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ORIGINAL PAPER
Stochastic Environmental Research and Risk Assessment (2024) 38:4485–4500
https://doi.org/10.1007/s00477-024-02814-z
deaths and a direct economic loss of 40.9 billion yuan. Fac-
ing with the increasingly severe problem of urban pluvial
ood disaster, it is necessary to explore the rapid and accu-
rate methods of urban ood prediction to eectively support
the real-time ood management and reduce the loss of urban
ood disasters.
Computational modeling of urban ood ow and trans-
port processes is instrumental to the guidance of urban
ood risk management (Mignot and Dewals 2022), which
can help the managers acquire the spatial distribution of the
ooding zones and calculate the potential economic losses.
A group of 2D hydrodynamic models can be used for simu-
lating urban ood inundation (Teng et al. 2017). However,
numerically simulating these hydrodynamic models is a
very time-consuming task. Anyway, a reliable ood warn-
ing system demands short simulation times to provide suf-
cient emergency response time (Chang et al. 2022). Hence,
these state-of-the-art 2D hydraulic models have still many
diculties in supporting real-time ood forecasting at suf-
ciently high resolution.
Building an oine database may be a feasible approach to
surrogate hydraulic model (Bhola et al. 2018). This method
draws dierent ood inundation maps to store in the database
1 Introduction
Floods are the most common natural disaster worldwide,
which can bring unimaginable consequences to human soci-
ety and environment (Zhang et al. 2022). More than half of
the global population lives in urbanized areas but most of
these people are increasingly at risk of ood (Hirabayashi et
al. 2013). Under the inuence of urbanization and extreme
weather (Heim and Richard 2015; Lu et al. 2019; Gu et al.
2022), many cities located in at area are susceptible to storm
events during monsoon, posing considerable threat to the
safety of residents and their property. For instance, in July
2021, the “7.20 rainstorm” in Zhengzhou, China caused 380
Minling Zheng
zhengml@zjhu.edu.cn
Xiang Zheng
gary.zx98@gmail.com
1 Nantaihu Design Institute of Water Conservancy and
Hydropower, Huzhou 313000, Zhejiang, China
2 School of Sciences, Huzhou University, Huzhou 313000,
Zhejiang, China
Abstract
Urban ood risk management has been a hot issue worldwide due to the increased frequency and severity of oods
occurring in cities. In this paper, an innovative modelling approach based on the Bayesian convolutional neural network
(BCNN) was proposed to simulate the urban ood inundation, and to provide a reliable prediction of specic water depth.
To develop the model, a series of historical rainfall data during the last 20 years were collected in Rushan China and
the responding ood events were reproduced using physically based hydraulic model. The ood condition factors used
in modeling include spacial factors and precipitation factors. The results showed that the BCNN model not only inherits
the powerful ability of aggregating spacial information from CNNs to perform high level of accuracy and computational
eciency in predicting 2D urban ood inundation maps, but also oers a measure of uncertainty in the form of predictive
variance, providing insights into the condence and reliability of its predictions. The proposed BCNN method oered a
new perspective for the analysis of surrogate model regarding real-time forecasting of ood inundation.
Keywords Deep learning · Bayesian convolutional neural network · Data-driven simulation · Rapid ood modelling ·
Hydrodynamic modelling
Accepted: 4 September 2024 / Published online: 17 September 2024
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024
Prediction of urban ood inundation using Bayesian convolutional
neural networks
XiangZheng1· MinlingZheng2
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Stochastic Environmental Research and Risk Assessment (2024) 38:4485–4500
based on previous disaster situation simulation. Before or
during storm, the inundation maps with a matching precipita-
tion are requested and retrieved from the database to provide
forecasts. However, due to environment change, like land-use
change, geomorphological change and engineering construc-
tion, old ood scenarios may not be appropriate for satisfying
current ood prediction requirements and should be updated
termly, creating extra eort for maintenance.
Beneting from advances in computer technology and
algorithms elds, it is possible to simulate and predict ood
variables using machine learning (ML) techniques, once they
are trained and calibrated appropriately. The broad applica-
tion of ML methods for rainfall-runo prediction (Gao et al.
2020), streamow predictions (Huang et al. 2016; Yaseen et
al. 2015), ood susceptibility prediction (Pham et al. 2020;
Wang et al. 2020), ood inuencing factors (Wu et al. 2021)
has been continued for a long time. As regards the urban
ood inundation depth prediction, due to the lack of over-
all monitoring ood data, how to build a reliable and avail-
able database for model development presents challenge to
researchers. Wu et al. (2020) established a data warehouse
with available structured and unstructured urban ood data,
integrated with a deep learning algorithm named gradient
boosting decision tree (GBDT) for urban ood modelling at
regional scale, giving ood condition maps under dierent
rainfall return periods in Zhengzhou city, China. Another fea-
sible approach to congure the ML models is based on ood
simulation by physically based models. For instance, Kabir et
al. (2020) proposed an CNN model for predicting ood inun-
dation maps induced by excessive ow in channels, using
ood simulations generated with a physically based model
as virtual reality. Chang et al. (2018) built a regional inunda-
tion early warning system through the self-organizing map
(SOM), combined with the recurrent nonlinear auto-regres-
sive exogenous (RNARX) networks to achieve ood fore-
casting up to 12 h ahead, where the ML models were trained
based on ood simulation data from physically-based mod-
els. Löwe et al. (2021) and Guo et al. (2021) developed the
U-Net CNN models to achieve predicting urban inundation
maps, regarding feature images of study site as input and max
ood depth from the 2D hydraulic models as output. These
results suggested that, compared to hydrodynamic models,
ML methods are more ecient to provide real-time ood risk
management under various climate change scenarios and can
be regarded as surrogate models for hydrodynamic models.
