This document describes a methodology for identifying critical time periods from hydrological observation data that contain important information for calibrating hydrological models. The methodology uses a statistical concept called data depth to identify unusual events in discharge or precipitation time series that lie near the boundary of the multivariate data set. These unusual events, which include extremes, long dry or wet periods, and periods of strong dynamics, are considered critical periods for model calibration. The methodology is tested on discharge and precipitation data from a catchment in Germany using two hydrological models. The results show that calibration using only the critical periods identified is only slightly worse than calibration using all the data, and the model parameters have similar transferability to different time periods.

1. Calibration of hydrological models on hydrologically unusual events
Shailesh Kumar Singh a,⇑
, András Bárdossy b
a
National Institute of Water and Atmospheric Research Ltd., Christchurch, New Zealand
b
Institute of Hydraulic Engineering, University of Stuttgart, Stuttgart D-70569, Germany
a r t i c l e i n f o
Article history:
Received 11 January 2011
Received in revised form 12 December 2011
Accepted 16 December 2011
Available online 4 January 2012
Keywords:
Critical events
Conceptual hydrological model
Data depth function
HBV
HYMOD
ICE algorithm
a b s t r a c t
The length of the observation period used for model calibration has a great influence on the identification
of the model parameters. In this contribution it is shown that a relatively small number of so called unu-
sual time periods are sufficient to specify the model parameters with the same certainty as using the
whole observation period. The unusual events are identified from discharge or precipitation observations
series using the statistical concept of data depth. The idea is to distinguish between model states which
are covered by previously observed states (interpolation case), and those for which no similar events
occurred (extrapolation case). Depth functions are used to identify unusual events from four days lagged
discharge or API (antecedent precipitation index) series. Data with low depth are near the boundary of
the multivariate set and are thus considered as unusual. The depth is calculated using the observations,
their natural logarithms, their rank and their first differences. Model calibration using the selected critical
periods is only slightly worse than using all data. The transferability of the parameters for different time
periods is equally good as using all the data and significantly better than random selection. Two different
models (HBV and HYMOD) are used to demonstrate the methodology for the Neckar catchment in
South-West Germany. The methodology developed in this study can be potentially useful for developing
monitoring strategies.
Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction
Hydrological models are used for forecasting and water man-
agement, to provide information for decision making. Due to the
simplification of the complex natural processes, the high spatial
and temporal variability, and the limited availability of observa-
tions, the identification of the model parameters is a difficult task.
Even physically-based models require parameter adjustments due
to differences between the observation and modelling scales, and
to a limited observability of certain variables and processes. Con-
ceptual models require model calibration to estimate parameter
values [1–3]. The non-uniqueness of the parameters makes this a
challenging task.
Past observations of discharge and weather (temperature, pre-
cipitation, etc.) are used for calibration of hydrological models.
The observation period might include floods, droughts and normal
flow periods. It is assumed that the calibration of the model will
only be considered successful if the observation period is represen-
tative of the hydrological behaviour of the catchment. The informa-
tion contained in the observations with respect to the parameters
is not uniformly distributed along the series. Certain time periods
might be useful for the identification of specific parameters while
others might be less important. For example, summer observations
are of no use to identify parameters related to snow accumulation.
Wagener et al. [4] shows that information contained in a data ser-
ies are non-homogeneous.
The length and information content of a time series play a vital
role in the parameter identification of a model. Several authors
have investigated data requirements for the identification of stable
model parameters [5–12,3]. Even so, it is very difficult to precisely
define what length of data is sufficient to identify model parame-
ters so that they can also be used for other time periods. Other
investigators have reported one year to eight years of data collec-
tion as being sufficient to obtain robust parameters. However, it
cannot be generalised because different models have different lev-
els of complexity and different catchments have different informa-
tion content in each year of hydrological record. Moreover, the
information content of hydrological data is generally not known.
Hence, we always use the whole data series so that the model uses
as much information as possible to identify its parameters. There is
a need to establish a method which can be used to identify the crit-
ical time periods in a given time series, which contain most of the
information need to identify the model parameters.
Time periods with high hydrological variability may useful for
calibration as they contain a lot of information for parameter iden-
tification [5]. In this study, a similar hypothesis was tested, assum-
ing unusual events in a given series represent most of hydrological
0309-1708/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.advwatres.2011.12.006
⇑ Corresponding author.
E-mail addresses: shaileshiitm@gmail.com, sk.singh@niwa.co.nz (S.K. Singh).
Advances in Water Resources 38 (2012) 81–91
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2. variability. These unusual events are also called critical events since
they contain most of the relevant (critical) information to help
identify model parameters.
The purpose of this study is to develop a simple and reliable
technique to identify those time periods (events) that can best be
used for model calibration. The hypothesis of the paper is that if
a model works well in critical time periods it also work well for
any other time periods. Critical events correspond to unusual cir-
cumstances such as the appearance of unusual sequences of rain-
fall amounts, or increases/decreases in discharge series or long
dry periods. These events can be identified from the observations
with the help of the statistical depth function.
