WATER RESOURCES RESEARCH, VOL. 49, 1–16, doi:10.1002/wrcr.20533, 2013
Exploring the hydrological robustness of model-parameter values with
alpha shapes
Jose-Luis Guerrero,1,2 Ida K. Westerberg,1,3,4 Sven Halldin,1 Lars-Christer Lundin,1,5 and Chong-Yu Xu6
Received 5 June 2013; revised 11 September 2013; accepted 12 September 2013.
[1] Estimation of parameter values in hydrological models has gradually moved from
subjective, trial-and-error methods into objective estimation methods. Translation of nature’s
complexity to bit operations is an uncertain process as a result of data errors, epistemic gaps,
computational deficiencies, and other limitations, and relies on calibration to fit model output
to observed data. The robustness of the calibrated parameter values to these types of
uncertainties is therefore an important concern. In this study, we investigated how the
hydrological robustness of the model-parameter values varied within the geometric structure
of the behavioral (well-performing) parameter space with a depth function based on shapes
and an in-depth posterior performance analysis of the simulations in relation to the observed
discharge uncertainty. The shape depth is a nonconvex measure that may provide an
accurate and tight delimitation of the geometric structure of the behavioral space for both
unimodal and multimodal parameter-value distributions. WASMOD, a parsimonious
rainfall-runoff model, was applied to six Honduran and one UK catchment, with differing
data quality and hydrological characteristics. Model evaluation was done with two
performance measures, the Nash-Sutcliffe efficiency and one based on flow-duration curves.
Deep parameter vectors were in general found to be more hydrologically robust than shallow
ones in the analyses we performed; model-performance values increased with depth,
deviations to the observed data for the high-flow aspects of the hydrograph generally
decreased with increasing depth, deep parameter vectors generally transferred in time with
maintained high performance values, and the model had a low sensitivity to small changes in
the parameter values. The tight delimitation of the behavioral space provided by the shapes
depth function showed a potential to improve the efficiency of calibration techniques that
require further exploration. For computational reasons only a three-parameter model could be
used, which limited the applicability of this depth measure and the conclusions drawn in this
paper, especially concerning hydrological robustness at low flows.
Citation: Guerrero, J.-L., I. K. Westerberg, S. Halldin, L.-C. Lundin, and C.-Y. Xu (2013), Exploring the hydrological robustness of
model-parameter values with alpha shapes, Water Resour. Res., 49, doi:10.1002/wrcr.20533.
1.
ability errors between spatial and temporal scales of model
and data hinder a derivation of model-parameter values
directly from basin traits [Beven, 2002]. Most model applications require calibration, an adjustment of the modelparameter values so that observed values of at least one
flux or state variable are adequately simulated. Historically,
calibration techniques have gravitated toward finding the
parameter-value combination (parameter vector) that gives
the best result, according to one, or multiple, goodness-offit measures. Today, the uncertainties in the model calibration as a result of the time period used for calibration
[Gupta and Sorooshian, 1985; Yapo et al., 1996; Wagener
et al., 2003; Singh and Bardossy, 2012], the selected objective function [Efstratiadis and Koutsoyiannis, 2010], and
the nature and magnitude of data errors [Bardossy and
Singh, 2008; Renard et al., 2010; McMillan et al., 2011;
Westerberg et al., 2011b], are increasingly recognized and
taken into account. The robustness of the calibrated parameter values to these types of uncertainties is therefore an
important concern.
Introduction
[2] Hydrological models are important tools for water
management but their representations of real-world hydrological systems are affected by significant uncertainties.
Problems such as inadequate conceptualizations of physical
processes, uncertainties in available data and commensur-
1
Department of Earth Sciences, Uppsala University, Uppsala, Sweden.
Civil Engineering Department, Universidad Nacional Aut
onoma de
Honduras, Tegucigalpa, Honduras.
3
IVL Swedish Environmental Research Institute, Stockholm, Sweden.
4
Department of Civil Engineering, University of Bristol, Bristol, UK.
5
Deceased 9 April 2012.
6
Department of Geosciences, University of Oslo, Oslo, Norway.
2
Corresponding author: J.-L. Guerrero, Department of Earth Sciences,
Uppsala University, Villav€agen 16, SE-752 36 Uppsala, Sweden. (joseluis.guerrero@hyd.uu.se)
©2013. American Geophysical Union. All Rights Reserved.
0043-1397/13/10.1002/wrcr.20533
1
GUERRERO ET AL.: HYDROLOGICAL ROBUSTNESS AND DEPTH
Figure 1. Illustration of Delaunay triangulation, convex hull and shape. Random points (black dots)
generated between two circles of different radius (Annulus), and corresponding Delaunay triangulation
(blue lines), convex hull (red line) and alpha shape (green lines)
region encompassing the well-performing parameter values. BS08 use the half-space depth [Tukey, 1975], which is
a convex depth measure and one of many depth functions
in computational geometry [Liu et al., 1999; Zuo and Serfling, 2000; Aloupis, 2006; Lee, 2006].
[6] Thapa [2010], Kiss et al. [2011], and Singh
[2010] are examples of ROPE implementations. Cullmann et al. [2011] compare ROPE to another calibration method, PEST [Skahill and Doherty, 2006], and
find that ROPE works better in evaluation of small to
medium-sized discharge events for a model applied to
a Swiss basin. Bardossy and Singh [2011] explore the
application of ROPE in 28 British catchments for
regionalizing model parameters and identifying catchment characteristics that can be used for regionalization, and find that deep parameters values perform
better than shallow ones. Half-space data depth is also
used to identify unusual events in precipitation and discharge time series [Singh and Bardossy, 2012] and to
compare weather states [Bliefernicht, 2010].
[7] Chebana and Ouarda [2011a] use the Mahalanobis
depth to identify extremes in a multivariate extreme-value
distribution and apply their method to flood-frequency
analysis with better results than when using canonicalcorrelation analysis [Chebana and Ouarda, 2011b]. They
also suggest the use of depth functions to calculate multivariate quantiles [Chebana and Ouarda, 2011b], and to
describe and visualize multivariate data [Chebana and
Ouarda, 2011c]. In the latter example they test multiple
depth functions to identify outliers and to describe bivariate
data that combine flood peaks, durations and volumes for a
Canadian basin.
[8] The half-space and the Mahalanobis depths are convex depth measures. Considering that a convex boundary
might be ill-suited to describe the shape of a point cloud
(Figure 1), we decided to test a nonconvex depth-measure,
based on shapes [Edelsbrunner, 1992]. The two main
advantages of such a depth measure are that it does not
require the assumption, implicit in most depth functions,
that the underlying multivariate distribution is unimodal
[Liu et al., 1999; Krasnoshchekov and Polishchuk, 2013],
and that it might give a tighter delimitation of the behavioral parameter space, i.e., less space to be explored. An
added advantage is that a depth function based on alpha
shapes has potential to detect different modes in a multivariate distribution [Krasnoshchekov and Polishchuk, 2013].
