Journal of Hydrology 446–447 (2012) 90–102
Contents lists available at SciVerse ScienceDirect
Journal of Hydrology
journal homepage: www.elsevier.com/locate/jhydrol
Temporal variability in stage–discharge relationships
José-Luis Guerrero a,b,⇑, Ida K. Westerberg a,d, Sven Halldin a, Chong-Yu Xu a,c, Lars-Christer Lundin a
a
Department of Earth Sciences, Uppsala University, Villavägen 16, SE-752 36 Uppsala, Sweden
Civil Engineering Department, Universidad Nacional Autónoma de Honduras, Ciudad Universitaria, Blv. Suyapa, Tegucigalpa, F.M., Honduras
c
Department of Geosciences, University of Oslo, P.O. Box 1047, Blindern, NO-036 Oslo, Norway
d
IVL Swedish Environmental Research Institute, Stockholm, Sweden
b
a r t i c l e
i n f o
Article history:
Received 7 April 2011
Received in revised form 12 March 2012
Accepted 18 April 2012
Available online 25 April 2012
This manuscript was handled by Andras
Bardossy, Editor-in-Chief, with the
assistance of Alin Andrei Carsteanu,
Associate Editor
Keywords:
Rating curve
Discharge
Uncertainty
Temporal variability
Alluvial river
s u m m a r y
Although discharge estimations are central for water management and hydropower, there are few studies
on the variability and uncertainty of their basis; deriving discharge from stage heights through the use of
a rating curve that depends on riverbed geometry. A large fraction of the world’s river-discharge stations
are presumably located in alluvial channels where riverbed characteristics may change over time because
of erosion and sedimentation. This study was conducted to analyse and quantify the dynamic relationship between stage and discharge and to determine to what degree currently used methods are able
to account for such variability. The study was carried out for six hydrometric stations in the upper
Choluteca River basin, Honduras, where a set of unusually frequent stage–discharge data are available.
The temporal variability and the uncertainty of the rating curve and its parameters were analysed
through a Monte Carlo (MC) analysis on a moving window of data using the Generalised Likelihood
Uncertainty Estimation (GLUE) methodology. Acceptable ranges for the values of the rating-curve parameters were determined from riverbed surveys at the six stations, and the sampling space was constrained
according to those ranges, using three-dimensional alpha shapes. Temporal variability was analysed in
three ways: (i) with annually updated rating curves (simulating Honduran practices), (ii) a rating curve
for each time window, and (iii) a smoothed, continuous dynamic rating curve derived from the MC analysis. The temporal variability of the rating parameters translated into a high rating-curve variability. The
variability could turn out as increasing or decreasing trends and/or cyclic behaviour. There was a tendency at all stations to a seasonal variability. The discharge at a given stage could vary by a factor of
two or more. The quotient in discharge volumes estimated from dynamic and static rating curves varied
between 0.5 and 1.5. The difference between discharge volumes derived from static and dynamic curves
was largest for sub-daily ratings but stayed large also for monthly and yearly totals. The relative uncertainty was largest for low flows but it was considerable also for intermediate and large flows. The standard procedure of adjusting rating curves when calculated and observed discharge differ by more than 5%
would have required continuously updated rating curves at the studied locations. We believe that these
findings can be applicable to many other discharge stations around the globe.
Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction
Discharge is a central variable when establishing the water balance for a basin, when calculating river routing and timing of
floods, and when managing the water resources of a region. Errors
in measurements of discharge and other water-balance terms can
make it hard to close the water balance using measured data
(Beven, 2001; Beven et al., 2011). Discharge measurements have
often been considered as relatively simple and certain in comparison to other water-balance terms. Because of this, the uncertainty
⇑ Corresponding author at: Department of Earth Sciences, Uppsala University,
Villavägen 16, SE-752 36 Uppsala, Sweden. Tel.: +46 18 4717164.
E-mail address: jose-luis.guerrero@hyd.uu.se (J.-L. Guerrero).
0022-1694/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.jhydrol.2012.04.031
of discharge measurements has received moderate attention compared to other hydrological measurement problems. It is generally
assumed that discharge is, and has been, measured indirectly at the
majority of all monitoring stations (although no global database
exists to substantiate how large this majority is). Discrete discharge measurements, commonly with the velocity-area method,
are used to derive a rating curve, the function relating discharge
to water stage (Herschy, 1994). An underlying assumption when
using a rating curve is that one and only one discharge corresponds
to each stage.
A wide range of conditions such as variable backwater, vegetation, ice, steadiness of the flow, variable channel storage, and channel shape affect the uncertainty of the stage–discharge relationship
(Boyer, 1964; Rantz, 1982; Schmidt, 2002). Pelletier (1988)
J.-L. Guerrero et al. / Journal of Hydrology 446–447 (2012) 90–102
provides one of the most comprehensive summaries of errors of
discharge determinations with the velocity-area method, giving
examples of total, combined uncertainties up to around 20%. Some
studies have addressed the problem of rating-curve uncertainties,
most of them in the last 10 years. Di Baldassarre and Montanari
(2009) present a rating-curve uncertainty analysis based on uncertainties in direct discharge measurements, interpolation and
extrapolation, unsteady flow conditions, and seasonal roughness
changes. Their analysis of a downstream stretch of the Po River
with moderate bed slope gives total rating-curve uncertainties
ranging from 6.2% at the lowest to 42.8% at the highest flows. Petersen-Øverleir et al. (2009) present a Bayesian methodology to assess rating-curve uncertainty at 581 Norwegian discharge stations.
For low flow at these stations, where ‘‘temporal channel changes
[...] typically affect the stage–discharge relationship with a magnitude smaller than the discharge measurement uncertainty, so that
unstableness may be ignored’’, only around one third of the stations have relative errors less than 40% for mean daily flow. Krueger et al. (2010) estimate discharge uncertainty using an algorithm
based on a fuzzy rating curve as part of an ensemble evaluation of
hydrological model hypotheses.
Di Baldassarre and Claps (2011) show that commonly used
methods ‘‘to interpolate river discharge measurements fail to
reproduce the stage–discharge relationship in the extrapolation
zone and can lead to results that are not physically plausible’’. They
recommended a hydraulic approach to derive stage–discharge
curves and the associated uncertainty.
None of the above-mentioned studies account quantitatively for
the rating-curve uncertainty caused by unstable riverbed geometry. We are only aware of five studies treating this aspect (McMillan et al., 2010; Shimizu et al., 2009; Westerberg et al., 2011;
Jalbert et al., 2010; Reitan and Petersen-Øverleir, 2011) despite
the widely recognised difficulties of estimating discharge in unstable channels (Dawdy, 1961; Andrews, 1979; Rantz, 1982; ISO,
1998; Léonard et al., 2000; Schmidt, 2002; Schmidt and García,
2003). Reitan and Petersen-Øverleir (2011) tackle the temporal
variability of the parameters of a rating model in three stations
in Norway. They use a Bayesian framework, considering that the
temporal evolution of the parameters could be modelled using
Ornstein–Uhlenbeck processes, which are the continuous-time
analogue of autoregressive models of order one. Also, they provide
a way to determine which, if any, of the parameters should be allowed to be time-dependent.
McMillan et al. (2010) present an ‘‘envelope-curve’’ method to
estimate discharge in a riverbed with significant uncertainty in
cross-section form because of deposition, transport, and sedimentation. On the basis of all known uncertainties, they estimate a
complete probability-density function of true discharge for a given
stage in order to evaluate the total error affecting a hydrological
model for the Wairau River basin in New Zealand. Shimizu et al.
