THE MULTI-SENSOR BAYESIAN COMBINATIONS Cinzia Mazzetti ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA...

THE MULTI-SENSOR BAYESIAN COMBINATIONS

Cinzia Mazzetti

ALMA MATER STUDIORUMALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNAUNIVERSITA’ DI BOLOGNA MUSIC ProjectMUSIC Project

CARPE DIEM Meeting – Helsinki, 24th June 2004


ALMA MATER STUDIORUMUNIVERSITA’ 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.



MULTI-SENSOR BAYESIAN COMBINATIONSMULTI-SENSOR BAYESIAN COMBINATIONS

RAINGAUGES + RADAR

RAINGAUGES + SATELLITE

RAINGAUGES + RADAR + SATELLITE

RADAR + SATELLITE



RAINGAUGES

BLOCK KRIGING

BLOCK KRIGING OF THE RAINGAUGES

KALMAN FILTER

RADAR

BLOCK KRIGING OF THE RAINGAUGES + RADAR (BKR)

POINTMEASUREMENTS

SPATIALESTIMATES

RAINGAUGES & RADAR BAYESIAN COMBINATIONRAINGAUGES & RADAR BAYESIAN COMBINATION



i

ii

S ZZ *0

*

0

000

* SS ZZE

2*

00 SS ZZE minimum

d VARIOGRAM It describes the spatial relation between measurement points and estimation points.

S0

Zn

Z1

Z2

x1

x2

xn

0

0

0

1SS dxxZ

SZ

BLOCK KRIGINGBLOCK KRIGING

10 i

i



BLOCK KRIGING: New Variogram fittingBLOCK KRIGING: New Variogram fitting

2

1 A

d

epd GAUSSIAN VARIOGRAM

p, , A VARIOGRAM PARAMETERS

Traditional estimation method for the Variogram

parameters(Matheron, 1970; De Marsily, 1986, Cressie, 1993)



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



10 j

j

j

ijij Sxxx 00 ,

T

lb

T uu

u

0

00 j

10 j

j

j

ijij Sxxx 00 ,

BLOCK KRIGING: New formulation with non-BLOCK KRIGING: New formulation with non-negativenegative weightsweights

New Block Kriging system

Block Kriging system

i

ii

S ZZ 00

S0

Zn

Z1

Z2

x1

x2

xn

No negative rain



llu

u

Cov

0

Covariance of the estimation errors:

llT

T

blT

Tbl

T

T

T

u

uu

u

Cov

*

*

*

*

*

**

0



BLOCK KRIGING: Error prone raingauge BLOCK KRIGING: Error prone raingauge measurementsmeasurements

T

bl

T uu

u

0 Block Kriging system

De Marsily (1986): 2

, iii

22, 2

1

2

1, jijiji xx

2

, 2

1, ijijibl xx

NEWFORMULATION:

2i Error Variance of gauge i



KALMAN FILTERKALMAN FILTER

Time Update(“Predict”)

Measurement Update(“Correct”)

Initial estimates for:

1kx

1kP

An optimal recursive data processing algorithm.

(Maybeck, 1979)

(Gelb, 1974)





A priori estimate Radar

Measurement BK raingauges

A posteriori estimate Raingauges and RadarBayesian combination

(Gelb, 1974)



Measurement Update (“Correct”)

Compute the Kalman gain:

Update estimate with measurement zk:

kkkkk xHzKxx ˆˆˆ

Update the error covariance:

kkk PHKIP

kkk Hxz Measurement equation:

RNp ,0

kx

A priori estimate:

Block Kriging + RADAR




Measurement equation:Gtt

Gt

Gt yZy

t

Rtt yy '

'''' tGtttt yyKyy

t

VKIPKIP tttt '''

Gt

V

kP

tV

A priori estimate:

1 Gttt

VVVK t 1 RHHPHPK Tk

Tkk



KALMAN FILTER: Update of the GAIN at KALMAN FILTER: Update of the GAIN at eacheach time steptime step