[3] The term robust has multiple interpretations. Robustness can be linked to the model structure that should be as
parsimonious as possible [Klemes, 1983; Dooge, 1986;
Son and Sivapalan, 2007], to data that should have small
errors and for which the error structure is either known or
can be approximated [Beven and Westerberg, 2011], or to
the calibration procedure [Bardossy and Singh, 2008; Peel
and Blöschl, 2011]. For the latter, Bardossy and Singh
[2008] (BS08 from now on) list the following criteria for
robust parameter vectors: 1) they lead to good model performance, 2) they lead to hydrologically reasonable representations of the corresponding processes, 3) they are not
sensitive to small changes in parameter values, and 4) they
are transferable in time, leading to good model performance for other time periods and perhaps also other catchments. Model performance can in summary be expressed as
the value of the performance measure used for model calibration and evaluation, and therefore depends on the choice
of this measure [Freer et al., 2003; Krause et al., 2005].
Model performance can also be analyzed in depth to determine the ability of the simulations to represent the relevant
hydrological processes in a posterior performance analysis
against observational data, given their estimated uncertainties, e.g., McMillan et al. [2010] and Westerberg et al.
[2011b]. In this paper we used a depth measure to analyze
how hydrological robustness of calibrated model-parameter
values varied within the geometric structure of the behavioral parameter space.
[4] Data depth is a concept in computational geometry, a
discipline engendered by Shamos [1978], with links to statistical analysis since its inception. If parameter vectors in
the behavioral space are considered samples from a multivariate distribution, then the deepest point (or region of the
space) is an estimator of the distribution’s median [Aloupis,
2006].
[5] The first application of data depth to hydrology is
found in the work of Chebana and Ouarda [2008], who use
the Mahalanobis [1936] depth function to find the weights
in a weighted linear least-squares regression for regional
flood estimation in Canada. Shortly after came BS08, who
build on the results of Bardossy [2007], and find that parameter vectors deep within the well-performing parameter
space are robust in terms of sensitivity to small changes in
parameter values and transferability to other time periods.
BS08 also propose a calibration method (robust parameter
estimation: ROPE) that iteratively refines the convex
2
GUERRERO ET AL.: HYDROLOGICAL ROBUSTNESS AND DEPTH
Figure 2. The Stanford bunny (Stanford University Computer Graphics Laboratory, 2012) for increasing radii. radius ¼ 0 on the left, radius ¼ 1 (convex hull) on the right. Points in red, shape in
green. Figure 2c shows a closed polyhedron and figure 2b an open one.
[14] 1. Their use is based on the assumption that the
deepest point, i.e., the median, is unique, which might be
inadequate if the parameter-value distribution is multimodal. This might be an issue when, e.g., there exist multiple
optima in the parameter-value distribution. Most depth
functions implicitly assume unimodality in the underlying
distribution [Zuo and Serfling, 2000].
[15] 2. The Mahalanobis and the half-space depths are
based on convex sets (see also next section) for which the
depth contours corresponding to a certain depth are convex
[Dyckerhoff, 2004] and might not adequately describe the
shape of a point set (Figures 1 and 2).
[16] 3. The Mahalanobis depth relies on strong assumptions about the normality of the underlying multivariate
distribution.
[17] In this paper an alternative depth function was used,
based on shapes, that has a potential to overcome some
of the limitations above. Alpha shapes [Edelsbrunner and
M€ucke, 1994] (Figures 1 and 2) are one possible formalization of the intuitive notion of the ‘‘shape of a set of points’’
and are a generalization of the convex hull concept.
[9] In this paper we used shapes to investigate how the
hydrological robustness of calibrated parameter values varied within the geometric structure of the behavioral parameter space. We used an in-depth posterior performance
analysis that accounted for observational uncertainties in
the data to analyze how the hydrological robustness varied
with depth for model simulations for six Honduran and one
UK catchment with different data quality and hydrological
characteristics. We also wanted to find out if the shape
depth was a useful estimator of parameter depth and under
which conditions deep parameters would be hydrologically
robust. Our concrete objectives were to:
[10] 1. Investigate how the hydrological robustness of
the behavioral parameter vectors changed within the behavioral parameter space.
[11] 2. Use shapes to detect multimodality in the behavioral parameter space.
[12] 3. Analyze the potential to reduce the computational
cost of ROPE.
2.
Background
2.2. Convexity and the Half-Space Depth
[18] A convex set in Euclidean space has the property
that a line between any two points of the set lays within the
set. The convex hull of a set of points S is the smallest possible convex set containing S (Figure 1).
[19] Convex sets are attractive in computational geometry because of the relative simplicity of operations on them,
valid in any number of dimensions [Gruber and Willis,
1993]. Convex sets play a central role in the half-space
depth. The half-space depth of a point x with respect to S
will be the minimum number of points of the set on one
side of any hyperplane passing through x. The contours
delimiting regions of equal depth are convex and the outermost is the convex hull of the set [Struyf and Rousseeuw,
1999].
[20] A conceptually simpler notion of depth, on which
the shape depth is based, is a successive peeling of points
in the boundaries of the distribution, until the center or
some limit is reached. In this study we used shapes
[Edelsbrunner and M€ucke, 1994] to define the successive
boundaries and compute depth.