(2009) simulate numerically the hysteresis effect on the rating
curve of the dune-flat bed transition in a riverbed that changes
with time as a function of discharge. Differences in discharge for
a given stage vary from zero to around 50% relative uncertainty
in the examples they provide. Westerberg et al. (2011) use a
non-stationary rating curve in a fuzzy-regression approach based
on estimated uncertainties in the measurements of stage and discharge for the Paso La Ceiba station in Honduras. Compared to a
constant rating curve, they obtain discharge estimate differences
of around 60% to +90% for low flows and around ±20% for intermediate to high flows. The final uncertainty they estimate for
mean daily discharge includes a temporal commensurability error
from having only three measurements of stage per day and vary
from 43% to +73% with the largest relative uncertainties occurring for low flows. Jalbert et al. (2010) analyse 1803 ratings from
19 discharge stations in the French Alps by statistically separating
91
‘‘initial’’ and ‘‘aging’’ rating curves. They exemplify their ‘‘temporal
uncertainty’’ at three stations where discharge at constant stages
vary over 25-year periods in ranges of 8.5–14.5 m3/s, 2–21 m3/s,
and 0.7–11 m3/s. Their temporal uncertainty is markedly higher
at low and high than at intermediate flows.
The traditional approach to tackle a changing river-channel
geometry calls for frequent measurements (weekly or more frequent) and shifts to the rating, as in Stout’s and Bolter’s methods
(Schmidt, 2002). Alternative approaches are, e.g. to use Manning’s
equation and take the evolution of the cross section into account
and to make use of ancillary data (Léonard et al., 2000). A generally
accepted procedure is to apply rating-curve shifts whenever a discharge measurement exceeds the accepted rating curve by more
than 5% (Rantz, 1982).
It is generally assumed that a large part of the world’s discharge-gauging stations are situated at alluvial rivers and that
many of them are subject to changes in their riverbed shapes
(although also here no global database exists to substantiate how
large this fraction is). A quantitative analysis of temporal change
and uncertainty in rating curves requires time series of very frequent ratings. Such time series were available from six discharge
stations in the Choluteca River basin in Honduras. We believe that
these gauging stations are representative both for the region and
for many other discharge stations in the world. The main purpose
of this study was to quantify the uncertainty associated with timevarying rating curves and also to investigate if the uncertainty
could be related to riverbed characteristics. We were also interested in relating the uncertainties to traditional curve-shifting approaches and to established Honduran monitoring practices.
2. Rating-curve theory
2.1. The rating equation
The Navier–Stokes equations are the accepted theoretical basis
for all fluid mechanics (Darrigol, 2002), of which water flow in
open channels is a special case. This set of equations is of little
practical use when directly applied to river-flow problems but
can be simplified into the equations of De Saint-Venant (1871).
The latter are widely used in flow-modelling problems (Moussa
and Bocquillon, 1996; Yen, 1979) and give acceptable results in
all but the most extreme conditions. The shear amount of data that
are needed in the calculation of discharge is one of the limiting factors when using a physically-based approach such as the Saint–Venant equations even in their simplified forms. Another limitation is
the stable-channel assumption that is violated for many alluvial
channels.
Because of these limitations, the rating curve is almost always
given as an empirical equation. One of the most commonly used
equations (ISO, 1998; Rantz, 1982; Schmidt, 2002) is:
Q ¼ aðx h0 Þn
ð1Þ
where Q is discharge, x is water stage or gauge height, and a, h0 and
n are empirical parameters (constants). Eq. (1) is a simplification of
the stage–discharge relationship, and can be conceptualised in two
ways (Schmidt, 2002). On the one hand, the discharge at a measuring station can be viewed as flow over a weir. When estimating discharge in weirs, the flow is considered to be rapidly varying from
sub- to supercritical flow, and Bernoulli’s equation (obtained
through an integration of Euler’s equation, in turn a simplification
of the Navier–Stokes’ equations) is used to derive a characteristic
equation for the weir, which is a power function of the stage. From
this viewpoint, parameters a and n in Eq. (1) can be viewed as factors related to the shape of the weir.
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On the other hand, the discharge can be viewed as open-channel flow described by uniform-flow formulae based on observational data, such as the Manning, Darcy–Weisbach, or Chézy
formulae (which can be traced back as simplifications of the Navier–Stokes equations). Flow in these equations is a function of
conveyance and energy slope. Assuming constant slope, the discharge is directly proportional to the conveyance. The conveyance
is expressed as a function of a resistance factor, the cross-sectional
area, and the hydraulic radius elevated to some power (Chow,
1959). The flow only depends on the channel shape if the resistance factor is constant.
In both conceptualisations, the parameter h0, or zero-flow
gauge height, represents a datum at which flow ceases. Empirical
adjustments of h0 are used to compensate for uncertainties caused
by the aforementioned disturbances (variable backwater, vegetation, ice, steadiness of the flow, variable channel storage and channel shape). The adjustment can be time-dependent, stagedependent, or given different values for different flow ranges
depending on the disturbance to be compensated for (Schmidt
and García, 2003). Such adjustments can be given a physical interpretation in the weir-flow, but not in the open-channel case, for
which the relation between water stage, hydraulic radius, and
cross-sectional area is non-linear. A change in h0 in the latter case
does not produce a parallel shift in the rating but a rating with different slope and shape.
2.2. Temporal variability in the stage–discharge relation
The theory of flow in alluvial channels establishes that the
channel will adopt different configurations, or flow regimes,
depending on the magnitude of the flow, going from a plane bed,
ripples, dunes, a transition phase in which a plane bed occurs
again, waves and finally antidunes (Dawdy, 1961; Engelund and
Fredsoe, 1982). Sediment transport in a river directly affects bedform evolution, through a process of scour and filling. Although a
simplification, Hjulström’s (1935) still-used diagram that delimits
areas of erosion, transport, and deposition in the plane given by
the average stream velocity and particle size captures the basic
mechanisms behind sediment transport and riverbed evolution
(Xu, 2004). All riverbed configurations have different flow resistances, hence different stage–discharge relationships. The resistance generally increases with flow velocity, with marked
discontinuities during transitions between channel configurations.
The changing boundaries generate a hysteretic effect on the ratings, depending on if the flow increases or decreases. While it is
possible to model the evolution of channel resistance to water flow
(Yen, 2002; Dottori et al., 2009), all methods require more data
than are normally available at hydrometric stations.
The alternating cycles of scour and fill in alluvial channels affect
the rating uncertainty at a different time scale than the hysteresis
effect (Andrews, 1979). River morphology constantly evolves over
time and rivers get incised in the landscape over large time spans
(Darby and Simon, 1999) but river shapes can also drastically
change on short time scales, especially during large floods (Rhoads,
1992), thus totally changing the ratings.
The traditional way to adjust for temporal changes in Eq. (1) is
to either apply shifts to the ratings for short-duration changes or to
change the rating-curve altogether if those changes are prolonged
(Rantz, 1982). An alternative way would be to consider the parameters of Eq. (1) as time-dependent variables. The traditional method requires ancillary data to determine the magnitude and
direction of the shifts. A method with variable parameters would
require both ancillary data and very frequent ratings. In the latter
case, Eq. (1) could be considered as a hydrological model and the
estimation of parameter values could be achieved through established calibration techniques (e.g., Choi and Beven, 2007; Wagener
et al., 2003). As far as we know, only Reitan and Petersen-Øverleir
(2011) have previously studied time-dependent rating parameters,
albeit with a dataset much sparser in time.
