Forecasting flash floods using Data BasedMechanistic models, online data assimilation and

meteorological forecastsPaul Smith & Keith Beven

Lancaster Environment Centre, Lancaster University, UKp.j.smith@lancs.ac.uk

1. Introduction

This poster focuses on the provision of short-range (up to 12 hours leadtime) predictions of water level in catchments whose natural responsetime is rapid. In such situations the hydrological forecasting problemis exacerbated by the meteorological forecasting problem which mustbe addressed to achieve useful lead times. As case study; the Gardond’Anduze (France) is presented (Section 2).COSMO-LEPS Numerical Weather Prediction (NWP) ensembles [3]are utilised. These give a sixteen member ensemble of precipitationforecasts with a three hour time step. A new ensemble is generatedevery twenty four hours and run for five and a half days.The coarseness of the NWP model grid relative to the catchment scalesuggests that a lumped hydrological model is most appropriate. The in-ductive Data based Mechanistic (DBM) modelling methodology [11, 12],is applied to create such a model.Real time data assimilation is used to improve the forecasts by:

• assimilating the observed water levels into the hydrological model;

• calibrating the NWP ensemble members.

The techniques used are outlined in Sections 3-4.

2. Gardon d’Anduze

The Gardon d’Anduze catchment is a 544 sq. km Mediterranean catch-ment located in the Cevennes Mountains of Southern France. It’s alti-tude ranges between 1567 m a.s.l. and 123 m a.s.l. The topographyis characterised by high mountain peaks, steep hill slopes and narrowvalleys [4]. High intensity rainfall, particularly in autumn, can producedevastating floods.Rainfall and water level observations are available at a number of sites(Figure 1). COSMO-LEPS ensembles for are available for 2008, whichincludes 2 flood events: in October (used for calibration) and November(used for validation)

Figure 1: A simplified representation of the Gardon d’Anduzecatchment showing the location of the observation sites; catchmentboundaries for the three river gauges and main river channels (black

lines).

The water level at Anduze is modelled. The modelling methodology isa three stage process:

1. identification and estimation of the DBM model using observed data(Section 3);

2. Calibration of the NWP ensembles (Section 3);

3. Coupling the corrected ensemble predictions and DBM model (Sec-tion 5).

3. The Anduze DBM model

For compatibility with the meteorological forecasts the DBM model isevaluated on a 3 hour time step. This is approximately the naturaltime delay of the system. Motivated by this, and the requirements offlood forecasting, the model predicts the maximum observed water levelwithin each time step (y = (y1, . . . , yT ), indexed by time).A series of catchment averaged precipitation values u = (u1, . . . , uT )derived from observations is used as the input series.Initial analysis of the system proceeded along standard lines [2, 5, 6,13, 10]. The resulting model is given by

pt = φ2pt−1 + ut−1 (1)vt = pφ1t ut (2)

yt −min (y) =b0 + b1z

−1 + b2z−2

1 + a1z−1 + a2z−2vt + ηt (3)

where ηt is a stochastic disturbance. Equation (1) represents a simpleantecedent precipitation index which is used in (2) to produce an effec-tive precipitation vt. This is filtered by the linear transfer function (LTF)in (3) to predict the water level. There is no time delay in the model,precipitation predictions are needed to provide forecast lead time.

The term min (y) represents a baseline output. The observed data indi-cates this value is not constant with time. A random walk representationis introduced in the state space model below.A state space model can be constructed from (3) [8, 10]. A partial frac-tion decomposition of (3) gives

b0 + b1z−1 + b2z

−2

1 + a1z−1 + a2z−2ftut =

(β1

1 + α1z−1+

β21 + α2z−1

+ κ

)ftut.

The two first order LTFs represent parallel fast and slow response paths[9]. The additive term κ represents a direct precipitation contribution tothe catchment dynamics. The first three components of the state vectorx relate to these three forms of precipitation response. The fourth statecorresponds to the random walk representation of the baseline value.The state transition matrix A, input partitioning vector B and observa-tion vector h along with realisations of stochastic disturbances ζt and ξtallow the state space model to be expressed as

xt+1 = Axt +Bft+1ut+1 + ζt+1 (4)yt+1 = h′xt+1 + ξt+1 (5)

The disturbances ζt and ξt are random draws from independent, zeromean symmetric distributions with variances Qσ2 and σ2 respectively.The noise variance ratio matrix Q in this formulation captures the differ-ent levels of noise on each response mode of the model.The state space model can be embedded in a linear Kalman Filter al-lowing efficient data assimilation and probabilistic prediction. The vec-tor of unknown parameters consisting of elements of of A and B andQ along with φ is optimised to minimise the sum of the squared fore-cast innovations up to 12 hours ahead (Figure 2). This optimisationis independent of σ2 which is computed for each forecast horizon.