2.1. Depth Functions
[13] The geometric concept of depth deals with the
ranking of data : ordering points in space from shallow to
deep. Such classification requires a frame of reference,
i.e., a direction along which depth increases, and defining
(at least) one deepest point. In one dimension, calculating
depth is straightforward, since there is a single axis along
which data can be ordered. Computing depth in a multidimensional setting is much more complicated. In two or
more dimensions the answer to the problem of ranking
data is not unique [Barnett, 1976 ; Small, 1990 ; Gil et al.,
1992 ; Aloupis, 2006] : there is no single axis along which
values can be ordered. The selection of a direction of
changing depth can be subjective and the a priori definition of a deepest point might be required. Depth functions
are mathematical constructs used to calculate the depth of
a point with respect to a multidimensional set, generally
through a center-outward ordering. Different depth functions might give different results and yield differing centers and/or depth profiles [Hugg et al., 2006]. The two
depth functions that have so far been used for parametervalue estimation in hydrology are the Mahalanobis depth
and the half-space depth [Chebana and Ouarda, 2011a,
BS08]. These two functions are based on some assumptions and have some properties that could potentially limit
their scope :
2.3. Alpha Shapes
[21] From an exploratory point of view, the convex hull
can be a poor descriptor of the shape of a set of points (Figures 1 and 2). Alpha shapes, on the other hand, are well
3
GUERRERO ET AL.: HYDROLOGICAL ROBUSTNESS AND DEPTH
Table 1. Characteristics of Catchments and Discharge Time Series
Elevation (m)
Station
Period
Number of
Measurements
Paso La Ceiba
Concepcion
Guacerique
Rıo Del Hombre
Quiebramontes
Tatumbla
Brue
5 May 1980 to 25 Aug 1997
25 Jan 1989 to 27 Oct 1998
27 Nov 1981 to 20 Oct 1998
5 May 1980 to 1 Oct 1998
14 Mar 1991 to 20 Oct 1998
3 Nov 1997 to 25 Oct 1998
1 Jan 1995 to 30 Jun 1998
Variablea
315
712
470
246
292
15 minb
Area (km )
Min
Max
Mean Annual
Runoff (mm)
1760
113
150
273
106
63
135
675
1170
1079
931
1096
1075
20
2318
2235
2037
2229
2037
2000
260
350
252
391
400
391
203
439
2
a
Three measurements per day were available most of the time. The given period includes some intervals without data.
15 min discharge data were integrated to daily values of discharge.
b
and were tested on these as well as on one catchment in
Great Britain. The six catchments in Honduras are located
in the upper reaches of the Choluteca River basin and are
characterized by shallow soils, with frequently occurring
surface runoff. The Brue catchment in Great Britain has a
different flow regime where runoff is dominated by subsurface processes.
suited for such a task [Bernardini and Bajaj, 1997; Teichmann and Capps, 1998; Giesen et al., 2006], since they
make no assumptions about the nature of the underlying distribution. This could be advantageous when exploring the
behavioral parameter space, since usually little is known a
priori regarding its shape. Krasnoshchekov and Polishchuk
[2013] discuss a depth function based on shapes [Edelsbrunner, 1992] that avoids the unimodality assumption.
[22] For three dimensions, Edelsbrunner and M€ucke
[1994] give the following analogy to describe shapes :
imagine a mass of ice cream with chocolate chip cookies.
Suppose that we set out to carve this mass using a scoop
and that the chips are unmovable. An shape would be the
structure left out after any possible carving and would
depend on the size of the scoop. The radius is a Euclidean distance (the size of the scoop) that is the only parameter needed to determine the shape. For an infinite (or
sufficiently large) radius, the shape is the convex hull
of the set (Figure 2d).
[23] The computation of an alpha shape, like the convex
hull, is based on the Delaunay [1934] triangulation of S. The
Delaunay triangulation (Figure 1), in two dimensions, is
such that the circumcircles of all the triangles conforming to
it do not contain a fourth point. Finding the shape starts by
computing this triangulation and consists of testing if the
distance between any two connected points is larger than a
certain value, called the radius, in which case the connection is removed. The shape is the set (or sets) of connected
points left after all such deletions (Figure 1). This basic concept can be extended to any number of dimensions. Note
that the connected points do not necessarily define a closed
polygon, and it is important to stress that the shape is not
necessarily a closed polytope (Figure 2b). The formal definition of shapes can be found in Edelsbrunner [1992].
[24] In our application, we limited ourselves to three
dimensions, because of code availability (CGAL-3.9, Computational Geometry Algorithms Library, 2012, http://
www.cgal.org), computational limitations, and for visualization purposes. We illustrated the concept using a simplified version (CGAL-3.9, 2012) of the classic Stanfordbunny data set available from the Stanford Computer
Graphics Laboratory online repository (Figure 2) (available
at http://graphics.stanford.edu/data/3Dscanrep/)
3.
3.1. The Choluteca River Basin
[26] The Choluteca River basin in the south of Honduras
has its outlet at the Gulf of Fonseca, in the Pacific Ocean
and a small part of its 7500 km2 encroaches on Nicaraguan
territory. Six mountainous catchments in the upper part of
the basin were used in this study with five of them situated
within the larger Paso La Ceiba catchment (Table 1 and
Figure 3).
[27] Two large dams within the Paso La Ceiba catchment
supply water to Tegucigalpa, the Honduran capital, but
only the flow at Paso La Ceiba is affected by them. The
characteristics of the Choluteca River basin and the Paso
La Ceiba catchment are described by Westerberg et al.
[2010, 2011a] and Guerrero et al. [2012]. The precipitation
regime over the area is complex and highly influenced by
the meridional migration of the Inter-Tropical Convergence
Zone (ITCZ) together with the configuration of the region’s
high mountain range, which contributes to a splitting of the
local climate into a dry and a rainy season during which
around 80% of the precipitation falls. The precipitation during the rainy season, which starts around April–May and
ends around November–December, is modulated by a relative low known as ‘‘El Veranillo,’’ or the midsummer
drought, in July–August. The ‘‘Veranillo’’ is associated
with a strengthening of the trade winds, which in turn is
related to variations in the position of the ITCZ over the
eastern Pacific [Maga~na et al., 1999; Hastenrath, 2002].
Furthermore, the regime is affected by interannual climate
variations related to El Ni~no Southern Oscillation [Giannini
et al., 2001].
[28] Shallow soils with limited water retention capacity,
steep topography and the intensity of the rainfall make for
a fast rainfall-runoff response in this area. Even at the outlet of the Paso La Ceiba, the largest catchment, the delay
between precipitation in the headwater areas and peak flow
at the outlet is less than 24 h [Westerberg et al., 2011b].
[29] The discharge data were available at varying time
periods (Table 1). The precipitation and potential evaporation were interpolated for each station so that there was 1
year of data for model warm-up.
Materials and Methods
[25] The methods presented here were developed for six
catchments within the Choluteca River basin, Honduras,
4
GUERRERO ET AL.: HYDROLOGICAL ROBUSTNESS AND DEPTH
(Table 1). The uncertainty in these discharge measurements
was assumed to be of 650%, similar to the range quantified
for daily average discharge at Paso La Ceiba [Westerberg
et al., 2011a].