The paper is organised as follows: After the introduction in Sec-
tion 2 the methodology and algorithm are described. The case
study area and the hydrological models used are presented in Sec-
tion 3. The results are summarised in Section 4. A possible use of
the methodology for ungauged catchments requires precipitation
forecasts which then are used to predict the occurrence of future
unusual events. This aspect is discussed in Section 5. The paper
closes with a discussion of the results and conclusions.
2. Methodology
It is obvious that in many cases only a part of the data collected
is important to identify model parameters. This is particularly true
for linear or monotonic models. In order to illustrate this idea con-
sider the following example. A linear regression is used to fit a
straight line y = ax + b to a data set Zn = {(xi,yi); i = 1,. . .,n}. This
fit can be described with the parameter vector h = (a,b) where a
and b are slope and intercept, respectively. We use three different
datasets to identify the parameters:
1. Case 1: All the n points
2. Case 2: Using a selected n⁄
critical points
3. Case 3: Using a selected n⁄
(same number as in Case 2) random
points
The above experiment was carried out M = 1000 times. Each
time a set of n = 100 points in two dimensional space was gener-
ated. For Case 2 (n⁄
= 10) the critical points were selected as those
points which have the five largest and the five smallest x value,
from the set of n = 100 points. In Case 3 n⁄
= 10 points were selected
randomly from the set with n = 100 points. For each set a regression
line was fitted using the least squares principle. This process was
repeated 1000 times, where the random set of points was always
generated from the same bi-variate normal distribution (with
r = 0.8). The slopes of the fitted regressions are shown for the three
cases in Fig. 1. It is clear that Case 1 (the whole dataset) and Case 2
(critical data) have very similar distributions, where as in Case 3 the
slopes show a much higher scatter. This shows that careful selec-
tion of a subset of the data may result in parameters which are sim-
ilar to those obtained from the whole dataset. The general concept
is that we can fit better regression lines if we use more data. Hence,
at first glance, the above example may appear to challenge this gen-
eral concept. From this example, we can see that only about 10% of
the data can give a regression almost as good as would be the case if
we selected the data carefully. This simulation example can be fur-
ther extended to real hydrological modelling.
2.1. Identification of critical time periods using data depth function
A time series of discharge or precipitation may have a large
number of sequences which look very similar and can be regarded
as a kind of repetition of previous events. Some, however, (among
them the maxima and minima) differ from previous observations.
As rainfall/runoff modelling is more complex than a simple func-
tion fitting, the critical events cannot be reduced to the biggest
and smallest events as in the previous example. Hydrological sys-
tems react to precipitation with a certain delay, and further the ac-
tual response of the catchment is related to past rainfall. Therefore,
instead of selecting critical events based on single daily values we
decided to use precipitation or discharge-related variables over a
period of a few days.
If, using critical events, we can select data that are hydrologi-
cally relevant then we may improve our calibration process [5].
To identify the critical time periods that may contain enough infor-
mation to identify model parameters, unusual sequences in the
series of discharges and precipitation have to be identified. The
intermittent nature of precipitation only allows the identification
of unusual events for high precipitation amounts. However, for
hydrological modelling, droughts might be of equal importance.
Thus, instead of precipitation P(t), the series of antecedent precip-
itation indices is used. The antecedent precipitation index API is
defined as [13]:
APIðt þ 1Þ ¼ aAPIðtÞ þ Pðt þ 1Þ; ð1Þ
where 0 < a < 1 is a constant. This index describes the wetness state
of a catchment and can be well related to discharge. The higher a,
the more influence is assigned to past precipitation. However the
value of a does not have significant influence on the identification
of critical events. a may vary from catchment to catchment based
on size, form or other characteristics of a catchment. In this study
a = 0.9 was used.
Another possible way to identify unusual hydrological events is
to use the observed discharge series Q(t). For simplicity, denote the
selected series (API or discharge) as X(t). Unusual events are
defined as a sequence of X(t) values that meet certain criteria.
Considering d consecutive time steps (in our case days) leads to a
d-dimensional set:
Xd ¼ fðXðt d þ 1Þ; Xðt d þ 2Þ; . . . ; XðtÞÞ t ¼ d; . . . ; Tg; ð2Þ
where T is the total number of observation time steps available.
Again, for simplicity, denote Xd(t) = (X(t d + 1),X(t d + 2),. . .,X(t)).
Fig. 1. Histogram for regression slopes in the three cases.
82 S.K. Singh, A. Bárdossy / Advances in Water Resources 38 (2012) 81–91

3. In the one dimensional space unusual events are the extremes.
The multivariate generalisation of extremes for our case is the
boundary of the multivariate set in the multidimensional space.