Variance of BK estimation errors

Rt

Gtt yy

tV

Variance of Radar 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)

GAIN: Ratio between the variances of the estimation errors 1 G

tttVVVK t



Grid: 20x20 Km = 400 Km2

9 Raingauges

O O

O

O O

O

O

O

O

O

TEST WITH SYNTHETIC DATATEST WITH SYNTHETIC DATA

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



ERROR FREE GAUGES

NormalDistribution

Bias Variance

Log-NormalDistribution

Bias Variance

Raw dataCombined data



ERROR PRONE GAUGES

NormalDistribution

Bias Variance

Log-NormalDistribution

Bias Variance

Raw dataCombined data



COMPARISON WITH OTHER METHODSCOMPARISON WITH OTHER METHODS

Brandes

Koistinen and Puhakka

Krajewski

The comparison is made on the basis of a common numerical example

Bayesian combination

COMPARING SOME EXISTING METHODS FORCOMBINING RADAR AND RAINGAUGEMEASUREMENTS TO THE NEW BAYESIAN METHOD.

SCOPE



GEN VARIO BIAS EX VAR BIAS EX VAR BIAS EX VAR BIAS EX VAR

GAU GAU EE 0.24 0.86 0.51 0.79 -0.53 0.71 -0.11 0.72EXP EXP EE 0.21 0.80 0.63 0.75 -0.36 0.73 -0.07 0.74MOD MOD EE 0.21 0.81 0.62 0.75 -0.42 0.71 -0.09 0.72GAU EXP EE 0.22 0.82 0.51 0.79 -0.53 0.71 -0.11 0.72GAU MOD EE 0.23 0.83 0.51 0.79 -0.53 0.71 -0.11 0.72EXP GAU EE 0.19 0.77 0.63 0.75 -0.36 0.73 -0.07 0.74MOD GAU EE 0.19 0.78 0.62 0.75 -0.42 0.71 -0.09 0.72

GAU GAU EE 0.06 0.88 0.48 0.80 -0.53 0.72 -0.19 0.72EXP EXP EE 0.04 0.81 0.57 0.76 -0.37 0.75 -0.13 0.74MOD MOD EE 0.03 0.82 0.59 0.77 -0.44 0.73 -0.14 0.73GAU EXP EE 0.04 0.84 0.48 0.80 -0.53 0.72 -0.19 0.72GAU MOD EE 0.05 0.84 0.48 0.80 -0.53 0.72 -0.19 0.72EXP GAU EE 0.01 0.78 0.57 0.76 -0.37 0.75 -0.13 0.74MOD GAU EE 0.03 0.79 0.59 0.77 -0.44 0.73 -0.14 0.73

GAU GAU EE -0.11 0.89 0.36 0.82 -0.50 0.73 -0.30 0.74EXP EXP EE -0.10 0.82 0.53 0.78 -0.32 0.77 -0.31 0.75MOD MOD EE -0.11 0.82 0.52 0.78 -0.38 0.75 -0.26 0.73GAU EXP EE -0.10 0.84 0.36 0.82 -0.50 0.73 -0.30 0.74GAU MOD EE -0.10 0.84 0.36 0.82 -0.50 0.73 -0.30 0.74EXP GAU EE -0.13 0.78 0.53 0.78 -0.32 0.77 -0.31 0.75MOD GAU EE -0.11 0.80 0.52 0.78 -0.38 0.75 -0.26 0.73

GAU GAU EE -0.26 0.88 0.24 0.84 -0.42 0.80 -0.63 0.76EXP EXP EE -0.25 0.81 0.37 0.81 -0.24 0.81 -0.50 0.78MOD MOD EE -0.26 0.82 0.36 0.81 -0.30 0.80 -0.55 0.77GAU EXP EE -0.25 0.84 0.24 0.84 -0.42 0.80 -0.63 0.76GAU MOD EE -0.25 0.84 0.24 0.84 -0.42 0.80 -0.63 0.76EXP GAU EE -0.28 0.77 0.37 0.81 -0.24 0.81 -0.50 0.78MOD GAU EE -0.28 0.79 0.36 0.81 -0.30 0.80 -0.55 0.77

No

rmal

dis

trib

uti

on

BAYESIAN COMBINATION

BRANDES (EP=125)

KOISTINEN KRAJEWSKIL

og

-no

rmal

dis

trib

uti

on

(s

kew

nes

s 0.

