ESTIMATION OF REGIONAL PARAMETERS IN A MACRO SCALE
HYDROLOGICAL MODEL
Kolbjørn Engeland1, Lars Gottschalk2 and Lena Tallaksen3
Department of Geophysics, University of Oslo, P.O. Box 1022 Blindern,
N-0315 Oslo, Norway, e-mail: 1kolbjorn.engeland@geofysikk.uio.no
2
lars.gottschalk@geofysikk.uio.no 3lena.tallaksen@geofysikk.uio.no
INTRODUCTION
Regional hydrological modelling or hydrological macro modelling implies a
repeated use of a model everywhere within a region using a global set of
parameters. Observations for calibration and validation of the model are only
available at a subset of sites where the model is applied. For sites without
observations the model application needs to be based on the global parameters.
In a distributed hydrological model the parameters are determined for each
fundamental unit (grid cell) of the model based on physiographic factors, e.g.
topography, soil type and vegetation class of the units. The same global
parameter values are used everywhere where the physiographic factors for a
fundamental unit falls into the same classes.
Parameters of lumped conceptual models can be regionalised by using
multiple regression to relate them to catchment characteristics. In the split
sample and proxy basin tests (Klemes, 1986), the model is calibrated on some of
the data and then validated on data from time periods and catchments not used in
the calibration. Neither the regression method nor the split sample and proxy
basin tests consider the non-uniqueness of acceptable parameter sets. A striking
result from Motovilov et al. (1999a; 1999b) was the variation in performance
criteria between different years and different catchments. Regional parameters
ought to be robust, physical based, and have a low dependence on the catchment
and time period used for calibration.
A conclusion that can be drawn from earlier studies is that more formal
procedures are needed for accepting a model for regional application as well as
in the search for regional parameters. In the hydrologic literature there are at
least two approaches that can serve as appropriate tools - the multi-objective
method (Gupta et al., 1998) and the Bayesian method (Binley and Beven, 1991).
In the first case the model is executed for several possible parameter sets and
catchments. On the basis of one or several error criteria it is possible to judge
which parameter sets give acceptable simulations and which do not. The method
provides a decision rule as how to select the parameter sets that perform
satisfactory for all catchments. The result will be several possible parameter
sets. The Bayesian method aims to estimate a probability distribution of the
parameters. Parameter sets are given likelihoods based on a quality measure
describing the goodness of fit between observed and simulated values. These
likelihoods can be used for updating the probability using additional data
following Baye’s theorem. The data can be streamflow from an additional year
or from an additional catchment. Both the multi-objective method and the
Bayesian method consider the uncertainty in the choice of parameter values.
Instead of letting the user or an optimisation routine decides the one and only
optimal parameter set, several parameter sets are considered as likely.
In this study the Bayesian method has been selected for analysing the
performance of the ECOMAG model for the NOPEX region. Streamflow data
from several catchments are used to update the probability distributions of the
parameters. For a model to "perform satisfactory" in a regional context it might
be expected that:
• The shape and the optima of the parameter distribution do not depend too
much on catchment or year used for estimating the distribution.
• The model result are sensitive to the parameters (objective criteria).
• The variance of the parameter distribution decreases as additional streamflow
data (from an additional year or catchment) are used for updating (follows
from the two points above).
• A performance criteria (here the Reff Nash-Sutcliffe coefficient (Nash and
Sutcliffe, 1970)) calculated for the optimal regional parameter set for each
catchment should be higher than some lowest acceptable value (here
minimum 0.75 is classified as a good result and between 0.75 and 0.36 as
satisfactory results).
An introductory part of the paper presents in brief the main features of the
study area, the ECOMAG model and the basic data sets used. It is followed by a
description of the theoretical concepts of the Bayesian method and the
application of this method for construction of two dimensional parameter
probability distributions. The purpose of this paper is to study the applicability
of the ECOMAG model as a macro scale hydrological model using the Bayesian
method as well as the quality and quantity of the NOPEX data for use in
regional hydrological modelling.