In order to identify unusual hydrological events the statistical con-
cept of data depth is used. Depth functions were first introduced by
Tukey [14] to identify the centre (a kind of generalised median) of a
multivariate dataset. Several generalisations of this concept have
been defined since it was first developed: see Rousseeuw and Stru-
yf [15], Liu et al. [16], Zuo and Serfling [17]. The points with high
depth are the points which lie in the interior of the data cloud
while those with low depth lie near the boundary of the set and
can be seen as unusual. In our study the main goal is to identify
unusual events, so we are not attempting to locate central points
of a series, but instead we intend to identify points near the bound-
ary having low depth. There are several depth functions suggested
in the literature. In this study, the Tukey half-space depth function
was used because of its simplicity and clear geometric interpreta-
tion. Formally, the half-space depth of the d-dimensional point p
with respect to set Xd is:
DXd
ðpÞ¼min
nh
ðminðjfx2Xhnh;xpi0gjÞ;ðjfx2Xhnh;xpi0gjÞÞ: ð3Þ
Here hx,yi is the scalar product of the d dimensional vectors, and nh
is an arbitrary unit vector in the d dimensional space representing
the normal vector of a selected hyperplane.
If the point p is outside the convex hull of X then its depth is 0.
Points on and near the boundary have low depth while points dee-
ply inside have high depth.
For each time step t the statistical depth of the corresponding
Xd(t) with respect to the set Xd is calculated and denoted by D(t).
The statistical depth is invariant to affine transformations; how-
ever non-linear transformations might have an effect on the depth.
Depth can also be calculated using transformed observations, using
their logarithms or their ranks.
For our purpose we defined low depth points using a threshold
D0 (in our case study D0 = 3 was used, but it can be in range of 1 to
5). Time steps t with D(t) D0 are considered to be unusual. Fig. 2
shows such a set Xd for d = 2, that we have calculated from the log-
arithm of the daily discharges. Points with low depth are marked
by circles, lying outside or near the boundary of the convex hull.
For modelling purposes we do not only select the day with a low
depth, but define a critical event as a sequence of m days before
and after it. The reason for this is that an isolated single daily dis-
charge does not reflect the corresponding hydrological processes.
An example of events which were selected by using discharge
and API is given in Fig. 3. It is important to see that critical events
selected by API and discharge are at almost the same time. This
suggests that we can use either API or discharge for critical event
selection. One can have the impression that only the high magni-
tude API and discharge can be unusual in nature. This is not true;
we found that low API or low discharge can also indicate critical
events. Typical unusual events are:
events containing extremes (extremes on any time scale from 1
to d day aggregations)
events corresponding to low flow periods
events with strong dynamics, such as a drought ending with
heavy rainfall
events with marked temporal intermittence
These events correspond to different hydrological processes
represented by the model. Thus these events enable a specific iden-
tification of the model parameters. The selection of a threshold
D0 1 leads to a slightly higher number of unusual events than
in the case of D0 = 1, which is advantageous as the effect of obser-
vation errors in the most extreme events is reduced.
2.2. Identification of Critical Events (ICE) algorithm
The algorithmic description of the previously described method
is:
Select a number of time steps d (as mentioned in the text) so
that it reflects the dynamics and the memory of the catchment
Using the above selected d and the observations (discharge or
API) create set Xd according to Eq. (2)
Calculate the depth D(t) of Xd(t) in Xd for each time t.
If for a time step t D(t) D0 then it defines a critical event which
consists of m (as mentioned in the text) time steps prior and m
time steps after t.
Define an objective function (according to the purpose of the
model), where the objective function can be calculated for the
critical events.
optimise the objective function only for the selected the critical
events using an appropriate optimisation algorithm (for exam-
ple by using the ROPE algorithm [18]).
For small catchments due to the quicker hydrological response
critical duration d should be shorter (three to four days) while for
larger catchments longer times are reasonable. The window con-
sidered for the event definition can be selected as m d. Note that
the critical events do not only consider the days before the low
depth day, but also those after. The reason for this is that the reac-
tion of the catchment to a critical event is only visible after the
event itself.
3. Case study
3.1. Application of the algorithm to hydrological models
To demonstrate the methodology in practical cases, two con-
ceptual hydrological models, namely HBV and HYMOD, were used
on the upper Neckar basin.
Fig. 2. Circles represent the boundary points or low depth points (critical events).
S.K. Singh, A. Bárdossy / Advances in Water Resources 38 (2012) 81–91 83

4. 3.2. Study area
The study was conducted on upper Neckar basin situated in the
region of Baden–Württemberg (south-west of Germany), using
data from the period 1961–2000. The rivers in the catchment are
not affected by larger hydro power plants or other water manage-
ment structures or navigations, which could influence the runoff
characteristics of the catchment. The upper Neckar catchment can
be considered to be a typical example of a mesoscale catchment
in this region. The Neckar is a right-bank tributary of the Rhine; it
is 367 km long. The area of the basin covers 4000 km2
. The elevation
in the catchment ranges from about 240 m a.s.l. to around 1010 m
a.s.l., with a mean elevation of 548 m a.s.l. The upper Neckar is
bounded by the north-western edge of the Swabian Jura on the
right bank side of the Neckar and by the Black Forest on its left bank.