3.1.3. Evaporation
[33] Data from the Toncontın station (Tegucigalpa’s
international airport), were used to calculate a baseline
daily time series of potential evaporation using the
Penman-Monteith [Penman, 1948; Allen et al., 1998] formula. The Toncontın evaporation series were spatially extrapolated over the study area by means of the long-term
averages of temperature from nine other stations in the basin, which were linearly regressed against elevation. Temperature changed with elevation at a rate of 0.0055 C/m
with a correlation coefficient of 0.79. The regression was
used to extrapolate daily temperature from Toncontın to the
SRTM [Farr et al., 2007] elevation grid. This temperature
field provided a basis for the spatial extrapolation together
with the Thornthwaite [1948] equation:
ETP ¼ C
a
10T
I
[34] This equation relates monthly potential evaporation
ETP (mm) to a constant C generally
equal to 16, to monthly
average temperature T C , to a temperature index I, and
to an exponent a that is a function of I. The Thornthwaite
equation is based on a month with 30 days and a day of 12
h, and was adjusted using the actual month length and the
average daylight derived with the formula recommended
by Forsythe et al. [1995].
[35] A separate constant C was used for every month,
ranging from 19 to 26. The Thornthwaite equation was calibrated to equate the Penman values at Toncontın. The
monthly C values were assumed constant over the area and
used to calculate the Thornthwaite potential evaporation
for all extrapolated temperatures. The baseline evaporation
was multiplied by the ratio between the Thornthwaite and
Penman yearly averages at each grid point to obtain daily
estimates that were then averaged to each catchment.
Figure 3. (top) The Paso La Ceiba and nested catchments
is located in the Pacific side of Honduras. (bottom) The
Brue catchment in southwest England has its outlet at the
Lovington flow gauging station (black dot), figure from
Westerberg et al. [2011b].
3.1.1. Rainfall
[30] Precipitation data from a network of 29 rain gages
in or nearby the Paso La Ceiba catchment were spatially
interpolated using the inverse-distance weighting method.
The lack of correlation in daily precipitation at different
stations prevented the use of more complex methods [Westerberg et al., 2010]. Precipitation was interpolated to a 900
grid, coincident to the Shuttle Radar Topography Mission
(SRTM) [Farr et al., 2007] grid, and then averaged to each
catchment.
[31] Precipitation readings are normally taken at 07:00.
Since most of the rain falls during the latter half of the day
and the delay time between precipitation and runoff is less
than 24 h, the registration of the precipitation was moved
to the day of the actual measurement to agree with the daily
time step in the model [Westerberg et al., 2011b].
3.1.2. Discharge
[32] Time series of discharge and upper and lower
bounds for the estimated discharge uncertainty, calculated
with a time-variable fuzzy rating-curve method according
to Westerberg et al. [2011a], were used for the Paso La
Ceiba catchment. For the remaining stations, a large set of
momentary discharge measurements taken to determine the
stage-discharge rating were used to calibrate the model
3.2. Brue Catchment
[36] The Brue catchment is located in southwest England
and has an area of around 135 km2 (Figure 3). The land use
is dominated by grasslands, underneath which clayey soils
are superimposed on alternating bands of permeable and
impermeable rocks. The topography is characterized by
low hills of up to 300 meters above mean sea level. An
extensive precipitation data set was available from the
HYREX (HYdrological Radar EXperiment) [Wood et al.,
2000; Moore et al., 2000] project, with 49 rain gages that
were previously used to calculate the areal precipitation
over the catchment with a nearest-neighbor technique by
Younger et al. [2009]. This areal precipitation was used as
a uniform input to the hydrological model. The average areal precipitation over the catchment was 770 mm per year
for the period 1 January 1995 to 31 December 1997. Based
on the results of Calder et al. [1983], potential evaporation
was calculated using a seasonal sine curve. Discharge and
the related uncertainty for the Lovington gaging station are
presented by Westerberg et al. [2011b] using the same
5
GUERRERO ET AL.: HYDROLOGICAL ROBUSTNESS AND DEPTH
Table 2. WASMOD Equationsa
Description
Available water
Active precipitation
Actual evaporation
Slow flow
Fast flow
Routing storage
Routed flow
Total runoff
Water balance
Equation
wt ¼ pt þ smt1
if ept > 1 then
else
Parameter
Unit
Interval
pt
nt ¼ pt -ept ð1 e ep Þ
nt ¼ pt ept
t
wt
et ¼ minðept ð1 Aet ept Þ; wt Þ
pffiffiffiffiffiffiffiffiffiffiffi
st ¼ Sf smt1
Aet
ft ¼ Ff smt1 nt
sct ¼ sct1 þ ft
rt ¼ Rf sct
dt ¼ minðst þ rt ; wt et Þ
smt ¼ smt1 þ ðpt et dt Þ
sct ¼ sct rt
½0; 1
Ff
pffiffiffiffiffiffi
mm
day
1
mm
½e7 ; e4 Rf
1
day
½0; 1
Sf
½e9 ; 1
a
In which t is the tth day, pt is precipitation, ept is potential evaporation, smt is soil moisture at the end of the tth day. Model parameters are indicated in
bold, input data are indicated in roman, and state variables and fluxes are indicated in italics. Input data and state variables and fluxes are given in mm
and mm/day.
in the discharge data, enables simultaneous calibration to
the whole range of flows, and is less sensitive to some types
of epistemic data errors. However, in some circumstances
the calibration provides little constraint for timing errors
and a posterior performance analysis should therefore be
performed to analyze if the FDC provides sufficient constraints for the particular model and data used. In calibration, the simulated FDC is compared to the uncertain
observed FDC, which is based on the estimated uncertainty
in the discharge data, at selected evaluation points on the
curve. Behavioral simulations are required to be inside the
uncertainty in the observed FDC at all these points and a
performance measure is calculated by weighting the deviations from the best-estimated discharge within the uncertainty bounds. Similarly to Westerberg et al. [2011b], the
evaluation points were calculated by dividing the area
under the curve (representing the total volume of water
contributed by flows smaller than or equal to a given magnitude) into 20 equal intervals, disregarding the extremes
(i.e., 19 evaluation points). If not specified otherwise, the
performance measures were calculated with all available
discharge data (Table 1). The FDC performance measure
ranges from zero to one, where one represents a perfect fit
to the best-estimated discharge at all evaluation points and
zero means that the simulation falls outside the uncertainty
bounds for at least one evaluation point, i.e., it is
nonbehavioral.
method as for the Paso La Ceiba catchment, but in this case
with a constant rating curve.
3.3. Hydrological Model and Simulations
[37] WASMOD [Xu, 2002] is a parsimonious, lumped,
conceptual, rainfall-runoff model based on the variablesource-area concept (Table 2). Originally a monthly waterbalance model [Xu et al., 1996], it has evolved and has
been applied at a global scale [Widen-Nilsson et al., 2009],
at a daily time step [Westerberg et al., 2011b], and in
regions of widely varying climate conditions [Xu et al.,
2006; Li et al., 2011; Kizza et al., 2011]. We used the
same version of the model as Westerberg et al. [2011b]
who present good results for the Paso La Ceiba catchment
with simulations that have high reliability and good precision for five different aspects of the hydrograph (peak
flows, base flows, rising limbs, falling limbs and troughs).