However, some recent studies have found that the pre-
dictive performance of individual approaches or models
is often constrained (Zhou et al. 2022; Liu and Merwade
2018), as the formation of pluvial ood in urban catchment
is a complex nonlinear process aected by a variety of fac-
tors, which is dicult to take all variables into account. Such
imperfect modeling creates uncertainty of model structure,
parameters during the prediction process. In the eld of
computer vision, the probabilistic method can help to deal
with heterogeneity in data and to account for uncertainty
induced by variables not included in the model (Abdar et
al. 2021). Bayesian inference is one of the most representa-
tive methods to estimate the uncertainty of forecasting by
using the mean and variance functions (Box and Tiao 1973).
Based on it, a novel data-driven approach called Bayesian
convolutional neural network (BCNN), which combines
the convolutional neural network and Bayesian inference
to generate a reliable deterministic model prediction, have
been successfully used in the eld of geosciences or meteo-
rology (Mo et al. 2021; Huang et al. 2022). The prediction
uncertainty is quantied through the probability distribution
of the model output. Furthermore, by integrating parameters
using prior probability distributions, the average of several
models is computed during training, which can be used to
regularize the network and avoid over-tting.
To the authors’ best knowledge, existing studies rarely
considered the eects of uncertainty existing in urban plu-
vial ood prediction, and no study has applied the BCNN
and data-driven approach to the prediction of local urban
ooding. Therefore, in this study, we aim to develop a
comprehensive urban ood prediction model using the
BCNN method. The urban ood inundation maps can be
acquired through the model. Meanwhile, the Bayesian train-
ing strategy enables the model to quantify the predictive
uncertainties.
2 Materials and methods
2.1 Study site
Rushan City is located in Shandong Province, eastern of
China (extended between
36◦41
–
37◦08
North and
121◦11
–
121◦51
East) (see Fig. 1), which occupies an area of
approximately 28.6
km2
with a permanent population of
about 155600, and the elevation of the city ranges from 11
to 86 m. Rushan City belongs to east Asian monsoon con-
tinental climate located in warm temperate zone, with mild
climate, small temperature dierence, abundant rainfall,
sucient sunshine and four distinctive seasons. Meanwhile,
ood, wind and other meteorological disasters also occur
frequently. According to local Bureau of Meteorology, the
average annual precipitation during 1956–2021 ranged
between 354.1 and 1506.7 mm, and the mean annual tem-
perature was
12.7
°C. The annual rainy season in this district
is from April to September, especially in July and August.
During rainy season prolonged intensive rainfall events
importantly created large volumes of surface runo, which
induced urban ood risk over the Rushan City. In addition,
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Stochastic Environmental Research and Risk Assessment (2024) 38:4485–4500
the overall low topography of the city, coupled with the lack
of drainage capacity, caused surface runo ow cannot be
discharged in time, generating urban inundation areas.
2.2 Flood simulation and target data
In order to facilitate ood forecast model construction, this
study utilized the MIKE from Danish Hydraulic Institute
(DHI 2015) to develop a coupled 1D/2D hydraulic model,
with the aim to simulate the urban ooding scenarios and
generate the raster grid maps of ood inundation depths.
The coupled 1D/2D model is an eective way to reect
the ow interchange between the ground and sewer system,
which contains two parts: a 1D conduit network model and
a 2D overland ow model. The essential data to construct
1D model was the municipal sewer network data, includ-
ing manholes, pipelines and outlets. The whole catchment
was divided into dierent small sub-catchments according
to the distribution of the manholes. The Model B module
can simulate the surface runo routing in each sub-catch-
ment, using Horton equation to process the soil inltra-
tion and restoration. The initial losses was set as 0.0006 m,
while the initial inltration rates were determined based on
the soil types in the subcatchments. For 2D overland ow
model, manholes are the important control points to link it
and the 1D model where surcharging ows are exchanged.
The overground ow is governed by two-dimensional dif-
fusive forms of Saint-Venant equations of continuity and
momentum, which can be calculated by Alternating Direc-
tion Implicit (ADI) method in MIKE. Buildings and roads
data, land-use data and DEM data are needed to model the
route of the overland ow. In order to reect changes of
surface elevation and illustrate the blockage eect of build-
ings on overland ows, the elevation of buildings was raised
10 m above original data (Löwe and Arnbjerg-Nielsen 2020;
Nguyen and Bae 2020). The value of the Mannings coef-
cient were determined by land-use types. More detailed
description about the MIKE FLOOD model can be found in
the literature (Tansar et al. 2020).
In this work, 33 intensive rainfall events were selected
during the period from the last 20 years, as shown in Table 1.
Each rainfall event was numbered from 1 to 33. The precipi-
tation data was provided by the local rainfall station with
5 min temporal resolution, which was deemed sucient to
reproduce the urban ood events using MIKE in the previ-
ous studies (Löwe et al. 2021). The related information on
all the data used in ood simulation is listed in Table 2.