3. Study area and data
The Choluteca River basin at the Paso La Ceiba discharge station
had an area of 1766 km2 prior to hurricane Mitch (Fig. 1). Two Honduran institutions, SANAA (Servicio Autónomo Nacional de Acueductos y Alcantarillados) and SERNA (Secretaría de Recursos
Naturales), have been in charge of hydrometric measurements in
the basin. All discharge stations were destroyed by hurricane Mitch
(end of October–beginning of November 1998) during its passage
through the basin. When the stations were restored, despite keeping the same name, the reference datum had been washed away,
leading to a distinct shift in the pre- and post-Mitch data. The Paso
La Ceiba station, considered by Flambard (2003) to be the best in
the basin, was relocated upstream around three km. Very few ratings are available for this station after Mitch (Table 1). The frequency and quality of ratings decreased after Mitch for all
stations. We only used pre-Mitch data in this study.
SANAA and SERNA have used the same measurement procedures. Flow speeds at different cross sections and depths have been
measured with Price Type AA current metres and have then been
integrated into discharge, using the velocity-area method. Stage
measurements have been taken visually from a staff gauge.
Although corollary data such as the hydraulic radius and the flow
speed at different points have been determined when using the
velocity-area method, much of the information was lost during
Mitch. Only measured discharge and its corresponding stage were
used for this study.
A total of seven stations were originally available, but one of
them was discarded after quality control. The six stations used
here have differing temporal coverage (Fig. 2). An important characteristic of the dataset is its high temporal density, with measurements taken at least every fifteen days on average, and sometimes
as often as every two days for certain periods and stations. The six
basins (Table 1) situated within the basin delimited by the Paso La
Ceiba station (Fig. 1) differ almost by two orders of magnitude in
size. Riverbed characteristics and cross sections were originally
unavailable and were surveyed for each of the six stations (Fig. 3
and Supplementary material).
Discharge in the Choluteca River basin has a considerable spatial (Table 1) and a strong temporal variability, reflecting the high
temporal and spatial variability of the region’s climate (Hastenrath, 1967; Aguilar et al., 2005; Westerberg et al., 2010). Droughts
in the dry season from November to April can create problems,
especially in the Pacific-coast part of the basin. The wet season
from May to October is interrupted by the so-called midsummer
drought, a precipitation minimum around July–August. The
rainfall-generation mechanisms that are active on the Pacific side
of Central America, where the Choluteca River is located are
complex and severe inundations are normally associated with
extreme weather events like the hurricane Mitch or the ‘‘temporales’’, atmospheric disturbances originated in the Pacific Ocean
that mostly affect the Central American Pacific side (Peña and
Douglas, 2002; Hastenrath, 1967).
In spite of their uniquely high frequency, the discharge data
suffer from several quality problems. The quality of hydrological
data in Honduras is not regularly controlled (Flambard, 2003;
Westerberg et al., 2010). Flambard (2003) points out lack of
equipment calibration, few vertical measurement points for velocity-area quantification of discharge and that historical high-flow
data are not used when calculating new rating equations as weaknesses in current praxis. The variability in the stage–discharge
J.-L. Guerrero et al. / Journal of Hydrology 446–447 (2012) 90–102
93
Fig. 1. Hydrometric stations and sub-basins in the upper Choluteca River basin.
relationship has been handled by Honduran hydrometric data providers by dividing the ratings into time periods and estimating the
parameters in Eq. (1) for each period. As an example, 14 different
curves were used between May 1980 and August 1997 at the Paso
La Ceiba station. However, the choice of breakpoint and length of
period have not been documented and do not seem to relate to
the procedure presented by Rantz (1982) of applying shifts when
a discharge measurement differs from the used rating curve by
more than 5% and changing the curve when the shift is deemed
permanent. Balairón Pérez et al. (2004) and Flambard (2003) explain these problems with lack of relevant data management and
scattered monitoring responsibilities.
4. Methods
Quality assessment of discharge data and removal of non-reliable data were the first step in the analysis. The temporal variation
and uncertainty of the ratings were evaluated through a Generalised Likelihood Uncertainty Estimation (GLUE) analysis (Beven
and Binley, 1992) applied on time series of data. The Monte-Carlo
(MC) simulations were run for the rating model (in this case Eq.
(1)) within a priorly defined parameter space. Since the ranges
for a, h0, and n were unknown initially, the first step in our GLUE
analysis was to delimit physically reasonable ranges for the three
parameters through field measurements. The second step consisted in choosing objective functions (performance criteria) to
be used as informal likelihood measures in GLUE.
When the temporal variation of the rating curve and its
uncertainty, had been analysed in GLUE, two different ways to
account for the variability were investigated. The first was to
assess to what degree the variability could be handled by traditional methods, especially those used by the Honduran monitoring authorities. The second was to elaborate possible relations
between a dynamic rating curve and riverbed characteristics to
find a way to estimate discharge at stations with dynamic
riverbeds.
4.1. Data-quality control
The Concepción-station data were split into three periods as the
station was relocated in 1990 and because of a sharp discontinuity
in the ratings in 1981.
Removal of outliers was done with a proximity algorithm based
on the concept of alpha shapes (Edelsbrunner and Mücke, 1994).
This quality control algorithm was based on the idea that ratingcurve points are grouped in time and should be removed if they
deviated too much from the other points in the group. The validity
of a measurement was confirmed on the basis of its Euclidean distance to other points in the three-dimensional Cartesian space defined by time, discharge and gauge-height vectors. This was a form
of density-based spatial clustering. Krasnoshchekov and Polishchuk (2008) further expand the concept for robust reconstruction
of curves.
As with all clustering methods, some parameter values needed
to be fixed (Charu and Philip, 2001). One of these parameters was
the radius a of the alpha sphere that defined the alpha shape (see
Edelsbrunner and Mücke, 1994, for definitions). This radius is a
measure of how close a data point needs to be to others to be considered part of the alpha shape, i.e., to validate the other data.
Other subjective choices were the units of the Cartesian-space vectors. The base-10 logarithm of discharge was used to compensate
for the wide range of measured discharges, spanning several orders
of magnitude. Data were also aggregated along the time dimension. Data were first grouped in pentads, i.e., measurements taken
between the first and fifth of January belonged to pentad 1, giving a
total of 73 pentads in a year. Gauge heights were not modified.
Since alpha shapes are a geometrical concept, results can vary a
lot depending on the units used. The values of stage and discharge
were linearly mapped to [0, 3] and the values of time to [0, 1]. The
radius a was set to either 0.1 or 0.12 depending on the density of
the points.
To illustrate the concept, let’s consider that three points have
the same gauge height and discharge. A fourth point would be considered as belonging to the same polytope if all four were within a
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Table 1
Basin characteristics for six hydrometric stations in the upper Choluteca River basin.
