applications, nonintrusive approaches such as the
Probabilistic Collocation Method (PCM) have been receiving major attention during the last years due to
their low computational cost (Webster et al., 1996). The advantage of nonintrusive methods is that they
can be applied in models with high complexity and do not demand any modification of the original code
(Loeven et al., 2007; Oladyshkin et al., 2011). PCM uses so‐called collocation points. Collocation points
are those sets of parameters ωfor which the original model is run in order to find the coefficients c
i
. The
selection of collocation points strongly influences the performance of the expansion in Equation 4 and dic-
tates the number of runs needed with the expensive original model. Hence, we will follow the optimal inte-
gration theory and choose the collocation points according to the roots of the polynomial one degree higher
than the order of expansion (Villadsen & Michelsen, 1978). Overall, the collocation points can be seen as
given N‐dimensional sets of parameters ω
i
where i¼1;2;…;Pfg, and where the number of selected colloca-
tion points is equal to the number Pof unknown coefficients c
i
. Once the collocation points have been com-
puted according to the considered prior distribution, the expansion coefficients are estimated from the
following matrix equation:
ϕω
ðÞ
×Vcðx;y;tÞ¼VZðx;y;t;ωÞ(5)
where ϕωðÞis the P×Pmatrix containing the polynomials evaluated at the Pcollocation points, V
c
(x,y,t)is
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the P× 1 vector containing the unknown expansion coefficients c
i
for each location (x,y) and time step t,
and V
Z
(x,y,t,ω)isaP× 1 vector that contains the model outputs of the original model Z(x,y,t,ω
i
) eval-
uated for each of the Pcollocation points ω
i
. The matrix ϕωðÞis time and space independent, meaning that
it can be generated once for a given expansion degree and number of calibration parameters. On the other
hand, the vectors V
c
(x,y,t) and V
Z
(x,y,t,ω) are time and space dependent, which means that they have to
be obtained for each location (x,y) and time step t. Once the expansion coefficients c
i
have been calculated
from Equation 5, we have constructed the surrogate model e
Zx;y;t;ωðÞin Equation 4. Technically, the
expansion coefficients of the polynomials c
i
are calculated for each node nin the mesh and for each cali-
bration (and validation) time step t.
2.8. Iterative Bayesian Updating of the Surrogate Model
To perform Bayesian updating, we replace the response from the original morphodynamic sediment trans-
port model Zx;y;t;ωðÞin the likelihood Equation 3 by its surrogate e
Zx;y;t;ωðÞ:
pZ
measjωk
ðÞ∝exp −
1
2e
Zx;y;t;ωk
ðÞ−Zmeas

TR−1e
Zx;y;t;ωk
ðÞ−Zmeas


(6)
Then, we draw a large number of Monte Carlo samples ω
k
with k=1,…,100,000 according to the considered
prior distributions of ω. Subsequently, we evaluate the surrogate model e
Zx;y;t;ωðÞfor each Monte Carlo
parameter combination ω
k
and use the measured data of riverbed geometry (Z
meas
) in Equation 6 to approx-
imate the corresponding likelihoods.
The surrogate model may, however, be imprecise and may produce incorrect outcomes in the parameter
ranges that show high likelihood because all approaches based on polynomial chaos expansion want to have
the smallest possible squared error on average over the prior, but not over the posterior (Oladyshkin &
Nowak, 2012). To overcome this issue, we employ an iterative Bayesian updating process that improves
the accuracy of the surrogate model by incorporating new collocation points ω
new
corresponding to the max-
imum a posteriori (MAP) value (Oladyshkin, Class, et al., 2013). In other words, we compare each realization
e
Zx;y;t;ωk
ðÞ(i.e., riverbed elevations) with the measured data Z
meas
and identify the MAP. Thus, each itera-
tion suggests a new collocation point at which we run the full morphodynamic model to assess the outcome
Zx;y;t;ωnew
ðÞ. Then, we update the expansion coefficients by solving Equation 5, which is now an overde-
termined system and becomes a least square scollocation problem (Moritz, 1978). Through this procedure,
we iteratively obtain a surrogate model that contains more accurate information about the model in all
alleged regions of interest where the likelihood to capture the data is higher. As stopping criterion for this
iteration, we repeat until the current posterior distribution shows only minor changes in comparison to
the previous one, indicating convergence.
