Hydrology for the Water Management of Large River Basins (Proceedings of the Vienna Symposium, August 1991). IAHS Publ. no. 201,1991.

HYDROLOGICAL FORECASTING AND UPDATING PROCEDURES

P. SERBAN Institute of Meteorology and Hydrology, Romania A.J. ASKEW World Meteorological Organization, Geneva, Switzerland

ABSTRACT Some of the inputs of greatest potential value for the management of rivers are forecasts of future flows. The heart of any flow forecasting system is a hydrological model. WMO has conducted a series of projects for the intercomparison of such models. The last of these was implemented from 1985 to 1989 and involved a total of 14 models from 11 countries. Central to the project was a study of the updating procedures used. This paper considers the role of flood forecasting in the management of large rivers, describes the updating procedures used in forecasting sys­tems and comments on their relative efficiency.

INTRODUCTION

The 1970s and 1980s have seen a great increase in the world's population and in the level of expectation of both individuals and nations. This has led to greatly increased demands on the world's limited resources of fresh water; demands not only for consumption, energy production and waste disposal, but also for the maintenance of a balanced and healthy aquatic environment. One response has been the refinement of a range of techniques for use in the development of water resources and the management of river basins (e.g. Yeh, 1985).

A NEED FOR DATA

Techniques for water-resource systems analysis can operate on a deterministic basis, assuming full knowledge of the past and future of all relevant processes, both physical and socio-economic. The results obtained can be very valuable, but no-one can doubt that the deterministic assumption is a major over-simplification. Efforts to introduce even a minimum of realistic uncertainty, however, lead immediately to greatly increased complexity and calls for far more data and computing power. There is still much to be done in refining the tools of water-resource systems analysis, particularly those that can operate with limited data.

Increasing effort is being channelled into the collection and interpretation of relevant socio-economic data. There is, however, a worrying lack of work

357

P. Serban &A. J. Askew 358

directed to the collection and processing of the geophysical data required. There is in fact ample evidence of a world-wide downward trend in such work, invol­ving both developed and developing countries (WMO, 1990a).

Past geophysical, more specifically hydrological, data are essential in assessing not only the magnitude of a body of fresh water and its extent in space, but also its variability with time. The frequency of past floods and droughts can be assessed with a fair degree of accuracy, but estimates of the future frequency of such events is clouded with considerable uncertainty. Yet it is for the future that water-resource systems are to be designed. Between the known past and the uncertain future, lies the present when existing systems must be operated as efficiently as possible. It is here that hydrological forecasting has a vital role to play.

THE ROLE OF HYDROLOGICAL FORECASTING

Hydraulic structures are usually operated in accordance with a set of rules which take account of the current state of the system, the demands being made upon it, and some estimates of future rainfall, streamflow, evaporation and the like. Sometimes these estimates are not explicitly required as input because the rules have been devised on the assumption that, for the current period in time, the various hydrological variables will take on their average values for the relevant month or season of the year. However, whether they are explicit or implicit, these estimates of future conditions are forecasts: Often very poor forecasts. It is well to recognize this fact and avoid the pit-fall of viewing them as characteristics of the natural system. The average daily discharge for November is a character­istic of the natural system; the assumption at 00 hrs on 1 November that the actual discharge for the coming 24 hours will equal this value is human judge­ment. Given some on-line information about the state of the natural system, in other words, some relevant hydrological data, it should be possible to improve the level of judgement and hence improve the operation of the structure concer­ned. At the very least, the wise operator will take note of the current level of flow and its rate of change and act accordingly. At the least, all important water-resource systems should incorporate the real-time collection of hydrological data and their use in developing forecasts of future conditions relevant to the operation of the systems. The costs of collecting the data and preparing the forecasts are a fraction of the cost of constructing and operating the system itself and will usually be far outweighed by their potential benefit in improved safety and operational efficiency. There is therefore a strong case to be made for the use of hydrological forecasts in the operation of water-resource systems. However, if they are to be used to their full potential, it is important that both the systems themselves and their operating rules be designed to take account of the availabili­ty of hydrological forecasts and of the anticipated precision of the forecasts. Whether explicitly or implicitly, the design of a water-resource system makes some assumption as to die manner in which it will be operated. Whether explicit­ly or implicitly, the design of operating rules makes some assumption as to the nature and precision of forecasts of future hydrological conditions. It is preferable for both sets of assumptions to be made explicitly and to be well founded.