Station name
Area
(km2)
Elevation (m)
Mean annual runoff (mm)
Mean
slope (%)
Measurement
period
Jun 1972–Oct
1998
Aug 1981–Oct
1998
May 1980–
Aug 1998
Mar 1991–Oct
1998
Jul 1972–Oct
1998
Nov 1987–Oct
1998
Lowest
Highest
Mean
Lowest
Highest
Mean
Concepción
116
1170
2235
1567
186
333
252
18
Guacerique
151
1079
2037
1477
275
493
371
19
1766
675
2318
1302
321
372
350
20
85
1096
2037
1504
367
413
391
19
272
931
2229
1399
359
471
400
18
64
1075
2000
1446
174
226
203
21
Paso la Ceiba
Quiebramontes
Rio del
Hombre
Tatumbla
Number of
measurements
Mean interval between
measurements (days)
813
12
718
10
1216
5
239
12
204
15
128
15
Numbers are given for pre-Mitch conditions. The Paso la Ceiba station was moved upstream around 3 km after Mitch so the new basin area is 1766 km2. Area and elevation
data are derived from SRTM data. The mean runoff is given for the period 1 May 1994–30 April 1997 and was calculated using both objective functions and the four different
rating methods. The lowest, highest and mean values of these different runoff values are given. The numbers of measurements are after quality control.
distance 2a from each other. With 73 pentads in year, a time vector
linearly mapped to [0, 1], and alpha equal to 0.1, a distance of 2a
along the time axis would be equal to 2 0.1(1 73) = 14.6
pentads, i.e. 73 days.
For the Paso La Ceiba station, a further visual inspection revealed more erroneously recorded or digitised data that were manually removed. Errors in gauge-height data could often easily be
detected when compared to the continuous (three times a day)
stage data. Data from surrounding stations were useful to aid the
removal decisions. The Sabacuante station, the 7th station originally considered, was discarded from further study as a result of
such a comparison. Westerberg et al. (2011) estimate the uncertainty in a discharge measurement at the Paso La Ceiba station
to ±25% caused by the lack of calibration of current metres, few
vertical measurements, and non-ideal station locations as analysed
by Flambard (2003). We assumed the same uncertainty for all discharge stations. The uncertainty in gauge height was assumed to
be of ±5%, also following Westerberg et al. (2011).
area and hydraulic radius. The constraint on h0 was then set so that
the average value of k for all stage–discharge data fell within a theoretically acceptable range. Chow (1959) states that plausible k
values range from 0.02 to 0.2795 if the most extreme conditions
are included: severe irregularity, frequently alternating crosssection, severe obstructions, very high vegetation and severe
meandering.
The range of values for n was then set on the basis of the established range of values for h0 through the following procedure. We
can express the product of the area A and the hydraulic radius Rh as
a function of gauge height x. By equating Eqs. (1) and (2) we get:
4.2. Delimiting parameter values
Differentiating Eq. (4) with respect to x and assuming that f(x)
and (x h0) are always positive yield:
Given the lack of metadata for the six stations, the delimitation
of parameter values had to be based on several simplifying
assumptions. We used Manning’s equation as a starting point:
Q¼
1 2=3 12
AR S
k h
ð2Þ
where Q is discharge, k a non-dimensional friction coefficient, A
cross-sectional area, Rh hydraulic radius, and S energy slope of the
channel, all given in SI units.
Area, hydraulic radius, and slope for each station were obtained
from the topographical survey (Fig. 3 and Supplementary material). The use of Manning’s equation implies that the channel was
assumed prismatic and stable with a constant friction slope and
steady flow. A further assumption was that the friction slope
equalled the riverbed slope. The value of k is normally calibrated
or obtained from tables. We used discharge measurements to
back-calculate its value.
Let’s call the measured gauge height minus the zero-flow gauge
height (x h0) the effective gauge height. The range of values for h0
was limited by two constraints. The first was that the effective
gauge height had to be above the lowest point in the cross section
for all measurements. The other was based on documented values
for the parameter k. The effective gauge heights were calculated for
different values of h0 and then used to determine cross-sectional
Cf ðxÞ ¼ aðx h0 Þn
ð3Þ
1
where C ¼ Sk2 and f ðxÞ ¼ AR2=3
h . Through logarithmic transformation,
we obtain:
lnðCÞ þ lnðf ðxÞÞ ¼ lnðaÞ þ n lnðx h0 Þ
f 0 ðxÞ
ðx h0 Þ ¼ n
f ðxÞ
ð4Þ
ð5Þ
Then, using the cross-sectional data, the value of n could be
determined for different water levels and values of h0. Since the
cross-sectional data could be rough, the values of n for a given
interval of the cross section were smoothed to avoid negative values. The final value of n was then taken as the average for all intervals up to the given gauge height.
In order to equate the discharge obtained from the rating equation to the actual measurements, it was necessary to allow parameter a to take different values. The entire range of values obtained
from all plausible values of h0 was used to delimit this parameter.
h0 was sampled within this range at 1-cm increments.
By combining the range of plausible effective gauge heights and
the average value of n it was possible to obtain a range of values for
a:
a¼
Q
ðx h0 Þn
ð6Þ
For each sampled value of h0, a value of n was obtained for each
stage/discharge measurement through Eq. (5). These values of h0
and n were finally used to calculate a through Eq. (6).
J.-L. Guerrero et al. / Journal of Hydrology 446–447 (2012) 90–102
40
20
40
0
20
0
100
50
40
0
20
0
40
20
40
0
20
0
1975
1980
1985
1990
1995
Fig. 2. Availability and number of measurements at the six hydrometric stations in
the upper Choluteca River basin. The y-axis is the number of measurements during
a 181-day period centred on the date given in the x-axis. The date of Mitch is
marked by the vertical line.
Fig. 3. River-bed and riverside topography at the Concepción station (466996,
1547374 UTM16N). Measurement points (left), cross section perpendicular to the
stream (upper right), and interpolated surface (bottom right). The units are in
metres. Datum is the water level on 23 June 2009.
95
points from the cuboid defined by the minima and maxima found
from the topographic analysis.
However, the plausible parameter-value sets needed not be uniformly distributed inside the cuboid and in order to further constrain the sampling space a polytope was constructed such that
it contained all the plausible parameter-value sets in its interior region. This polytope was defined to be the alpha shape with the
largest possible alpha radius containing all plausible points, without neither dangling faces nor edges, i.e., a single closed polyhedron. The parameter-value sets for the MC simulations were then
sampled from the interior region of this polyhedron.
An objective function used as an informal likelihood measure in
GLUE should increase monotonically with increasing goodness of
fit and likelihoods of non-behavioural parameter-value sets should
be equal to zero. We used two objective functions, a time-weighted
Nash–Sutcliffe efficiency (NSE) and a Score (SC) criterion. NSE was
used as a baseline and defined as:
NSE ¼ 1 m
wi ðQ o;i Q c;i Þ2
1X
W i¼1 ðQ o;i Q o Þ2
ð7Þ
P
i
where wi ¼ ðem Þ2 1, andW ¼ m
i¼1 wi .
Here, m is the number of ratings in each time window, and Q
discharge. Indices o and c stand for observed and calculated,
whereas i signifies a specific rating. Data were weighted exponentially within each time window to put more emphasis on the most
recent measurements. Only positive values were considered for
analysis. The choice of the weighting exponent was a subjective
measure of the importance of past measurements.