The slopes in the catchments are generally mild; approximately
90% of the area has slopes varying from 0 to 15 degrees, although
some areas of the Swabian Jura and the Black Forest may have val-
ues as high as 50 degrees [19]. The upper Neckar basin was divided
into 13 subcatchments depending on the available discharge
gauges (Fig. 4). Three of the headwater subcatchments having dif-
ferent features were used for this study. The dataset used in this
study includes measurements of daily precipitation from 151
gauges and measurements of daily air temperature at 74 climate
stations. The meteorological input required for the hydrological
model was interpolated from the observations with External Drift
Kriging [20] using topographical elevation as external drift. The
mean annual precipitation is 908 mm/year. Land use is mainly agri-
cultural in the lowlands and forest in the medium elevation ranges.
Hydrological characteristics of the three selected subcatchments
are given in Table 1. Table 2 shows runoff and precipitation charac-
teristics for different time periods. For further details please refer to
Samaniego [19], Bárdossy et al.[21] and Singh [3].
3.3. Hydrological model
3.3.1. HBV model
A modified version of the HBV conceptual model has been used
for this study (Fig. 5). The HBV model [22] was developed by the
Swedish Meteorological and Hydrological Institute (SMHI) in the
early 1970’s and modified at the Institute of Hydraulic Engineering,
University Stuttgart. It has conceptual routines for calculating
snow accumulation and melt, soil moisture and runoff generation,
runoff concentration within each subcatchment, and flood routing
of the discharge in the river network. The snow routine uses the
degree-day approach. Soil moisture is calculated by balancing pre-
cipitation and evapotranspiration using field capacity and perma-
nent wilting point. Runoff generation is simulated by a nonlinear
function of the soil moisture state and the precipitation. The runoff
concentration is modelled by two parallel nonlinear reservoirs rep-
resenting the direct discharge and the groundwater response.
Flood routing between the river network nodes uses the Muskin-
gum method. The physical meaning of model parameters and their
ranges are given in Table 3. Additional information about the HBV
model can be found in Bergström [22], Hundecha and Bárdossy
[23] and Bárdossy and Singh [18].
3.3.2. HYMOD IWS model
A modified version of the HYMOD conceptual model has been
used for this study. HYMOD is a simple conceptual model. This
300 400 500 600 700 800
Time(Days)
0
20
40
60
80
Discharge(m
3
/s)
Event selection by API
Observed Discharge
EVENT 1
EVENT 2
300 400 500 600 700 800
Time(Days)
0
20
40
60
80
Discharge(m
3
/s)
Event selection by Qobs
Observed Discharge
Event 1
Event 2
Fig. 3. Example of event selection from Neckar Catchment (Rottweil).
84 S.K. Singh, A. Bárdossy / Advances in Water Resources 38 (2012) 81–91

5. model has two main components namely, rainfall excess (two
parameters), runoff partitioning between quick and slow response
(one parameter) and runoff routing (two parameters). The rainfall
excess model is based on the characteristics of the runoff produc-
tion process at a point in a catchment and then a probability distri-
bution which describes the spatial variation in the catchments is
derived by an algebraic expression given in [24]. This model makes
an assumption that the soil structure and texture, and water stor-
age capacity varies across the catchment, therefore, the distribu-
tion function of different storage capacities is described as
FðCÞ ¼ 1 ðC=CmaxÞb
0 6 C 6 Cmax: ð4Þ
The model structure is shown in Fig. 6. The five parameters of this
model are:, the maximum storage capacity in the catchment (Cmax),
the degree of spatial variability of the soil moisture capacity within
the catchment (b), the factor distributing the flow between the two
series of reservoirs (a), and the residence times of the linear reser-
voirs (Rq) and (Rs). The modification to the original HYMOD was
done by adding a snow routine. The snow accumulation or melting
is calculated based on the degree day method that HBV uses. The
physical meaning of model parameters and their ranges are given
in Table 4. Additional information about the HYMOD model can
be found in Moore [24], Wagener et al. [25], Boyle et al. [26] and
Singh [3].
4. Results and discussion
Critical events were selected using discharge and API for differ-
ent time periods, by applying the above mentioned ICE algorithm.
Both the HYMOD and HBV models were calibrated using the ROPE
algorithm [18] for the three cases given below.
1. Case 1: The whole series of observations of the given time per-
iod were used.
2. Case 2: Only the selected critical events of the given time period
were used.
3. Case 3: Randomly selected events (same number and the same
length as in Case 2) of a given time period were used.