CATCHMENT AND MODEL DESCRIPTIONS
The NOPEX Area
The NOPEX area (Halldin et al., 1995) is situated in the southern part of
Sweden, north-west of Uppsala. It is an area of low relief with altitude ranging
from 5 to 145 m.a.s.l.. Till and clay are the most common soil types, and
coniferous forest and agricultural land are the dominating surface covers
(Seibert, 1994).
Data from the regular climate and discharge observation network of the
Swedish Meteorological and Hydrological Institute in the NOPEX area were
used for the period 1981-1990. The data set contains daily values from 10
streamflow stations (Fig. 1), 25 precipitation stations, 7 temperature stations and
5 stations for vapour pressure deficit. Temperature and vapour pressure deficit
were interpolated to a regular 2km grid by inverse distance weighting and the
precipitation was interpolated by kriging (Motovilov et al., 1999a).
Figure 1. NOPEX area and the ten gauged catchments.
The ECOMAG Model
The ECOMAG model (Motovilov et al., 1999a; 1999b) describes the main
processes of the land surface hydrological cycle: infiltration, evapotranspiration,
heat and water regime of the soil, snowmelt, formation of surface, subsurface,
groundwater and river runoff.
The region is divided into grid cells, and the same model algorithms are
applied on each cell. In this application a grid size of 2x2 km is used. The
vertical structure of the model includes two layers, an upper layer called horizon
A and a lower groundwater zone. Each grid cell is assigned a soil class and a
vegetation class, and some of the parameters are determined from the respective
classes, whereas the rest of the parameters are common for the whole region. To
reduce the number of calibrated parameters, the relative differences between
corresponding parameters for soil and vegetation classes are assumed to be
constant, and the standard parameter values of a class are instead multiplied by a
common calibration factor.
Application of ECOMAG to the NOPEX region
The present study is based on the work by Motovilov et al.(1999a; 1999b)
where a regional calibration and validation of ECOMAG was done following the
proxy basin scheme suggested by Klemes(1986). As a first step the model was
calibrated using standard meteorological and hydrological data for seven years
for three basins. An additional adjustment of the soil parameters was performed
using soil moisture and groundwater level data from five small experimental
catchments. The model was validated against streamflow for 14 years from six
other catchments, synoptic streamflow and evapotranspiration measurements
performed during two concentrated field efforts in 1994 and 1995.
THE BAYESIAN METHOD
The Bayesian approach used here is mainly as described by Binley and Beven
(1991), but the notation from Tanner (1996) has been adopted. The method aims
to establish a multi-dimensional probability density distribution for the
parameters dependent on observations. The posterior probability, p, of the
parameter set θi, given observations Y is:
P(θ i Y ) =
L(θ i Y )p (θ i )
C
(1)
where p(θi) is the prior probability for the parameter set, L( θi |Y ) the likelihood
function calculated from the observations Y, and C is a scaling constant making
the cumulative sum equal one. Streamflow data from an additional period or a
neighbouring area are used for updating the probability distribution p(θi |Y ). An
empirical, non-parametric probability distribution is established by performing a
sampling of Eq. (1) in the parameter space.
Estimation of Likelihood and Prior Probability Distribution
First a likelihood function for the parameter sets has to be estimated. The
formula used here is:
 σ i2
L(θ i Y ) ~ exp − 2
 σ obs



(2)
where L(θi|Y) is the likelihood of the ith parameter set θi, given the observations
Y, σi2 is the sum of squared simulation errors divided by number of time steps in
a period and σobs2 is the observed variance over a period. In this study the
likelihood function is calculated for a period of one year, 1. June - 31. May.
Repeated use of equations 1 and 2 results in the following expression for the
posterior probability function given n set of observations:
  σ2 
 σ 2 
p(θ i Y1 ....Yn ) = exp −  2i ,1  − .... −  2i ,n 


σ

  σ obs ,1 
 obs ,n 
(3)
The prior probability distributions are chosen to be uniform except for the
conductivity parameters. Since the conductivity often follows a log-normal
distribution, a log-uniform distribution is chosen. The optimised parameters
from Motovilov et al. (1999a) were restricted to be within the upper and lower
limits of the uniform or log-uniform distributions.