The MIKE software was run 33 times using dierent pre-
cipitation series above to produce dierent ood conditions,
generating target output ood data of the study area. The
simulation results consist of raster format les with 5 ms in
spatial resolution and 5 min in time resolution. Consider-
ing that the simulation results usually contain more shal-
low-water areas than deep-water areas under low-intensity
precipitation, the value of the depth below 0.05m were
Fig. 1 The location of the study area and the maps showing the layout of Rushan City
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Stochastic Environmental Research and Risk Assessment (2024) 38:4485–4500
eliminated in this paper, as these areas are not relevant for
neither economic damages nor losses of life, that is,
Y
=
0,x≤0.05,
x, x>0.05, (1)
where x represents simulation water depth and Y represents
output water depth used in this paper. The value 0.05 is a
threshold value. This value is selected based on the ood
hazard thresholds for dierent objects as suggested by
Kramer et al. (2016). According to the results of ood simu-
lation,
33 ×1567 ×1834
water depth values were obtained
as the total sample set.
2.3 Flood triggering factors
Urban ood events tend to exhibit signicant spatial aggre-
gation patterns because ood water generation and ow
routing are highly aected by land use distribution and
topography (Li et al. 2022). Spatial factors play a crucial
role in urban ood model development. According to lit-
erature review (Zhang et al. 2020; Pham et al. 2020) and
expert advice, eight spatial variables considering the topo-
graphical, geological and hydrological characteristics of
the catchment were chosen for model development: eleva-
tion, slope, aspect, curvature, topographic humidity index,
imperviousness, terrain ruggedness index, stream power
index. All spatial variables were set the same resolution as
the ood inundation maps produced by hydraulic model to
accomplish end-to-end model training process. The detailed
descriptions of these spatial variables are as follows:
●Digital elevation model (DEM): DEM directly reects
changes of surface elevation. In most cases, urban wa-
terlogging events occur in the region with low elevation
(Zhang et al. 2020). A DEM of study area with 5 m reso-
lution ranging from 4 to 79 m was used here to represent
altitude of each grid unit.
●Slope (SLO): Slope is a critical factor for ood sus-
ceptibility management as it represents the structure of
terrain and thus aects runo velocity and intensity of
ow. It was calculated on the basis of the focal average
of terrain altitude within a 100 m radius to ensure that
the hydrological behavior of an area.
●Aspect (ASP): Aspect is dened as the direction of the
projection of the slope normal on the horizontal plane,
which is also an important index to describe topographic
information.
●Curvature (CUR): Curvature characterizes the concave-
ness or convexity of terrain, contributing the heteroge-
neity of overland ow.
Table 1 The selected rainfall events used in this study
No. Rainfall
event
Duration
(h)
Max-rainfall Max-
depth
(m)
Aver-
depth
(m)
1 19960715 10 311.004 3.47 0.09
2 19960725 5 116.400 2.20 0.03
3 19960726 6 109.872 2.59 0.04
4 20020725 11 242.004 3.32 0.07
5 20030806 9 197.160 3.17 0.07
6 20030823 8 128.880 2.77 0.04
7 20030827 12 136.200 3.15 0.05
8 20040711 15 187.440 2.82 0.07
9 20040725 5 116.520 2.21 0.02
10 20050731 11 87.960 2.25 0.04
11 20050803 8 139.800 2.86 0.05
12 20050804 5 115.800 2.35 0.03
13 20050808 11 115.800 2.41 0.04
14 20070701 3 125.400 1.86 0.03
15 20070719 3 104.400 1.92 0.02
16 20080815 5 99.252 2.12 0.03
17 20080817 12 297.192 3.29 0.07
18 20090709 10 124.992 2.99 0.06
19 20090712 6 109.164 2.53 0.04
20 20090714 8 129.372 2.81 0.04
21 20100805 7 157.440 2.63 0.04
22 20110718 18 98.760 1.88 0.06
23 20110727 9 170.400 2.94 0.05
24 20120705 21 116.880 2.88 0.06
25 20120710 19 134.040 2.91 0.06
26 20140722 7 137.640 2.50 0.03
27 20140725 24 263.400 3.45 0.11
28 20190811 20 98.760 2.88 0.05
29 20200723 22 213.720 3.41 0.07
30 20200804 4 108.960 2.08 0.03
31 20210703 12 108.240 2.23 0.04
32 20210831 21 198.240 3.35 0.06
33 20220627 12 145.440 3.20 0.06
The data are provided by the local rainfall station. Here, the fourth
column means maximu m hourly rainfall (mm), and the th and
sixth col um ns mea n th e ma xi mu m in un dation de pt h (m) an d th e ave r-
age inundation depth (m) respectively
Table 2 Overview of the data used in ood simulation
Data Format Resolution Source of data
Digital Elevation
model
TIFF 5 m ZY-satellite
Land use TIFF 10 m Sentinel-2A
satellite
Urban drainage
network
Shapele The urban admin-
istration Bureau
of Rushan
Building and
road geospatial
datasets
Shapele
Rainfall Text 30 min Shandong Meteo-
rological Bureau
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Stochastic Environmental Research and Risk Assessment (2024) 38:4485–4500
are dened to measure the mass distribution on hyetographs:
m1
,
m2
,
m3
,
m4
,
m5
, which calculate dierent distribution
ratios on a hyetograph. These precipitation indicators were
transformed to 2D formation and concatenated at the special
characteristic as suggested by Löwe et al. (2021).
2.4 Bayesian convolutional neural network (BCNN)
Convolutional Neural Network (CNN) is a widely used and
highly eective deep learning technique. It has attained
remarkable success within the realm of computer vision
tasks, due to their ability to extract unknown features and
learn compact representations. However, CNN is usu-
ally regarded as structure with deterministic parameters,
which can not provide any information about associated
uncertainty.
