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THE MULTI-SENSOR BAYESIAN COMBINATIONS

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1 THE MULTI-SENSOR BAYESIAN COMBINATIONS
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project THE MULTI-SENSOR BAYESIAN COMBINATIONS Cinzia Mazzetti CARPE DIEM Meeting – Helsinki, 24th June 2004

2 ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project MUSIC Project developed new techniques for combining weather radar, weather satellite and rain gauge derived precipitation estimates in a Bayesian framework. MUSIC Project (Multi Sensor precipitation measurements Integration Calibration and flood forecasting) SCOPE of the BAYESIAN COMBINATIONS: Eliminating the BIAS and producing MINIMUM VARIANCE precipitation estimates. CARPE DIEM Meeting – Helsinki, 24th June 2004

3 MULTI-SENSOR BAYESIAN COMBINATIONS
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project MULTI-SENSOR BAYESIAN COMBINATIONS RAINGAUGES + RADAR RAINGAUGES + SATELLITE RAINGAUGES + RADAR + SATELLITE RADAR + SATELLITE CARPE DIEM Meeting – Helsinki, 24th June 2004

4 RAINGAUGES & RADAR BAYESIAN COMBINATION
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project RAINGAUGES & RADAR BAYESIAN COMBINATION POINT MEASUREMENTS RAINGAUGES SPATIAL ESTIMATES BLOCK KRIGING BLOCK KRIGING OF THE RAINGAUGES RADAR KALMAN FILTER BLOCK KRIGING OF THE RAINGAUGES + RADAR (BKR) CARPE DIEM Meeting – Helsinki, 24th June 2004

5 CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project BLOCK KRIGING Zn xn Z1 S0 x1 Z2 x2 minimum It describes the spatial relation between measurement points and estimation points. VARIOGRAM CARPE DIEM Meeting – Helsinki, 24th June 2004

6 BLOCK KRIGING: New Variogram fitting
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project BLOCK KRIGING: New Variogram fitting GAUSSIAN VARIOGRAM p, , A VARIOGRAM PARAMETERS Traditional estimation method for the Variogram parameters (Matheron, 1970; De Marsily, 1986, Cressie, 1993) CARPE DIEM Meeting – Helsinki, 24th June 2004

7 CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project The Variogram parameters are updated at each time step using a Maximum Likelihood estimator. (Todini, 2001 parte 1) The characteristic of the Maximum Likelihood estimator is that the Log-Likelihood function is independent of the Kriging weights , depending only on the observations, the semi-variogram model and its parameters. BK on the Reno river basin CARPE DIEM Meeting – Helsinki, 24th June 2004

8 CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project BLOCK KRIGING: New formulation with non-negative  weights Block Kriging system S0 Zn Z1 Z2 x1 x2 xn New Block Kriging system No negative rain CARPE DIEM Meeting – Helsinki, 24th June 2004

9 CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project Covariance of the estimation errors: CARPE DIEM Meeting – Helsinki, 24th June 2004

10 BLOCK KRIGING: Error prone raingauge measurements
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project BLOCK KRIGING: Error prone raingauge measurements Block Kriging system Error Variance of gauge i De Marsily (1986): NEW FORMULATION: CARPE DIEM Meeting – Helsinki, 24th June 2004

11 CARPE DIEM Meeting – Helsinki, 24th June 2004 Initial estimates for:
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project KALMAN FILTER An optimal recursive data processing algorithm. (Maybeck, 1979) Time Update (“Predict”) Measurement Update (“Correct”) CARPE DIEM Meeting – Helsinki, 24th June 2004 Initial estimates for: (Gelb, 1974)

12 CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project Time Update (“Predict”) Measurement Update (“Correct”) (Gelb, 1974) A priori estimate Radar Measurement BK raingauges A posteriori estimate Raingauges and Radar Bayesian combination CARPE DIEM Meeting – Helsinki, 24th June 2004

13 Measurement Update (“Correct”) Block Kriging + RADAR
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project Measurement Update (“Correct”) Block Kriging + RADAR Measurement equation: Measurement equation: A priori estimate: A priori estimate: Compute the Kalman gain: Compute the Kalman gain: Update estimate with measurement zk: Update estimate with measurement zk: Update the error covariance: Update the error covariance: CARPE DIEM Meeting – Helsinki, 24th June 2004

14 KALMAN FILTER: Update of the GAIN at each time step
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project KALMAN FILTER: Update of the GAIN at each time step GAIN: Ratio between the variances of the estimation errors Variance of Radar estimation errors (?) Variance of BK estimation errors Modeled using an exponential Variogram. The estimation of the Variogram parameters is performed at each time step using the Maximum Likelihood estimator. (Todini, 2001) CARPE DIEM Meeting – Helsinki, 24th June 2004