We can afford a plain rejection sampling technique to estimate the posterior distribution pωZmeas
j
ðÞat each
iteration step. We use the rejection sampling method (Smith & Gelfand, 1992) because the surrogate model is
cheaper to evaluate, and so allows drawing a large number of parameter combinations ω
k
with k=1,
…,100,000 from the prior distributions and the evaluation of the surrogate model output e
Zx;y;t;ωk
ðÞfor
each set of ω
k
. Rejection sampling represents the posterior distribution according to the following impor-
tance weights W
k
:
Wk¼pZ
measjωk
ðÞ
max ðpZ
measjωk
ðÞÞ (7)
Importance weights W
k
are the probabilities of accepting prior realizations as posterior realizations; all other
ones are rejected. A realization ω
k
is accepted when W
k
≥U
k
, where U
k
is a random number drawn from the
uniform distribution U(0,1), and otherwise rejected.
2.9. Assessing the Quality of Bayesian Calibration and Validation
Monte Carlo simulations of the aPC iterative Bayesian updating process estimate the prior and the posterior
distribution of the model output. From this, we compute the expected riverbed evolution for the entire
Lower River Salzach. Standard deviations from these expected values and also the MAP estimate are
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available. In this case, with a uniform prior distribution, the MAP estimate is the realization of the Monte
Carlo sample with the highest likelihood value (Equation 6) and corresponds to the deterministic best fit
solution. The MAP can thus be compared to the parameter combination found during conventional (i.e.,
manual) calibration. To display the morphodynamic development (riverbed evolution) for calibration
(2002–2010) and validation (2010–2013), we roll out the mean value ⟨Δe
Z⟩(x,y,t,ω
k
) of the aPC model results
on the entire movable riverbed in the section of interest, that is, at the nodes of the original mesh (n= 1,138).
⟨Δe
Z⟩n;tis calculated for a particular node nand time step tby considering the measured riverbed elevation
Z
meas
from the year 2002 (calibration) or 2010 (validation) as initial riverbed:
⟨Δe
Z⟩n;t¼∑S
k¼1f
Zk
S
−Z2002;2010 (8)
where e
Zis the corresponding aPC model output for a combination ω
k
taken from the prior PDFs or for an
accepted combination ω
k
taken from the posterior PDFs. The sample size is represented by S(for example
100,000 in the prior simulations). Subsequently, we derive statistical quantities and estimate the spatial
mean and the spatial standard deviation of the riverbed evolution. With this, we can evaluate and quantify
the prior and posterior uncertainties and assess the quality of Bayesian calibration and validation.
2.10. Verification of Surrogate Model
The aPC results are verified against the results of the original morphodynamic model to test the approxima-
tion made by Equation 4. For this purpose, we compute the root‐mean‐square error ϵbetween the outputs Δ
e
Zn;tof the surrogate model and the outputs ΔZ
n,t
of the original model obtained with the MAP parameter
combination ω
MAP
as follows:
ϵ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
n∑
n
i¼1
ðΔZn;tðωMAPÞ−Δe
Zn;tðωMAPÞÞ2
s(9)
2.11. Validation of Model Results and Comparison to the Earlier Manual Calibration
As a final step, we calculate the residuals between the measured and simulated riverbed evolution for each
calibration node (n= 204) as follows:
e¼ΔZmeas −
ΔZn;t
Δe
Zn;tðωMAPÞ
ΔZn;tðωMAPÞ
8
>
<
>
:9
>
=
>
;(10)
where ΔZ
meas
is a vector that contains the measured riverbed evolution, ΔZ
n,t
contains the results of the
manually calibrated full model, and e
Zn;tðωMAPÞand Z
n,t
(ω
MAP
) contain the results of the surrogate and the
full model obtained with the ω
MAP
parameter set, respectively.
In doing so, we are able to derive the distribution of the residuals efor the calibration (2002–2010) and vali-
dation (2010–2013) period as well as the corresponding statistical quantities, that is, mean error (ē), standard
deviation (σ
e
) and root‐mean‐square error (ϵ
e
). Consequently, we can assess the overall quality of the sto-
chastically calibrated models and eventually relate our findings to the previously manually calibrated full
complex model.
3. Results and Discussion of the Bayesian Calibration and Validation of the
Sediment Transport Model
3.1. Iterative Bayesian Updating of the Surrogate Model
We construct a surrogate model using a second‐order aPC expansion according to the prior distributions of
the calibration parameters ω¼fωθcrit ;ωk′
St ;ωdg(Table 3). For this purpose, the surrogate model requires the
model output Zx;y;t;ωðÞof the original morphodynamic sediment transport model obtained for the para-
meter combinations defined by 10 collocation points. Figure 2 illustrates these original collocation points
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within the parameter space of the calibration parameters θ
crit
,k′
St;j,d
i
. We use Equation 5 to find the aPC
surrogate in Equation 4.