To solve this problem, Blundell et al. (2014) introduced
a new principled algorithm by learning the probability
distribution for the weights of neural networks, which is
called Bayes by Backprop. Given a set of training dataset
D=(Xi,Y
i)
, with
Xi
denoting the input variables and
Yi
denoting the output variable, the goal is to construct a pos-
terior distribution p(w|D) to achieve the following posterior
predictive distribution of the target y for a new input x,
p(y
|
x, D)=
p(y
|
x, w)p(w
|
D)dw, (2)
where w denoting all trainable network parameters. Due to
innite possibilities for the weights w, the integral above
cannot be achieved directly, leaving the true posterior dis-
tribution p(w|D) intractable. To solve this problem, various
attempts have been made at approximating this integral,
such as Markov Chain Monte Carlo (Kupinski et al. 2003;
Salakhutdinov and Mnih 2008). The most common way is
through variational inference, assuming a new
θ
-parameter-
ized distribution
q(w|θ)
as the approximation of p(w|D). The
degree of approximation can be measured by the Kullback–
Leibler (KL) divergence. Hence, the question of calculating
intricate integral turns into seeking optimal parameters
θopt
as
●Topographic humidity index (TWI): TWI is dened as
log(α/ tanβ)
with
α
being the contributing area per unit
contour length and
β
the local terrain slope (Beven and
Kirkby 1979), which is used to simulate the dry and wet
conditions of soil water in a watershed and reect the
hydrological response of topography to the watershed.
●Imperviousness (IMP): Imperviousness in each pixel af-
fects the inltration process of runo, ranging from 0 to
1. The part of a watershed contributing to surface runo
is proportional to the amount of impervious areas (Boyd
et al. 1994; Huang et al. 2008).
●Terrain Ruggedness Index (TPI): Terrain Ruggedness
Index, which quanties the vertical dierence between
a focal cell and the average elevation of its neighboring
cells (Riley et al. 1999), is a major factor employed for
evaluating the intricacy and robustness of the terrain.
●Stream Power Index (SPI): Stream power index repre-
sents the erosive power of surface runo, encompassing
slope degree and upstream contributing area
α∗tanβ
to controll channel widening and thereby inuences
ooding probability (Burrough and McDonnell 1998).
In this study, the Band Collection Statistics tool in ArcGIS
10.1 was chosen to analyze the autocorrelation of the spatial
variables, which output Pearson correlation coecients as
the result (Table 3). The analysis revealed that there were no
statistically signicant correlations between spatial variables
(p > 0.05).
Data transformation was required in order to eliminate
dimensional dierences. All features were scaled to [0, 1]
or
[−1,1]
while not change the linearity of the original data.
In the subsequent learning process, it can also signicantly
accelerate the speed of gradient solution and improve the
learning speed of the model. The distribution maps of all the
spatial characteristics are shown in Fig. 2.
On the other hand, in order to describe the time and space
variability of real local precipitation, we utilized the indica-
tors proposed by Wartalska et al. (2020). The unevenness of
precipitation over time can be characterized by two geomet-
ric indicators r and
rcg
, which dene the gravity and center
position of the hyetograph. Moreover, ve mass indicators
Table 3 Pairwise correlation matrix of ooding spatial factors in the study area
Spatial variable DEM SLO ASP1 ASP2 CUR TWI IMP SPI TPI
DEM 1.00 0.34 − 0.01 0.01 0.19 − 0.33 0.00 0.24 0.19
SLO 0.34 1.00 0.00 0.01 0.59 − 0.39 0.00 0.34 0.02
ASP1 − 0.01 0.00 1.00 0.00 0.00 0.01 0.00 − 0.01 0.00
ASP2 0.01 0.01 0.00 1.00 0.01 − 0.01 0.00 0.01 0.00
CUR 0.19 0.59 0.00 0.01 1.00 − 0.24 0.00 0.08 − 0.03
TWI − 0.33 − 0.39 0.01 − 0.01 − 0.24 1.00 0.00 − 0.07 − 0.26
IMP 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00
SPI 0.24 0.34 − 0.01 0.01 0.08 − 0.07 0.00 1.00 − 0.11
TPI 0.19 0.02 0.00 0.00 -0.03 − 0.26 0.00 − 0.11 1.00
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Stochastic Environmental Research and Risk Assessment (2024) 38:4485–4500
Fig. 2 Spatial variable distributions of the study area. They are TWI, SLO, SPI, TPI, DEM, IMP, CUR, ASP respectively, from the top left corner
to the bottom right corner
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Stochastic Environmental Research and Risk Assessment (2024) 38:4485–4500
Applying Bayes by Backprop to construct a BCNN is
composed of setting a probability distribution on the weights
of every lter, using variational inference to calculate the
hard-to-solve true posterior probability distribution, and
obtaining aleatoric and epistemic uncertainty. Specically,
we use
Ns
samples of
w(i)
to approximate the posterior dis-
tribution p(w|D). The predictive mean and standard devia-
tion of BCNN for an arbitrary input x can be then computed
using
Ns
samples.
2.4.1 Model structure
The BCNN network architecture is depicted in Fig. 3. The
2D arrays of spacial features with the resolution of
H×W
are set as the inputs to the networks. The precipitation fea-
tures are pre-processed and concatenated at the input special
features. They are passed through an alternating cascade of
convolutional/transposed convolutional layers, producing
the maximum ood inundation map with the same resolu-
tion as input data.
