Introduction
For many applications in hydraulics, hydrology and water
management, reliable river discharges are crucial. A commonly used practice
for the estimation of these discharges is the use of rating curves. Through
the calibration of a relation between stage and discharge measurements (i.e.,
a rating curve), high-frequency stage measurements can be transformed into
high-frequency discharge measurements. As this relation is based on only a
limited number of simultaneous stage–discharge measurements, it is a
relatively budget-friendly method for discharge assessment in rivers.
The use of rating curves requires attention for the consistency of the
measured stage–discharge data set. Several causes (e.g., geometric changes of
the river bed, infrastructure works, weed growth) can alter the hydraulic
behavior of the river in the considered measurement location temporarily or
permanently and thus limit the validity of a calibrated rating curve.
Information about this temporal (in)consistency is hence critical to prevent
additional errors and biases from occurring in the determined river
discharges. Moreover, a correct assessment of (in)consistent periods can
enhance other applications based on the investigated data. For instance,
errors in hydrological or hydraulic model results that are caused by changes
in a river's situation can (if they lead to inconsistency of the rating curve
data as well) be avoided by using known consistent stage–discharge time
periods for model evaluation. Methods to detect and describe this temporal
(in)consistency have been studied by several authors (see
, for a more extensive review).
assumed that changes in rating curve behavior are mainly
caused by floods and that periods in between are consistent. Other methods
described the variation of rating curve parameters in time
. Within predefined consistent periods,
accounted for an aging error toward an initial rating
curve, which expresses the increasing risk of a change of the river bed in
time. expanded this method with two preliminary steps.
First, the stage–discharge data set is segmented into consistent periods, and
subsequently hydraulic analogues of each stage–discharge measurement are
selected within these periods. Although measurement uncertainties are
considered in the latter step, the first and thus defining data segmentation
does not account for them.
discuss that in the methods mentioned above, the
assessment of temporary or permanent changes of the hydraulic regime requires
assumptions or decisions that more or less influence eventual results.
Moreover, most of the methods start from a definitive choice of a rating
curve model and comprehend an assessment of its parameters distribution.
However, if data consistency is assessed prior to a definitive in-depth
analysis (as in ), it can provide an increased
understanding that contributes to the selection of an appropriate, more
definitive rating curve model. An important criterion for this preliminary
consistency analysis is that results minimally depend upon choices and
decisions made by users.
Therefore, have developed a methodology to enable
the detection of consistent periods in stage–discharge data. It is called
Bidirectional Reach (BReach) and considers a period to be consistent if no
consecutive and systematic deviations from a current situation occur that
exceed observational uncertainty. This definition of consistency is commonly
used in operational hydrology . It requires the assessment
of (1) observational uncertainty, (2) a current situation and (3) the
consecutive and systematic character of nonacceptable deviations.
Observational uncertainty is estimated for each country using regional
information (Sects. and ). The
assessment of a current situation is done by evaluating the capability of a
rating curve model to describe a subset of the data in each pair of
stage–discharge measurements (Sect. ). This capability is
defined by a degree of tolerance, i.e., a definition of satisfactory behavior
for the rating curve model in a series of gauging points
(Sect. ). By combining multiple degrees of tolerance,
complementary information is provided that allows for the exclusion of causes
for model failure other than data inconsistency. Hence, changes throughout
time in the combined model performance indicate possible changes in data
consistency. This information is used for the last requirement in the
definition of consistent periods, i.e., the assessment of the consecutive and
systematic character of nonacceptable deviations (Sect. ). In
Sect. , the different steps of the methodology are briefly
explained and all necessary choices are discussed.
In , one observed and several synthetic data sets
are used to evaluate and test the robustness of the methodology. The
methodology was shown to perform well, with robust results despite decreased
data availability, erroneous estimations of measurement uncertainty and even
a partially deficient rating curve model. All investigated data sets in this
study belong to the same geographical location. Therefore, the objective of
the current paper is to perform an additional analysis with more diverse
measured data sets in order to further explore the methodology's
applicability. For this purpose, several gauging stations in the United
Kingdom (UK), New Zealand and Belgium are selected based on their
well-documented history and their specific characteristics related to rating
curve consistency. For each country, regional information is maximally used
to estimate observational uncertainty. Based on this uncertainty, a BReach
analysis is performed and, subsequently, results are validated against
available knowledge about the history and behavior of the site. In a
selection of the investigated stations, results of the BReach methodology are
additionally compared with results of a classical residual analysis.
Study sites and their characteristics.
Station name
River
Country
Upstream
Mean
First h–Q
Last h–Q
hmin,h-Q-
hmax,cont-
Amount
Characteristics related
catchment
flow
measurement
measurement
hmin,cont
hmax,h-Q
of h–Q
with rating curve behavior
(km2)
(m3s-1)
(m)
(m)
measurements
and data consistency
Colsterworth
Witham
UK
51.3
0.23
1984
2012
0.08
0.78
99
Flat V weir, stable rating curve.
Taw Bridge
Taw
UK
71.4
1.85
1968
2012
0
1.15
412
Original situation: unstable bed control.
Stable rating curve since installation
of a Flat V weir
(September–November 1998).
Clog-y-Fran
Taf
UK
217.3
7.6
1961
2012
0
0.16
1166
Considerable overspill over the right bank
at a stage of approximately 3.2-3.4 m.
Weed growth affecting low flow behavior.
Channel prone to build-up of silt.
Mais
Pohangina
New Zealand
488
19a
1975
2014
0.13
1.13
598
Unstable (gravel) bed control.
Barnett's Bank
Wairau
New Zealand
3825
100b
1999
2015
0
0.98
270
Unstable (gravel) bed control.
Aarschot
Demer
Belgium
2146
14.3
1980
2014
0.14
0.02
299
Small backwater effect at high flows.
Important deepening and widening of
the river bed until July 1982.
Introduction of new measurement
devices since February 2006. Maintenance
works affecting the river bed
between May 2007 and December 2010.
Zichem
Demer
Belgium
13.4
1980
2016
0.03
0.02
185
Important deepening and widening of
the river bed in 1988. Completion of
deviation of the mouth of a small
tributary at the location of
the measurement station in 2003.
Introduction of new measurement devices
since March 2008. Occasional weed
growth (registered since 2011).
Diest
Demer
Belgium
13.3
2001
2012
0.43
0.01
34
Introduction of new measurement
devices since March 2008.
Occasional weed growth
(registered since 2011).
Hulshout
Grote Nete
Belgium
443.5
4.83
1998
2015
0.16
0.13
38
Weed growth affecting low flow behavior.
Maaseik
Meuse
Belgium
21787
249.5
1993
2013
0.07
0.26
109
Downstream movable weir affecting
low flow behavior. Dredging of
the winter bed and change of the local
flow situation under the bridge at
the measurement station in 2008.
hmin,h-Q (hmax,h-Q): minimum (maximum)
stage value of all available stage–discharge measurements.
hmin,cont (hmax,cont): minimum (maximum) value of
all available continuous stage measurements.
a .
b .