Figure 3 illustrates the randomly drawn combinations of parameters ω
k
(k=1…N
MC
;N
MC
=10
5
) occurring
during the Monte Carlo procedure and the corresponding outputs from the surrogate model e
Zðx;y;t;ωkÞfor
one exemplary node in the mesh (node id: 13,319). The color scheme indicates the values of the resulting
river bed elevation e
Zat a time step t= 2010 (end of calibration period).
Following the Bayesian framework, each realization is weighted through the likelihood upon comparison
with the measured riverbed elevation at the calibration nodes (n= 204) according to Equation 6. Then, we
iterate according to section 2.8. Therefore, Figure 2 contains, in addition to the 10 original collocation points
used to construct the zeroth‐iteration surrogate model (blue dots), also 10 new collocation points obtained
during the iterative Bayesian updating (green dots). The new iteratively chosen collocation points corre-
spond to regions with higher likelihood values. The iterative Bayesian updating is successful if the posterior
distributions of the calibration parameters narrowed and stabilized. This condition has been reached after 10
iteration steps. Each iteration requires approximately 4 hr of computation time (updating the aPC, comput-
ing the likelihood and rejection sampling on the 10
5
MC candidates).
Figures 4a–4d show the individual obtained posterior PDFs for the calibration parameters after the first,
second, ninth, and tenth iterations. In each subfigure, the posterior combinations are sorted into 20 equally
spaced bins. The vertical dashed lines indicate the values obtained via manual calibration. It can be seen that,
for all three calibration parameters, the PDF has narrowed and stabilized towards the end (between the ninth
and tenth iterations). However, in the univariate views of Figures 4a–4d, it is not easy to see that the cali-
brated parameters have a strong nonlinear dependence on each other. To visualize this nonlinear depen-
dence, we plot in Figure 5 the importance weights W
k
in a multivariate scatter plot after the tenth iteration
of Bayesian updating. Parameter combinations with likelihood values near zero are not shown. The red color
represents the N= 857 accepted (posterior) Monte Carlo samples with the highest likelihood values, that is,
those that result in the most acceptable match to the measured data Z
meas
. The maximum a posteriori (MAP)
combination obtained after the tenth iteration of Bayesian updating is given in Table 4. Referring the MAP
combination back to the absolute values given in Table 2, this results in an increase in critical Shields
parameter from 0.04 to 0.044, a reduction in grain roughness from 25.03 m
1/3
s
−1
to 23.69 m
1/3
s
−1
(weighted
average over all elements in the section of interest), and in an increase in the grain size classes to
d
i
= {2.13,4.13,10.13,18.13,33.63,65.13,92.13, 142.13} mm.
Although the parameter combination with the highest likelihood can be derived (MAP), Figure 5 reveals a
variety of parameter combinations resulting in a high likelihood, that is, they are all possible combinations.
Thus, the outcomes demonstrate that one single deterministic solution does not exist, given the
Figure 2. Representation of original collocation points for the construction of the surrogate model in the parameter
space of the calibration parameters and the new collocation points obtained by iterative Bayesian updating.
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interdependence of the calibration parameters. This underlines the need for stochastic calibration
procedures in morphodynamic sediment transport modeling since the commonly applied models are
highly parameterized (e.g., Merritt et al., 2003), including multiple parameter interdependencies (e.g., the
relation between grain roughness and grain size, (see Strickler, 1923) or the relation between critical
shear stress and grain size (see Buffington & Montgomery, 1997)). This interdependency of the three
calibration parameters is clearly reflected in Figure 5.
3.2. Bayesian Model Calibration With the Surrogate Model
Figure 6 illustrates the results of the aPC surrogate model for the calibration period (2002–2010) for the sec-
tion of interest in the Lower River Salzach (km 59.3 to km 52.4). The results are separated into those obtained
with the prior and posterior distributions of the considered calibration parameters. Figures 6a and 6b show
the mean riverbed evolution ⟨Δe
Z⟩n;tobtained with (a) the N
MC
= 100,000 candidates from the prior and (b)
with the N= 857 candidates that were accepted after rejection sampling (posterior) in the last iteration.
Figures 6c and 6d show the standard deviation σ
n,t
among all aPC results for (c) the prior distribution and
(d) the posterior distribution.