The mean Nash–Sutcliffe coefficients of the hydrological models
corresponding to the whole time period obtained by model calibra-
tion using the above three calibration cases was compared. The re-
sults are similar for all the three selected catchments chosen for this
study. Hence, results from only one catchments (Rottweil) is pre-
Fig. 4. Study area: upper Neckar catchment in south-west Germany.
Table 1
Summary of the different subcatchments in the study area.
Subcatchment Subcatchment
size (km2
)
Elevation
(m)
Slope
(degree)
Mean
Discharge
(m3
/s)
Annual
Precipitation
(mm)
1 Rottweil
(Neckar)
454.65 555–
1010
0–34.2 5.1 968.16
2 Tübingen
(Steinlach)
140.21 340–880 0–38.8 1.7 849.84
3 Süssen (Fils) 345.74 360–860 0–49.3 5.9 1003.45
S.K. Singh, A. Bárdossy / Advances in Water Resources 38 (2012) 81–91 85

6. sented here. Tables 5 and 6 show the calibration results by HYMOD
and HBV respectively for time period 1961–1970. From the tables it
is clear that the performance of the model calibration based on crit-
ical events is comparable with model calibration on the whole data
series. Both models show very similar performance measured in
terms of Nash–Sutcliffe coefficient. Most noticeably only 6% of the
whole data set is enough to achieve the same performance (Case
2) as using the whole data set (Case 1). When the events were se-
lected randomly (Case 3) the performance was slightly worse than
other cases in calibration. This shows that the selection of the
events using the depth function is useful to identify those events
which contain hydrologically important information.
The robustness of the methodology was tested on a different
time period. Tables 7 and 8 show that, if the model is calibrated
on critical events, the parameters are transferable in the same
way as when the whole data series is used for calibration. The
transferability was tested for three different time periods (1971–
1980, 1981–1990, 1991–2000). In all these time periods both the
model dynamic was similar and the performance of calibrated
parameters over time period was as good as if we had used the
whole data series. At the same time, when the parameters were ob-
tained by calibration on randomly selected events; their transfer to
other time periods leads to poor results. To aid the visual appraisal
of the results, Figs. 7 and 8 present observed and modelled dis-
charge for various cases for the Rottweil for the validation time
period 1971–1980 from the HYMOD and HBV models respectively.
Overall, the match is the best for Case 1 followed by Case 2 and
Case 3, the difference between Case 1 and 2 being minor. The graph
for Case 3 appears to be consistently under performing. These fig-
ures affirm the interpretation of results from Tables 7 to 8 that
both the model calibrated by using the whole data and model cal-
ibrated on critical events, show a nearly similar match with the ob-
served hydrograph, whereas calibrating using a random selection
of events is inferior.
In some previous studies [6–9,11] the part of continuous series
was selected or events were selected randomly from a given
hydrological time series (same as Case 3 in this study) to study
the performance of models calibration. The advantage of the ICE
algorithm is that it identifies very different specific time periods
which are useful for model calibration.
To see the effect of the length of the data on the model calibra-
tion, the HYMOD model was calibrated on a single year using crit-
ical events over that year. Subsequently, one year of data was
added to the calibration, thus each calibration has one parameter
sets (e.g the calibration using 1971 data has one parameter set,
the second calibration is over a period of 1971–1972, two year of
data, has another parameter set and so on). This procedure was
also repeated for randomly selected events. Fig. 9 shows model
performance of each parameter set obtained by each calibration
and validated over periods of selected ten years (1961–1970).
The X axis shows the number of years used for calibration and Y
axis shows the performance over the selected ten years. From
Fig. 9, it can be seen that when only one year of data was used
for calibration, the validation over 1971–1980, the Nash Sutcliffe
coefficient is 0.65, where as for randomly selected events from
one year, it is 0.51. When the first two years of data were used
for calibration, the performance in validation over 1961–1970
got worse, where as for randomly selected events performance it
got better. This is by chance (because random selection process
may have selected good events from first year). In general, it can
be seen from the figures that performance increases as the number
of calibration years increases, but after some period of time, there
is practically no more improvement in the performance. The per-
formance for the critical event selected by the ICE algorithm was
always better then randomly selected events. On average seven
to eight years of data (critical events) seems to be reasonable for
model calibration.
Note that the cumulative frequency (cumulated over time) of
unusual events can be calculated from both the precipitation and
discharge series. These cumulative frequencies are calculated
sequentially from a starting point increasing the function by one
when a critical event occurs. These cumulative curves increase
Table 2
Runoff characteristics for different time periods.