5)

Lo

g-n

orm

al d

istr

ibu

tio

n

(ske

wn

ess

1.0)

L

og

-no

rmal

dis

trib

uti

on

(s

kew

nes

s 2.

0)

BIAS EX VAR BIAS EX VAR

0.22 0.81 0.26 0.84

0.34 0.72 0.36 0.77

0.31 0.74 0.35 0.78

0.22 0.81 0.26 0.84

0.22 0.81 0.26 0.84

0.34 0.72 0.36 0.77

0.31 0.74 0.35 0.78

Brandes (EP=10) Brandes (EP=50)

GAUGES:Bias = 0.00Exp. Variance = 0.90

RADAR:Bias = 5.00Exp. Variance = 0.70



RAINGAUGES RADAR & SATELLITE BAYESIAN RAINGAUGES RADAR & SATELLITE BAYESIAN COMBINATIONCOMBINATION

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

y yP~

yUx T ˆˆ UPUP yT

x ~~



KALMAN FILTER at the SATELLITE SCALEKALMAN FILTER at the SATELLITE SCALE



A priori estimate Aggregated BK+RADAR estimate

Measurement Satellite

A posteriori estimate Raingauges+Radar+Satellite Bayesian combination(Sat. scale)



xzx ˆ R

1~~

RPPK xxx

Estimation errors variance

of Satellite estimate

Estimation errors variance of aggregated

BK+RADAR 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)

GAIN: Ratio between the variance of the estimation errors

KALMAN FILTER: Update KALMAN FILTER: Update ofof the GAIN at the GAIN at eacheach time steptime step



BLOCK KRIGING OF THE RAINGAUGES + RADAR

UPSCALING

BK GAUGES + RADAR at satellite scale SATELLITE

KALMAN FILTER

BK GAUGES + RADAR + SATELLITE at satellite scale


BLOCK KRIGING OF THE RAINGAUGES + RADAR + SATELLITE

y yP~

yUx T ˆˆ UPUP yT

x ~~

xxP~~



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)

DOWNSCALING (KALMAN SMOOTHER)DOWNSCALING (KALMAN SMOOTHER)



Nkx NkP

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).

On scale

N

k

k-1

Up

scalin

g

Dow

nscali

ng



xz

BK and RADAR scale Satellite scale

xzKxx xx ˆˆˆ

xxxPKIP ~~~

yT

yyPUKIP ~~~

xzKyy xy ˆˆˆ

yUx T ˆˆ

UPUP yT

x ~~

y

yP~

1~~

RUPUUPK y

Tyy

xxUPUUPyy yT

y ˆˆˆˆ 1~~

yT

yT

yyPURUPUUPIP ~

1~~~~



Grid: 20x20 Km = 400 Km2

Radar Pixels 1x1 Km

Satellite pixels 5x5 Km

9 raingauges

O O

O

O O

O

O

O

O

O


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

WE GENERATED A NOISE FIELD FOR THE GAUGES AND WE ADDED IT TO THE TRUE RAINFALL FIELD TO GET SATELLITE LIKE ESTIMATES





ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC ProjectBIAS EX. VAR BIAS EX. VAR BIAS EX. VAR