[38] A first round of 106 Monte-Carlo (MC) simulations
was performed for all stations using all available discharge
data for calibration, except when performing split–sample
testing. The empirical cumulative distribution functions of
behavioral parameter values were compared and the parameter Aet was considered fixed to 0.7 in a second round of
107 simulations that were the basis for the geometrical
analyses since the software only allowed analysis in three
dimensions or less. In order to assess the impact of fixing
the Aet parameter on streamflow simulations, 107 random
MC simulations with a free, randomly varying, Aet parameter were also performed, but not used for depth
calculations.
3.5. Alpha Shapes Calculations
3.5.1. Multimodality Detection
[41] A depth function based on shapes can give an
idea about the median of each mode in a multimodal distribution [Krasnoshchekov and Polishchuk, 2013]. Multimodality can be reflected as disconnected components in
the shape. Since the shape depends on the Euclidean
distance between points, we first scaled the sampling
ranges for all parameters (Table 2), to [0 ;1]. Multimodality was analyzed by examining the shape with radius
equal to the median of the distances between the connected points of the Delaunay [1934] triangulation of all
(107) MC parameter vectors. The subjective selection of
the radius to detect multimodality was based on two
competing factors : (i) the radius should be large enough
to avoid excessive fragmentation, and (ii) the radius
3.4. Performance Measures
[39] The Nash-Sutcliffe efficiency (NSE) [Nash and Sutcliffe, 1970] is one of the most widely used performance
measures in hydrology, despite being much debated [see
Westerberg et al., 2011b, and references therein], and was
included as a baseline in our model evaluations using subjectively a behavioral threshold of 0.6.
[40] Model performance was also evaluated using flowduration curves (FDC) within the extended Generalised
Likelihood Uncertainty Estimation (GLUE) limits-ofacceptability framework, according to the procedure
described by Westerberg et al. [2011b]. This allows modelrejection criteria to be based on the estimated uncertainty
6
GUERRERO ET AL.: HYDROLOGICAL ROBUSTNESS AND DEPTH
the vertices of BAS1 to the average performance of the circumcenters of the boundary triangles.
[53] 2. How sensitive is model performance to displacements of parameter vectors near the boundary between the
behavioral and the nonbehavioral parameter space ? The
positive and negative unit normal vectors of the oriented
triangles forming BAS1 were found and defined a direction
inward and outward the behavioral space. The model performance for parameter vectors at different distances along
both normal vectors was evaluated.
[54] The model’s ability to represent the hydrological
processes in the catchments was then evaluated by comparing observed and simulated hydrographs for each catchment
in a posterior performance analysis. For the Brue and Paso
La Ceiba catchments where continuous data were available,
six different aspects of the hydrograph were analyzed : base
flows, troughs, small peaks, large peaks, rising limbs, and
falling limbs. These aspects were identified similarly to
Younger et al. [2009] and Westerberg et al. [2011b]. Base
flows were defined as flows smaller than 1.7 m3/s at Brue
and 5 m3/s at Paso La Ceiba, large peaks defined as peak
flows larger than 10 m3/s at Brue and 100 m3/s at Paso La
Ceiba, and the time-window parameter for the identification
of the other aspects set to 1 day at Brue and 3 days at Paso
La Ceiba. The posterior performance analysis consisted of:
[55] 1. Calculation of summary-evaluation measures :
Reliability (percentage of time the observed and simulated
uncertain bounds overlapped) and Precision (the width of
the overlapping range as a percentage of the width of the
simulated bound and then calculated as an average for the
whole time series). The range for both measures is between
zero and 100%, and the measures were calculated both for
the entire time series and for the different hydrograph
aspects. The uncertainty bounds for the simulated discharge
were taken to be the 5% and 95% percentiles of the
likelihood-weighted distributions of the simulated discharge for every time step for all behavioral parameter vectors. The Reliability and Precision measures were
calculated both with a freely varying Aet parameter and
with a constant Aet. This provided a basis to assess the
impact of a constant Aet on the discharge simulations.
[56] 2. Analyses of scaled scores for different depths and
different hydrograph aspects [Westerberg et al., 2011b]. The
scaled scores describe the deviation between a simulated time
series from a single parameter vector and the observed discharge with its uncertainty bounds. A simulated value that is
exactly at the best-estimate discharge has a scaled score of
zero, one that is at the lower observed uncertainty bound a
scaled score of minus one, and one that is at the upper observed
uncertainty bound has a scaled score of one. Values inside/outside the bounds are linearly interpolated/extrapolated.
[57] Finally, we analyzed how the temporal transferability of the behavioral parameter vectors varied with depth
by performing split-sample tests at Paso La Ceiba and
Brue, where sufficient discharge data were at hand
(Table 1). At Paso La Ceiba the model parameter values
were calibrated in 1980–1988 and evaluated in 1989–1997
and vice versa. At Brue the model parameter values were
calibrated in January 1995 to June 1997 and evaluated in
July 1997 to June 1998 as well as calibrated in January
1996 to June 1998 and evaluated in January 1995 to December 1995.
should be small enough to allow multimodality detection
given the sampling density.
3.5.2. Data-Depth Calculations
[42] From a finite set S of points in Rd , where d is the
number of dimensions, a finite number of shapes can be
derived and all are subsets of the underlying Delaunay triangulation (Figure 1). The number of radii for which the
shape experiences topological change is finite, and can
be determined from the triangulation. Whereas a convex
hull divides space into an interior and an exterior region, an
shape does not necessarily do so (Figure 2b). The boundary shape (BAS) of S was defined to be, in R3 , the shape with a single connected component with smallest
possible radius such that the shape had neither dangling edges (1–simplex) nor faces (2–simplex), thus forming a closed polyhedron (Figure 2c). The BAS was found
through a binary search on the ordered set of topologically
relevant radii until a polytope fulfilling the given conditions was identified : a closed polyhedron.
[43] The depth of a single parameter vector with respect
to the behavioral set was determined as follows:
[44] 1. Let S be the set of behavioral parameter vectors
and BASn the BAS for depth n.
[45] 2. Find the BAS1 of S. Parameter vectors belonging
to BAS1 have depths equal to one and all parameter-vectors
on or inside BAS1 have a depth 1.
[46] 3. Eliminate the parameters belonging to the BAS1
from S.
[47] 4. Increase depth by one.