Convolutional block attention module (CBAM) (Woo
et al. 2018) was added into the network framework, as
depicted in Fig. 4 in detail. The CBAM module consists of
two submodules: Channel Attention Module (CAM) and
Spartial Attention Module (SAM), which was used to com-
press the channel and spatial dimension of the input fea-
tures respectively. In this work, the CBAM are used here to
identify potential features that contribute to high ood risks,
θopt
=arg min
θ
KL
[q(w|θ)||p(w|D)]
=arg min
θ
KL[q(w|θ)||p(w)]
−
E
q(w|θ)
[logp(D
|
w)
−
logp(D)],
(3)
where
KL
[q(w|θ)||p(w)] =
q(w|θ)log q(w
|
θ)
p(w)dw.
This derivation form, which is called as variational free
energy, is built upon three terms. As
logp(D)
is a constant
term, only the rest two terms need to be optimized. Since
the KL divergence is still dicult to calculate precisely, a
random variational method is used here to sample w from
the variational distribution
q(w|θ)
, constructing a computa-
tionally tractable Monte-Carlo approximation as
F
(D, θ)≈
N
i=1
logqw(i)|θ−logpw(i)−logpD|w(i), (4)
where N is the number of the Monte-Carlo samples
w(i)
drawn from the distribution
q(w|θ)
and
p(w
(i)
)
is the prior
on the weights. The Eq. (4) denes the loss function
LELBO
for optimization. Through computing it and backpropagat-
ing the gradients
∇θ
L
ELBO
the latent variables
θ
can be
updated. More details about Bayes by Backprop can refer to
(Blundell et al. 2014).
Fig. 3 The structure of the BCNN. Spatial inputs with size of
H×W
are processed through a sequence of 2D convolutional blocks that are
linked through CBAM connection and max pooling connection. through represents the number of lters
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Stochastic Environmental Research and Risk Assessment (2024) 38:4485–4500
Nf=36
. Similarly, we set the convolutional kernels size
k=3
using this scheme.
2.4.2 Model development and making predictions
Taking the whole research area as the input of the model will
generate excessive memories which slows down the speed
of training. In order to limit the size of the input and at the
same time to generate enough sample data, a method similar
to “data augment” was adopted in this paper by dividing the
study area into dierent patches of the same size.
During network training, the spatial variables corre-
sponding to each patches in the training set, together with
the rainfall variables corresponding to rainfall events, were
input into the network, and the maximum inundation depth
corresponding to the patch is taken as the output layer. At
each variational layer, we considered a Gaussian distribu-
tion for the prior distribution of the weights. Applying Bayes
by Backprop for each training epoch produced an optimal
value of the variational parameters, which minimized the
loss function
LELBO
and dened the approximated posterior.
signicantly improving the feature extraction ability of the
model without increasing the amount of computation.
The residual and skip connections were also adopted in
our model, concatenating the output from the spatial convo-
lution convolutional layers to the input of the corresponding
transposed convolutional layers. This unique mode of con-
catenation, which is widely applied for image segmentation,
can eectively resolve the the problem of gradient explo-
sion and gradient disappearance in the process of training.
In this work, considering that large scale resolu-
tion of input arrays will reduce the training data, we put
H=W=256
for the resolution of the spacial features, that
is,
256 ×256
spacial data are input into the networks.
Nf
represents the number of lters in each convolutional layers.
Since there is no well-known theory about how many con-
volution layers should be used, the most common approach
is to measure the performance of the neural network by
changing the size of the kernel. We varied the value of
N
f
from 24 to 48 to seek for the optimum value that lead to
the best performance as measured on the training data and
found that the model got the best tting consequence when
Fig. 4 The detail about Convolutional Block Attention Module. Panel
A: The overview of the CBAM. Panel B: The channel sub-module
utilizes both max poling outputs and average pooling outputs with a
shared network to tell “what to pay attention to”. Panel C: The spatial
sub-module utilizes same outputs that are pooled along and forward
them to a convolutional layer to tell “where to pay attention to”
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Besides, the critical success index (CSI) was also used in
this paper, which presented the success of forecasting based
on the hit rate and was dened by
CS
I=
H
H+FA+M
, (6)
where H, M, and FA denote the quantities of hits, misses,
and false alarms respectively for a given threshold. Here,
this index was employed to evaluate the performance of dis-
tinguishing ood areas with varying water depths (thresh-
olds). A higher score signied superior predictive capability.
3 Results
3.1 Model performance
The accuracy of the BCNN model was assessed using
validation dataset including rainfall event “19960715”,
“20050803” and “20080815”. Taking the limitation of the
length into consideration, we random chose the patches
located in urban and suburb of the city respectively where
ooding occurred to show the performance of BCNN model.
Figure 5 depicts the maximum water depths for validation
rainfall events obtained by BCNN and MIKE 21 model
respectively, in which a threshold of 0.05m is used to screen
the depth values, i.e. the predicted depth is truncated to be 0
when it is smaller than 0.05m.
Besides, the total submerged areas under validation
scenarios were calculated and the results are presented in
Table 4. Also given in this table are the ratios indicating the
proportions of the submerged areas. The calculations were
performed separately for all of these validation scenarios,
considering only the areas with a maximum water depth
greater than 0.05m.