Value of Additional Data
The normalised Shannon entropy measure H (Eq. (4)) describes the variance
of a multi-dimensional probability function. It is used as a numerical measure of
the advantage of using additional data in the estimation of the probability
distribution of the parameters:
M
H =−
∑p
i =1
i
log p i
log M
(4)
where the probabilities pi, are scaled making their sum equal to one, and M is the
total number of simulations. This function has a maximum at 1 when all the
realisations have the same probability, and a minimum at zero when one
realisation has a probability of 1 and all others are zero.
The way H develops as more observations are used is dependent of the
probability function in Eq. (1) and how the additional observations contribute in
Eq. (3).
RESULTS
Calculations were performed in two-dimensional parameter spaces to avoid
too extensive computing time. To have a common reference, the horizontal
conductivity of horizon A was chosen as one dimension in the parameter spaces
for all the calculations. The other dimensions were in turn vertical conductivity
of horizon A, horizontal conductivity of the groundwater zone, thickness of
horizon A, evaporation parameter, critical temperature snow/rain precipitation,
degree-day-factor and threshold temperature for start of snowmelt. These nine
parameters proved to be the most sensitive to simulation of streamflow. The
likelihood function was computed for a time period of one year, 1. June - 31.
Mai, for each catchment for the period 1981-1990.
The probability distributions updated on all catchments and years for two
dimensional parameter spaces are shown in Fig. 2. The entropy measure of the
marginal probability distributions as additional streamflow data are used for the
updating (10 years for each station) is seen in Fig. 3. The entropy decreases as
additional data are used for updating the probability distributions, and quite well
defined areas, with two exceptions, appear to be good parameter sets for the
NOPEX region (Fig 2). The parameter values found by Motovilov et al. (1999a)
appear to be close to the modal values found here. Also the Reff Nash-Sutcliffe
coefficients calculated for the optimal parameter values indicate satisfactory or
good model performance for all catchments. These results suggest that regional
parameters, as defined in this paper, exist.
−3
x 10
0.035
0
8
10
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10
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Horizontal conductivity of horizon A*
10
10
0.5 1 1.5 2 2.5 3 3.5
Surface retension storage*
0
0.03
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0
10
0.025
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10
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0.015 10−2
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10
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1
1.5
2
2.5
Degree−day−factor*
−3
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1
1.5
2
2.5
Evaporation parameter*
3
x 10
10
0
10
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0.005
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10
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10
Vertical conductivity of horizon A*
10
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0
1
−1 −0.5 0 0.5 1 1.5 2 2.5
10
10
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10
Conductivity of groundwater zone*
Critical temperature precipitation oC
0.05
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0
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10
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10
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10
0
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Critical temperature start of snowmelt oC
−1
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10
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1
1.5
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2.5
Thickness horizon A*
3
* Factor multiplied with the parameter values found by Motovilov et al. (1999a, 1999b)
Figure 2. The posterior probability distribution for two dimensional parameter
spaces updated on data from all years and catchments. The gray shades reprecent
the probability.
The two parameters that do not follow this trend are the vertical conductivity
of horizon A and the surface depression storage. The model results are relatively
insensitive to the vertical conductivity of horizon A if the value is high enough.
As all precipitation is likely to infiltrate in unsaturated areas in typical Nordic
catchments this is a reasonable result. Providing a sufficient high value has been
chosen it could therefore be excluded from the parameter space to save
computing time. The optimal value of the surface depression storage changes
much between years and catchments and thus suggests that the parameterisation
of the grid cells does not represent the physical properties that are most
important for this parameter. It is also possible that this process description
compensates for other physical processes, e.g. interception. A change in the
description and/or parameterisation of the process is necessary to determine a
regional parameter value.