The prior and posterior mean results show a similar pattern of erosion and deposition along the Lower River
Salzach, with alternating gravel bars and deepenings of up to ±3 m. However, local deviations exist, leading
to a spatial mean riverbed evolution of −0.17 m in Figure 6a and −0.05 m in Figure 6b. Calibration demon-
strates that there is less erosion in comparison with the prior assumptions. This can be attributed to the
updated posterior parameter sets, in which θ
crit
and d
i
have increased and k′St has decreased (cf. Figure 5,
Table 4). In the applied equation of Hunziker (1995) as well as in the original equation of Meyer‐Peter
and Mueller (1948), the parameters d
i
and k′St both appear in the denominator as part of the dimensionless
bed‐forming shear stress. The dimensionless bed‐forming shear stress is then compared to the critical shear
stress by subtracting the latter from the former value and so an increased value for θ
crit
induces higher resis-
tance of the riverbed towards erosion. Since erosion‐mitigating effects dominate the calibration results, less
erosion is the consequence.
The corresponding standard deviations of all model outputs have an average of 0.73 m in the prior
(Figure 6c) and 0.15 m in the posterior results (Figure 6d). When spatially integrated, this leads to a volu-
metric standard deviation of 414,610 m
3
in the prior and 83,174 m
3
in the posterior. The prior results have
regions with an increased prior standard deviation up to 1 m in many parts of the river. Critical spots are
mainly located in the upstream section and close to the outflow. In the upstream section of the Lower
River Salzach, the sediment input from the Saalach River dominates the morphodynamic behavior. The
Figure 3. N
MC
combinations (N
MC
=10
5
) and surrogate model outputs e
Zðx;y;t;ωkÞfor one exemplary node in the
mesh (node id: 13,319) and the calibration time step t = 2010.
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three‐dimensional effects in the mixing zone close to the confluence are highly complex, and so are affected
in a nonlinear fashion and to a high degree by parametric uncertainties. In the downstream section, the
effect of the lower boundary condition impacts the prior results, explaining the uncertainties in the prior
simulations. When looking at the posterior standard deviation obtained with the calibrated posterior
Figure 4. Posterior distributions of the three selected calibration parameters θ
crit
,k′
St;j, and d
i
during the iterative
Bayesian updating obtained after (a) first iteration, (b) second iteration, (c) ninth iteration, and (d) tenth (final)
iteration. The vertical dashed lines refer to the manually calibrated value.
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parameter distributions (Figure 6d), it can be seen that the uncertainties have significantly decreased. This
can be quantified by the decrease in the mean spatial standard deviation of 0.58 m (331,436 m
3
). Thus, the
posterior parameter distributions obtained from Bayesian updating have significantly increased the
quality of the aPC surrogate model. Only a few remaining spots, again close to the edges of the model
domain, show higher remaining uncertainties. The remaining (now reduced) uncertainties indicate that
the Bayesian calibration revealed a spread of simulations with well‐fitting solutions. Vice versa, it implies
that a manually obtained deterministic calibration lacks robustness. This generally confirms the results of
previous studies (Schmelter et al., 2011, 2012; Schmelter & Stevens, 2013) and emphasizes the significant
contribution of Bayesian updating in morphodynamic and sedimentological studies to capture more
accurately and realistically the underlying processes (Mohammadi et al., 2018; Wu & Chen, 2009).
3.3. Results of the Surrogate Model for the Validation Period
We validate the distribution of parameter combinations found during Bayesian calibration with an indepen-
dent set of measured data from the year 2010 to 2013. Figure 7 illustrates the results of the aPC surrogate
model for the validation period (2010–2013) along the section of interest in the Lower River Salzach. The
results are again separated into those obtained with the prior and posterior distributions. Figures 7a and 7b
show the mean riverbed evolution ⟨Δe
Z⟩n;tobtained with (a) the prior and (b) the posterior parameter
distribution.
Figure 5. Likelihood dependency of the three calibration parameters after the tenth Bayesian update.
Table 4
Maximum a Posteriori (MAP) Set of Parameter Combinations Found During Bayesian Updating of the aPC
Surrogate Model
Maximum a
Calibration parameters Prior assumption posteriori (MAP)
Critical θ
crit
[‐]U(0.033,0.047) 0.044
Shields
parameter
Grain k′
St;jm1=3s−
 k′
St;jþU−4;þ4ðÞ k′
St;j−1:34
roughness
Grain size d
i
[mm] d
i
+U(−0.2, + 4) d
i
+ 2.13
distribution
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The prior and posterior mean results show a similar pattern for the simulated riverbed morphology.