359 Hydrological forecasting and updating procedures

WMO PROJECTS

The importance of hydrological forecasting, for both water-resource management and disaster mitigation, has long been recognized by the World Meteorological Organization (WMO). Over the years it has undertaken a series of projects designed to encourage the development of improved forecasting techniques and to provide information to those faced with a choice of techniques. Central to any hydrological forecasting system is a model which takes as input the relevant data on the past and present state of the natural system and produces an estimate of its future state. During the past twenty years WMO has undertaken three projects for the intercomparison of flood forecasting models (WMO, 1975, 1986 and 1990b).

The WMO projects each included a numerical intercomparison of the performance of the various models. In the first two cases, these intercomparisons were based on the ability of the models to simulate periods of flow extending over a number of years, given the relevant input (rainfall, temperature, etc.), but without any opportunity to update the forecasts at any time. There were important technical reasons why updating could not be allowed. Nevertheless, all concerned were aware of the lack of reality that this engendered. If, as said earlier, the wise operator of a hydraulic structure would never in practice ignore the current water level or discharge rate, it is even less likely that a forecast office would issue forecasts for years at a time without ever correcting them for current conditions.

The third WMO project (WMO, 1987 and Askew, 1989) was therefore undertaken explicitly to take advantage of developments in micro-computers which allowed an intercomparison of models which permitted updating of forecasts. The project was implemented from 1985 to 1989 under the title "real­time intercomparison of conceptual models" (WMO, 1990b). A total of 14 mo­dels from 11 countries were compared using four standard data sets. As was expected, the inclusion of updating had a major impact, not only on the organiza­tion of the project, but also on the results obtained. There is no doubt that, when used in operational practice, all hydrological models are associated with some form of updating procedure and any description of a model is incomplete without a description of the updating procedure used with it.

The various types of hydrological models have been described at length elsewhere (e.g. Becker & Serban, 1990), but updating procedures are not so widely discussed. This is the topic of the remainder of this paper.

FORECAST UPDATING

A major problem which occurs when applying a hydrological model in real-time is the fact that, in general, the simulated hydrograph is different from the measured hydrograph up to the time of the forecast. This difference is due to: (a) errors induced by the model input data; (b) imperfections in the model, or in some of its component parts (computation

of snowmelt or effective rainfall, flood routing, etc.); (c) model calibration with only a limited amount of data; (d) changes in time in certain characteristics of the basin;

P. Serban &A. J. Askew 360

(e) errors in the determination of the discharge hydrograph at the gauging station. Errors between simulated and measured hydrographs are, generally, of three

types (Fig. 1): (a) volume or amplitude errors, generally due to the modelling of infiltration, or

to errors in model input data; (b) errors in the timing of the simulated and measured hydrograph or phase

errors mainly induced by the routing component of the model; (c) hydrograph shape errors mainly introduced by the transfer component of the

model: for instance, a unit hydrograph.

3 - amplitude errors b - phase errors C - shape errors

Tin» TITO Timt

Fig. 1. Definition of types of error between measured (-) and simulated (...) hydrographs (a) amplitude errors; (b) phase errors; (c) shape errors.

Various combinations of these three types of error can occur in operational practice.

Hydrological forecasting models consist of a "simulation model" and a procedure for forecast updating (Fig. 2). The "simulation models" make use of input data, either measured or estimated, relating to precipitation (P) and other meteorological factors (M), such as air temperature, wind speed, humidity, etc., in order to compute discharges (QS). These models do not take into consideration the discharges (QM) recorded at the river station up to the time that the forecast is derived.

For application in real-time, updating procedures have been developed which take account of the errors between the computed and measured discharges in order to correct the values derived by the "simulation" models over the entire forecasting interval.

Continuous updating procedures can be applied for forecasts at each time step. It is important to note that these procedures differ from the periodic recalibration of models - referred to as "periodic updating" by some authors. Periodic recalibration of model parameters is necessary when the characteristics of a basin or of a river bed change in time, in particular as a result of man's influence. It is quite independent of the continuous updating that is the subject of this paper.

Fond M.