Methods
Study areas and data
The BReach methodology is applied to three stage–discharge data sets in the
UK, two in New Zealand and five in Belgium. These stations are selected based
on their particular properties with regard to data consistency. Their
well-documented history enables a verification of the results of a BReach
analysis. An overview of these stations and their main characteristics is
given in Table . The UK data are provided with a quality
indication and hence only stage–discharge measurements marked as “good” are
used in this research. The New Zealand data were preprocessed by the Horizons
Regional Council and the Marlborough Regional Council and are assumed to have
a sufficient quality level. For the Belgian stations, raw (unprocessed)
gauging data are available. Therefore, stage–discharge measurements with
recorded stages that deviate more than 5 cm from the nearest
continuous value are treated as outliers and not used in the analysis. These
continuous stage data have a temporal resolution of 1 h (before 2003)
and of 15 min (after 2003). Taking into account the estimated
95 % uncertainty boundaries of the (gauging) stage measurements
(±2 cm) and assuming a similar magnitude for those of the
continuous measurements (Sects.
and ), this difference of 5 cm guarantees that only
measurements with large errors are excluded from the analysis.
BReach methodology: description and practical application
The aim of the BReach methodology is to identify consistency in rating curve
data based on a quality analysis of model results. The methodology consists
of several consecutive steps
:
Step 1: selection of a model structure for the analysis.
Step 2: sampling of the parameter space.
Step 3: assessment of acceptable model results.
Step 4: assessment of different degrees of tolerance.
Step 5: assessment of the bidirectional reach for all degrees of tolerance.
Step 6: identification of consistent data periods.
In this section, all steps and their practical application in this paper are
briefly described.
Step 1: selection of a model structure for the analysis
A first step is the choice of a rating curve model that appropriately
approximates the relation between discharge and stage for an important part
of the measured range. In this paper, the chosen rating curve model depends
on the characteristics of the measurement station.
For the station of Colsterworth (UK, Table ), a flat V weir
controls the flow and thus a power law can be used to describe the
stage–discharge relationship:
Q=c(h-h0)n,
where Q is the discharge (m3s-1), c is a scale coefficient
(m3/ns-1), h is the stage (m), h0 is a location
parameter (m) that expresses the stage of zero flow and n is an exponent
(–) that is a function of the type and the shape of the considered cross
section.
For all other analyzed stations (except from the station of Clog-y-Fran, UK),
a segmented rating curve with two segments is used
e.g.,:
Q=c0(h-h0)n0h<hbr,1c1(h-h1)n1h≥hbr,1,
where each segment describes a different flow situation and
hbr,1 is the breakpoint between two consecutive segments. In
this breakpoint, continuity between both segments must be provided. By using
this rating curve model with a breakpoint at low flow conditions, it is
possible to account for two different situations. First, the model is able to
describe a change in flow situation. In many cases, flow at low stages is
locally controlled (e.g., by one or several riffles). At higher stages, the
flow situation at these riffles becomes drowned and the flow is controlled by
a longer river reach e.g.,. Second, a
two-segment rating curve allows the effect of geomorphological changes
throughout time to be accounted for. If the river bed deepens, the value of
h0 in Eq. () is expected to decrease. It is thus possible
that in certain periods of the measured stage–discharge data, the stage of
zero flow is higher than the lowest measured stage within the complete period
and hence the sampling range of h0 (Sect. ) is too
narrow. For these periods, the use of a second segment can overcome this
shortcoming and the role of the first segment will thus be small(er).
Although both model structures are simple, this approach is satisfactory for
nearly all stations. By analyzing well-chosen subsets of the data (e.g.,
winter data if the influence of weed growth can be expected, as in the river
Grote Nete at Hulshout, Belgium) or by performing an analysis on the data
after sorting them by stage instead of chronologically (e.g., to assess the
influence of a downstream movable weir in the river Meuse at Maaseik,
Belgium), the chosen rating curve models also satisfy for less
straightforward flow situations (see
Sects. , and ).
The complex flow situation in the river Taf at Clog-y-Fran (UK), however,
requires a rating curve model with increased complexity. In this station, the
hydraulic behavior is influenced by the combination of weed growth affecting
low flow behavior, a considerable overspill over the right bank at higher
stages and an unstable bed control. For these reasons, a segmented rating
curve with three segments is used. The second segment overcomes similar
difficulties, as described for the two-segment rating curve. The third
segment is representing the flow for stages higher than bank overspill.
Generally, the choice of rating curve model should maximally be based on the
existing flow situation at the rating curve station. In case more complex
flow situations (e.g., hysteresis or backwater effects) can be observed and
described, it is possible to apply the BReach methodology with an adapted
rating curve model
e.g.,. In
case there is little or no knowledge of the flow situation, it is tempting to
use a rating curve model with multiple segments and wide sample ranges for
the breakpoints. If the amount of samples is sufficiently large, the
possibility of obtaining nearly identical values for the parameters of two
adjacent segments theoretically enables an excess of segments in
the chosen model to be eliminated. As shown in the example at Clog-y-Fran
(Sect. ), the parameter sets that result in a model
structure with the largest maximum reaches will be decisive for eventual
BReach results. This approach however involves the risk of overfitting the
model to the available gauging data, mainly in case of small and inconsistent
stage–discharge data sets. It is not implausible that in such a case of
sparse gauging data, eventual BReach results are obtained by a model
structure that is not capable of describing the real flow situation at the
site, but instead incidentally fits a series of consecutive gauging points
that belong not only to different height ranges but also to different
consistent periods. Therefore, and similar to in other rating curve
applications, the choice of an appropriate rating curve model should
preferably be based on a hydraulic analysis of the measurement site
.
It is important to mention that all decisions to be made in the BReach
methodology, such as the assessment of the measurement uncertainty
(Sects. and ) and of different degrees
of tolerance (Sect. ) are made independently of the
appropriateness of the chosen rating curve model. Despite the
methodology's ability to account for a limited model deficiency
(Sect. and ), this additionally
advocates a well-considered choice of a model structure.
Step 2: sampling of the parameter space
The sampling of the power law parameters (Eq. ) is
similar to , where sampling intervals are bounded to
a physically realistic order of magnitude; h0 is sampled from the
interval [hbed – 40 cm,
hmin,cont – 2 cm], where hbed is the
lowest bed level of the available reliable cross section measurements. If no
cross section data are available, it is the local datum toward which the
measured stages are expressed. The value of hmin,cont is the
lowest measured stage in the continuously measured data series. Samples of
n are taken from the interval [0.5, 3.5]. The outermost values obtained
when applying the power law function for all gauging points with the upper
and lower limits for h0 and n are used to define the sampling interval
for the coefficient c. The lower limit is obtained by halving the resulting
lowest value for c, and for the upper limit the highest value is doubled.
Both parameters h0 and n are sampled from a uniform distribution. For
parameter c, a more dense sampling is aimed at for smaller values. Hence,
this parameter is sampled from a uniform distribution after
log-transformation.
The two-segment rating curve (Eq. ) has seven parameters,
of which h1 is computed to obtain continuity between the two consecutive
segments of the rating curve. Sampling of h0, n0, n1, c0
and c1 is the same as for the single power law. The sampling interval
for hbr,1 is [hmin,cont – 2 cm,
hbr,max]. For all stations, the height range in which the
lowest flows occur is assessed visually from the stage–discharge plots. An
upper limit of this height range is estimated and taken as a value for
hbr,max (Table ).
Values for hbr,max.
Station
hbr,max
(m)
Taw Bridge
0.4
Mais
1
Barnett's Bank
3
Aarschot
10
Zichem
16.4
Diest
17.2
Hulshout
7.5
Maaseik
23.4
The three-segment rating curve used at Clog-y-Fran has 11 parameters, of
which h0, n0, n1, n2, c0, c1 and c2 are
sampled similarly to for the two-segment power law and h1 and h2
are again computed to obtain continuity between two consecutive segments.