The SC criterion was based on estimated uncertainties in the
stage and discharge measurements. Similar to Pappenberger
et al. (2006) we used a fuzzy membership function, given as a rectangular frustrum and delimited by the estimated errors in both
stage and discharge measurements. We assumed that calculated
discharges within ±5% of the observed values were equally good,
and that calculated discharges within ±25% were acceptable. For
stages we assumed that calculations within ±5 mm of the observed
were equally good, and that calculated discharge values within ±5%
were acceptable (Eq. (8)):
Q 0:75Q
1:5Q Q
; 1;
;0
f1 ðQ Þ ¼ max min
0:95Q 0:75Q
1:25Q 1:5Q
ð8aÞ
x 0:95x
1:05x x
;0
f2 ðxÞ ¼ max min
; 1;
ðx 0:005Þ 0:95x
1:05x ðx þ 0:005Þ
ð8bÞ
4.3. Assessing the temporal variability of the stage–discharge relation
The values of parameters in Eq. (1) were studied through MC
analysis, performed on moving windows of data. The moving windows were defined as having a length of 30 data points, with an
overlap of 29 data points between successive windows. Data within each window were analysed using the GLUE methodology. The
three largest measured discharges were also included in the window, if they were not already part of it, and given a weight of
one to compensate for the possible lack of high discharge measurements. From a geometric perspective, sampling parameters for the
MC analysis could be described as sampling points in a space with
as many dimensions as there were model parameters. The coordinates of a point inside this space defined a parameter-value set.
The rating model (Eq. (1)) we used had three parameters. The values the rating parameters could take were bound using Manning’s
equation combined with topographic information and plausible
parameter-value sets were found (Table 2). Sampling for all rating
parameters from an assumed prior uniform distribution in this
three-dimensional space was equivalent to uniformly sample
The SC value within each fuzzy frustrum was calculated as
s = f1(Q) f2(x).
The results of every behavioural simulation were weighted by
its associated likelihood and uncertainty limits were derived from
the cumulative, likelihood-weighted distribution of simulations of
Eq. (1), using the GLUE procedure. One hundred thousand MC simulations were performed for both objective functions. There is no
generally accepted threshold for which a Nash–Sutcliffe efficiency
is non-behavioural, so we did not set the behavioural threshold to a
specific value in this case, but instead considered the top 1000 (1%)
as behavioural. Since the GLUE approach was used separately on
each moving window of data, different uncertainty bounds were
obtained at every time step. Similarly, for SC, the top 1000 simulations were also taken as behavioural. SC was finally calculated for
all the behavioural equations as a normalised, total score for all ratings within the given time frame.
For every time window, an uncertainty range was defined for
every parameter using the 5% and 95% discharge prediction quantiles. The relative width of the uncertainty range reflected the
relative identifiability of the different parameters for different
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J.-L. Guerrero et al. / Journal of Hydrology 446–447 (2012) 90–102
periods. Finally, to evaluate the effect of the temporal variability of
the parameter values on the estimation of discharge, the effect of
the parameter variability was tested on the minimum, the median
and the maximum measured gauge heights.
4.4. Including time variability in the rating equation
The standard Honduran praxis of updating the rating curves
was compared to the results of a dynamic rating curve. Since the
Honduran authorities could not provide the dates when the official
rating curves were changed, their procedure was approximated by
deriving a rating curve for each hydrological year (1 May–30 April).
These annual rating curves provided benchmarks against which
our variable rating curves could be assessed.
The dynamic rating curve was based on Eq. (1), but instead of
changing all three parameters whenever required, two of them, n
and h0, were set to the values derived from the cross-section
whereas the third, a, was allowed to vary freely. Three steps were
taken in the process of determining which values these parameters
should take:
1. We fixed the values of h0 using the measured cross section and
Manninǵs equation.
2. We fixed the values of n based on the previously calculated h0
values and the measured cross section.
3. Finally for each h0 and n combination, the value of a yielding
equality between the observed and calculated discharge was
obtained.
Time series a(ti) were thus obtained where ti signifies dates at
which the measurements were taken. A different a(ti) was obtained
for each behavioural parameter-value set. As in the MC movingwindow analysis, 100,000 different parameter-value sets were
tested. The data were not limited to 30 measurements but the entire dataset was used and an equal weight was given to all
measurements.
Continuous discharge estimates were obtained through Eq. (1)
by interpolating a(ti) between measurements. Each time series
was interpolated using cubic splines of order four (Craven and
Wahba, 1978). Within every interval (four points in this case), a
third-order polynomial g(ti) was selected that minimised the given
function:
X
ðgðt i Þ aðti ÞÞ2 þ ð1 pÞ Z
ðgðsÞÞ2 ds ¼ 0
ð9Þ
and fulfilled the condition that successive polynomials are joined at
the data points in such a way that the first and second derivatives
are continuous. Summation and integration were done over the entire time period. With smoothing parameter p = 1, the spline passed
through all points, and with p = 0 a straight line was obtained. Values falling between these two extremes were a compromise between good fit and smoothness. Predicted parameter values
sometimes fell outside the plausible range (Table 2) but only values
falling within that range were used when deriving the spline function. The choice of smoothing parameter value was subjective. For
this study, it was given a small value (105), which eliminated
high-frequency fluctuations in a.
4.5. Assessing the effect of temporal rating-curve variability on
streamflow estimations
Discharge was calculated at all stations for three hydrological
years: 5 May 1994–30 April 1997 using four methods: (i) a benchmark method with a single rating curve for the entire period, (ii)
separate rating curves for each hydrological year (approximating
Honduran routines), (iii) a moving-window analysis, using constant rating parameters in the interval between successive windows, (iv) Eq. (1) with the time variable a(t) with values
interpolated between measurements using p = 105.
Continuous time series of gauge heights from each station were
approximated from manual stage measurements at Guacerique at
0700, 1100 and 1600 daily, digitised to provide a near continuous-baseline time series for the given 3-year period. Gauge heights
at the other stations were approximated using Guacerique data.
The Generalised Extreme Value (GEV) distribution is defined by
three parameters: n, r and l or the shape, scale and location
parameters (Kotz and Nadarajah, 2001). Maximum-likelihood estimates of the parameters nb, rb and lb of a GEV distribution from
the Guacerique gauge heights of the 3-year period defined a baseline distribution. The parameters rs and ls of a second distribution,
using only gauge heights from the stage–discharge measurements,
were then calculated with the same procedure. The remaining
parameter necessary to define a second GEV distribution was considered equal to nb. Each record from the gauge-height time series
at Guacerique was then translated to a cumulative frequency using
the baseline distribution. The same frequency was finally used to
estimate, from the second distribution, the corresponding gauge
heights at the other station, for the given point in time. These
gauge heights were then used to estimate the discharge at each
station.
As a last step we investigated possible relations between a(t)
and riverbed characteristics. There is clear seasonality in discharge,
precipitation, and gauge height in the Choluteca River basin so
these entities are correlated. We concentrated our analysis on possible relations between a(t) and gauge heights.
5. Results
5.1. Delimiting the rating-curve parameters
After the quality control, 36 measurements were deleted for
each station on average, with a maximum of 128 for Concepción
and a minimum of 5 for Río Del Hombre (Table 1). The station survey (Fig. 3 and Supplementary material) along with the data-quality control and parameter-delimitation analysis limited h0 at all
stations to physically reasonable values (Table 2), using values of
k considered feasible by Chow (1959) as limits for the average of
all measurements. These then allowed reasonably well constrained, behavioural ranges to be defined also for a and n (Fig. 4).
The spread in k values was large for small gauge heights but became well constrained for large gauge heights, as exemplified for
Table 2
Parameter-value ranges for six hydrometric stations in the upper Choluteca River basin.