Subcatchment Rottweil (Neckar) Tübingen (Steinlach) Süssen (Fils)
Time period Annual precipitation
(mm)
Annual discharge
(mm)
Annual precipitation
(mm)
Annual discharge
(mm)
Annual precipitation
(mm)
Annual discharge
(mm)
1961–1970 997.53 375.26 851.84 400.36 1007.94 575.55
1971–1980 908.48 309.36 808.14 366.62 960.02 512.62
1981–1990 997.21 385.66 888.84 404.86 1041.72 541.81
1
0
Runoff
Response
Routing
K2
Q1
Q0
K1
K0
Q2
QPer
FC
PWP
Snow
accumulation
and
melt
routine
DD
Metl
TT
T
Rain
ET
a
L
S1
S2
Kper
P
eff
/(P+Melt)
SM
1
FC
(SM
/ FC)
B
0
Soil
Moisture
accounting
Precipitation
Snow
Q
Peff
Qout
MAXBAS
Fig. 5. Schematic representation of HBV model[3].
Table 3
Model parameter ranges for HBV model.
Parameter Description Max Min
Tcrit Threshold temperature for snow melt initiation 2 2
DD Degree-day factor 5 0.5
Dew Precipitation/degree-day relation 2 0
b Model parameter (shape coefficient) 6 1
L Threshold water level for near surface flow 30 1
k0 Near surface flow storage constant 20 0.5
k1 Interflow storage constant 50 5
kperc Percolation storage constant 100 20
k2 Baseflow storage constant 1000 10
86 S.K. Singh, A. Bárdossy / Advances in Water Resources 38 (2012) 81–91

7. sharply at the beginning but their rate of increase decreases with
time (see Fig. 10). Note that events which were considered as crit-
ical at the beginning of a period might become usual due to the
occurrence of other critical events. These curves indicate the rate
with which new critical events are occurring and thus can be con-
sidered as learning curves of the hydrological model. They might
be used to objectively compare the calibration period required
for a model under different meteorological circumstances.
5. Use for developing monitoring strategies
For many modelling purposes data are collected during specific
measurement campaigns. Here it is of interest to chose appropriate
times when measurements can deliver the maximum possible
information. The previous sections showed that critical events pro-
vide more information than randomly chosen times, thus it is rea-
sonable to perform a measurement if an unusual event is likely to
occur. For discharge measurements required for rainfall runoff
modelling one might use the past precipitation and precipitation
forecasts, to predict whether on the next day an unusual situation
is likely to occur. For this purpose, using the ICE algorithm one can
calculate the depth of the d day period ending on the following day
for any precipitation amount occurring on that day. If the meteoro-
logical forecasts indicate a precipitation for which the d-day depth
is low than it is reasonable to plan to measure on that day. This
procedure can help to measure hydrologically critical events in
data scarce regions, and the model can be calibrated on these
events. The strategy suggested is given in following section.
Excess
α
Storage
Model
Discharge
Rs
β
P ET
Cmax
Rq
Rs
Rq
Rq
SOIL MOISTURE ACCOUNTING ROUTING
Fig. 6. Schematic representation of HYMOD model.
Table 4
Model parameter ranges for HYMOD model.
Parameter Description Max Min
Cmax Maximum storage capacity 600.00 150.00
Beta Degree of spatial variability of the soil
moisture capacity
8.00 3.00
Alpha Flow distributing factor 0.80 0.20
RS Residence times of the slow reservoirs 0.20 0.01
RQ Residence times of the quick reservoirs 0.70 0.30
Th Threshold temperature for snow melt
initiation
2.00 2.00
DD Degree-day factor 5.00 0.50
Dew Precipitation/degree-day relation 2.00 0.00
Table 5
Calibration of HYMOD model for period 1961–1970 for Rottweil. The case 1 is using whole series for given time period. The case 2(a) when events section is based on discharge
series and case 2(b) when event selection is based on API. The case 3 is using only randomly selected events (same number as in case 2) for given time period. (the statistics shown
in the table are for 1000 parameter sets).
mean NS max NS min NS Std Nr. of days used Percentage of days used
Case 1 0.694 0.701 0.692 0.00185 3652 100
Case 2(a) 0.677 0.699 0.628 0.01313 238 6.51
Case 2(b) 0.681 0.699 0.643 0.01002 237 6.48
Case 3 0.641 0.665 0.614 0.00850 242 6.62
Table 6
Calibration of HBV model over period 1961–1970 for Rottweil. The case 1 is using whole series for given time period. The case 2(a) when events section is based on discharge
series and case 2(b) when event selection is based on API. The case 3 is using only randomly selected events (same number as in case 2) for given time period. (the statistics shown
in the table are for 1000 parameter sets).
mean NS max NS min NS Std Nr. of days used Percentage of days used
Case 1 0.699 0.714 0.698 0.00135 3652 100
Case 2(a) 0.697 0.709 0.682 0.00613 238 6.51
Case 2(b) 0.690 0.710 0.680 0.00484 237 6.48
Case 3 0.657 0.672 0.624 0.00703 242 6.62
Table 7
Validation of HYMOD model over period 1971–1980 for Rottweil. The case 1 is using
whole series for given time period. The case 2(a) when events section is based on
discharge series and case 2(b) when event selection is based on API. The case 3 is
using only randomly selected events (same number as in case 2) for given time
period. (the statistics shown in table is for 1000 parameter sets).
mean NS max NS min NS std
Case 1 0.624 0.644 0.586 0.00909
Case 2(a) 0.627 0.647 0.579 0.01273
Case 2(b) 0.625 0.649 0.597 0.01029
Case 3 0.521 0.571 0.479 0.01856
S.K. Singh, A. Bárdossy / Advances in Water Resources 38 (2012) 81–91 87

8. 5.1. Practical application: start measuring important events
For measuring important events, a conditional depth with re-
spect to the next day’s unknown precipitation curve was prepared.