[48] 5. Repeat steps 2–4, until no points are left or BASn
cannot be found.
[49] All parameter vectors outside BASn ¼ 1 have zero
depth. Iterations were stopped with 200 points remaining.
This can be thought of as peeling the layers of an onion,
where each successive layer is an shape and each peel a
unit increase in depth.
3.6. Hydrological Robustness and Depth
[50] The hydrological robustness of parameter vectors at
different depths was analyzed from four different aspects
based on the criteria given by BS08: model performance,
sensitivity to parameter-value changes, hydrologically reasonable process representation, and transferability in time.
These analyses were performed within an uncertainty estimation framework, accounting for estimated uncertainties
in discharge. Where the number of behavioral Monte Carlo
simulations was larger than 100,000, the depth computations were limited to the first 100,000 randomly sampled
behavioral simulations because of computational limitations related to memory constraints.
[51] First we explored how the performance-measure values varied with depth in the behavioral parameter space,
both concerning average performance and variability in performance. Does performance increase when moving from
shallow to deep parameter vectors? The performance for the
points on the vertices of BASn was the baseline. Then we
examined the sensitivity of the model performance to a
change in the parameter vector. Two aspects were tested:
[52] 1. Is the performance sensitive to lateral displacements (as opposed to displacements toward the center)? In
R3 the boundary simplices of the BAS are triangles and
this test consisted in comparing the average performance of
7
GUERRERO ET AL.: HYDROLOGICAL ROBUSTNESS AND DEPTH
4.
Results
4.1. Simulations, Multimodality, and Depth
Computations
[66] The number of behavioral parameter vectors differed
from station to station and no behavioral parameter vectors
were found at all for NSE at one of the stations (Figure 4). It
is not possible to directly compare the number of behavioral
simulations between NSE and FDC, since the behavioral
threshold was subjectively selected to 0.6 for NSE.
[67] The behavioral parameter spaces were spatially
compact and no signs of multimodality were found for the
selected criteria. The space occupied by the behavioral parameter vectors was partly different for the two performance measures (e.g., Figure 5).
[68] The number of depths that could be computed was
related to the number of behavioral simulations. All random parameter vectors inside BAS1 were found to be behavioral, i.e., there were no regions with nonbehavioral
results inside BAS1, for the given sampling density, thus
showing that the behavioral space was well structured in
three dimensions.
4.2. Hydrological Robustness and Depth
4.2.1. Model Performance
[69] Average performance generally increased with
depth (Figures 6 and 7). The increase in performance was
accompanied by a decrease in the standard deviation of the
values of the performance measures. Deep parameter vectors did not always include the best-performing ones from
the MC simulations. This confirmed the BS08 results that
average model performance increases while performance
variability decreases going from shallow to deep parameter
vectors.
Figure 4. Histograms of the performance-measure values
for the behavioral parameter vectors for calibration using all
available data. The values inside the plot-boxes are the number of behavioral parameter vectors from 107 simulations.
3.7. Potential Efficiency of Alpha Shapes
[58] Alpha shapes can be better descriptors of the shape
of a set of points than their convex hull. The hyper-volume
of an shape is, if it exists, at most equal to the hypervolume of the convex hull. The ROPE (BS08) calibration
technique requires iterative MC simulations of points inside
a convex hull. The smaller volume of shapes therefore
has the potential to speed up the ROPE process:
[59] 1. Define a hyper-rectangle bounding the parameter
space.
[60] 2. Perform MC simulations with parameter vectors
sampled inside the rectangle.
[61] 3. Select well-performing parameter vectors.
[62] 4. Find the convex hull of the well-performing parameter vectors.
[63] 5. Perform MC simulations with parameter vectors
sampled from the inside of the convex hull.
[64] 6. Repeat steps 3–5 until the improvement in performance does not change more than expected from observation errors.
[65] Step 4 of the ROPE procedure could be replaced by
‘‘select the BAS of the well-performing parameter vectors,’’ seeking to diminish the search space for the MC simulations by using alpha shapes. Consequently, step 5 would
be modified to ‘‘Perform MC simulations with parameters
inside the BAS.’’ In order to assess the improvement potential, we compared BAS volumes at different depths to the
corresponding convex-hull volumes.
Figure 5. BAS1 at Paso La Ceiba for NSE (green/transparent) and FDC (red/solid) for calibration using all available data. Axis-aligned bounding rectangle shown. Rt—
routing, Sf—slow flow, and Ff—fast flow are WASMOD
parameters. The shape shown here was obtained by scaling the behavioral range (instead of the sampling range) to
[0;1] to aid visualization. The limits of the bounding rectangles are: Rt [0.11–1], Sf [0.0001–0.19], and Ff [0.0009–
0.0056].
8
GUERRERO ET AL.: HYDROLOGICAL ROBUSTNESS AND DEPTH
Figure 6. Average and standard deviation of the performance-measure value versus depth for calibration using all available data.
at equal depth exhibited a similar degree of sensitivity
(Table 3).
4.2.3. Process Representation
[73] The process representation was analyzed for the
Brue and Paso La Ceiba catchments where continuous
measurements of discharge were available.
[74] The reliability of the simulations varied between
73% and 83% as an aggregated measure, but could span a
wider range for specific hydrograph parts (Figure 9). In
particular both performance measures led to parameter
vectors for which the model did a poor work in reproducing low flows (Figure 10). This was related to the fixed
evaporation parameter. The low-flow simulations were
poorer with a fixed parameter than with a varying one for
both catchments and objective functions (Figure 9). However, a poorer model performance in the end of the dry
season was also observed as a probable model-structural
error in a previous study when using a variable evaporation parameter [Westerberg et al., 2011b]. For other parts
of the hydrograph (large peaks, small peaks, troughs, rising and falling limbs) the difference between the fixed
and the varying evaporation parameter was small. The
performance of WASMOD was better for Paso La Ceiba
than for Brue and the other Honduran catchments. It is
reasonable to believe that the three-parameter model was
too simplistic at Brue and that the much larger precipitation and discharge uncertainties for the small Honduran
4.2.2. Sensitivity to Parameter-Vector Changes
[70] The sensitivity to parameter-vector changes was
generally found to decrease with depth (Figure 7). Changes
in performance tended to become smaller with depth and
were mostly positive, reflecting a performance increase
with depth (Figure 7). Even in the worst cases, the change
in performance with depth was not significantly different
from zero (Figure 7). Deep parameter vectors tended on average to perform better than shallow ones and the model
was also less sensitive to changes in their values.