Parameter
a
h0
n
Station
Concepción
Guacerique
Paso La Ceiba
Quiebramontes
Río Del Hombre
Tatumbla
1.11–202.43
0.19–0.43
1.00–2.45
0.38–37.68
0.28–0.52
1.00–2.25
0.67–71.07
0.64–0.06
1.80–2.96
0.46–21.22
0.17–0.34
1.25–2.12
0.39–42.87
0.07–0.32
1.00–2.71
0.12–23.01
0.13–0.23
1.00–2.40
97
2.6
320
1.3
160
Concepción
Guacerique
Paso La Ceiba
Quiebramontes
Río Del Hombre
Tatumbla
100
10
ARh2/3
Mean n
J.-L. Guerrero et al. / Journal of Hydrology 446–447 (2012) 90–102
1
(i)
0
0
2
4
Effective Gauge Height (x − h0)
6
0.1
0
h0
0.23
Difference in %
1
100
0.18
0.5
10
1
(iii)
(ii)
0
0
0.1
10
20
30 0.5
1
a
1.5
n
2
0.13
2.5
Fig. 4. Bounding of the rating-curve parameters for the Tatumbla discharge station.
(i): AR2=3
from Eq. (3) and average value of n vs. effective gauge height (in metres).
h
Empirical cumulative distribution functions (ecdf) for given values of h0 (thin lines)
and for all plausible values (thick line): (ii) for parameter n (iii) for parameter a. The
greyscale shows the different values of h0 used when calculating the ecdf’s of a and
n for single values of h0 (thin lines).
100
10
1
0.1
1990
10
0
ho = 0.28
Manning´s k
ho = 0.52
10
10
−1
h0
−3
0.3
0.5
1998 1990
1994
1998
Fig. 6. Absolute difference (in %) between the last observed discharge in the
moving-window and the discharge calculated from a rating curve, determined
through a least-squares fit to all points in the window. The horizontal line is the 5%
difference.
0.5
−2
0.4
10
1994
0.6
0.7
0.8
0.9
1
1.1
1.2
Gauge Height (m)
Fig. 5. Values of k at the Guacerique station obtained from cross-sectional river-bed
data and a measured slope of 1% with all plausible values of h0, highlighting the
possible minimum and maximum h0 values.
the Guacerique station (Fig. 5). This translated into a larger uncertainty in the ratings for low than for high flows. The value of n was
related to the shape of the cross-section and showed a larger rate
of change for small than for large gauge heights (Fig. 4). Changes
in the cross-section were reflected in the value of n.
5.2. Assessing variability of the stage–discharge relation
It would have been necessary to update the rating curve almost
continuously if the Honduran authorities would have followed the
commonly accepted rules stated by Rantz (1982) (Fig. 6).
The benefits of the sampling scheme used to generate parameter vectors were twofold. Firstly, only points close to, in the Euclidean sense, the plausible parameter vectors were sampled.
Secondly, compared to assuming uniform distributions for the rating parameters, this made for a more efficient sampling scheme, in
the sense that fewer simulations were required to produce an
equal number of behavioural results, resulting in reduced computational demands.
The moving-window analysis showed that the rating-equation
parameters were constantly changing over time (Fig. 7). The
change tended to be most pronounced and systematic for parameter a that, given an open-channel conceptualisation, depends on
the scale and shape of the cross-section as well as the hydraulic
resistance (Petersen-Øverleir et al., 2009). Both hydraulic resistance and scale can be expected to change with stage. Parameter
identifiability, as reflected by the width of the uncertainty bounds
was also non-stationary. The temporal variation displayed depended on the selected objective function (Figs. 8 and 9). For any
given period, the discharge for a constant gauge height could be
similar, as for Río Del Hombre station in 1985–1990 for either
objective functions, or very dissimilar, as for Guacerique station
where the discharge could either be increasing or decreasing
depending on the objective function (Fig. 8).
The effect of temporal variability of the rating parameters on
the estimation of discharge was clearly visible at all gauge heights
and stations (Figs. 8 and 9). A seasonal component seemed to be
present at all stations, at least part of the time, and was most pronounced at the Guacerique and Paso La Ceiba stations. The seasonality was more visible when SC, as compared to NSE, was used to
assess behavioural parameter values. The decrease at most stations
in number of measurements in the early 1990s (Fig. 2) seemed to
be reflected as less variable ratings (Fig. 9). Except for the seasonal
variability, there was no common trend or frequency in the change
of the ratings at the six stations. The discharge for a given gauge
height could vary by a factor of more than two during a single year.
Also, the predicted discharge varied considerably with the likelihood measure. The discharge was not stationary at any station
for a given gauge height and all stations showed differing longterm trends. Overall, the magnitude of the discharge was generally
lower when SC was the objective function (Figs. 8 and 9).
Four different methods were compared to assess the impact of
the temporal variability of the stage–discharge relationship on
the calculated discharge, combined with two different objective
functions. The results were different depending on the objective
function and the method used (Table 2). Comparing the mean discharge to the minimum and maximum one obtained from all
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J.-L. Guerrero et al. / Journal of Hydrology 446–447 (2012) 90–102
40
20
a
0
0.53
h0
0.47
0.41
2.3
1.85
1.4
n
1982
1985
1988
1991
1994
1997
Fig. 7. Behavioural parameter values, using SC as objective function on a moving window of measurements at the Guacerique station. A darker shade indicates a higher
likelihood.
0.034
Min
0.017
0
Discharge (m3/s)
0.66
0.38
Med
0.12
14
7.5
Max
1
1982
1985
1988
1991
1994
1997
3
Fig. 8. Temporal variation of the discharge (m /s) for the minimum, median and maximum gauge heights (m), using two objective functions (SC is the grey line and the NSE
the black hashed line) at Guacerique station.
methods (Table 1), the difference between mean and extremes ranged from 32% to +36%.
5.3. Including variability in the rating equation
5.3.1. A time-variable rating equation
Since Eq. (1) is ultimately an empirical function, there is no
obvious way to assign temporal variability to one or more of the
three parameters. We decided to contain all temporal variability
into parameter a since it showed the most pronounced and systematic variability (Fig. 7). The smoothed a(t) (Fig. 10) reflected,
if not always the magnitude of the discharge change (to which a
is linearly related), at least the shape of the change shown by the
moving-window MC analysis. The magnitude of the change was related to the choice of smoothing parameter value. With p = 105
and SC as objective function, the magnitude was close to the one
shown by the MC analysis (Fig. 9). It is important to note that for
the 3-year period when the effect of the smoothed a(t) was tested,
the average interval between measurements was twelve days for
all stations (maximum of seventeen and minimum of five) and
the average longest-interval was of 56 days (maximum of 84 and
minimum of 28).