This means that the depth of the last d 1 days plus the next day is
calculated for different possible precipitation amounts for the next
day.
If a given day’s precipitation makes the statistical depth low for
the d day period ending that day, then it is important to measure
the discharge, which may be useful for model calibration. Fig. 11
shows a conditional depth with respect to the precipitation curve
for four different days. Fig. 11(a) shows that if next day’s precipita-
tion is more than 10 mm, it will be an unusual event and, if precip-
itation is less than that, then it is not. Similarly we can see from Figs.
11(c and d) different, (much higher) amounts of precipitation can
make an event unusual. Thus for an ungauged catchment we have
a better chance to find a few critical events, which might give a more
robust result than taking some randomly selected measurements.
To illustrate the concept,measured precipitation for the time
period 1991–2000 from the Upper Neckar catchment is taken as
the forecasted precipitation. In reality we get the forecasted pre-
cipitation from a quantitative precipitation forecast. As this is fore-
Table 8
Validation of HBV model over period 1971–1980 for Rottweil. The case 1 is using
whole series for given time period. The case 2(a) when events section is based on
discharge series and case 2(b) when event selection is based on API. The case 3 is
using only randomly selected events (same number as in case 2) for given time
period. (the statistics shown in table is for 1000 parameter sets).
mean NS max NS min NS Std
Case 1 0.708 0.730 0.682 0.01014
Case 2(a) 0.700 0.711 0.684 0.00545
Case 2(b) 0.699 0.723 0.666 0.01201
Case 3 0.630 0.650 0.593 0.01038
1040 1050 1060 1070 1080 1090 1100
0
5
10
15
20
25
30
35
40
45
50
Time (days)
Discharge
(m
3
/s)
Observed Discharge
Case 1
Case 2(a)
Case 2(b)
Case 3
Fig. 7. Validation of HYMOD model over period 1971–1980 for Rottweil. The case 1
is using the whole series for given time period. The case 2(a) when event selection is
based on discharge series and case 2(b) when event selection is based on API. The
case 3 is using only randomly selected events (same number as in case 2) for given
time period.
1040 1050 1060 1070 1080 1090 1100
0
5
10
15
20
25
30
35
40
45
50
Time (days)
Discharge
(m
3
/s)
Observed Discharge
Case 1
Case 2(a)
Case 2(b)
Case 3
Fig. 8. Validation of HBV model over period 1971–1980 for Rottweil. The case 1 is
using the whole series for given time period. The case 2(a) when event selection is
based on discharge series and case 2(b) when event selection is based on API. The
case 3 is using only randomly selected events (same number as in case 2) for given
time period.
1 2 3 4 5 6 7 8 9 10
0.5
0.52
0.54
0.56
0.58
0.6
0.62
0.64
0.66
0.68
Years
Mean
NS
Nr. of Years Vs Performance for validation year 1961−70
Case 2
Case 3
Fig. 9. Sequentially year addition in calibration of HYMOD and validated over
10 years (1961–1970) (red line from case 3 and blue from case 2). (For interpre-
tation of the references to colour in this figure legend, the reader is referred to the
web version of this article.)
0
5
10
15
20
25
30
35
40
45
1 2 3 4 5 6 7 8 9 10
years
Number
of
events
Fig. 10. The cumulative frequency (cumulated over time) of unusual events
calculated from discharge series over time period 1970–1980 for Rottweil.
88 S.K. Singh, A. Bárdossy / Advances in Water Resources 38 (2012) 81–91

9. casted precipitation, the API is calculated using next day’s precip-
itation. The ICE algorithm was then used to decide if the event is
critical or not. In this case depth was calculated with respect to
previous observation period. This series changes each time as
new forecasted precipitation comes in. Events were selected for
the above mentioned time period using the ICE algorithm. The pro-
cess was also repeated for random selection. Hence, three cases are
given below:
1. Case 1: Using the whole series for a given time period, where
the API is based on known precipitation
2. Case 2: Using only selected critical events for a given time per-
iod, API is based on forecasted precipitation
3. Case 3: Using only selected (same number as in Case 2) random
events for a given time period, where the API is based on fore-
casted precipitation
Fig. 11. Conditional depth with respect to precipitation curve for different days. The number on top of the figure shows the nth day in a series (e.g precipitation-depth 3660
means conditional depth with respect to precipitation curve on 3660th day in a series).