361

'ÔIMULATION MOOFL

Hydrological forecasting and updating procedures

C/M

UPDATING

p/focsùwe QF_

J JW Time

Fig. 2. Block diagram of a forecasting procedure (a) and the principle effect of updating. Legend: P rainfall j time of preparation of forecast M other meteorological factors QF forecasted hydrograph QS simulated hydrograph e simulation error QM measured hydrograph e forecasting error

TYPES OF UPDATING PROCEDURE

Forecast updating procedures differ depending on the variables modified (Table 1 and Fig. 3). These are: (a) procedures which update the model input variables; (b) procedures which update the variables of state in the model of the system; (c) procedures which update the model output variables.

UPDATING OF MODEL INPUT

VARIABLES

MANUAL -INTERACTIVE

AUTOMATED

.TRIAL - ERROR PROCEDURE

TYPE OF UPDATING PROCEDURE

UPDATING OF STATE VARIABLES OF THE MODELLED SYSTEM

MANUAL-INTERACTIVE

AUTOMATED

UPDATING OF MODEL OUTPUT VARIABLES

MANUAL-INTERACTIVE

AUTOMATED

KALMAN FILTER OR

EXTENDED KALMAN FIUER

AUTOREGRESSIVE MODELS

Fig. 3. Types of updating procedure.

P. Serban & A. J. Askew 362

Table 1. Characteristics of forecast updating procedures (information based on WMO (1987 & 1990b)).

MODEL

1. UBC - Canada

2. CEQUEAU Canada

3. CRM Czechoslovakia

4. GAPI Hungary

5. SMAR Ireland

6. CLS Italy

7. VIDRA Romania

Updating procedure

Automated

X

X

X

X

X

X

-

Manual-interactive

X

X

Variables to be directly updated

Inputs State variables

Water equivalent of snow cover

Rainfall and snowmelt

Baseflow indicator

Unit hydrograph

Outputs

Discharges

Discharges

Discharges

Discharges andflood volume Discharges

Flood volume and shape of the hydro-graph

8. HBV Sweden

9. SRM Switzerland/USA

10. TANK Japan

11. SSARR USA

12. HFS USA

13. NAMS11 Denmark

14. NAMKAL Denmark

Air temperature and precipitation

Snow covered area Discharge

Rainfall and snowmelt Air temperature and precipitation

Amount of water in the model reservoirs

Discharges of tributaries (lateral inflow)

Amount of water in the model reservoirs

Certain models make use of updating procedures for both input and output variables, whereas others use the updating of both the state and output variables. Among the models taking part in the WMO intercomparison project three updated only the model inputs, three only the state variables and four only the outputs, whereas one of them updated bodi the inputs and outputs and three of them updated the state variables and the model outputs as well.

It is noted that a few models also update the model parameters. In general, this procedure is not to be recommended because in most models the parameters

363 Hydrological forecasting and updating procedures

are not independent and the modification of one parameter would require the modification of other parameters. The parameters are usually determined by calibrating the model and any adjustment of them amounts to a re-calibration. It is useful to study the interaction between parameters when calibrating the model, but such a study is not part of the model's operational application.

Updating procedures can be classified as "automated" or "manual-interacti­ve", depending on the way they are applied. Automated procedures are complete­ly reproducible and objective. In most instances, they are preferred to manual procedures (Table 1). Manual-interactive procedures are based on the practical experience of the forecaster and therefore include a certain amount of subjectiv­ity. The application of manual procedures, such as was done in the WMO project, can involve the use of interactive programmes which are based on a real "man-machine" dialogue.

Some models use both automated and manual procedures. In these cases the automated adjustment schemes are used only as instruments for assisting the user in adjusting the model to the observed conditions. After applying the automated procedures, the forecaster reviews the state of the system and compa­res it with the data measured over the basin. If any unanticipated errors or unusual circumstances have arisen, the user changes certain variables on a subjective basis and runs the automated procedure once again.

DESCRIPTION OF UPDATING PROCEDURES

Updating procedures for model input variables

The primary input variables that may be updated are precipitation and air temperature. In some cases snowmelt is updated as an intermediate parameter, being snowmelt model output and rainfall-runoff model input.