Based on the stage–discharge data, the sampling interval for
hbr,1 is chosen [hmin,cont – 2 cm,
1 m]. Based on the information about out-of-bank flow at higher
stages, hbr,2 is sampled from the interval [2.9 m,
3.5 m]. Both parameters are sampled from a uniform distribution.
For all types of rating curves, the parameter space is sampled using a Latin
Hypercube sampling. For the single power law, 1.3×106 samples are
taken. For the two-segment and the three-segment power law, 6.5×106 and 1.3×107 samples are taken, respectively.
Step 3: assessment of acceptable model results
Following , a result of a rating curve model and a
parameter set is classified as acceptable if it fits in a rectangular
acceptance zone that is enclosed by the 95 % uncertainty boundaries of
the accompanying stage and discharge measurement.
An estimation of these measurement uncertainties is made by many authors. A
good literature overview with a summary of the major findings is given in
and in for both stage and
discharge measurements. In these studies, errors on stage measurements are
generally indicated as relatively small. Most of the estimated 95 %
uncertainty boundaries lie within ±10 mm, although values up to
±40 mm are also mentioned for more uncertain locations. Error
distributions are mostly assumed to have a negligible bias and to be
independent of the value of the measured variable (homoscedastic).
Discharge measurements are more uncertain and their errors are subjected to
heteroscedasticity, i.e., error distributions vary with changing discharge
values. Therefore, uncertainty on discharge measurements is typically
expressed as a percentage of the occurring discharge. In nearly all studies,
it is assumed to have a negligible bias. reports
95 % uncertainty boundaries between ±4 and ±17 % for 5–35
verticals with the velocity–area method. report the same
order of magnitude for the velocity–area method with various techniques.
found that despite expressing errors as a percentage of
occurring discharge, the value of the scale parameter in the assumed error
distribution depends on the value of the normalized discharge (i.e., measured
discharge divided by mean discharge), with 95 % boundaries up to
±25 % for low normalized flows and ±13 % for the highest
normalized flows.
Although most of these studies share some general considerations, eventual
uncertainties on stage and discharge measurements can depend upon location,
flow conditions and measurement technique, and hence estimated uncertainty
boundaries are subjected to a relatively large variation. Therefore, this
paper maximally uses available local information for the estimation of
observational uncertainty boundaries. Nevertheless, the ranges provided in
literature offer a valuable framework to validate these local findings. For
each country separately, an estimation of local uncertainty boundaries is
described in Sect. .
Based on the estimated observational uncertainty, results of a model (i.e., a
rating curve model with a sampled set of parameters) are categorized as
acceptable or nonacceptable. The result of this step is a binary matrix with
classification results for each parameter set and each data point.
Step 4: assessment of different degrees of tolerance
As mentioned in the introduction, the BReach methodology evaluates the
capacity of a rating curve model to describe a subset of the data in each
observation. For this evaluation, a definition of satisfactory behavior of a
rating curve model is necessary. In this paper, this definition is called the
degree of tolerance and expresses the percentage of model results that are
allowed to be nonacceptable in a sequence of data points.
discuss that possible causes of a rating curve
model being nonacceptable in a data point are (1) the occurrence of a higher
observational error than estimated for the definition of acceptable results,
(2) model deficiency in certain ranges of the investigated variables and
(3) data inconsistency. Due to the random occurrence in time of causes (1)
and (2), corresponding nonacceptable results tend to be singularities in a
chronologically sorted series of stage–discharge measurements. If, on the
contrary, a nonacceptable model result is caused by the occurrence of data
inconsistency, it can be expected that for the same model, nonacceptable
results will also occur in other, neighboring (in time) data points. Hence,
these causes of failure will be highlighted when using higher degrees of
tolerance (i.e., relaxation of the amount of points that can be nonacceptable
in a sequence of data). As different degrees of tolerance provide
complementary information, degrees of 0, 5, 10, 20 and 40 % are used for
this research.
Step 5: assessment of the bidirectional reach for all degrees of tolerance
Before assessing the bidirectional reach, all stage–discharge measurements
are sorted chronologically and their index within this sorted data series is
used to refer to them. Subsequently, a degree of tolerance is selected and
the binary classification matrix is used to evaluate a model and its results
from the perspective of one data point. The temporal span for which this
model behaves satisfactorily is assessed both in the direction of the previous
and the following data points using a directional search, that stops as soon
as the required conditions are not met. Within these spans, the index of the
outermost observation with an acceptable result is referred to as the left
(previous points) or right (following points) reach. This information is
aggregated for all parameter sets by taking the outermost left and right
reaches. They are called the maximum left and right reach and represent the
indices beyond which none of the sampled parameter sets is acceptable within
a data series with satisfactory behavior. Assessment of the maximum left and
right reach is repeated for all data points and for all degrees of tolerance,
and results are summarized in a combined BReach plot (e.g.,
Fig. a). In this plot, each gray tint represents results
for a specific degree of tolerance. For each data point on the x axis, the
gray zone represents the span between the index of the maximum left reach
(under the bisector) and the maximum right reach (above the bisector). The
vertical distance between the bisector and the index of the maximum left
reach represents the maximum amount of data points before the investigated
data point that can be described with at least one set of parameters under
the chosen degree of tolerance. Similarly, the vertical distance between the
index of the maximum right reach and the bisector represents this maximum
amount for the data points after the investigated data point.
Step 6: identification of consistent data periods
Combined BReach plots (e.g., Fig. a) provide a visual
means to evaluate the capability of the rating curve models to describe a
subset of the data in each point. Changes of this capability throughout time
result in discontinuities of a BReach plot, and each degree of tolerance
provides complementary information. In accordance with the discussion in
Sect. , discontinuities in the maximum reaches for stringent
degrees of tolerance provide information about the diversity of measurements
caused by (1) the occurrence of a higher observational error than estimated
for the definition of acceptable results, (2) model deficiency in certain
ranges of the investigated variables or (3) data inconsistency. The
resulting BReach plots show changes in model performance precisely, but
include too wide a variety of possible causes to detect data inconsistency.
For a higher degree of tolerance, a model is allowed to generate
nonacceptable results in a larger percentage of the data points. Therefore,
discontinuities caused by (1) and (2) will disappear from the plots due to
their random character. As a result, changes in consistency will be
emphasized in the plot, but the larger tolerance does not facilitate a precise
location of these changes. If plots that combine all degrees of tolerance
indicate consistent data periods (i.e., periods without important
discontinuities), plots with higher degrees of tolerance are used to assess
the amount and indicative locations of consistency changes, and, based on this
information, plots with stringent degrees of tolerance are used to locate
these possible consistency changes more precisely .
Alternative analyses
In this paper, a BReach analysis is performed for all stations. If a seasonal
variation in the rating curve behavior (due to weed growth) is presumed, a
second analysis is performed on a subset of data measured during winter
months (between December and March). In the UK and in Belgium, such a set of
winter data is not expected to be influenced by weed growth. The combination
of a BReach analysis on all data that shows no consistency and an analysis on
only winter data that indicates consistent periods can confirm the influence
of weed growth.
If it can be assumed that the behavior of the rating curve changes with
changing stages, an additional BReach analysis is performed. For the latter,
the data are sorted by stage instead of chronologically. Results of such an
analysis can reveal the height ranges in which the rating curve behavior alters.
As multisegmented rating curves aim to overcome these alterations, it is not
interesting to use them in this context. Therefore, a single power law is
used for all BReach analyses on data sorted by stage.