Table 9
Calibration of HYMOD for Rottweil for time period 1991–2000, event selection is
based on forecasted precipitation and known precipitation. The case 1 is using whole
series for given time period. The case 2 is using only selected critical events from a
given time period and events selection is based on API. The case 3 is using only
randomly selected events (same number as in case 2) for given time period. (the
statistics shown in table is for 1000 parameter sets).
mean NS max NS min NS Std
Case 1 0.620 0.640 0.613 0.0054
Case 2 0.615 0.645 0.536 0.0193
Case 3 0.615 0.646 0.559 0.0186
Table 10
Validation of HYMOD for Rottweil for time period 1981–1990, event selection is
based on forecasted precipitation and known precipitation. The case 1 is using whole
series for given time period.The case 2 is using only selected critical events from a
given time period and events selection is based on API. The case 3 is using only
randomly selected events (same number as in case 2) for given time period. (the
statistics shown in table is for 1000 parameter sets).
mean NS max NS min NS Std
Case 1 0.734 0.759 0.708 0.0100
Case 2 0.731 0.758 0.672 0.0140
Case 3 0.713 0.749 0.615 0.0183
S.K. Singh, A. Bárdossy / Advances in Water Resources 38 (2012) 81–91 89

10. The HYMOD model was calibrated on all the three cases men-
tioned above. Table 9 shows results of the model calibration, when
the events were selected using forecasted precipitation and mea-
sured (known) precipitation. The results show that the calibration
from predicted API event selection is slightly worse than the cali-
bration using the whole time period for known precipitation. In
calibration, Case 3 is as good as Case 2. Table 10 shows the trans-
ferability to other time periods for all the three cases. It is very
clear from Table 10 that Case 2 performance was as good as Case
1, where as Case 3 has not shown the same performance as in cal-
ibration. This shows that we can use forecasted precipitation to de-
cide which events should be considered for observation.
We have shown that it is possible to calculate the conditional
depth with respect to precipitation curves for ungauged catch-
ments of interest. Based on this curve, and given forecast or mea-
sured precipitation, one can decide to measure the stream flow
event or not. Once a certain number of events is reached one can
calibrate the model for that catchment. Comparing the set of the
selected events to that of the whole precipitation observation per-
iod, one can see if the selected events are representative or not. By
doing so we can either avoid or complement the use of transfer
functions for regionalisation for prediction in ungauged catch-
ments. This measurement strategy aims to maximise the informa-
tion content in the data, for a given sampling effort.
6. Summary and conclusions
In this paper, we investigated the information contained in a
time series with respect to hydrological model parameter iden-
tification. We found that a small number of specifically selected
events is sufficient to identify the parameters of a hydrological
model. Indeed, the calibration is as good as that arising from the
calibration using the whole time period.
A novel ICE algorithm for identification of critical events based
on data depth was developed. Data points with low data depth
are critical for model calibration.
Critical events contain only a small portion ( 10 %) of the
observation period. The same number of randomly selected
events does not enable a good model calibration.
The methodology developed in this paper can help to decide
when to recalibrate a forecasting model. The occurrence of
new critical events indicates the necessity of model updates.
The ICE algorithm helps to decide which events should be to be
measured. Hence it could be extended to ungauged catchment
sampling problems.
The critical events selected by the ICE algorithm not only can be
used for calibration of a simple conceptual model, but can be
used for a complex model. An example can be seen in a study
by [27], where WaSiM-ETH model was calibrated on critical
events with promising results. In an another study [28] use
the ICE algorithm to calibrate the TopNet model in a New Zea-
land catchment. The concept of this paper was also found to be
very useful in improving the training efficiency of a data driven
model [29].
The major limitation of the applicability of the concept of this
paper is the time required for pre-processing of the data to
identify the critical events. On the other hand, the calibration
effort can be substantially reduced.
Further research is needed to use the concepts developed in this
paper for other purposes and models.The ICE algorithm might be
useful for regionalization and could contribute to better prediction
in ungauged basins. The Authors in their further work intend to
investigate the transferability of model parameters not only in
time but also in space. The suggested methodology is not only
applicable for discharge modelling but may be useful for other pur-
poses such as water quality modelling.
Acknowledgments
The work described in this paper is supported to S. K. Singh by a
scholarship program initiated by the German Federal Ministry of
Education and Research (BMBF) under the program of the Interna-
tional Postgraduate Studies in Water Technologies (IPSWaT). Its
contents are solely the responsibility of the authors and do not
necessarily represent the official position or policy of the German
Federal Ministry of Education and Research. The authors thank
Dr. Ross Woods and anonymous referee for the constructive com-
ments. The helpful comments of Dr. D. Andrew Barry, editor of this
paper, are gratefully acknowledged.
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