5.3.2. The effect of temporal variability of ratings on total volume of
discharge
The estimated total water volume varied considerably depending on both the objective function and the rating method (Tables 1
and 3). The magnitude of the change was not consistent from station to station. The total water volume was larger for NSE except
when the variable parameter a was used (Table 3). Furthermore,
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J.-L. Guerrero et al. / Journal of Hydrology 446–447 (2012) 90–102
1
Concepción
0.5
0
0.7
Guacerique
0.35
0
14
7
0
0.4
Quiebramontes
0.2
Median Discharge (m3/s)
Paso La Ceiba
0
1.2
Río Del Hombre
0.8
0.4
0.14
Tatumbla
0.07
0
1973
1976
1978
1982
1985
1988
1991
1994
1997
Fig. 9. Discharge (m3/s) at six hydrometric stations in the upper Choluteca River basin for their median gauge heights, using NSE (hashed black line) and SC (grey line) as
objective functions.
can go from 0.5 to 1.5. With a smoothed time series of a, the ratingcurve change is continuous. With the moving-window and a-newrating-curve-per-year methods change is triggered by new stage–
discharge measurements. This explains the delays in the variations
between the methods (Fig. 11).
a
35
20
6. Discussion
5
1994
1995
1996
Optimum NSE
p=10−5, NSE
Optimum SC
p=10−5,SC
Fig. 10. Time series of parameter a and the smoothed time series a(t) with NSE
(hashed black line) and SC (grey line) as objective functions at Guacerique station
The MC simulation giving the highest NSE was taken as the ‘‘Optimum NSE’’.
Similarly the MC simulation with highest SC was taken as the ‘‘Optimum SC’’.
the differences in total volume between the two objective functions were smallest when the time variable parameter a was used.
In spite of these dissimilarities, all results demonstrated that the
use of a static rating curve can induce considerable discharge error
even for the long-term average discharge (Table 3). This error is
much more pronounced for daily and monthly (Fig. 11) discharge
and the quotient between static and dynamic discharge estimates
Hydrological modelling has four main uncertainty sources (Refsgaard and Storm, 1996): (i) input-data errors, (ii) output-data errors, (iii) parameter uncertainty, and (iv) structural errors in the
selected model. This study had a focus on uncertainties related to
(iii) and (iv) as a way to quantify and understand the temporal variation of rating curves in six Honduran alluvial river channels. It
complements the one of Westerberg et al. (2011) that focuses on
the quantification of discharge uncertainty.
With more than 7300 river-discharge stations in 156 countries,
the global runoff database at GRDC (Global Runoff Data Centre,
2011) is the globally most complete collection of river data. But
not even this database provides metadata (i) on location of stations
on alluvial or non-alluvial rivers, or (ii) on the means by which discharge is measured (Looser, 2010, personal communication). We
can thus only assume that a large fraction, maybe a majority, of
all discharge stations globally are located at places where riverbeds
are changing over time. We can also assume that the majority of
these stations base their discharge data on stage measurements
and rating curves that are updated according to criteria that are
not commonly reported. If these assumptions are correct, which
we believe, the analysis of time-variable rating curves deserves
more attention than it has had so far.
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J.-L. Guerrero et al. / Journal of Hydrology 446–447 (2012) 90–102
Table 3
Relative difference in total volume of discharged water for the period 1 May 1994–30 April 1997 for the six hydrometric stations. The four rating methods in combination with
both objective functions were used. Results are based on the median discharge of all behavioural rating curves. The basis for the comparison was obtained using the Score as
objective function and one single rating for the entire period and given a base value of 100.
Station
Total volume error
One single rating
Concepción
Guacerique
Paso la Ceiba
Quiebramontes
Río del Hombre
Tatumbla
One rating per year
NSE
SC
NSE
SC
NSE
SC
161
161
105
99
118
120
100
100
100
100
100
100
173
179
106
101
131
122
100
117
100
99
101
103
169
170
108
98
120
117
96
117
103
94
100
94
123
116
95
103
107
108
121
119
110
106
114
112
Ratio
1
1994
Smoothing
SC
1.5
0.5
Moving window
NSE
1995
1996
One rating/year
Moving Window
Smoothing
One Single Rating
Fig. 11. Average monthly discharge ratios between discharge from three timevarying rating curves and the discharge from a single rating curve for the period 1
May 1994–30 April 1997 at the Guacerique station. The three time varying rating
are: one rating/year, moving-window results, constant h0 and n parameters
combined with a time varying parameter a. Results are based on the median of
all behavioural simulations for SC. Monthly discharge values are the average of the
discharges calculated from sub-daily gauge heights.
We are only aware of five studies that analyse rating curves on
the basis of their temporal variability (McMillan et al., 2010; Shimizu et al., 2009; Westerberg et al., 2011; Jalbert et al., 2010; Reitan
and Petersen-Øverleir, 2011). Jalbert et al. (2010) and Reitan and
Petersen-Øverleir (2011) provide data comparable to our analysis.
For Jalbert et al. (2010), their time series of discharge at constant
stage heights from three stations in the French Alps support our
findings of large temporal rating-curve variability. Whereas we
found that discharge for a given stage could differ by a factor two,
the examples of Jalbert et al. (2010) provide even larger variability.
Our finding that rating-curve uncertainty is largest for low flows
(Fig. 5) is also corroborated by Jalbert et al. (2010) who find that a
much larger proportion of their ratings are outside of their
confidence limits at low than at intermediate flows. Even greater
similarities are found with the work of Reitan and Petersen-Øverleir
(2011). They focus on the temporal variability of the rating parameters in a Bayesian framework and obtain similar results regarding
the long-term variability. Additionally they provide a way of
determining which, if any, of the rating parameters should be considered time-dependent. Although they discuss the possibility of a
seasonal variation, it is not reflected in their results, presumably because their stations are stable on a seasonal scale or the temporal
sparsity of their data: 208 measurements from 1908 to 2010 and
44 measurements from 1969 to 2010 in the two, out of a total of
three, stations where they found the ratings to be time-dependent.
All five studies concentrate on finding practical ways of incorporating rating-curve uncertainty caused by temporal variability.
This study, along with the ones of Jalbert et al. (2010) and Reitan
and Petersen-Øverleir (2011) provide examples of what this temporal variability looks like for a number of discharge stations. Like
them, we were unable to find a single variability pattern that applied to all stations. Some stations showed long-term increasing
or decreasing trends whereas others provided a more cyclic behaviour. There was a tendency that all our stations reflected some seasonality to their rating curves. This effect seems more pronounced
at those stations and time periods that had frequent ratings. It
would, therefore, be interesting to find out if the seasonality effect
is a common trait to other stations where stage and discharge are
measured continuously or at a high temporal resolution.
Modelling the gauge heights at other stations given the data at
Guacerique implied strong assumptions in terms of spatial homogeneity that could not be verified, the calculated discharges at the
other stations may therefore not be representative for shorter periods whereas the calculation of the total water volumes is probably
less affected.
The discharge estimates in this study pointed to a high degree
of uncertainty caused by time-variable rating curves. The variability in the ratings was more pronounced than what current estimation techniques allow for. The variability was such that the ratings
changed with practically every new measurement. The traditional
recommendation to modify ratings when use of the present rating
curve creates more than 5% difference with actual measurements
did not seem reasonable for Honduran conditions. It was especially
inappropriate for low flows when the percentage was based on
small and uncertain numbers. In the absence of experimental data
or further information about the uncertainty in discharge measurements in Honduras, this uncertainty was estimated to ±25% by
Westerberg et al. (2011), based on literature values and knowledge
of local measurement conditions, but as discussed in this study, the
assumption of a constant measurement error might not always be
appropriate. This estimated uncertainty is larger than the 5% limit
but no attempt was made here at separating the uncertainty in the
data from the variability in the results.