To avoid confusion between both a temporal BReach analysis and an analysis on
data sorted by stage and between several types of rating curve models,
results of the analyses will be referred to as BReachx_ys. In
this formulation, x is the type of analysis (t, on chronologically sorted
data, or h, on data sorted by stage) and y is the amount of segments in
the chosen rating curve model.
Assessment of uncertainties on stage and discharge measurements
The assessment of 95 % uncertainty boundaries of the stage and discharge
data is based on available local information. This information availability
differs for each country and hence, in this section, the followed approach is
described for each country.
(a) ECDF and corresponding KS statistic of both
ΔQm,% and ΔQϵ,% with μ=0 % and σ=4 %. (b) KS statistics with μ=0 % and different values for σ. (c) ECDF of both
ΔQm,% and ΔQϵ,% with μ=0 % and σ=3.12 %. All plots are made for low flow data and
a Gaussian error distribution.
UK measurement stations
For the UK stations, have analyzed the relative rating
curve residuals from 26 measurement stations with very stable rating curves.
A relative residual is defined as the ratio of the deviation (between
discharge measurement and derived rating curve) and the measured discharge.
The distribution of these residuals is investigated for different bins of
normalized flow Qn (i.e., measured flow divided by mean flow).
Results of this investigation show that logistic distributions with a zero
location parameter (i.e., μ=0) and a scale parameter (σ) that
varies exponentially with normalized discharge (Eq. ) fit the
residuals well for all bins.
σ=4.18e(-3.051Qn)+3.531
The 95 % uncertainty boundaries of discharge measurements for the UK data
used in this paper are derived from these distributions and vary between
±28 % for the lowest normalized flows and ±13 % for the
highest normalized flows. For stage measurements, that typically have smaller
measurement errors than discharges, a uniform error of ±5 mm was
assumed by . Again, 95 % boundaries of this error
(±4.875 mm) are used for the definition of the acceptance zone in
the BReach methodology.
New Zealand measurement stations
In , the uncertainty on measured discharges in the
measurement station of Barnett's Bank is assumed to follow a Gaussian
distribution with zero mean (μ) and a standard deviation (σ) of
4 %. Errors on stage measurement are considered Gaussian with zero mean
and a standard deviation of 2 cm. Uncertainty boundaries of 95 %
are thus ±8 % for discharges and ±4 cm for stages. These
estimations are based on literature data and local expertise.
However, assume error distributions for Barnett's Bank
similarly to those described in Sect.
. In this case, 95 % uncertainty boundaries for
discharge measurements vary again between ±28 % for the lowest
normalized flows and ±13 % for high normalized flows and are thus
substantially higher than in the abovementioned approximation with a normal
distribution. Stage uncertainty boundaries, on the contrary, are estimated
smaller by (±4.875 mm versus
±4 cm). Therefore, two different BReach analyses are performed
for all New Zealand data, each based on one of these uncertainty estimations,
and results are compared.
Belgian measurement stations
For the Belgian measurement stations, no prior information concerning
measurement uncertainties was available. Nevertheless, it is possible to gain
insight into plausible characteristics of measurement errors by analyzing
simultaneous measurements. Although, in this paper, a BReach analysis is
performed on only five Belgian measurement stations, simultaneous
measurements of nine different stations are used for a preliminary
uncertainty assessment of discharge measurements in order to maximize the
amount of (scarce) data.
A pair of simultaneously measured discharges consists of two discharge
measurements that are measured with the same type of device within a time
span of 2 h and for which the corresponding measured stages are identical.
Combining this information for nine Belgian stations results in a set of 42
simultaneous pairs that are all measured with an OTT QLiner. The restriction
to only one type of measurement device prevents a mixture of possibly
different error distributions, each corresponding with a different
measurement technique. The errors of two simultaneous measurements are
assumed to be independent.
To overcome the heteroscedastic character of discharge measurement errors,
they are expressed as a percentage of the real discharge. Nevertheless,
different authors find that parameters of error distributions change with
changing discharges e.g.,. To investigate
this, the simultaneous discharge measurements are sorted according to their
normalized discharge (see Sect. ). Subsequently, two
subsets of this data set are created, containing the 21 lowest and highest
pairs of measurement. They are referred to as low flow and high flow data.
These subsets are assumed to be unbiased (error distribution with zero mean,
see Sect. ).
Neither data set allows for a direct assessment of measurement errors.
However, if an error distribution is assumed, it is possible to test equality
between the distributions of both the relative differences of the
simultaneously measured discharge pairs and a created set of relative
differences based on two equally sized samples of measurement errors from the
assumed distribution. For instance, a Gaussian measurement error with zero
mean and a standard deviation of 4 % is assumed for the low flow data set
(see Sects. and ). From this
distribution, two samples ϵ1 and ϵ2 are taken, each
with size m (in this paper m=106), and they pairwise represent the
assumed errors of two simultaneous flow measurements. As these errors are
expressed as a percentage of the real discharge, a measurement (for both
j=1 and j=2) can be written as follows:
Qmeas,j,i=1+ϵj,iQtrue,i,
with i∈[1,m], Qmeas,j,i one of both measured
discharges in measurement pair i and Qtrue,i the real
discharge that occurred during the measurements. Combining
Eq. () for both measurements in a pair leads to the following:
Qmeas,1,i-Qmeas,2,iQmeas,1,i=ϵ1,i-ϵ2,i1+ϵ1,i.
Independent of the real discharge, this relative difference of two
simultaneous measurements can thus be expressed by their measurement errors.
If the assumed error distribution (Gaussian, μ=0 % and σ=4 %) is correct, a data set calculated from the measurement pairs using
the left-hand side of Eq. () (further called
ΔQm,%) will have the same distribution as a data set
calculated from the two sets of sampled errors using the right-hand side of
Eq. () (further called
ΔQϵ,%). When applying a two-sample nonparametric
Kolmogorov–Smirnov (KS) test on these data sets, the resulting p value
is 0.62, which is much higher than the commonly used 5 % level for
rejection of the hypothesis that both data sets are equally distributed. The
corresponding value of the Kolmogorov–Smirnov statistic is
0.16. This is the maximum vertical distance between the empirical cumulative
distribution functions (ECDFs) of both tested data sets (Fig. a).
The same analysis is repeated for both low and high flow data and for
Gaussian and logistic error distributions with different values of the scale
parameters, equidistantly taken from the interval [1 %, 6 %] and
[0.35 %, 4.4 %], respectively. As an example,
Fig. b shows the resulting values of the KS statistic
against the corresponding value of the scale parameter for the low flow data
set using a Gaussian distribution. A p value resulting from a KS test
depends both on the value of the KS statistic and on the number of points in
the investigated data sets. As the latter remains constant for all tests,
p values and values of the KS statistic will show a similar (although
inverse) pattern. Figure b clearly shows the occurrence
of the lowest value of the KS statistic (and corresponding highest p value)
for a standard deviation of 3.12 %. In Fig. c, the
ECDF of ΔQm,% corresponds well with the ECDF of
ΔQϵ,% for this latter distribution. However, the
occurrence of a high p value (and corresponding small value of the KS
statistic) provides no confirmation of the null hypothesis, and it is possible
that many other hypotheses lead to similar p values. Nevertheless, the
value of the KS statistic provides information not only about differences in
central tendency but about any difference in the ECDFs. From this
perspective, , and
compared the ECDFs of both behavioral and
nonbehavioral parameter values and used the KS statistic as a measure for the
sensitivity of a parameter. In this research, it is used to evaluate the
behavior of error distributions that are a priori chosen based on currently
available knowledge, without excluding the plausibility of other, unexplored
error distributions.