15 TEST WITH SYNTHETIC DATA CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project TEST WITH SYNTHETIC DATA Grid: 20x20 Km = 400 Km2 9 Raingauges WE GENERATED THE “TRUE” RAINFALL FIELD ON THE GRID AND ON THE GAUGES WE GENERATED A NOISE FIELD ON THE GRID AND WE ADDED IT TO THE TRUE RAINFALL FIELD TO GET RADAR LIKE ESTIMATES WE GENERATED NOISES FOR THE GAUGES AND WE ADDED THEM TO THE TRUE RAINFALL FIELD ON THE GAUGES Distributions used in the data generation: Normal Distribution Log-Normal Distribution Variograms used in data generation: Gaussian Exponential Modified spherical CARPE DIEM Meeting – Helsinki, 24th June 2004

16 CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project ERROR FREE GAUGES Bias Variance Normal Distribution Bias Variance Log-Normal Distribution Raw data CARPE DIEM Meeting – Helsinki, 24th June 2004 Combined data

17 CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project ERROR PRONE GAUGES Bias Variance Normal Distribution Bias Variance Log-Normal Distribution Raw data CARPE DIEM Meeting – Helsinki, 24th June 2004 Combined data

18 COMPARISON WITH OTHER METHODS
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project COMPARISON WITH OTHER METHODS COMPARING SOME EXISTING METHODS FOR COMBINING RADAR AND RAINGAUGE MEASUREMENTS TO THE NEW BAYESIAN METHOD. SCOPE Brandes Koistinen and Puhakka Krajewski Bayesian combination The comparison is made on the basis of a common numerical example CARPE DIEM Meeting – Helsinki, 24th June 2004

19 CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project GAUGES: Bias = 0.00 Exp. Variance = 0.90 RADAR: Bias = 5.00 Exp. Variance = 0.70 CARPE DIEM Meeting – Helsinki, 24th June 2004

20 RAINGAUGES RADAR & SATELLITE BAYESIAN COMBINATION
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project RAINGAUGES RADAR & SATELLITE BAYESIAN COMBINATION BLOCK KRIGING OF THE RAINGAUGES + RADAR UPSCALING BK GAUGES + RADAR at satellite scale SATELLITE KALMAN FILTER BK GAUGES + RADAR + SATELLITE at satellite scale DOWNSCALING (KALMAN SMOOTHING) BLOCK KRIGING OF THE RAINGAUGES + RADAR + SATELLITE CARPE DIEM Meeting – Helsinki, 24th June 2004

21 KALMAN FILTER at the SATELLITE SCALE
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project KALMAN FILTER at the SATELLITE SCALE Time Update (“Predict”) Measurement Update (“Correct”) A priori estimate Aggregated BK+RADAR estimate Measurement Satellite A posteriori estimate Raingauges+Radar+Satellite Bayesian combination (Sat. scale) CARPE DIEM Meeting – Helsinki, 24th June 2004

22 KALMAN FILTER: Update of the GAIN at each time step
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project KALMAN FILTER: Update of the GAIN at each time step GAIN: Ratio between the variance of the estimation errors Estimation errors variance of aggregated BK+RADAR estimate (?) Estimation errors variance of Satellite estimate Modelled using an exponential Variogram. The estimation of the Variogram parameters is performed at each time step using the Maximum Likelihood estimator. (Todini, 2001) CARPE DIEM Meeting – Helsinki, 24th June 2004

23 MUSIC Project CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project BLOCK KRIGING OF THE RAINGAUGES + RADAR UPSCALING BK GAUGES + RADAR at satellite scale SATELLITE KALMAN FILTER BK GAUGES + RADAR + SATELLITE at satellite scale DOWNSCALING (KALMAN SMOOTHING) BLOCK KRIGING OF THE RAINGAUGES + RADAR + SATELLITE CARPE DIEM Meeting – Helsinki, 24th June 2004

24 DOWNSCALING (KALMAN SMOOTHER)
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project DOWNSCALING (KALMAN SMOOTHER) Reich–Tung-Striebel (RTS) Kalman fixed-interval smoother Smoothing produces the best estimate at epoch k using the observations up to the latter time N. (Gelb, 1974) CARPE DIEM Meeting – Helsinki, 24th June 2004

25 CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project On scale N k k-1 Upscaling Downscaling Raingauges Radar and Satellite Bayesian Combination Smoothing produces the best estimate at SCALE k (Radar and BK scale) using the observations up to the latter SCALE N (Satellite scale). CARPE DIEM Meeting – Helsinki, 24th June 2004

