can be used for real time operation. Further ... In the development and application of these ... perform real time updating of flood and reservoir inflow forecasts.
Scientific Procedures Applied to the Planning, Design and Management of Water Resources Svstems (Proccfdines of the Hamburg Symposium, August 1983). IAHSPubl. no. 147.
Application of the Kaiman filter to real time operation and to uncertainty analyses in hydrological modelling JENS CHRISTIAN REFSGAARD*, DAN ROSBJERG & LARS M. MARKUSSEN Institute of Hydrodynamics and Hydraulic Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark ABSTRACT The NAM rainfall-runoff model (a lumped, conceptual model developed in Denmark) has been reformulated in a state space form, and the Kaiman filtering algorithm has been incorporated. Uncertainties on rainfall input and on the measured discharges are taken into account, as well as the uncertainties on the most important model parameters. When the Kaiman filtering algorithm is applied as an updating procedure, the model can be used for real time operation. Further, due to the inclusion of the most important sources of uncertainty, the state space model can be used for calculation of uncertainty bands on the simulated streamflows. For instance, the effects of parameter uncertainty and rainfall uncertainty, respectively, can be evaluated and compared. The general approach and the fundamental principles of the modelling are described. Furthermore, the functioning of the updating procedure and the uncertainty analyses are illustrated by simulation results. Application du filtre de Kaiman â 1'exploitation en temps réel et aux analyses d'incertitudes dans la mise au point de modèles hydrologiques RESUME Le modèle pluie-débits NAM (modèle global conceptuel mis au point au Danemark) a été formule dans une nouvelle structure pour opérer dans l'espace et l'algorithme du filtre de Kaiman y a été incorporé. On a tenu compte des incertitudes sur les entrées: précipitations et sur les débits mesurés aussi bien que des incertitudes sur les paramètres les plus importants du modèle. Lorsque l'algorithme du filtre de Kaiman est appliqué à une opération de mise à jour, le modèle peut être utilisé pour une exploitation en temps réel. En outre grâce à la prise en compte des plus importantes sources d'incertitudes, le modèle spatial peut être utilisé pour le calcul de bandes d'incertitudes sur les débits simulés. On décrit les principes généraux de l'approche et les principes fondamentaux de la mise en Also at: Denmark
Danish
Hydraulic
Institute,
Agern Allé
5, DK-2970
Hgfrsholm, 273
274 Jens Christian
Refsgaard
et
al.
modèle. Puis le fonctionnement du procédé de mise à jour et les analyses d'incertitudes sont illustrés par des résultats de simulation.
INTRODUCTION For many years the sciences of deterministic hydrology and stochastic hydrology have developed more or less independently. However, within the last few years the two basically different methods have more and more often been combined into a hybrid deterministic-stochastic description of the hydrological phenomena - often resulting in improved hydrological understanding and improved hydrological tools. Within the last two decades modelling of the rainfall-runoff process has been performed predominantly with purely deterministic methods, first the lumped, conceptual models (e.g. the Stanford watershed models) and later the more complicated, distributed, physically based models. In the development and application of these models the various sources of uncertainties (errors) have been discussed but have not been quantified and taken computationally into account. The state space theory and the Kalman filtering technique are powerful mathematical tools for treating various uncertainties in mathematical modelling. They are a well-proven engineering tool for linear systems. During the last decade the application of the Kalman filter within hydrology has increased considerably, especially in connection with real time operation of hydro systems. Kalman filtering combined with a traditional deterministic rainfall-runoff model has so far only been applied by a few people to perform real time updating of flood and reservoir inflow forecasts. Kitanidis & Bras (1978) together with Goldstein & Larimore (1980) have combined the Kalman filter with the Sacramento Soil Moisture model. Fjeld & Aam (1980) have combined the Kalman filter with the Swedish HBV model. In the present paper results from combining Kalman filtering with the NAM rainfall-runoff model are shown. The combined model can be used both for updating and for uncertainty analyses. The present paper is based on the studies of Jçfrgensen et al. (1982) and Markussen (1982).
SOURCES OF UNCERTAINTY IN RAINFALL-RUNOFF MODELLING In rainfall-runoff modelling the following fundamentally different sources of uncertainty exist (see e.g. Fleming (1975) for a discussion): (a) error in input data to the model; (b) error in measurement of output from nature; (c) error in model structure; (d) non-optimal values of model parameters. Of these four sources only the first two are usually accounted for in Kalman filtering. In connection with the input data, error (a), usually only the main variable, namely the mean areal precipitation, is treated as uncertain, while the other climatological input variables (such as temperature and potential évapotranspiration) are
Kalman filter
in hydrological
modelling
275
not. The error in measurement of output, error (b), is the uncertainty in the observed streamflow which, like error (a), is modelled as a white noise with a given variance. Furthermore, in this study the errors (c) and (d), due to model and parameter values, are accounted for in a heuristic manner, as described in the next section. Thus by accounting for both model and data uncertainties in a quantitative way the importance of the two can be compared, and the usual modeller postulate that the main uncertainty in rainfall-runoff modelling is due to the data (and not due to the model) can be evaluated.
THE
MODEL
NAM is a deterministic rainfall-runoff model of the lumped, conceptual type. It has been developed at the Institute of Hydrodynamics and Hydraulic Engineering (ISVA) at the Technical University of Denmark by Nielsen & Hansen (1973). NAM simulates the rainfall-runoff process in rural catchments on a daily basis. It operates by accounting continuously for the content in four different and mutually interrelated storages representing physical elements in the catchment (see Fig.l). ^
SNOW
RAIN
J
OVERLAND FLOW
P
L2
P N
for L/L* > C „ L2 for L/L* < C
L/L*
SNOW I STORAGE
te
L/L* - C L 2
IF
1 - C,
U
for L/L* > C
for L/L* £ C SURFACE STORAGE
IDL „
„
_ _ _ _ _
______
LOWER ZONE STORAGE
INTERFLOW
a
) _
FLOW L/L*