BLOCK KRIGING 0.03 0.88 0.01 0.74 0.01 0.74

RADAR 4.95 0.75 4.94 0.74 4.94 0.74

BK+RADAR 0.02 0.90 0.01 0.82 0.01 0.82

BK+RADAR aggr 0.02 0.92 0.01 0.88 0.01 0.88

SATELLITE 9.99 0.70 9.99 0.70 9.99 0.70

BK+RAD+SAT aggr 0.12 0.93 0.46 0.89 0.37 0.89

BK+RADAR+SAT 0.12 0.90 0.46 0.84 0.37 0.84

BLOCK KRIGING -0.06 0.88 -0.08 0.74 -0.08 0.74

RADAR 4.95 0.75 4.95 0.74 4.95 0.75

BK+RADAR -0.06 0.90 -0.09 0.82 -0.09 0.82

BK+RADAR aggr -0.06 0.92 -0.09 0.88 -0.09 0.88

SATELLITE 10.00 0.70 10.00 0.70 9.99 0.70

BK+RAD+SAT aggr 0.04 0.93 0.35 0.89 0.28 0.89

BK+RADAR+SAT 0.04 0.90 0.35 0.84 0.28 0.84

BLOCK KRIGING -0.13 0.87 -0.16 0.73 -0.16 0.73

RADAR 4.96 0.75 4.95 0.75 4.96 0.75

BK+RADAR -0.13 0.89 -0.16 0.82 -0.17 0.82

BK+RADAR aggr -0.13 0.92 -0.16 0.88 -0.17 0.87

SATELLITE 10.00 0.70 10.00 0.70 10.00 0.70

BK+RAD+SAT aggr -0.02 0.93 0.30 0.90 0.23 0.89

BK+RADAR+SAT -0.02 0.90 0.30 0.84 0.23 0.84

BLOCK KRIGING -0.22 0.87 -0.25 0.71 -0.25 0.71

RADAR 4.96 0.75 4.96 0.75 4.97 0.75

BK+RADAR -0.23 0.89 -0.26 0.81 -0.26 0.81

BK+RADAR aggr -0.23 0.92 -0.26 0.87 -0.26 0.87

SATELLITE 10.00 0.70 10.01 0.69 10.00 0.70

BK+RAD+SAT aggr -0.10 0.93 0.21 0.89 0.15 0.89

BK+RADAR+SAT -0.10 0.90 0.21 0.83 0.15 0.83Lo

g-n

orm

al d

istr

ibu

tio

n

(sk

ew

ne

ss

2.0

)

GAUSSIAN EXPONENTIAL SPHERICAL

No

rma

l d

istr

ibu

tio

nL

og

-no

rma

l d

istr

ibu

tio

n

(sk

ew

ne

ss

0.5

)

Lo

g-n

orm

al d

istr

ibu

tio

n

(sk

ew

ne

ss

1.0

)