26 CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project BK and RADAR scale Satellite scale CARPE DIEM Meeting – Helsinki, 24th June 2004

27 TEST WITH SYNTHETIC DATA CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project TEST WITH SYNTHETIC DATA Grid: 20x20 Km = 400 Km2 Radar Pixels 1x1 Km Satellite pixels 5x5 Km 9 raingauges WE GENERATED THE “TRUE” RAINFALL FIELD ON THE GRID AND ON THE GAUGES WE GENERATED A NOISE FIELD ON THE GRID AND WE ADDED IT TO THE TRUE RAINFALL FIELD TO GET RADAR LIKE ESTIMATES WE GENERATED A NOISE FIELD FOR THE GAUGES AND WE ADDED IT TO THE TRUE RAINFALL FIELD TO GET SATELLITE LIKE ESTIMATES WE GENERATED NOISES FOR THE GAUGES AND WE ADDED THEM TO THE TRUE RAINFALL FIELD Variograms used in data generation: Gaussian Exponential Modified spherical Distributions used in the data generation: Normal Distribution Log-Normal Distribution CARPE DIEM Meeting – Helsinki, 24th June 2004

28 CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project GAUGES: Bias = 0.00 Exp. Variance = 0.90 RADAR: Bias = 5.00 Exp. Variance = 0.75 SATELLITE: Bias = 10.00 Exp. Variance = 0.70 CARPE DIEM Meeting – Helsinki, 24th June 2004

29 CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project TRUE BK RADAR BK+RADAR CARPE DIEM Meeting – Helsinki, 24th June 2004

30 CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project BK+RADAR SATELLITE BK+RAD+SAT BK+RADAR BK+RAD+SAT TRUE CARPE DIEM Meeting – Helsinki, 24th June 2004

31 RAINGAUGES & SATELLITE BAYESIAN COMBINATION
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project RAINGAUGES & SATELLITE BAYESIAN COMBINATION RAINGAUGES BLOCK KRIGING BLOCK KRIGING OF THE RAINGAUGES UPSCALING BK GAUGES at satellite scale SATELLITE KALMAN FILTER BK GAUGES + SATELLITE at satellite scale DOWNSCALING (KALMAN SMOOTHING) BLOCK KRIGING OF THE RAINGAUGES + + SATELLITE CARPE DIEM Meeting – Helsinki, 24th June 2004

32 TEST WITH SYNTHETIC DATA CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project TEST WITH SYNTHETIC DATA Grid: 20x20 Km = 400 Km2 Satellite pixels 5x5 Km 9 raingauges WE GENERATED THE “TRUE” RAINFALL FIELD ON THE GRID AND ON THE GAUGES WE GENERATED A NOISE FIELD FOR THE GAUGES AND WE ADDED IT TO THE TRUE RAINFALL FIELD TO GET SATELLITE LIKE ESTIMATES WE GENERATED NOISES FOR THE GAUGES AND WE ADDED THEM TO THE TRUE RAINFALL FIELD Variograms used in data generation: Gaussian Exponential Modified spherical Distributions used in the data generation: Normal Distribution Log-Normal Distribution CARPE DIEM Meeting – Helsinki, 24th June 2004

33 CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project GAUGES: Bias = 0.00 Exp. Variance = 0.90 SATELLITE: Bias = 10.00 Exp. Variance = 0.70 CARPE DIEM Meeting – Helsinki, 24th June 2004

34 CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project TRUE BK BK aggr. SATELLITE BK+SATELLITE CARPE DIEM Meeting – Helsinki, 24th June 2004

35 CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project BK+SATELLITE aggr. BK+SATELLITE TRUE CARPE DIEM Meeting – Helsinki, 24th June 2004

36 APPLICATION TO THE RENO RIVER BASIN
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project APPLICATION TO THE RENO RIVER BASIN BASIN AREA: 1081 Km2 NUMBER OF RAINGAUGES: 25 RADAR: C band Doppler Double Polarization RADAR PIXELS: 1 x 1 Km SATELLITE: Meteosat SATELLITE PIXELS: 5 X 5 Km CARPE DIEM Meeting – Helsinki, 24th June 2004

37 CARPE DIEM Meeting – Helsinki, 24th June 2004
15th April 1998 – ore 15 2.6 BLOCK KRIGING RADAR SATELLITE BK + RADAR BK + SATELLITE BK+RADAR+ SATELLITE CARPE DIEM Meeting – Helsinki, 24th June 2004 0.0