1
Normalised entropy
0.8
0.6
Fyrisån
Lillån
Hågaån
Sagån
Örsundaån
Surface retension storage
Degree−day−factor
Evaporation parameter
Vertical conductivity of horizon A
Conductivity of groundwater zone
Critical temperature precipitation
Critical temperature start of snowmelt
Thickness horizon A
0.4
0.2
1
11
Sävjaån
Sävaån
Stabbybäcken
Stalbobäcken
21
31
41
51
61
71
Number of data sets used for updating of probability
81
Figure 3. The entropy measure for the marginal probability distributions, as
additional data are used for the updating.
Fig. 3 reveals that for three parameters (critical temperature for start of
snowmelt, degree-day-factor and critical temperature for phase of precipitation)
the entropy stops decreasing when the data sets from the last three catchments
are used for updating the probability distributions. This suggests that a limit is
reached for how accurate the parameters can be defined. When the limit is
reached, additional data do not provide more information about the parameters.
CONCLUSIONS
The aim of this work was to demonstrate that it is meaningful to define
regional parameters for the macro scale hydrological model ECOMAG.
The Bayesian method has proved to be useful for an objective estimation of
regional parameters. For the NOPEX region a regional parameter set exists
according to some predefined criteria. The use of additional data implied a
decrease in the variances of the marginal distribution for most of the parameters,
and a relatively narrow area in parameter space appeared to contain good
parameter sets for simulation of streamflow in the nine catchments studied in the
NOPEX region.
Computing time is a limitation for the number of dimensions in the parameter
space and the sampling density. Possible solutions could be to reduce the
dimensions of the parameter space by searching for relationship between
parameters or to use a Monte Carlo Marcov Chain resampling procedure e.g. the
Metropolis-Hastings Algorithm.
An other issue for further investigations is the choice of likelihood function.
The statistical assumptions for the likelihood function used here, i.e. the
simulation errors are independent normally distributed with zero mean and
constant variance, are not fulfilled. The simulation errors of a dynamical model
are most likely autocorrelated, and the variance is dependent on the streamflow
value, climatic conditions and physical properties of the catchments.
The Bayesian method also provide the possibility to include internal variables,
e.g. soil moisture and groundwater in the updating of the probability
distributions.
REFERENCES
Binley, K. and Beven, K. (1991) Physically-based modelling of catchment
hydrology: a likelihood approach to reducing predictive uncertainty. In D. D.
Farmer and M. J. Rycrift, Eds. Computer modelling in the Environmental
Sciences, The Institute of Mathematics and its Applications Conference Series,
Claredon Press, pp. 75-88.
Gupta, H. V., Sorooshian, S. and Yapo, P. O. (1998) Towards improved
calibration of hydrologic models: Multiple and noncommensurable measures of
information, Water Resources Research 34 (4), pp. 751-763.
Halldin, S., Gottschalk, L., Van de Girend, A. A., Gryning, S-E., Heikinheimo,
M., Högstrom, U., Jochum, A. and Lundin, L-C. (1995) Science plan for
NOPEX, NOPEX Technical report No. 12, Institute of Earth Sciences, Uppsala
University.
Klemes, V. (1986) Operational testing of hydrological simulation models,
Hydrological Sciences Journal 31 (1), pp. 13-24.
Motovilov, Y.G., Gottschalk, L. Engeland, K. and Rodhe, A. (1999a) Validation
of a distributed model against spatial observations, Nopex special issue of
Journal of Agricultural and Forest Meteorological Research.
Motovilov,Y.G., Gottschalk, L., Engeland, K. (1999b) ECOMAG - a physically
based hydrological model - application to the NOPEX region, Technical report,
Department of Geophysics, University of Oslo.
Nash, J. E. and Sutcliffe, J. V. (1970) River flow forecasting through conceptual
models part 1 - A discussion of principles, Journal of Hydrology 10, pp. 282290.
Seibert, P. (1994) Hydrological characteristics of the NOPEX research area,
Nopex Technical Report No. 3, Instutute of Earth Sciences, Uppsala University.
Tanner, M.A. (1996) Tools for Statistical Inference: Methods for Exploration of
Posterior Distributions and Likelihood Functions, Springer, New York