According to Table 4, the submerged area predicted by
the BCNN model aligns closely with the simulation results
of the hydrodynamic model, with the errors controlled
within
5%
. The results reveal that the BCNN model devel-
oped in this study, consistently predicts the distribution of
ood inundation in the study area across dierent rainfall
scenarios. The model successfully captures the spatial dis-
tribution pattern of the ood and accurately identies most
areas where waterlogging occurs. Additionally, the model
demonstrates the ability to correctly distinguish areas that
do not experience stagnant water.
In order to illustrate the accuracy of the model in pre-
dicting water depth, the predicted values and target values
of ood water depth under dierent rainfall scenarios are
drawn into scatter plots, and the results are shown in Fig. 6.
The X-axis in the gure represents the target water depth
We set
Ns=20
to sample w from the distribution as sug-
gested by Zhu and Zabaras (2018), and the predictive mean
and standard values corresponding to both samples were
calculated as output.
We set Adam as our model’s Optimizer with an initial
leaning rate of 0.001 and a batch size of 126. The mean
squared error (MSE) was used to match all the data obtained
from the prediction of the neural networks and those pro-
duced by the hydrodynamic simulations. The value of
iterations for training was not xed as we applied ‘early
stopping’ in the course of model training, which means that
the training process will stop automatically when the model
performance does not improve after a certain number of
iterations.
The proposed models were implemented in Python pro-
gramming language using Torch 1.1 framework. In order to
speed up the computational calculation, the training tasks
were accomplished on an NVIDIA RTX A4000 GPU with
16 GB memory.
2.5 Model assessment
To assess the performance of the model in emulating the
urban pluvial circumstance, the predictions of the deep-
learning model are directly compared with the outputs
simulated by 2D hydraulic model in terms of water depth.
We chose root mean squared error (RMSE) (Barnston 1992)
and Nash-Sutclie eciency (NSE) (Nash and Sutclie
1970) as assessment matrices to evaluate the capability of
the deep-learning model and the prediction errors, which
dened by
RM
SE =
N
i=1(Ypre−Ysim)2
N,
NS
E=1−
N
i=1(Ypre−Ysim)2
N
i=1
(Y
pre−
¯
Y
sim
)2,
(5)
where
Y
pre and
Ysim
represent water depth produced by
BCNN model and hydraulic model respectively, N the sam-
ple size, and
¯
Ysim
represents mean of the water depth simu-
lated by hydraulic model. If the value of RMSE is closer to
0, it proves that deep-learning model shows good stability
in predicting urban pluvial water depth. The value of NSE
varies between 0 and 1, and
NSE=1
represents a perfect
t between the predicted and reference data. This metric is
widely applied for the assessment of hydrological models,
considering NSE scores larger than 0.9 to be “very satisfac-
tory”, between 0.8 and 0.9 to be “fairly good”, and smaller
than 0.8 to be “unsatisfactory” (Huang et al. 2016; Gao et
al. 2020).
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Stochastic Environmental Research and Risk Assessment (2024) 38:4485–4500
Fig. 5 The maximum ood inundation maps produced by BCNN and
MIKE 21 during dierent rainfall events. Results on the left column
are the predictions of ood inundation by the BCNN model, and on the
right column by MIKE 21. The rst line to the third line illustrate the
rainfall events of “19960715”, “20050803”, “20080815”, respectively.
The background of the picture is the topography of the area added
shadows
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Stochastic Environmental Research and Risk Assessment (2024) 38:4485–4500
(MAE) is also computed here. It can be seen from Fig. 6
that the MAE values are all less than 0.02m, suggesting that,
on average, the predicted water depth is close to the actual
values. Additionally, the prediction errors of water depth in
most grids are controlled within a range of
−0.1
m to 0.1m.
This implies that the predicted values closely align with
the target values, with no signicant regions showing large
dierences between the predictions and the actual observa-
tions. However, it has to be noted that the predicted depths
are generally underestimated.
For the aim to reect the detail performance of the model
in predicting ood depths, we calculate the NSE values of
all ood areas and randomly select two
256 ×256
patches
to show the results, as shown in Fig. 7. For each grid with
the ood depth more than 0.05m, its NSE value is calcu-
lated separately and plotted on the map. A higher NSE
value indicates a closer match between the predicted and
observed ood depth on the grid, as depicted by red color in
the gure. Conversely, a lower NSE value suggests a poor
values which were simulated by the 2D hydrodynamic
model, whereas the Y-axis represents the predicted water
depth value generated by the BCNN model. The black base-
line with 45 degrees is included in the gure to provide the
reference, corresponding to the most ideal result. According
to Fig. 6, it can be observed that the predicted water depth in
most grids demonstrates a good tting eect with the target
values. The evaluation metrics further support this observa-
tion. Specically, for the three rainfall events considered, the
RMSE values are all less than 0.05m, indicating relatively
small overall prediction errors. In order to better reect the
true situation of prediction error, the mean absolute error
Table 4 Proportion of submerged area for validation events
Rainfall event Hydrodynamic model BCNN
Submerged (km
2
)Ration Submerged
(km
2
)
Ratio
20080815 0.75 1.09 0.89 1.29
20050803 1.62 2.35 1.42 2.06
19960715 3.75 5.45 3.34 4.85
Fig. 7 The NSE values of dierent patches in the study area
Fig. 6 The scatter plots produced by BCNN during dierent rainfall events
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Stochastic Environmental Research and Risk Assessment (2024) 38:4485–4500
variances are shown for four random regions during the rain-
fall event “19960715”. From the gure, it can be observed
that the regions with uncertainty in the prediction of water
depth roughly align with the areas experiencing ooding.