Alternating gravel bars and deepenings are clearly visible along the Lower River Salzach. Deviations are
present in the downstream and upstream section. The corresponding spatial mean is −0.03 m for
Figure 7a and −0.02 m for Figure 7b. The similar values can be explained by the overall decreased magnitude
of erosion and deposition in the posterior results, leading on average to almost the same net evolution.
Figures 7c and 7d show the standard deviation σ
n,t
among all aPC results obtained from (c) the prior and
(d) the posterior distribution. The corresponding spatial mean now decreased from 0.67 to 0.15 m. When spa-
tially integrated, this results in a volumetric reduction from 379,043 m
3
in the prior to 86,408 m
3
in the
Figure 6. Simulated riverbed evolution ( ⟨Δe
Z⟩n;t) obtained with the aPC surrogate model in the Lower River Salzach
(km 59.3 to km 52.4) for the calibration period (2002–2010) and the corresponding standard deviation (⟨σ⟩n;t): (a) prior
mean result, (b) posterior mean result, (c) prior standard deviation, and (d) posterior standard deviation. Please note that
the river width is artificially stretched by a factor of 3; white regions indicate no riverbed change.
Figure 7. Simulated riverbed evolution (⟨Δe
Z⟩n;t) obtained with the aPC surrogate model in the Lower River Salzach (km
59.3 to km 52.4) for the validation period (2010–2013) and the corresponding standard deviation (⟨σ⟩n;t): (a) prior mean
result, (b) posterior mean result, (c) prior standard deviation, and (d) posterior standard deviation. Please note that the
river width is artificially stretched by a factor of 3; white regions indicate no riverbed change.
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posterior. It can be concluded, that the posterior parameter combinations found during Bayesian calibration
also increase model quality and robustness for an independent set of data. Thus, we rate the validation with
the surrogate model as successful.
3.4. Verification of the Surrogate Against the Full Sediment Transport Model
In order to verify the aPC surrogate model, we apply the maximum a posteriori (MAP) set of calibration
parameters obtained from the tenth Bayesian iteration (see Table 4) to both, the surrogate and the original
Hydro_FT−2Dsediment transport model. The results obtained with this parameter set for calibration and
validation can be interpreted as the best deterministic scenario simulated with the aPC surrogate model
and with the original morphodynamic sediment transport model.
Figure 8 shows the resulting riverbed evolution for the calibration period (2002–2010). Figure 8a contains
the measured riverbed evolution (only important in section 3.5), Figure 8b the riverbed evolution simulated
with the surrogate model (Δe
Zn;tðωMAPÞ) and Figure 8c the corresponding riverbed evolution simulated with
the full morphodynamic model (ΔZ
n,t
(ω
MAP
)). As expected, the simulation result of the aPC surrogate model
(Figure 8b) shows very good agreement with the simulated riverbed evolution of the full morphodynamic
model (Figure 8c). This is due to the iterative Bayesian updating process to improve the surrogate. The
root‐mean‐square error between the two models results in ϵ
Cal
= 0.31 m.
Figure 9 shows the riverbed evolution for the validation period (2010–2013). Figure 9a contains the mea-
sured riverbed evolution (only important in section 3.5), Figure 9b the riverbed evolution simulated with
the aPC surrogate model ðΔe
Zn;tðωMAPÞÞ, and Figure 9c the corresponding riverbed evolution obtained with
the morphodynamic model (ΔZ
n,t
(ω
MAP
)) using the MAP set of parameters and being initialized from the
simulated river bed morphology of the calibration period. Again, the simulated riverbed evolution of the
aPC surrogate model (Figure 9b) shows very good agreement with the riverbed evolution simulated with
Figure 8. Riverbed evolution in the Lower River Salzach (km 59.3 to km 52.4) for the calibration period 2002–2010
obtained from (a) measurements, (b) stochastically calibrated aPC surrogate model (MAP), and (c) stochastically
calibrated morphodynamic model (MAP). Please note that the river width is artificially stretched by a factor of 3; white
regions indicate no riverbed change.
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the original morphodynamic model (Figure 9c). The root‐mean‐square error between the two models is
ϵ
Val
= 0.42 m.
Model verification by means of the root‐mean‐square error indicates that our constructed aPC surrogate
model approximates in a reliable manner the fully deterministic sediment transport model (ϵ
Cal
= 0.31 m
and ϵ
Val
= 0.42 m). Although a performance loss is seen from calibration to validation, this is a typical degree
and not to be blamed on the surrogate. The aPC surrogate model can thus be used to study the complex mor-
phodynamic behavior in the Lower River Salzach since the data fit for calibration and validation is of very
good quality (Figures 8b, 8c, 9b, and 9c).