Figure 2: Twelve hour ahead predictions of water level at Anduze (witherror bounds) during a sequence of flood events in the calibrationperiod. Predictions generated using the (unavailable) observed

precipitation data.

4. Precipitation Calibration

To improve the hydrological forecasts the NWP precipitation predictionsrequire calibration to the observed catchment averaged precipitation.For each ensemble a catchment averaged precipitation rt,i is computedfor each member (indexed by i) at each time step. Thus a single en-semble member is summarised by a time series of forty-four catchmentaveraged precipitation values.Each ensemble member has its own correction; a multiplicative gain gt,igoverned by

gt+1,i = gt,i + εt+1,i (6)ut+1 = rt+1,igt+1,i + νt+1,i (7)

The zero mean random variables εt+1,i and νt+1,i have variances qλ2 andλ2 respectively. Similar corrections have in the past been used for longterm drift in process based models [10] and space-time radar precipita-tion adjustments [1].The parameter q is does not vary between ensemble members or en-sembles; λ2 alters only with the forecast lead time. The model in Equa-tions 6 and 7 can be embedded in a Kalman filter. The value of qis optimised by minimising the sum of the squared forecast innovations4∑

n=1

∑t

16∑i=1

(ut+n − rt+n,igt+n,i)2. The temporal summation is over those en-

sembles which are between 1 and 2 days old to allow for stabilisationafter initialisation. Figure 3 shows the expected value of the predictionsfor the October 2008 (calibration) flood.

Figure 3: Spaghetti plot of calibrated precipitation ensemble membersfor the October 2008 flood event. Each black line represents the

expected 12 hour ahead prediction for a given ensemble member. Reddots show the observed precipitation.

5. Coupling the Meteorological and Hydrological modelling

Assimilation of the hydrological data can always be performed using ob-served precipitation values. The additional uncertainty arising throughthe use of predicted precipitation is only present in forecasts of waterlevel.A solution to the resulting non-linear forecasting problem is utilisedwhich is based on two approximations:

•When forecasting, the antecedent precipitation pt+1 is computed us-ing the expected value of the predicted precipitation;

• The n-step ahead prediction variance takes the form

σ2nωn

(1 + h′P̂t+n|th

)where ωn > 1 is constant dependant upon the forecast horizon andP̂t+n|t is the forecast state covariance matrix computed using the ex-pected values of the predicted precipitation.

Figures 4 and 5 summarises the 12 hour ahead predictions for the Oc-tober 2008 (calibration) and November 2008 (validation) event.

Figure 4: Spaghetti plot of the twelve hour ahead predictions of waterlevel at Anduze (with error bounds) during a flood events in October

2008 (calibration). Predictions generated using the ensembleprecipitation predictions shown in Figure 3.

Figure 5: Spaghetti plot of the twelve hour ahead predictions of waterlevel at Anduze (with error bounds) during a flood events in November

2008 (validation).

6. Discussion and Conclusions

Comparison of Figure 3 and Figure 4 highlights that the timing of thepredicted precipitation is a significant cause of hydrological forecast er-ror. A more complex on-line precipitation calibration methodology whichrecognises the timing errors is under development.The derivation of the uncertainty bounds in Figures 4 and 5 also re-quires revision. The inclusion of state dependant formulations for ωn, σ2

and Q [5, 7, 10] is the subject on on-going work.A state dependant formulation for ωn may be particularly important.Recognition of when a precipitation event is poorly forecast by a givenensemble or ensemble member should reflect directly on the confi-dence in the hydrological predictions derived from it.Despite these reservations the prediction methodology does appear tohighlight time periods where high water levels can be expected.

References

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AcknowledgementsThe first author is funded by the EU as part of IMPRINTS FP7 project. The authors wish to thank Arthur Marchandise (SCHAPI) and FlorianPappenberger (ECMWF) for the provision of the data used in this study and helpful discussions as to its use.

IMPRINTS Workshop, Barcelona, June 2010