This suggests that the model’s uncertainty estimation cor-
responds to the areas where ood events are occurring. Fur-
thermore, the gure also indicates that regions with large
prediction errors are associated with higher uncertainty.
This implies that the model’s uncertainty estimation is con-
sistent with the accuracy of its predictions. In areas where
the model’s predictions have higher errors, the uncertainty
is also larger.
3.4 Comparison with CNN
To prove BCNN’s superiority in predicting urban pluvial
ood, here we made a comparison with traditional CNN
networks. CSI was chosen as the comparative index. For
dierent models, separate CSI score values were computed
for thresholds of 0.1 m, 0.3 m, 0.8 m and 1 m respectively,
with the aim to distinguish prediction accuracy in areas
with distinct water depth. The structure of the CNN model
referred to Guo et al. (2021), while the other settings (input
factors, data augmentation, and so on) were the same as in
this paper.
Table 6 shows the CSI score values computed for the
water depth predictions generated by the dierent neural
networks in the validation experiment. The results suggest
that the BCNN method successfully improves the predic-
tive performance of the network when the thresholds were
0.1 m and 0.3 m. Compared to CNN model, the values of
CSI0.1m
and
CSI0.3m
for BCNN were slightly increased by
about 0.05 and 0.02 respectively. Those two models both
achieved acceptable performance in distinguish ood areas
with low values. BCNN got the best score for the thresh-
old of 0.1 m, with the value of 0.635, while CNN got the
most favourable result for the threshold of 0.3 m with the
value of 0.593. However, Both two networks were not able
to accurately predict the ood with high water level. This
behaviour was particular obvious when the water level was
higher than 1 m. The
CSI1m
values of those two models
were only 0.472 and 0.409 respectively. The limited rainfall
event datasets could be one of the reasons for this phenom-
enon of underestimation, as the extreme high water levels
only appear in heavy storms and our dataset during training
only contain few samples, resulting the poor predictive per-
formance on the high values of water level.
prediction accuracy for that grid, as illustrated by blue color
in the gure.
According to Fig. 7, the NSE values in most of the ood
areas are larger than 0.8, indicating that the model can not
only identify the places that are prone to produce waterlog-
ging, but also can predict the specic ood depth accurately.
However, it should be noted that although the model has
demonstrated satisfactory predictive performance in the
central ooding areas, it cannot accurately predict shallow
ood extending along the runo path. Besides, the perfor-
mance of the model is poor in the ood periphery, which
explains why the model’s prediction of ood area is gen-
erally lower than the target value, as well as the insu-
cient prediction of water depth in the scatter plots. Given
that data-driven models do not rely on physical laws, these
errors are inevitable.
3.2 Computation time
The computational time required by 2D hydrodynamic
model and BCNN model to reproduce the max ood inun-
dation maps under dierent ood scenarios is illustrated
in Table 5. The calculations of two modelling approaches
were performed on the same computing platforms, utilizing
high-performance professional-grade GPUs for accelera-
tion. According to Table 5, the results suggest that compared
with the 2D hydrodynamic model, the BCNN model can
signicantly reduce the calculation costs required to cal-
culate these three ood events, once trained and calibrated
appropriately. For the long-duration storm (such as rain-
fall event “19960715”), the computational eciency of
the BCNN model is more than 600 times than that of the
hydrodynamic model. This dramatic reduction in computa-
tion time can eectively avoid any type of numerical insta-
bility or oscillation problems encountered in the process of
numerical calculation, and be computationally ecient for
real-time applications.
3.3 Predictive uncertainty
In addition to predict the mean value of the maximum ood
depth at specic locations, the BCNN model also provides
a measure of uncertainty in the form of predictive variance.
According to Fig. 8, the predicted water depth errors and
Table 5 Evaluation of computational eciency by implementing the
numerical inundation model and the proposed Bayesian convolutional
neural networks (BCNN) model
Rainfall event Duration Computational cost (h)
HD BCNN
20080815 5 h 4.03 h 0.0147 h
20050803 8 h 6.52 h 0.0152 h
19960715 10 h 7.43 h 0.0161 h
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Stochastic Environmental Research and Risk Assessment (2024) 38:4485–4500
the uncertainties caused by the factors not mentioned to pro-
vide more guidance.
Various ood triggering factors related to the morpho-
logical, geological and hydrological conditions are applied
in constructing ood susceptibility models (Andaryani et
al. 2021; Wang et al. 2020; Rijal et al. 2018). Morpholog-
ical plays an crucial role in the initial stage of the ood-
ing process because the inundations of urban ooding are
mainly occurred in local depressions (Jamali et al. 2018),
where the elevation is below the elevated terrain and can be
identied through topographical information of surrounding
environments.
Dierent ways of connecting rainfall input to the neural
network may also lead to distinct performance of the model.
The formation of 1D rainfall data has been widely con-
nected to the model in terms of realizing time series predic-
tions like LSTM or RNN (Nguyen and Bae 2020; Hu et al.
2018). In that way, the rainfall intensities varying time inter-
vals are calculated and passed through 1D connection layer.