3.5. Validation and Comparison to the Earlier Manual Calibration
Now we look again at Figures 8 and 9. The simulations conducted with the stochastically calibrated models
(Figures 8b and 8c) reproduce well the measured evolution pattern of the calibration period (Figure 8a).
Deviations are visible in the upstream section close to the Saalach confluence where the simulated evolution
amplitude is increased and in the downstream section where the simulations underestimate the measured
riverbed erosion and predict emerging gravel bars. The riverbed evolution measurements for the validation
period (Figure 9a) are less accurately reproduced by both models since the evolution amplitude close to
the Saalach confluence is still increased and gravel bars clearly emerge in the downstream section of the
Lower River Salzach (Figures 9b and 9c). Moreover, the measurements in vicinity of the outflow indicate
low but wide deposition. This is not reflected in the simulation results as they also predict progressing gravel
bars for this region. A general observation of the measured evolution pattern indicates, however, a trend
toward deposition and an existence of gravel bars in the section of interest. It should be further noted, that
the largest deviations occur in regions where the highest uncertainties can be found. These uncertainties,
however, remain significantly reduced through Bayesian calibration (Figure 6 and 7). Nevertheless, it is likely
that both models overestimate the river dynamics in the Lower River Salzach. We attribute this to the numer-
ical implementation of the sediment input from the River Saalach in the full morphodynamic model.
Although the total volumes are well documented, the intermittent character that stems from the operation
Figure 9. Riverbed evolution in the Lower River Salzach (km 59.3 to km 52.4) for the validation period 2010–2013
obtained from (a) measurements, (b) stochastically calibrated aPC surrogate model (MAP), and (c) stochastically
calibrated morphodynamic model (MAP). Please note that the river width is artificially stretched by a factor of 3; white
regions indicate no riverbed change.
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of a hydroelectric power plant and respective flushing events (Beckers et al., 2018) cannot be accurately
approximated with the river‐specific rating curve used in the full model (Beckers et al., 2016, 2018).
Instead, the more continuous sediment supply due to the rating curve applied resembles the historic
condition of the Lower River Salzach with distinct gravel bars moving down the river. From a 2‐D
numerical perspective, it is also conceivable that the full model might underestimate transverse bed slope
effects on sediment transport. In general, the effect and consideration of slope effects in 2‐D models are
widely discussed (e.g., Recking, 2009; Siviglia et al., 2013). More specifically, for a braided river it has been
shown that an underestimation of transverse bed slope effects may cause an overestimation of river
dynamics and results in deeper/narrower channels (Williams et al., 2016). Consequently, the aPC surrogate
model cannot capture these effects either as it was built based on the full complex morphodynamic model
(Equation 4). Further studies that test the sensitivity of transverse bed slope effects on sediment transport
and focus in particular on possibilities to optimize the sediment input numerically using the full complex
morphodynamic model in combination with problem‐based surrogates
are thus recommended.
In a final step, we carry out a residual analysis to assess the overall quality
of the models and finally compare the stochastically calibrated results
using the MAP parameter combination (Table 4) with those obtained
through manual calibration. The residuals eare calculated according to
Equation 10 for each calibration node (n= 204) and their distribution is
shown in Figures 10a–10f. The results for the calibration period are shown
in the upper panels for (a) the stochastically calibrated aPC surrogate
model (MAP), (b) the stochastically calibrated sediment transport model
(MAP), and (c) the manually calibrated sediment transport model. The
lower panels contain, accordingly, the results for the validation period
for (d) the stochastically calibrated aPC surrogate model (MAP), (e) the
stochastically calibrated sediment transport model (MAP), and (f) the
Figure 10. Comparison of differently calibrated models by means of a residual analysis showing the difference between the measured and simulated riverbed
evolution for all calibration nodes (n= 204). The upper panels contain the results for the calibration period (2002–2010) whereas the lower panels contain the
results for the validation period (2010–2013).
Table 5
Statistical Quantities of Residuals e Obtained for Differently Calibrated
Models Divided Into Mean Error (ē), Standard Deviation (σ
e
) and Root‐
Mean‐Square Error (ϵ
e
)
Period Model and calibration strategy ē[m] σ
e
[m] ϵ
e
[m]
Calibration aPC surrogate (MAP) −0.14 0.54 0.56
(2002–2010) Hydro_FT‐2D (MAP) −0.15 0.67 0.68
Hydro_FT‐2D (manual) −0.11 0.67 0.68
Validation aPC surrogate (MAP) −0.08 0.69 0.70
(2010–2013) Hydro_FT‐2D (MAP) −0.06 0.88 0.88
Hydro_FT‐2D (manual) −0.08 0.91 0.91
Note. The full distribution is given in Figure 10.