Most procedures which update input variables are interactive and of the "trial-error" type because with most models it is difficult to solve the reverse problem, that is, to determine the model input when the model output and parameters are given, while taking into account the errors which influence computation quantities and the non-linearity of the hydrological system itself. A block-diagram for a "trial-error" updating procedure is given in Fig. 4.

The most important stages in these types of procedure, to be carried out at each forecasting moment j , are: (a) computation of the error "e" between the measured and simulated hydro-

graphs; (b) comparison of the error with a pre-defined acceptable level of error; (c) selection of the input variables to be adjusted, plus the adjustment incre­

ment for each variable and the maximum number of increments of change allowed in any computation period;

(d) rerun of the model using the adjusted input variables.

P. Serban & A. J. Askew 364

Time of forecastj

Model run

I 1--0

ML 3 Vs/j Error e;r

Computation J=Jtf

Com,

yss

imputed tiydrqgraph 'Inin the admit tea error //'mitt

A/O

À in température and snow covered area (or ônowmett) are adjusted 6y i A M; and t/ie precipitation (if any) 6y i à Pi Prec/pitat/on

cd/'asfed iylAP/

i-if-/

yes '<«W NO

~@

T/ie made/ /j rerun aver the interval j-k -f-J

Fig. 4. Block diagram of updating procedure of the "trial-and-error" type for input variables. ef error between computed QSV during the ith iteration and the corresponding measured Qmt hydrograph TEj accepted error threshold (typically 5%) i iteration counter nm max. number of iterations K modelled system memory

Updating procedures for the state variables of the modelled system

State variables that may be adjusted include: areal extent of snow cover, snow depth, snow water equivalent, the amount of water contained in model reservoirs simulating water storage at the surface and in the unsaturated and saturated zones and in water courses.

One justification for state variable updating is that errors in model inputs are accumulated and appear as errors in the water content of the stores of the conceptual rainfall-runoff model which, if not corrected, will in turn lead to erroneous output variables. The water equivalent of the snow cover is generally

365 Hydrological forecasting and updating procedures

updated using snow course data. The amount of water stored in the conceptual reservoirs of the model is often updated by means of a Kalman filter. This can be applied to linear models or, with the aid of an extended version of the Kalman filter, to nonlinear models.

The Kalman filter may be integrated with transfer models of the ARMA type (Wood & Szôllôsi-Nagy, 1980), or with conceptual hydrological models, such as: the HFS model (Georgakakos et al, 1988) and the NAMKAL model (Refsgaard et al., 1983). The application of the linear Kalman filter to the study of a physical system (Fig. 5) requires (Gelb, 1974): (a) a description of the system dynamics as a system of linear equations of the

form:

Xj + , = #Xj + rUj + Wj (i)

(b) a definition of the measurement equation relating the measurements generally carried out on the system output with the variables of state:

Zj = HXj + Vj (2)

where: X is the vector of the variables of state, describing the system evolution; U a control vector containing the input variables; # the transition matrix; T the input adjustment matrix; W the modelling error vector; Z the measurement vector; H the measurement selection matrix; V the measurement error vector.

The matrixes # , V and H defining the characteristics of the modelled system can be time-constant or variable in time.

The V and W errors are considered independent and normally distributed.

V~N(0,R); W~N(0,Q); E [V*., ^ 1 = 0

input

Uj

Randcm unmeasured disturbance Wj

Initial conditions

•Xltol-Xo

System

Xj*1=<}>Xj^ Tuj +Wj

Variables of state

Previous state

Xj

X M ^

Xj.1

Xj

Xj

Unit delay

Fiandom measurement noise

'Vj

Measurement

Zj - H x j +V j Output ,

Zj

Fig. 5. Sketch representing a physical system.

P. Serban &A. J. Askew 366

As the X state quantities at any time step are only estimates of the "true" X values, the covariance matrix for the estimation errors is defined as:

P j ^ E ^ X j - X j ^ M V X j i j - . n (3)

Once the initial values for X,, and Pc are pre-established, the equations of the linear Kalman filter for the forecasting and the updating stages are the following:

Forecast at moment i

- the vector of state:

xJ + 1|J = #x j l j+ru j (4)

- the matrix of estimation errors:

Pj + 1 | J = # P j l J # T + Q1 (5)

Forecast updating using the measurement at moment i +1

- the correction matrix Kj + [

Kj + 1 = P j + 1 | j Hr [ H P j + 1 | j H r + Pj + 1]-" (6)

- the vector of state

A A A

Xj + i | j + i = Xj + 1 | j + Kj + 1 [Z j + 1 - HXj + 11 j] (7)

- the matrix of estimation errors

P j + 1 | j + 1 = P J + 1 | j - K j + 1HPJ + 1 | j (8)

The algorithm is repeated by substituting into eqs. (4) and (5) the estima­ted values of the state vector Xj + , | j + , and of the matrix of errors Pj + , | s + x

obtained through equations (7) and (8).