However, this strategy is not suitable for our model as the
1D convolutional layers only handled the data towards the
xed direction and lead to homogenization in the same side
of the space. The transformation of rainfall characteristics
4 Discussions
With regard to urban pluvial ood analysis, the sample in
the study area is usually composed of a set of factor vec-
tors in the form of two-dimensional data (Darabi et al. 2021;
Falah et al. 2018; Pourghasemi et al. 2020). Most research-
ers focus on the relationship between these factor vectors
and ood susceptibility, which belong to a point-wise clas-
sication problem. Nevertheless, the research directly dis-
cussing the relationship between the ood triggering factors
and 2D ood inundation maps remains very limited as so
far. A quick urban ood inundation forecasting is helpful for
disaster mitigation by providing accurate inundation maps
and ood early warnings to the public. This paper contrib-
utes a novel data-driven methods to predict urban pluvial
water depth based on ood triggering factors and quantify
Table 6 The CSI scores values for dierent thresholds as compared to
the CNN model
CSI0.1m
CSI0.3m
CSI0.8m
CSI1.0m
CNN 0.579 0.593 0.516 0.472
BCNN 0.635 0.610 0.497 0.409
Fig. 8 The scatter plots produced by BCNN during the rainfall events
“19960715”. The rst row of images is the target water depth value,
and the second row of images is the prediction error, and the third
row of images is the prediction variance, namely the uncertainty of
the model prediction results. a–d respectively represent four dierent
patches in the study area
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Stochastic Environmental Research and Risk Assessment (2024) 38:4485–4500
Results showed that the proposed BCNN model per-
formed well in predicting urban pluvial ood. The main
outcomes of this study in dierent aspects of evaluation are
elaborated in the following items: In terms of prediction
accuracy, the ood submergence map of the BCNN model
proposed in this paper is basically consistent with the ood
submergence map generated by the hydrodynamic model.
For the prediction of the submerged area, the verication
results show that the error between the two predicted water-
logging area can be controlled within 5%. For the predic-
tion of specic water depth, RMSE values of verication
set results are all less than 0.05m, and NSE values of most
waterlogging areas are greater than 0.9, indicating that the
prediction results of water depth are ideal. In terms of com-
putational eciency, the traditional hydrodynamic model
needs complex computational process to generate high-res-
olution ood inundation map, which limits its application
in real-time ood prediction. The data-driven prediction
model proposed in this paper can overcome the limitations
brought by computational eciency and reduce the compu-
tational cost. Besides, the performance of BCNN method
was also compared with CNN model. The results revealed
that the BCNN model could achieve better performance in
distinguish ood areas with low values, with the CSI values
decreasing by about 0.02–0.05.
The proposed innovative modelling approach provides a
new perspective for similar in-depth analysis surrounding
real-time forecasting of ood inundation. The results can
support the manger of government and the resident for real-
time ood disaster management and protection.
Acknowledgements The rst author would like to thank Prof. Zong-
kun Li and Wei Ge from Zhengzhou University for their help, sup-
port and valuable guidance. We also wish to acknowledge the Urban
Administration Bureau and Bureau of Meteorology of Rushan for pro-
viding the basic data that formed our research, and the referees for
their careful reading of the paper and many constructive comments
and suggestions.
Author contributions Xiang Zheng: writing the manuscript, meth-
odology and programming; Minling Zheng: preparing the data, and
reviewing the manuscript.
Funding This work is supported in part by the National Natural Sci-
ence Foundation of China (11771005).
Data availability The sewer system data and the selected rainfall
events are collected from the open public website of Shandong prov-
ince.
Declarations
Competing interests The authors declare no competing interests.
into 2D spatial arrays is not a common processing mode but
it’s suitable for the image-based BCNN model.
Beneting from the exibility architecture design and
powerful feature extraction capability, CNN gradually
exhibits outstanding performance in the eld of image pro-
cessing. The BCNN used in our work inherits these char-
acteristics. It can exibly use the up/down sampling layers
to process complex spatial features. Besides, residual and
skip connections can also be used to avoid the problem of
gradient vanish during the training. In addition, the atten-
tion mechanism make the network focus on the important
information and accelerate the speed of model training and
convergence. However, as we need to sample parameters
(w and b) from its prior distribution, the total number of
parameters of BCNN are much larger than CNN. Consider-
ing it, we did not arrange and optimize all possible combina-
tions of hyper-parameters. We only considered to optimize
Nf
, which has the greatest inuence on the result. The set-
tings of other hyper-parameters, such as kernel size or initial
learning rate, referred to other relevant literature. The ulti-
mate error of the BCNN was controlled within 10 cm, which
is an ideal and acceptable result as any additional optimiza-
tion work may lead to overtting.
This study realized the prediction of urban pluvial ood
in spatial scale. However, in fact, the problem of urban plu-
vial ooding is linked to large spatial scale and short time
scale. How to incorporate changes over small time scales
into the model, and realizing the prediction of spatio-tem-
poral distribution of urban ood remains further research.
Meanwhile, it should be noted this study utilized hydraulic
simulation results to represent reality ooding scenarios.
Due to the lack of monitoring data, the parameters of the
hydraulic model were not calibrated. Therefor, in a way, our
study can serve as an alternative model for hydraulic model
and cannot fully encapsulate reality.
5 Conclusions
Due to rapid expansion of population in civic region, the
ood under extreme rainfall events occurring in urban area
will pose a huge threat to human life and property. There-
fore the need for evidence-based urban ood risk manage-
ment is greater than ever before. In this study, we develop
a deep learning BCNN model driven by spacial variables
and rainfall variables to predict urban ood maps caused
by rainstorms. The framework of BCNN inherits the pow-
erful ability of processing spatial information from CNN,
which enable it to eciently extract and aggregate infor-
mative features to recognize the place prone to arise ood.
Meanwhile, BCNN utilize Bayesian variational inference to
quantify the predictive uncertainties.
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