When the temporal variability of the ratings was expressed as
changes in behavioural parameter-value sets, they were considerable and depended on the selected objective function. The uncertainty bounds obtained from GLUE analysis are directly
interpretable when the behavioural threshold is defined from an
estimation of the uncertainty in the data. Such estimation is made
for mean daily discharge by Westerberg et al. (2011). Here, the
behavioural limit was the top 1000 simulations and the uncertainty bounds obtained for SC and NSE were thus difficult to compare. This was especially true since only SC took input-data
uncertainty into account and placed an equal weight on all gaugings, contrary to NSE that focused on high flows. When the top
1000 simulations were deemed behavioural, the uncertainty
bounds given by the SC were generally larger than the ones
obtained by NSE. The differences between the rating methods
J.-L. Guerrero et al. / Journal of Hydrology 446–447 (2012) 90–102
reported here could be even larger, since only comparisons between the median of the behavioural results was analysed here,
because of the ad hoc manner in which the behavioural threshold
was set. No use was made of the uncertainty limits obtained from
GLUE when comparing methods and objective functions.
The SC criterion explicitly includes an estimation of data uncertainty. An objective way to define a behavioural threshold would
be to consider a parameter-value set behavioural only if the rating
curve it defines crossed all the frustrums. This method was tested
but resulted in either no behavioural parameter-value sets or having a few stage–discharge measurements (those furthest away
from the rating curve) dominating the parameter-value set
selection.
It was deceptively simple to obtain high NSE values for the rating-curve fit. Even when only a single rating curve was used for the
entire period, values of NSE superior to 0.95 were consistently obtained. Increasingly better efficiencies were obtained when using
one rating curve per year, the moving-window approach, and finally the smoothing approach. 100% efficiencies were possible, as expected, when using a smoothing parameter p equal to 1. The
selection of p = 105 allowed variations shown during the MC analysis to be reflected and the high-frequency variations to be limited.
The use of a time-dependent parameter in the rating equation offered an alternative to quantify the time rate of change of the rating curve. The high-frequency filter applied to the time series of a
yielded a very good fit between the observed and measured data,
albeit at the cost of adding new degrees of freedom to the analysis.
The ranges of parameter values for the MC simulations were obtained from station morphology and by assuming Manning’s equation to be the physical basis of the ratings. This implied a series of
assumptions that were not always fulfilled, furthermore supporting the suspicion that rating variability might have been even larger than illustrated here. The magnitude of the smoothing
parameter p reflects implicit assumptions in the physics behind
the ratings. By using an interpolating spline (p = 1), one assumes
fast changes in the rating but only when there are measurements.
On the other hand, the larger the smoothing, the more one assumes
changes to be gradual. Gradual changes can be a conceptualisation
of an even and continuous sedimentation, whereas scouring tends
to be abrupt.
Discharge calculated with MC moving-window parameters assumes that rating-curve changes only occur when there is a new
measurement. A time-dependent rating-curve parameter yields
continuous change. The effectiveness of both methods depends
on the temporal density of measurements since a long period without measurements can lead to discharge bias. An alternative to
overcome this shortcoming in the MC moving-window method
would be to allow for a to be a dynamic parameter, and the other
two parameters to change following the values from the MC moving-window analysis. This was tested as a first attempt to relate
the time variability of a to climate and basin properties. There
was a tendency for the optimal a values and gauge heights to be
linearly related. This relationship and the usefulness of the dynamic rating curve should be subject to future research. The length
of the moving window was selected to 30 data points, similarly to
Westerberg et al. (2011) who evaluated the effect of using 20, 25
and 35 data points instead of 30 at the Paso La Ceiba station and
found that it gave similar results (see Westerberg et al. (2011)
for further discussion on this and other assumptions in the moving-window approach).
Existing Honduran praxis regarding rating curves can lead to
large biases in the estimation of discharge (Table 3, Figs. 8–11).
The seasonal variations must be added (Figs. 10 and 11) to the
long-term biases (Figs. 8 and 9). It was hard to pinpoint a single
cause for the variability. Alternating cycles of scour and fill, with
deposition during the dry season and erosion during the wet
101
season can change the channel’s conveyance. Vegetation could play
a role since vegetation might grow during the dry season on river
banks that are flooded during the wet season, yielding a totally different friction slope. Backwater effects might come into play when
the water level rises. We did not attempt to clarify what process, or
combination of processes, gave rise to the variability, but instead
explored the variability that the data revealed. Exploring these issues should be prioritised before proposing a technique that could
yield more accurate results. In this sense, and despite the high temporal density, there are other sources of variability that the dataset
was not fit to elucidate. For instance, the passage of a flood wave
could give a different relation between stage and discharge during
the rising or falling limbs. This phenomenon works at much shorter
timescale than current measurements schemes. Another avenue of
future research would be set up continuous measurements of discharge and stage to deepen the present analysis and reduce measurement errors. Even then, these measurements alone could
turn out to be insufficient to reveal which processes are most
important, but would probably give a clearer picture of the temporal variability in the stage–discharge relationship.
7. Conclusions
The high frequency of ratings and their high temporal variability in six hydrometric stations in the upper Choluteca River basin
allowed a unique analysis of the discharge uncertainty caused by
temporal variability of the rating curve. The variability was so large
that traditional rating techniques were insufficient to account for
it. A time-invariant rating curve can result in large errors in the
estimation of discharge. This study investigated the temporal variability and uncertainty of the rating curve by GLUE methodology
with NSE and SC as objective functions, as well as four different
rating methods. When both objective functions and all rating
methods were combined and compared the difference between
the 3-year water volume and the minimum and maximum volume
was 32% and +36% respectively. The differences were increasingly
larger for monthly and daily discharge. It should be stressed that
such comparisons were obtained from testing different rating
models, that the uncertainty bounds yielded by GLUE were not taken into account and only the medians of the behavioural results
were compared.
Changing the rating parameters with each new measurement,
as in the MC moving-window analysis or allowing the a parameter
to change continuously in time, partially solved the problem, but
even then the variability might have been underestimated, especially when there were large time gaps between measurements.
We believe that these findings could be applicable for many other
alluvial river-discharge stations and that the uncertainty associated with the temporal variability in the rating curve might generally have been underestimated.
Preliminary results have indicated that it might be possible to
express one of the rating parameters as a function of stage, under
a moving-window analysis, but further research is needed to relate
all parameters to riverbed, basin or climate characteristics. We
consider that continuous measurements of discharge are the first
step needed to reach that goal.
Acknowledgements
This work was part of the projects Research cooperation in
hydrology and geotechnology – subproject Hydrology and water management of the upper Choluteca River basin and Effects of climate
change and extreme weather on water-resources sustainability in Central America funded by Sida (Swedish International Development
Cooperation Agency) under grants 75007349 and SWE-2005-296.
102
J.-L. Guerrero et al. / Journal of Hydrology 446–447 (2012) 90–102
We are grateful to SANAA (Servicio Autónomo Nacional de Acueductos y Alcantarillados) and SERNA (Secretaría de Recursos
Naturales) for access to data and their staff for collection of this
large amount of rating-curve data. The computationally demanding calculations were performed at Uppmax, Uppsala Multidisciplinary Centre for Advanced Computational Science under project
p2008030. We are grateful to Ms Diana Carolina Fuentes Andino
for help in collecting cross-section and organising some of the discharge data.
Appendix A. Supplementary material
Supplementary data associated with this article can be found,
in the online version, at http://dx.doi.org/10.1016/j.jhydrol.2012.
04.031.
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