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manually calibrated sediment transport model. Table 5 presents the corresponding statistical quantities of
the residuals.
The distributions of the residuals for calibration (2002–2010) and validation period (2010–2013) suggest an
improvement of the stochastically calibrated surrogate (Figures 10a and 10d) over the stochastically cali-
brated full model (Figure 10b and 10e) to the manually calibrated morphodynamic model (Figure 10c and
10f). This ranking is confirmed by the statistical quantities and the root‐mean‐square errors ϵ
e
given in
Table 5. While the manually calibrated morphodynamic model performs as good as the stochastically cali-
brated morphodynamic model in the calibration period, the stochastically calibrated morphodynamic model
outweighs the manual calibration in the validation period. This can be explained with the significantly
reduced posterior uncertainty (Figures 6 and 7) and the applied maximum a posteriori (MAP) parameter
combination (Table 4) resulting from Bayesian calibration. As expected, the MAP parameter combination
is more robust compared to the manually calibrated parameters and thus performs better on an independent
data set, that is, for the validation period.
In summary, all calibrated models (stochastic and manual) produce sufficiently accurate results when con-
sidering the total dimensions of the investigated river section (6.9 km long, on average 82 m movable riv-
erbed width) and the limitations given by the original morphodynamic Hydro_FT−2D sediment transport
model. Moreover, the consistent negative mean errors of the residuals (Table 4) confirm that all models over-
estimate the deposition in the Lower River Salzach and underline the need for future studies on the numer-
ical implementation of the sediment input from the River Saalach.
3.6. Overall Assessment of Bayesian Calibration and Validation Using the Surrogate
It has been shown, that the stochastically calibrated full morphodynamic model has the same accuracy and
performs better than the manually calibrated full morphodynamic model. However, the main difference
between both calibration methods is on the computational effort required to achieve an acceptable fit with
the measured riverbed evolution. While the computational effort for one run of the full morphodynamic
model is 7.14 hr for calibration and 2.0 hr for validation, one call of the aPC surrogate model, now merely
a polynomial, is 13.52 ms (for calibration and validation). Given this, obtaining the N
MC
model outputs by
testing 100,000 parameter combinations requires 0.38 hr. Considering that the aPC surrogate model was con-
structed with 20 runs of the full complex morphodynamic model (definition of initial and updated collocation
points) and required ∼4 hr for each of the 10 iterations (Gaussian likelihood and rejection sampling of the
100,000 MC candidates) during Bayesian calibration, ∼222.8 hr are required to obtain the final accepted para-
meter combination. During manual calibration, more than 200 parameter combinations were tested with the
full morphodynamic model (Beckers et al., 2016, 2018). This corresponds to approximately 1,828 hr of com-
putation time. In terms of total time requirement, the final accepted parameter combination, that is, the max-
imum a posteriori (MAP) combination, was obtained via Bayesian calibration in about one‐eighth of the time
compared to the manual calibration. This result makes clear the significant time reduction.
For the presented case, we could define three most sensitive parameters based on the previous conducted
numerical studies on the Lower River Salzach. In the event that more calibration parameters shall be con-
sidered during stochastic calibration, the total computational time scales solely with the additionally
required full model runs to define the collocation points to construct the surrogate (Equation 4). An increase
to four or five parameters leads to 15 or 21 full model runs for constructing the initial aPC surrogate, respec-
tively. Using the evidence from this study, the same number of iterations (Bayesian updating of the surrogate
against measured data) is approximately required to refine the surrogate. This leads to 15 and 21 supplemen-
tary full model runs to find the new collocation points. Therefore, the total time requirement would result in
314.2 hr (four calibration parameters) and 423.9 hr (five calibration parameters) and indicates roughly a
linear increase in time for each additionally considered calibration parameter.
4. Summary and Conclusions
The presented work provides a stochastic calibration and validation for a morphodynamic sediment trans-
port model of the Lower River Salzach. A Bayesian framework is employed with a surrogate model con-
structed via arbitrary polynomial chaos expansion (aPC). Considering the strong computational
limitations of the physical‐deterministic sediment transport model (7.14 hr for one calibration and 2.0 hr
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for one validation run) and the therefore very limited expansion order of the aPC (second), we employ itera-
tive Bayesian updating of the surrogate model. Through the iterations, we identify the most probable region
in the space of three calibration parameters (θ
crit
,k′
St;jd
i
) and refine the surrogate accordingly. The combina-
tion of strict Bayesian principles with model reduction assures that the constructed surrogate model, based
on only 20 runs of the original sediment transport model, still captures effectively the morphodynamic beha-
vior and the sediment transport processes in the Lower River Salzach. One model call of the final aPC sur-
rogate model requires only 2 × 6.76 ms = 13.52 ms for both calibration and validation. Hence, we could
afford a brute‐force Monte Carlo simulation (100,000 realizations) treatment of the surrogate to quantify
parametric and predictive uncertainty.