Kolmogorov–Smirnov test results (simultaneous discharge
measurements).
Data set
Error distribution
Minimum
Maximum adjusted
Scale
95 % uncertainty
type
KS statistic
p value
parameter
boundaries
(–)
(–)
(%)
(%)
Low flow data
Gaussian distribution
0.14
0.79
3.12
±6.12
Low flow data
Logistic distribution
0.14
0.79
1.80
±6.59
High flow data
Gaussian distribution
0.13
0.82
1.90
±3.72
High flow data
Logistic distribution
0.13
0.81
1.10
±4.02
Low flow data have values for Qn between 0.72 and
3.64 and high flow data have values for Qn between 3.72 and
8.41.
The pattern of the other results (Gaussian distribution with high flow data,
logistic distribution with both low and high flow data) is very similar.
Table shows the characteristics of both flow classes and
both distribution types that correspond with a minimum KS statistic. There is
a good correspondence between the ECDFs of ΔQm,% and
ΔQϵ,% for both distribution types. Adjusted
p values (i.e., p values after Benjamini–Hochberg correction, that
accounts for a false discovery rate; ) and KS
statistics are also very similar and hence prohibit making a distinction in
favor of one single distribution type. Like in other studies
(Sect. ), this table clearly indicates that high flow data
correspond with lower values of the scale parameters (and thus smaller
uncertainty boundaries) than low flow data. For each flow class, the 95 %
uncertainty boundaries of the two distributions do not differ strongly, but
they are relatively small compared with the uncertainty boundaries applied in
Sects. and and with
literature data e.g.,. A tentative
explanation for these low uncertainty values could be the relatively tranquil
flow situations in the investigated measurement stations due to low slopes.
Moreover, most of the investigated locations are situated at a bridge,
facilitating discharge measurements in controlled conditions.
The limited amount of data prohibits a more precise description of this
tendency toward lower uncertainties for higher normalized discharges.
Moreover, the lowest normalized flow in the set of simultaneous discharge
measurements is 0.72. Results of show that an increase of
measurement uncertainties can be expected for lower normalized flows. As more
than 80 % of all investigated Belgian stage–discharge data have
normalized discharges within the range of the low flow data subset or lower,
it was decided to assume 95 % uncertainty boundaries to be ±6.4 %
for all investigated Belgian discharge measurements. Although these values
originate from the investigated low flow data measured with QLiners, they are
applied for all discharge measurements, independent of their measurement
technique. It can be expected that discharge uncertainties will differ for
different techniques e.g.,, but
a lack of simultaneous discharge measurements prohibits a similar
analysis for other measurement devices. Extra measurement campaigns might
augment insight for these devices.
For the assessment of uncertainties on stage measurements, the data
availability is different. Two simultaneous stage measurements are provided
for each stage–discharge measurement. However, the first type of measurement
is recorded from a staff gauge during a discharge measurement and the second
type is registered by a continuous measurement device. Hence, it can be
expected that error distributions of both data types differ and a similar
approach to that for discharge measurements is not justified. Therefore, 95 %
uncertainty boundaries are estimated to be ±2 cm. This value is
based on literature data and on local expertise.
(a) Combined BReacht_1s plot (all data) for
Colsterworth. For each index on the x axis the gray area
indicates the span between the index of the maximum left reach (under the
bisector) and the maximum right reach (above the bisector). Each gray tint
represents a different degree of tolerance (i.e., percentage of data points
allowed to have nonacceptable model results). (b) Stage–discharge
data for Colsterworth.
Residual analysis as a benchmark
In order to benchmark the results of the BReach analyses, a residual analysis
is performed for several of the investigated measurement stations. An
analysis of the relative deviations from an “average” rating curve is
frequently used in operational hydrology, as their behavior can be used as an
indication of the stability of a measurement station or of a shift in the
rating curve e.g.,.
The performed analysis is based on a set of parameters that results from the
minimization of the root mean square error (ERMS) of the chosen
rating curve model in all data points (which is further referred to as “the
ERMS optimized model”). This approach assumes that relative
errors of discharge measurements are homoscedastic and follow a Gaussian
distribution . At the
stations that are selected for this analysis (Maaseik, Aarschot and Barnett's
Bank), similar conditions are assumed when assessing discharge measurement
uncertainty boundaries (Sect. ).
Results and discussion
For each measurement station, results of the BReach analyses are validated
using the available local information.
River Witham at Colsterworth, UK
In Fig. a, a combined BReacht_1s plot is
shown for Colsterworth. The BReach plot shows data consistency during the
entire measured period. This corresponds with the nature of the measurement
station, a flat V weir with a stable stage–discharge relationship. Even a
0 % degree of tolerance shows no discontinuities within the entire data
period. Figure b shows the available stage–discharge
measurements at Colsterworth.
(a) Combined BReacht_2s plot (all data) and
(b) combined BReachh_1s plot (data between November
1998 and August 2012) for Taw Bridge. In all subplots, for each index on the x axis the gray area indicates the span between the index of the maximum
left reach (under the bisector) and the maximum right reach (above the
bisector). Each gray tint represents a different degree of tolerance (i.e.,
percentage of data points allowed to have nonacceptable model results).
(c) Stage–discharge data for Taw Bridge.
River Taw at Taw Bridge, UK
In Fig. a, a combined BReacht_2s plot is
shown for Taw Bridge. In this plot, time instants near the peak discharges
(return period ≥ 2 years) are indicated with a red mark on the
bisector. This plot shows changes in data consistency that correspond with
historical information. An important change in consistency occurs at index
299 and stage–discharge points after this time instant are likely to belong
to one single consistent period. This starting point corresponds with the
moment of installation of a flat V weir (October 1998). Before this date, the
data series shows many discontinuities, also for higher degrees of tolerance.
The time instants of these discontinuities often coincide with those of the
highlighted peak floods. Hence, the plot suggests that these flood events
caused geomorphological changes of the river bed that induced changes in
consistency and that periods in between were relatively stable.
The English Environment Agency uses a segmented power law to assess
discharges in this measurement station. Rating curve changes generally imply
changes of the rating curve coefficients for the lowest and medium flows. The
time instants of these official changes are indicated with cyan lines that
depart from the bisector. If the change involves also the flood rating curve,
an asterisk and (if available) some background information is added to the
date indication. Although many of the rating curve changes correspond with
discontinuities, the BReach plot sometimes suggests different or fewer moments
of change.
In Fig. b, results of a BReachh_1s analysis
on the stage–discharge data measured after installation of the weir is shown.
As can be expected, the plot shows consistency for nearly the complete height
range. Only for the highest stages and the lower degrees of tolerance, some
discontinuities occur in the plot. Figure c shows the
available stage–discharge measurements at Taw Bridge. Data measured after
installation of the weir are indicated separately.
Combined (a) BReacht_3s plot (winter data),
(b) BReacht_1s plot (winter data) and
(c) BReacht_2s plot (winter data) for Clog-y-Fran. In
all subplots, for each index on the x axis the gray area indicates the span
between the index of the maximum left reach (under the bisector) and the
maximum right reach (above the bisector). Each gray tint represents a
different degree of tolerance (i.e., percentage of data points allowed to have
nonacceptable model results).