38 CARPE DIEM Meeting – Helsinki, 24th June 2004
15th April 1998 – ore 16 4.4 BLOCK KRIGING RADAR SATELLITE BK + RADAR BK + SATELLITE BK+RADAR+ SATELLITE CARPE DIEM Meeting – Helsinki, 24th June 2004 0.0

39 CARPE DIEM Meeting – Helsinki, 24th June 2004
15th April 1998 – ore 18 8.7 BLOCK KRIGING RADAR SATELLITE BK + RADAR BK + SATELLITE BK+RADAR+ SATELLITE CARPE DIEM Meeting – Helsinki, 24th June 2004 0.0

40 CARPE DIEM Meeting – Helsinki, 24th June 2004
15th April 1998 – ore 19 2.8 BLOCK KRIGING RADAR SATELLITE BK + RADAR BK + SATELLITE BK+RADAR+ SATELLITE CARPE DIEM Meeting – Helsinki, 24th June 2004 0.0

41 BAYESIAN COMBINATIONS & TOPKAPI
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project BAYESIAN COMBINATIONS & TOPKAPI RADAR OVERESTIMATION 13-22 November 2000 CARPE DIEM Meeting – Helsinki, 24th June 2004

42 CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project THANK YOU CARPE DIEM Meeting – Helsinki, 24th June 2004

43 CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project REFERENCES: Alberoni, P.P., Nanni, S., Application of an adjustment procedure for quantitative rainfall evaluation, Advances in hydrological applications of weather RADAR. Proceedings of the 2nd International Symposium on Hydrological Applications of weather RADAR. Barnes, E.A., A technique for maximizing details in numerical weather map analysis. J. Appl. Meteor., 3: Brandes, E.A., Optimizing rainfall estimates with the aid of RADAR, J. Appl. Meteor., 14: Cressie, N.A., Statistics for Spatial Data. Wiley, New York. Creutin, J.D., Delrieu, G., Lebel, T., Rain measurement by raingauge-radar combination: a geostatistical approach. J. Appl. Atmos. Ocean. Technol., 5:102–115. De Marsily, G., Quantitative Hydrogeology, Academic Press. Fieguth, P.W., Karl, W.C., Willsky, A.S., Wunsch, C., Multiresolution optimal interpolation and statistical analysis of TOPEX/POSEIDON satellite altimetry. IEEE Trans. Geosci. Remote Sensing, 33(2): Gelb, A Applied Optimal Estimation, MIT Press, Cambridge, MA. Kalman, R.E A New Approach to Linear Filtering and Prediction Problems. Transaction of the ASME—Journal of Basic Engineering, Koistinen, J., Puhakka, T., An improved spatial gauge-RADAR adjustment technique, 20th Conference on RADAR Meteorology, AMS Boston USA, Krajewski, W.F., Cokriging Radar-Rainfall and Rain Gage Data. Journal of Geophysical Research, 92(D8): CARPE DIEM Meeting – Helsinki, 24th June 2004 Matheron, G., La théorie des variables regionaliséés et ses applications, Cah. Cent. Morphol. Math., 5.

44 ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project Maybeck, P.S., Stochastic Models, Estimation, and Control, Volume 1, Academic Press, Inc. Rauch, H., Tung, F., Striebel, C., Maximum likelihood estimates of linear dynamic systems. AIAA J. 3(8): Todini, E., Bayesian conditioning of radar to rain-gauges, Hydrol. Earth System Sci., 5: Todini, E., 2001 (Part 1). Influence of parameter estimation uncertainty in Kriging. Part 1. Theoretical development, Hydrol. Earth System Sci., 5(2):

45 ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project

46 CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project BK RAINGAUGES RADAR BK and Radar estimate are combined on the basis of the local relative uncertainty, which is updated at each time step on each pixel. CARPE DIEM Meeting – Helsinki, 24th June 2004

47 Measurement Update (“Correct”) Block Kriging + SATELLITE
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project Measurement Update (“Correct”) Block Kriging + SATELLITE Measurement equation: Measurement equation: Satellite A priori estimate: A priori estimate: Compute the Kalman gain: Compute the Kalman gain: Update estimate with measurement zk: Update estimate with measurement zk: Update the error covariance: Update the error covariance: CARPE DIEM Meeting – Helsinki, 24th June 2004

48 BAYESIAN COMBINATIONS & TOPKAPI
ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA MUSIC Project BAYESIAN COMBINATIONS & TOPKAPI RADAR UNDERESTIMATION SATELLITE UNDERESTIMATION CARPE DIEM Meeting – Helsinki, 24th June 2004 13-18 April 1998

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