In the morphodynamic sediment transport model, the initial riverbed geometry is from the year 2002. For
calibration, available riverbed measurements from the years 2005 and 2010 have been used. The findings
have been validated using riverbed measurements from the year 2013. Bayesian calibration and validation
of the aPC surrogate model with Monte Carlo simulation provides detailed statistical information about
the predicted riverbed behavior along the Lower River Salzach. The presented results show that automated
Bayesian calibration helps to significantly improve the fit to data and to reduce the remaining uncertainty
along the entire river. The standard deviation reduces on spatial average from 0.73 to 0.15 m during the cali-
bration period and from 0.67 to 0.15 m during the validation period. The largest reductions in uncertainty
occurred at critical spots such as the upstream region where the River Saalach mouths into the River
Salzach and the region close to the downstream boundary condition.
We verified the surrogate model against the physical deterministic sediment transport model. For this, we
used the best deterministic scenario corresponding to the maximum a posteriori (MAP) response of the
aPC surrogate. The test indicates very good agreement between the surrogate and the full Hydro_FT−2D
sediment transport model for both calibration (ϵ
Cal
= 0.31 m) and validation (ϵ
Val
= 0.42 m) especially when
considering the spatial domain of the investigated river section of interest (6.9 km long, on average 82 m
movable riverbed width). The verification results reveal that the aPC surrogate can be used to study the mor-
phodynamic sediment transport processes in the Lower River Salzach.
In a final step, we conducted a residual analysis between the riverbed measurements and simulation results
to asses the overall quality of the models for the calibration and validation period. We calculated the resi-
duals for the stochastically calibrated surrogate model and the morphodynamic model using the MAP para-
meter set as well as for the manually calibrated morphodynamic model. We conclude that the
surrogate‐based Bayesian approach is at least as good as a manual calibration conducted in an earlier study,
but requires only a fraction of the computational time (more than 8 times faster) for obtaining the results.
Overall, it can be concluded that Bayesian calibration for physical‐deterministic sediment transport models,
such as Hydro_FT−2D, by means of an aPC surrogate model, offers a significant contribution to morphody-
namic river modeling. The key conclusion is that, within a significantly reduced computational time, we
quantified the uncertainties, increased the robustness, and finally improved the overall quality of the calibra-
tion and validation of a large‐scale and time‐demanding physical‐deterministic sediment transport model.
This confirms the belief of Schmelter et al. (2011), that Bayesian modeling provides a tool for innovation
in sediment transport research, especially when being combined with surrogate techniques to address com-
putational time constraints (see also Mohammadi et al., 2018). The applied framework in this study is uni-
versally applicable and not confined to any physical‐deterministic sediment transport model. Although aPC
is a well selected specific choice (see also Köppel et al., 2019), there are more surrogate techniques, many of
which will serve the same goal: speeding up the model, such that uncertainty quantification and Bayesian
updating of large‐scale applications becomes feasible.
Data Availability Statement
Input data to the morphodynamic sediment transport model were previously published in Beckers et al.
(2016, 2018). The methods for model reduction and Bayesian updating were published in the references
(Oladyshkin & Nowak, 2012, 2018; Oladyshkin, Class, et al., 2013). The codes to achieve the model reduction
using Data‐driven Arbitrary Polynomial Chaos (aPC) are available at this site (https://mathworks.com/
matlabcentral/fileexchange/72014-apc-matlab-toolbox-data-driven-arbitrary-polynomial-chaos). The codes
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that combine aPC with strict Bayesian principles for stochastic model calibration and parameter inference
are available at this site (https://mathworks.com/matlabcentral/fileexchange/74006-bapc-matlab-toolbox-
bayesian-arbitrary-polynomial-chaos). The model outputs as well as the calibration and validation data
are accessible at Zenodo (Beckers et al., 2019).
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Acknowledgments
The authors would like to thank the
German Research Foundation (DFG)
for their financial support of the project
through the Cluster of Excellence
“Data‐Integrated Simulation Science”
(EXC 2075) at the University of
Stuttgart. The data used in this study
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