River Taf at Clog-y-Fran, UK
In Fig. a and a, combined
BReacht_3s plots based on only winter data, and all
data are shown for Clog-y-Fran. Although the plot with all data
(Fig. a) indicates many discontinuities, the maximum
reaches of the more tolerant degrees cover a large part of the data set for
several points. They are sometimes alternated by data points with more
limited reaches. The plot based on only winter data
(Fig. a) indicates larger consistent blocks. In this
latter plot, time instants near the peak discharges (return period
≥ 5 years) are indicated with a red mark on the bisector. The time
instants of the discontinuities often coincide with those of the highlighted
peak floods. Hence, the plots do not only confirm the influence of weed
growth, but also suggest that high flood events cause geomorphological
changes of the river bed that induce changes in consistency and that periods
in between often are relatively stable. Nevertheless, not all floods in
Fig. a cause discontinuities and not all discontinuities
can be linked with the occurrence of large floods. Besides erosion due to
large floods, the cross section is also known to be prone to the (more
gradual) build-up of silt. This and other unknown processes might influence
the result of this BReach analysis to some extent. Natural Resources Wales,
who manage this gauging station, use a segmented power law to assess
discharges in this measurement station. In Fig. a, the
available time instants of these official changes are indicated with cyan
lines that depart from the bisector. Although many of the rating curve
changes correspond with discontinuities, the BReach plot sometimes suggests
different or fewer moments of change.
Although the flow situation in Clog-y-Fran is complex, the available
information about the station can be linked with results of a BReach
analysis. These results indicate the need for an in-depth analysis that
should lead to an appropriate modeling approach for periods with weed growth.
For the remaining (winter) data, an assessment of consistent periods is
possible. However, the choice of an appropriate rating curve model is crucial
for success. Figure b and c show
results of a BReacht_1s and a
BReacht_2s analysis on winter data in Clog-y-Fran, respectively. The
two-segment rating curve has only a breakpoint at the stage of overspill
over the right bank. These two figures do not mutually differ a lot, showing
that a difference in the rating curve model that affects higher flows has a
minor effect on eventual BReach results. This corresponds with earlier
results of based on synthetic data. However,
comparison of Fig. b and c with Fig. a
reveals that BReach results alter importantly when adding an extra segment
(with a breakpoint at low stages) to the rating curve model. In all other
stations where a segmented rating curve (two segments) is used, there is a
more limited or even a negligible difference with BReach results resulting
from a simple power law. Therefore, it is plausible to assume that in
Clog-y-Fran, the flow situation changes from a locally controlled flow (e.g.,
caused by a riffle affecting the lowest flows) towards a flow situation
controlled by a longer river reach for higher flows. It is plausible that
this importance of an appropriate modeling of low flow stage–discharge
relations on BReach results corresponds with a higher distinctive capacity of
these data toward temporal consistency.
(a) Combined BReacht_3s plot (all data) and
(b) combined BReachh_1s plot (winter data) for
Clog-y-Fran. In all subplots, for each index on the x axis the gray area
indicates the span between the index of the maximum left reach (under the
bisector) and the maximum right reach (above the bisector). Each gray tint
represents a different degree of tolerance (i.e., percentage of data points
allowed to have nonacceptable model results). (c) Stage–discharge
data for Clog-y-Fran.
In Fig. b, results of a BReachh_1s analysis
of the winter data is shown. The plot shows a relatively large consistency
for stages above index 172 (1.33 m). This is linked with the
influence of geomorphological changes on river stages, that is expected to
decrease for increasing discharges due to the corresponding increase of the
conveyance of the river cross sections. For high discharges, the order of
magnitude of these influences will not exceed the width of the observational
uncertainty boundaries anymore and will thus result in more consistent BReach
results. Figure c shows the available stage–discharge
measurements at Clog-y-Fran. Winter data are indicated separately.
In the stage–discharge data of Clog-y-Fran (Fig. c), four
gaugings have discharge values that are smaller than 50 % of all other
available discharge measurements with a similar stage. These four
observations are all measured on the same day and there is no indication of
similar deviations in the months before and after this date. Although it is
plausible that these deviations are caused by an erroneous registration of
the discharge, there was not enough information to consider these gaugings as
outliers. These data occur near the end of the time series (January 2007) and
have only a minor effect on the BReach results.
River Pohangina at Mais, New Zealand
In Fig. a, a combined BReacht_2s plot based on
measurement uncertainties as applied in is shown for
Mais. In this plot, time instants near the highest measured stages (return
period ≥ 1 year) are indicated with a red mark on the bisector.
Throughout the whole data set, many discontinuities occur in the plot. The
time instants of these discontinuities often coincide with those of the
highlighted peak floods. Hence, this plot confirms that, in this gravel-bed
river, most of these flood events cause geomorphological changes of the river
bed that induce changes in consistency and that periods in between are
relatively stable.
The Horizons Regional Council interpolates rating curves based on
stage–discharge measurements. As these interpolations are changed up to a few
times a year, it is not informative to plot these official rating curve
changes on the BReach plot.
Combined BReacht_2s plot (all data) for Mais using
uncertainty boundaries from (a) McMillan et al. (2010) and
(b) Coxon et al. (2015). In all subplots, for each index on the x axis the gray area indicates the span between the index of the maximum
left reach (under the bisector) and the maximum right reach (above the
bisector). Each gray tint represents a different degree of tolerance (i.e.,
percentage of data points allowed to have nonacceptable model results).
(c) Stage–discharge data for Mais.
Figure b is a combined BReacht_2s plot based
on measurement uncertainties described by . There is a high
resemblance with Fig. a and general conclusions are
identical. There are a few reasons for this high resemblance. First,
show that a limited misjudgment of observational
errors does not alter the conclusions of a BReach analysis fundamentally.
Moreover, the classification of results of a rating curve model as acceptable
or nonacceptable is based on the assessed uncertainties on both stage and
discharge measurements (see Sect. ). As mentioned in
Sect. , uncertainty boundaries for discharge
measurements in are substantially larger than in
while stage uncertainty boundaries are smaller. These
opposite differences average the final results. Figure c
shows the available stage–discharge measurements at Mais.
Combined BReacht_2s plot (all data) for Barnett's Bank
using uncertainty boundaries from (a) McMillan et al. (2010) and
(b) Coxon et al. (2015). In all subplots, for each index on the x axis the gray area indicates the span between the index of the maximum
left reach (under the bisector) and the maximum right reach (above the
bisector). Each gray tint represents a different degree of tolerance (i.e.,
percentage of data points allowed to have nonacceptable model results).
(c) Stage–discharge data for Barnett's Bank.
Combined BReacht_2s plot (all data) for Aarschot. For
each index on the x axis the gray area indicates the span between the index
of the maximum left reach (under the bisector) and the maximum right reach
(above the bisector). Each gray tint represents a different degree of
tolerance (i.e., percentage of data points allowed to have nonacceptable model
results).
River Wairau at Barnett's Bank, New Zealand
In Fig. a and b, combined BReacht_2s plots are
shown for Barnett's Bank, with measurement uncertainties of
and of , respectively. Again, both
plots are very similar and general conclusions are identical.
Figure c shows the available stage–discharge measurements
at Barnett's Bank.
In Fig. a, time instants near the highest measured stages
(return period ≥ 0.5 years) are indicated with a red mark on the
bisector. suggest a 0.5-year return period as a
threshold that induces consistency changes in this gravel-bed river. This is
partly confirmed in the BReach plot, in which discontinuities often (but not
always) coincide with the highlighted peak floods that cause geomorphological
changes of the river bed. Periods in between these consistency changes are
relatively stable. The Marlborough Regional Council interpolates rating
curves based on stage–discharge measurements. As these interpolations are
changed up to a few times a year, it is not informative to plot these
official rating curve changes on the BReach plot.