The term HXj + 1 | i of relation (7) represents the estimation Zj + t | j of the observed value, Zj + l, using the equation:

Zj + 1 | J = HXJ + I | J (9)

The use of a Kalman filter helps by considering the most significant error sources (input and output variables, non-optimal parameter values) and because of this the discharges can be computed within given confidence limits.

367 Hydrological forecasting and updating procedures

Updating procedures for the model output variables

Output variables that may be updated include: discharge, flood volume, hydro-graph shape and lateral inflow.

The discharge updating procedures most widely used in practice (e.g. UBC, CEQUEAU, SMAR and NAMS models) are those based on autoregressive models, AR, that were fitted to the errors "e" between the computed and measured hydrographs:

ej = a ^ . , + 8363.2 + ... + a ^ + bj (10)

where a„ % ... a,, are coefficients of the autoregressive model and the bj are residual (uncorrelated) errors.

The order and coefficients of the autoregressive model are determined by means of the "e" error series over a relatively short period of time before the time of forecast. The forecast error («j + k | j) at time j +k using the AR model is given by:

(ej + klj) — a I^ +k-l |j + «2^ +k-2|j + a3ej + k-3 |j + •••• (11)

Updating procedures applied to state variables and based on the Kalman filter require the linearization of the model and its re-writing in state space form. The programming is difficult and, in addition, the memory occupied by the program is large and it takes quite a long time to run. This procedure is optimum from a mathematical standpoint and yields satisfactory results with short lead times. The efficiency of the procedures is still open to discussion, however, for longer lead times. There is a particular difficulty with respect to flood crests, because the hydrological conditions in the region of the crest differ considerably from the conditions that apply at the time of forecasting.

Updating procedures based on Kalman filters are advantageous in as much as they make it possible to determine confidence limits for forecast discharges, which are very useful in assessing the likely accuracy of the forecasts.

Procedures which update model output variables and are based on autore­gressive models are easy to program and apply and do not take much computati­on time. Their efficiency depends on the degree of error persistence between the measured and computed hydrographs. Unfortunately, it is in the vicinity of flood crests that error persistence is least and errors show a tendency to oscillate both rapidly and widely. Likewise, an error in the time of occurrence of forecast discharge can give rise to larger errors where an autoregressive model is applied.

Updating procedures based on autoregressive models or Kalman filters yield very good results for hydrological forecasting as regards amplitude errors. The efficiency of these procedures is still open to discussion, however, witii respect to phase and shape errors (Fig. 1).

In order to eliminate the deficiencies of the AR models and of the Kalman filter in the case of phase and shape errors, procedures have been developed for identifying error types: amplitude, phase and shape. It is claimed that these procedures can update a hydrograph as successfully as an experienced forecasting hydrologist (Sittner & Krouse, 1979; Serban & Plesa, 1986; Rungo et al., 1989).

P. Serban&A. J. Askew 368

CONCLUDING REMARKS

The freshwater resources of the world are limited and under pressure, but there need be no limitation on mankind's ingenuity in finding ways of maximizing the use of these resources. Essential to any such efforts are hydrological data: historical data on which to design new water-resource systems and real-time data for use in operating the systems to maximum efficiency.

Operational efficiency can be greatly aided by the accurate forecasting of future hydrological conditions, essential to which is the application of a hydrolo­gical model in conjunction with an effective updating procedure. The choice and correct application of an updating procedure can be as important as the choice of model. The WMO project completed in 1989 went a long way in describing and evaluating the various updating procedures in current use, but there is more that can be done in refining these techniques in the years ahead.

ACKNOWLEDGEMENTS The authors express their gratitude to the Secretary-General of WMO for his permission to publish this paper.

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369 Hydrological forecasting and updating procedures

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