[71] The change in performance for lateral parametervalue changes showed that some of the circumcenters were
not behavioral and the performance at the circumcenters
was found to be, on average, different than the performance
at the vertices, meaning that BAS1 was only an approximation of the behavioral limit. The performance change was
small but not negligible for lateral parameter-vector
changes but the small magnitude of these changes in comparison to the change in performance with depth showed
that BAS1 was a good approximation of the behavioral
boundary (Table 3).
[72] The performance dropped when moving away from
BAS1 whereas it increased moving inward (Figure 8). The
drop was another indicator of the ability of BAS1 to capture
the behavioral limit. In conclusion, deep parameter vectors
appeared less sensitive than shallow ones to small
parameter-value changes (Figure 7) and parameter vectors
9
GUERRERO ET AL.: HYDROLOGICAL ROBUSTNESS AND DEPTH
during the 1980–1988 period in combination with the fixed
evaporation parameter. Calibrating/evaluating during periods with different characteristics when the low-flow behavior of the model was constrained might explain the drop in
performance with depth during evaluation. The FDC performance measure requires adequate simulation of the
whole flow range (which NSE does not), which would lead
to lower performance if the low-flow parts of the FDCs for
the two periods have little overlap.
[77] The different parts of the hydrograph where the
deviations increased/decreased with depth were mostly
similar between calibration/evaluation periods for both performance measures, although in a few cases there was a
decrease in deviations with depth (i.e., better results) in calibration but not in evaluation.
4.3. Delimitation of the Behavioral Parameter Space
[78] Alpha shapes constrained the behavioral parameter
space better than the corresponding convex hulls (Figure
13). The reduction in volume between the space delimited
by a BAS could be as low as 29% of the volume of the corresponding convex hull. The average volume of the shapes was around 83% of the convex-hull volume for
NSE and 74% for FDC (Figure 14).
5.
Discussion and Conclusions
[79] This study supported results from previous studies
that hydrological robustness of parameter vectors appears
to increase with depth in the behavioral parameter space.
Deep parameter vectors for six Honduran and one UK
catchment were found to be more robust than shallow parameter vectors with regards to both average model performance and sensitivity of the performance to changes in
parameter vectors. The increase in performance with depth,
as reflected by the numerical value of the two tested performance measures was undeniably clear, but the posterior
performance analysis showed that simulated flows could
actually become worse or show little change with depth for
some parts of the hydrograph. This was especially the case
for low flows, where the conclusions that could be drawn in
this study were affected by the fixed evaporation
parameter.
[80] To the best of our knowledge only two depth functions have previously been used for parameter estimation in
hydrology, (a) the Mahalanobis distance that makes strong
assumptions regarding the data distribution, and (b) the
half-space depth. Both are convex depth measures that
assume unimodality. Here we have introduced a third depth
Figure 7. Change in performance with depth (n) for calibration using all available data, Dper ðnÞ ¼ per ðBAS nþ1 Þ per ðBAS n Þwhere per ðBAS n Þ is the average performance
of the model evaluated on the vertices of BASn. The black
lines show a linear fit with the slope significantly (95%
confidence interval) different from zero (continuous line)
or not (dash-dot line). The horizontal gray lines are Dper
(n) ¼ 0.
catchments compared to Paso La Ceiba caused this
difference.
[75] The deviations (measured by the scaled scores) in
general decreased with depth for the same parts of the
hydrographs (large peaks, small peaks, and falling limbs)
for both performance measures, sometimes even under
split-sample testing (Figure 11). For other parts of the
hydrograph (rising limbs, troughs and base flows), there
was no clear decrease in the deviations or instead worse
results with depth. This may have been a result of the fixed
evaporation parameter. It was therefore difficult to draw
conclusions about how the hydrological robustness varies
with depth for low flows.
4.2.4. Transferability in Time
[76] Deep parameter vectors transferred better than shallow ones with NSE as performance measure, but not for
FDC, except in one case (Figure 12). In two of the tests
with FDC as performance measure, the average performance increased with depth initially but ended up decreasing.
At Paso La Ceiba, the comparatively poorer results
obtained with the FDC performance measure compared to
NSE might have been related to generally lower discharge
Table 3. Change in Average Efficiency From Vertices to Circumcenters of the Triangles Forming BAS1, the Approximate Boundary of the Behavioral Space
10
Station
NSE
FDC
Paso La Ceiba
Concepci
on
Guacerique
Rıo Del Hombre
Quiebramontes
Tatumbla
Brue
0.06
0.07
0.04
0.01
0.05
0.06
0.10
0.12
0.06
0.11
0.03
GUERRERO ET AL.: HYDROLOGICAL ROBUSTNESS AND DEPTH
Figure 8. Change in average performance near BAS1 for calibration using all available data. The median length of the edges of the Delaunay triangulation of the Monte Carlo parameter vectors was taken
as the unit distance d. V was defined as the set of positive normal vectors to the facets of BAS1 with
norm d whose tails are at the circumcenter of the facets. The average performance was evaluated by scaling V and calculating the average performance of the heads of the vectors. The scaling factor multiplier
is given in the logarithmic y-axis
Figure 9. Reliability and precision for the total time series and for different parts of the hydrograph.
The different hydrograph aspects are defined in Westerberg et al. [2011b].
11
GUERRERO ET AL.: HYDROLOGICAL ROBUSTNESS AND DEPTH
Figure 10. (top) Observed and simulated discharge at Paso La Ceiba in January 1983 to February
1984. The uncertainty in the observed daily discharge is shown in gray shading. The lower and upper
limits for the simulated bounds for the NSE and FDC performance measures (black and blue lines) are
the 5% and 95% percentiles of the likelihood-weighted distribution of the simulated discharge for each
day for all behavioral parameter vectors. The mean monthly observed discharge (gray line, bottom left,
and zoomed for the dry season, bottom right) is the average of the best-estimate daily observed discharge
for 1980–1997. The monthly simulated discharge (black and blue lines, bottom plots left and right) is the
averaged median discharge from all behavioral simulations.
of depth, when prediction reliability for another time period
is of interest.
[83] The temporal transferability was lower when calibrating/evaluating for periods with differing flow characteristics. This was particularly the case when using the FDC
performance measure, which, in contrast to the NSE, constrains the whole flow range. The evaporation parameter
was fixed to 0.7, likely impacting the performance more
negatively when evaluating the model with the FDC criterion than with NSE since the evaporation parameter primarily affected low-flow simulations.
[84] No multimodality was detected within any behavioral parameter space. However, multimodality could be an
issue in other cases where different performance measures
are used. Further studies should be undertaken to see if
multimodality could be an issue for nonparsimonious and
complex models with many parameters and/or a markedly
nonlinear model structure.