River Demer at Aarschot, Zichem and Diest, Belgium
In , stage–discharge measurements in the river
Demer at Aarschot are used to validate the BReach methodology. Although in
the current research, a different rating curve model is used and
observational uncertainties are assessed slightly different (see
Sects. and ), the resulting
BReacht_2s plot (Fig. ) shows similar
results and indicates no consistency before index 29 (August 1982) due to a
deepening and widening of the river's cross section and a heightening of the
river dikes. After that time instant, a more consistent period starts that
lasts until index 233 (February 2005). During the last decade, the data show
again a lack of consistency. This is possibly a joint effect of the
occurrence of large floods, the introduction of new measurement devices and
local maintenance works that affect the cross section of the river bed
. In this figure, time instants near the highest
measured stages (return period ≥ 5 years) are indicated with a red mark
on the bisector.
Combined BReacht_2s plot (all data) for
(a) Zichem and (b) Diest. In all subplots, for each index
on the x axis the gray area indicates the span between the index of the
maximum left reach (under the bisector) and the maximum right reach (above
the bisector). Each gray tint represents a different degree of tolerance
(i.e., percentage of data points allowed to have nonacceptable model
results).
As data are available in two other measurement stations on the river Demer, a
comparison between the results of these stations is interesting.
Figure a shows combined BReacht_2s results
at Zichem, situated 16 km upstream of Aarschot. Moments near the highest
measured stages (return period ≥ 5 years) are indicated with a red mark
on the bisector. In Zichem, an important change in consistency is shown at
index 72 (December 1988). This corresponds with historical information. In
1988, the river bed near Zichem was deepened and widened and the dikes were
heightened, causing the detected consistency change. Before that time
instant, the plot shows several discontinuities that possibly suggest changes
in consistency. Unfortunately, it was not possible to verify these changes
due to a lack of information about this time period. After 1988, the plot
suggests the start of a new consistent period until index 144 (March 2008).
However, it is difficult to pinpoint the end of this second consistent period
precisely. In Zichem, the stage–discharge measurements of March 2008 are the
first available measurements since October 2002 and thus this change may
already be situated within this period. Since then, the stage–discharge data
show nearly no consistency. Again, it is likely that this is a joined effect
of several different causes (occurrence of floods, change of measurement
device, deviation of the mouth of a small tributary at the location of the
measurement station, occasional observations of weed growth in the river).
Stage–discharge data for (a) Aarschot, (b) Zichem
and (c) Diest.
Figure b shows combined BReacht_2s results
in Diest (5 km upstream of Zichem). Moments near the highest measured stages
(return period ≥ 5 years) are indicated with a red mark on the bisector.
In this station, only 34 stage–discharge gaugings measured during 1 decade
are available. Nevertheless, a similar tendency to that in the recent data of
Aarschot and Zichem can be noticed in the plot. The data are consistent until
index 24 (March 2008). Again, it is plausible that this consistency change is
linked with the occurrence of peak discharges, with a change in measurement
device and with the occasional occurrence of weed in the river bed.
The Flemish Hydrological Information Centre uses a segmented power law to
assess discharges in this measurement station. In Figs.
and a, b, the time instants of these official changes of
the rating curves are indicated with cyan lines that depart from the
bisector. Many of the rating curve changes correspond with discontinuities or
with the start of a year with a major flood. Nevertheless, the BReach plot
sometimes suggests different moments of change. In
Fig. a–c, a plot of the available stage–discharge
measurements are given for Aarschot, Zichem and Diest. These plots show that
for low stages in Aarschot, recently measured discharges (black) are higher
than discharges during the long consistent period (red). In Zichem and Diest,
however, recent discharges tend to be smaller. This latter effect is possibly
caused by the observed weed growth in these two stations.
Combined BReacht_2s plot with (a) all data and
(b) only winter data for Hulshout. In all subplots, for each index
on the x axis the gray area indicates the span between the index of the
maximum left reach (under the bisector) and the maximum right reach (above
the bisector). Each gray tint represents a different degree of tolerance
(i.e., percentage of data points allowed to have nonacceptable model results).
(c) Stage–discharge data for Hulshout.
River Grote Nete at Hulshout, Belgium
Figure a shows a combined BReacht_2s plot
for Hulshout. Although the plot indicates no consistent periods, the maximum
reaches of the most tolerant degree cover almost the complete data set for
several data points. They are alternated by data points with very limited
reaches. Figure b shows a combined
BReacht_2s plot of the winter data in Hulshout. For this subset
of data, the plot indicates a high consistency for almost the complete
period. These results indicate the influence of weed growth and the need for
an in-depth analysis that should lead to an appropriate modeling approach for
periods with weed growth. Figure c shows the available
stage–discharge measurements at Hulshout. Winter data are indicated
separately.
Although the data set is limited to only 38 points, BReach results offer
insight into the situation of the measurement station. However, it is likely
that a more elaborate data set will result in more robust conclusions.
(a) Combined BReacht_2s plot (all data),
(b) BReachh_1s plot (all data) for Maaseik. In all
subplots, for each index on the x axis the gray area indicates the span
between the index of the maximum left reach (under the bisector) and the
maximum right reach (above the bisector). Each gray tint represents a
different degree of tolerance (i.e., percentage of data points allowed to have
nonacceptable model results). (c) Stage–discharge data for
Maaseik.
River Meuse at Maaseik, Belgium
In Maaseik, BReacht_2s plots of all data points with high
degrees of tolerance (Fig. a) show an alternation of data
points with nearly no reach and data points that have maximum reaches that
cover a large part of the data set. In Fig. b, results of
a BReachh_1s analysis on the same stage–discharge data are
shown. This plot shows no consistency for the lower stages, but indicates a
relatively high consistency for stages beyond index 31 (23.46 m).
This corresponds with the local situation in Maaseik. Stage–discharge
measurements at lower stages are influenced by the downstream movable weir in
Linne. For higher stages, this influence is smaller. Moreover, the effect of
dredging the river bed and of the installed guiding dam on the occurring
stages decreases for increasing discharges due to the corresponding increase
of the conveyance of the river cross sections. For high discharges, the order
of magnitude of these influences will not exceed the width of the
observational uncertainties boundaries anymore and will thus result in more
consistent BReach results. An in-depth analysis should lead to an appropriate
modeling approach for low flow data. Figure c shows the
available stage–discharge measurements at Maaseik.
Combined BReacht_2s plot (all data) and relative
residuals of the ERMS optimized rating curve for
(a) Maaseik, (c) Aarschot and (d) Barnett's Bank.
Combined BReachh_1s plot (all data) and relative residuals of
the ERMS optimized rating curve for Maaseik. In all BReach plots,
for each index on the x axis the gray area indicates the span between the
index of the maximum left reach (under the bisector) and the maximum right
reach (above the bisector). Each gray tint represents a different degree of
tolerance (i.e., percentage of data points allowed to have nonacceptable model
results).
Results of the residual analysis
In Fig. , results of the BReach analysis are plotted
together with the relative residuals of the ERMS optimized model
for Maaseik, Aarschot and Barnett's Bank.
In Maaseik, data points with limited maximum reaches in the temporal BReach
plot correspond with points with more extreme values for the residuals
(Fig. a). When sorting the stage–discharge data along
height, the residuals show the same pattern as the BReach plot
(Fig. b) with large absolute values and a high variability
of the residuals (and thus no data consistency) for low stages and small
absolute values and lower variability (and thus large consistency) for higher
stages. In both subplots, the two approaches thus provide comparable
information.
Also, in the results for Aarschot (Fig. c), a period that is
indicated as consistent in the BReach results corresponds with smaller
absolute values and a lower variability of the residuals, while inconsistent
periods coincide with larger absolute values and a high variability of the
residuals. Again, the information content of both methods can be compared.
In the station of Barnett's Bank, however, both approaches show a different
amount of information (Fig. d). This station is subjected to
many geomorphological changes that are mainly caused by floods. The BReach
results suggest the existence of different consecutive consistent periods and
provide information about the floods that are situated at discontinuities in
the plot (and thus probably related to an important change in the river's
geometry). The plot with residuals, on the contrary, does not provide clear
periods with small absolute values and low variability. The reason for this
lack of information is the general character of the ERMS
optimized model, that is fitted to the complete data set. If a data set
mainly consists of a long consistent time period (as in Aarschot), the model
fit will be dominated by this period and thus residuals in this period will
be small. In case the data set consists of different consecutive situations
that mutually differ (as in Barnett's Bank), this general fit will be
insufficient to meet the characteristics of individual consistent time
periods, and a residual plot will thus be uninformative. The approach of the
BReach methodology, that evaluates the performance of a chosen model from the
perspective of each data point separately, does not suffer from this
generalization and is thus capable of revealing these smaller consistent
periods.
General considerations regarding the use of the BReach methodology
In this section, some general thoughts about the use of the BReach
methodology for rating curve data are given. It is obvious that the quality
of results is related with the gauging frequency of the stage–discharge data.
The stations analyzed in this paper vary from densely measured (up to a mean
amount of 22 gaugings a year) to rather poorly measured (2 gaugings a year).
Stations with a more complex flow situation are measured more frequently. In
many cases, local hydrological services decide to apply a similar
differentiation in the gauging frequency that depends on the station's
complexity. Based on the available data, it was possible to recognize the
history and characteristics of each analyzed station in the BReach results.
Nevertheless, it is difficult to pinpoint a minimum required gauging
frequency to guarantee a successful application. If a large time gap occurs
in the measured data, this can introduce uncertainty about the exact moment
of a consistency change. In extreme situations, a temporary change can even
disappear from the data, resulting in a (misleading) apparently consistent
period. The bar (with indication of the years) above a BReach plot permits
detection of these noninformative periods. If more detail is wanted, it can be
interesting to create an additional BReach plot in which the absolute time is
used in both axes (and thus the indices used in the current plots are
projected on these time axes) and with an indication of the moments of the
available gaugings on the bisector.
Results of show that stage–discharge data for
higher stages have a smaller distinctive capacity in the BReacht
analysis. This corresponds with results of , who
confirm the validity of one single flood rating curve throughout a period
with different geometric situations (affecting the rating curve for lower
flows). A limited deficiency of the rating curve model for the highest flows
leads to satisfying BReach results as the effects of the model deficiency
disappear from the plots with higher degrees of tolerance
. In the current paper, results in Clog-y-Fran
(Sect. ) confirm these findings. It is, however, important
to emphasize that these results are site-specific and are expected to depend
on the extent to which the higher parts of the cross section contribute to
changes in the flow situation. On the contrary, a model deficiency in a
height range that contributes significantly to changes in the flow situation
will lead to important changes in BReacht results. This is shown
in this paper for low stages at Clog-y-Fran (Sect. ).
In any case, it is necessary to be informed about the specific situation of
the analyzed rating curve station. Not only is it important for an adequate
choice of a rating curve model, it is also required for a correct
interpretation of the BReach results and the design of possible alternative
BReach analyses (Sect. ). For instance, it would not be
possible to distinguish between the BReacht results of all
available data at Hulshout and Maaseik (Sect.
and ) without any knowledge of the local situation.
The computational load of the BReach methodology depends on several aspects.
First, it increases linearly with the size of the sample of the parameter
space (and is thus larger for more complex rating curves). Second (and more
important), the necessary calculation time strongly depends on both the
amount of stage–discharge data points and the degree of consistency of the
data set. The principle of the BReach algorithm is that for each data point,
a maximum left and right reach must be searched. If a data set is highly
consistent, the length of these searches increases significantly. Doubling
the amount of data points can (for consistent data sets) hence result in
8 times the original calculation time. In the research for this paper,
all calculations are performed on a personal computer with a 3.4 GHz CPU
Core I7 and 8 GB RAM. For most stations, a BReach analysis took a few
minutes to a few hours. In the most complex case (Clog-y-Fran, with 1166 data
points and 1.3×107 samples), calculation of BReach results required
72 h.
At the moment, interpretation of BReach results is done manually by the
user. The availability of a (semi)automatic routine that identifies possible
consistent data periods would improve the BReach methodology. As the degree
of squareness of a BReach plot within a certain period expresses the lack of
important discontinuities, it might play a role in the decision process for
assessing consistent periods.
Conclusions
The objective of this paper was to test the BReach methodology
for assessing temporal consistency in rating curve data on various
stage–discharge data set in the UK, New Zealand and Belgium. This led to
successful results for all tested sites.
For each country, local information is maximally used to estimate
observational uncertainties that serve as an input for the methodology. In
this context, a new approach is proposed for the Belgian data using relative
differences between simultaneous discharge measurements to test the
plausibility of several a priori assumed error distributions. This approach
offers promising insights in the plausible character of measurement error
distributions in addition to a more general use of existing literature data
about observational uncertainties. However, the limited size of the data set
with simultaneous measurements is an important restriction. In order to
investigate the possibilities of the proposed approach more profoundly, a
more elaborate data set with large spread in time, measurement stations,
measurement device and flow conditions is necessary. Such an enlarged data
set would not only increase the reliability of a KS test, but would also
enhance the possibility to use more bins with smaller ranges of normalized
discharge (replacing the current two arbitrary subgroups) and to investigate
other measurement devices.
Overall, results of the BReach analyses correspond with site-specific
situations. Nevertheless, the investigated cases show that knowledge about
the local situation of a measurement station is crucial to design the
necessary BReach analyses and to interpret their results correctly. Results
show consistency in locations that are known as stable. Where human
interventions (e.g., installation of a weir, deepening of a river) altered
the rating curve behavior, results show corresponding consistency changes. In
locations influenced by weed growth, a higher consistency can be assessed
after isolating winter data. Similarly, consistency can be assessed for
higher stages in a station where a downstream weir influences low flow
behavior. Stations that are prone to geomorphological changes caused by flood
events show discontinuities in the BReach plots at the time instants of the
highest floods. Moreover, the plots can also indicate which peak floods do
not cause consistency changes. The return period that serves as a threshold
for consistency changes varies from station to station. These results provide
extra insight into the rating curve behavior and confirm the added value of
the proposed BReach methodology as a preliminary assessment of data
consistency prior to an in-depth determination of discharges and their
uncertainty. Moreover, this assessment of (in)consistent periods can enhance
other applications based on the investigated data (e.g., by informing
hydrological and hydraulic model evaluation design about consistent time
periods to analyze).
A comparison between the results of both a residual analysis and a BReach
analysis shows that the latter mainly provides additional information in case
of a data set that consists of different, consecutive consistent time periods
that mutually differ.
In the BReach methodology, the chosen rating curve model is required to
appropriately approximate the relation between discharge and stage for an
important part of the measured range. In this paper, analyses with only a
subset of the data or with stage–discharge data sorted by stage
(BReachh) enable a part of a known model deficiency to be overcome.
Nevertheless, it is advisable to select a best possible model structure based
on the available knowledge about flow conditions in the investigated
measurement site.