[85] The spatial structure of the behavioral parameter
space was shown to be compact and the volume efficiently
delimited by the shapes algorithm. Volume reduction
was significant between the shapes and the convex-hull
methods. The potential improvement in computational efficiency from this volume reduction depends on the benefits
of sampling from a smaller space, i.e., having to perform
less simulations to attain the same sampling density or,
alternatively, sampling the space more densely with the
same number of simulations. However, computing an shape is more demanding than computing a convex hull,
and sampling inside an shape also requires more
measure, which is not affine invariant, but makes no
assumptions about the underlying distribution and is better
suited to describe the shape of the behavioral parameter
space. Affine invariance means that the results are independent of the underlying coordinate system under affine
transforms of space, i.e., those preserving linearity.
[81] The study was limited by computational restraints
since only a three-parameter model could be used, and the
evaporation parameter had to be fixed. This affected model
simulations and large deviations were sometimes found
between observations and simulations, especially for the
Brue catchment that has a more complex rainfall-runoff
relationship than the Honduran catchments. Better simulation results for low flows were found in a previous study
[Westerberg et al., 2011b] for the Paso La Ceiba catchment
using the same model with an unconstrained evaporation
parameter, whereas there was less difference for peak
flows. The possible model-structural errors imposed by this
restriction were therefore significant and hindered conclusions to be drawn about hydrological robustness for low
flows.
[82] The exploration of the behavioral parameter space
shed light on how robustness of different parameter vectors
varied in space and could aid in model identification.
Because of the different types of model and data errors
affecting the model calibration, different parameter vectors
will affect simulations of various aspects of the hydrograph
differently. In prediction, the realization of errors will often
be different to the calibration period and it could be recommended to use all behavioral parameter vectors, regardless
12
GUERRERO ET AL.: HYDROLOGICAL ROBUSTNESS AND DEPTH
Figure 11. Scaled scores for different parts of the hydrograph for different depths for Paso La Ceiba in
the evaluation period 1989–1997 with NSE as performance measure. The gray patch shows the [1;1]
interval that defines the uncertainty in the observed data (1 at the lower bound, 0 at the best-estimate
discharge and 1 at the upper bound) and the colored bars show the change in the distribution of scaled
scores with depth, going from blue to green with increasing depth.
Furthermore, the detection of multimodality as disconnected components in the shape does not necessarily take
into account the performance within disconnected regions
and should therefore not be viewed as a method to detect
local optima.
[88] 2. Not being affine invariant also means that different results might be obtained if the underlying coordinate
system is changed, e.g., by a different sampling range for
MC simulations.
[89] 3. The results presented here were limited to three
dimensions, but can be extended to higher dimensional settings. Alpha shape codes are freely available (CGAL-3.9,
2012), up to three dimensions. Alpha shapes are subsets of
Delaunay triangulations, for which high-dimensional codes
have been developed, which could facilitate further developments of the concept. Half-space depth and Mahalanobis
depth algorithms are also freely available and can be found
as, e.g., modules to the R language.
[90] BAS1 was only an approximation of the behavioral
boundary. It depended on the density of the sampling and
the coordinate-system scale. In order to get more consistent
estimates, a tactic other than random sampling should be
applied. For instance, the parameter space could be evenly
sampled through, e.g., a tessellation (a partition of the sampling space in a countable number of closed sets), and the
operations than sampling inside a convex hull. Therefore,
the computational advantage will depend on the advantages
of sampling inside a smaller space compared to the additional costs. Computation of data depth is increasingly
demanding when the parameter-space dimension increases
and only brute-force algorithms are presently available for
many cases. The operational complexity of geometric
methods generally increases faster than exponentially with
the number of dimensions. We believe that this will be a
limiting factor to the practical application of methods like
the shapes and are currently testing a computationally
lighter depth function, based on convex hulls, that allows to
increase either the dimensionality or the number of sample
points.
[86] Besides computational considerations, other factors
should be taken into account when selecting a depth
function:
[87] 1. The ability of the shape depth function to detect
multimodality but not to be affine invariant. Zuo and Serfling [2000] suggest that a choice should be made, regarding modality, before selecting a depth function. If
multimodality is not expected then functions such as the
half-space depth have added inferential power. The ability
to detect multimodality comes at the cost of not fulfilling
all conditions desirable in a statistical depth function.
13
GUERRERO ET AL.: HYDROLOGICAL ROBUSTNESS AND DEPTH
Figure 12. Change in performance with depth for splitsample tests at Paso La Ceiba (first row) and Brue (second
row). The calibration/evaluation periods are defined in section 3.6.
density of the sampling could be increased near the iteratively refined boundary in order to get a better delimitation
of the behavioral space. Deep parameters could then be
found through a geometric study of the resulting structure,
instead of the random-sampling approach used here.
[91] This paper has elucidated the link between data
depth in the behavioral space and hydrological model
robustness. BS08 have previously demonstrated the existence of this link. The reason behind it remains unclear.
Deep parameters are robust in a geometrical sense: the median, which is the deepest point in a given space, is a robust
estimator of location. Deep parameter vectors, i.e., close to
Figure 14. Box plots of the ratios between the volumes
of BAS and the corresponding convex-hull volumes at different depths for calibration using all available data at the
stations.
the median, have again appeared to be more hydrologically
robust than shallow parameter vectors. Elucidating the reason for this should be the subject of further research. Since
the functions in our model are monotonic, we speculate
that parameter vectors on the edge of the behavioral space
relate to extremes while deep parameters might provide a
more balanced fit.
[92] Acknowledgments. 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. The
computationally demanding calculations were performed at Uppmax, Uppsala Multidisciplinary Centre for Advanced Computational Science under
project p2008030 (flyttad mening). We are grateful to SANAA (Servicio
Aut
onomo Nacional de Acueductos y Alcantarillados) and SERNA (Secretarıa de Recursos Naturales) for access to the Honduran data and to Philip
Younger for providing the areal precipitation data for the Brue catchment,
as well as the figure describing the catchment. Professors Hoshin Gupta
and Alberto Montanari and five anonymous reviewers helped to improve
the quality of this article. The Editorial Office speeded up publication with
their prompt responses.
Figure 13. Convex hull (gray shading) and BAS1 (green
solid) at Quiebramontes, for calibration on the whole time
period using FDC as the performance measure. Axisaligned bounding rectangle shown. Rt—routing, Sf—slow
flow, and Ff—fast flow are WASMOD parameters. The shape shown here was obtained by scaling the behavioral
range (instead of the sampling range) to [0;1] to aid visualization. The limits of the bounding rectangles are: Rt [0–
1], Sf [0.0001–0.13], and Ff [0.0009–0.0136]
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