RADAR:Bias = 5.00Exp. Variance = 0.75

SATELLITE:Bias = 10.00Exp. Variance = 0.70



TRUE BK RADAR BK+RADAR



BK+RADAR

SATELLITE BK+RAD+SAT

BK+RAD+SAT

TRUEBK+RADAR



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


BLOCK KRIGING OF THE RAINGAUGES ++ SATELLITE



O O

O

O O

O

O

O

O

O


WE GENERATED NOISES FOR THE GAUGES AND WE ADDED THEM TO THE TRUE RAINFALL FIELD

WE GENERATED A NOISE FIELD FOR THE GAUGES AND WE ADDED IT TO THE TRUE RAINFALL FIELD TO GET SATELLITE LIKE ESTIMATES




Grid: 20x20 Km = 400 Km2

Satellite pixels 5x5 Km

9 raingauges



BIAS EX. VAR BIAS EX. VAR BIAS EX. VAR

BLOCK KRIGING 0.03 0.87 0.01 0.74 0.01 0.74

BK aggr 0.03 0.91 0.01 0.87 0.01 0.86

SATELLITE 9.99 0.70 9.99 0.70 9.99 0.70

BK+SATELLITE aggr 0.59 0.92 0.93 0.88 0.78 0.89

BK+SATELLITE 0.59 0.89 0.93 0.76 0.78 0.76

BLOCK KRIGING -0.06 0.88 -0.02 0.73 -0.08 0.74

BK aggr -0.06 0.91 -0.02 0.86 -0.08 0.86

SATELLITE 10.00 0.70 10.00 0.70 9.99 0.70


BK+SATELLITE 0.50 0.89 0.85 0.76 0.67 0.76

BLOCK KRIGING -0.13 0.87 -0.19 0.72 -0.16 0.73

BK aggr -0.13 0.91 -0.19 0.86 -0.16 0.86

SATELLITE 10.00 0.70 10.00 0.69 10.00 0.70


BK+SATELLITE 0.44 0.89 0.70 0.75 0.66 0.75

BLOCK KRIGING -0.22 0.87 -0.24 0.71 -0.25 0.71

BK aggr -0.22 0.91 -0.24 0.86 -0.25 0.86

SATELLITE 10.00 0.70 10.01 0.69 10.00 0.70


BK+SATELLITE 0.39 0.89 0.75 0.74 0.63 0.74Lo

g-n

orm

al d

istr

ibu

tion

(s

kew

nes

s 2.

0)

GAUSSIAN EXPONENTIAL SPHERICAL

No

rmal

dis

trib

utio

nL

og

-no

rmal

dis

trib

utio

n

(ske

wn

ess

0.5)

Lo

g-n

orm

al d

istr

ibu

tion

(s

kew

nes

s 1.

0)


SATELLITE:Bias = 10.00Exp. Variance = 0.70



TRUE BK BK aggr. SATELLITE

BK+SATELLITE



BK+SATELLITE aggr.

BK+SATELLITE

TRUE



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

APPLICATION TO THE RENO RIVER BASINAPPLICATION TO THE RENO RIVER BASIN


15th April 1998 – ore 15 BLOCK KRIGING

BK + RADAR BK + SATELLITE BK+RADAR+SATELLITE

RADAR SATELLITE2.62.6

0.00.0


4.44.4

0.00.0

BLOCK KRIGING RADAR SATELLITE


15th April 1998 – ore 16


8.78.7

0.00.0





2.82.8

0.00.0






Discharge at CASALECCHIO

0

500

1000

1500

2000

2500

3000

1 6

11

16

21

26

31

36

41

46

51

56

61

66

71

76

81

86

91

96

10

1

10

6

11

1

11

6

12

1

12

6

13

1

13

6

14

1

time (h)

Dis

ch

arg

e (

m3

/s)

OBSERVED

BLOCK KRIGING

RADAR

BK + RADAR

BK + SATELLITE

BK + RADAR + SATELLITE


0

100

200

300

400

500

600

700

1 4 7

10

13

16

19

22

25

28

31

34

37

40

43

46

49

52

55

58

61

64

67

time (h)

Dis

ch

arg

e (

m3

/s)

RADAR OVERESTIMATION

BAYESIAN COMBINATIONS & TOPKAPI

13-22 November 2000



THANK YOU



REFERENCES:REFERENCES:

Alberoni, P.P., Nanni, S., 1992. 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., 1964. A technique for maximizing details in numerical weather map analysis. J. Appl. Meteor., 3:396-409.

Brandes, E.A., 1975. Optimizing rainfall estimates with the aid of RADAR, J. Appl. Meteor., 14:1339-1345. Cressie, N.A., 1993. Statistics for Spatial Data. Wiley, New York.

Creutin, J.D., Delrieu, G., Lebel, T., 1988. Rain measurement by raingauge-radar combination: a geostatistical approach. J. Appl. Atmos. Ocean. Technol., 5:102–115.

De Marsily, G., 1986. Quantitative Hydrogeology, Academic Press.

Fieguth, P.W., Karl, W.C., Willsky, A.S., Wunsch, C., 1995. Multiresolution optimal interpolation and statistical analysis of TOPEX/POSEIDON satellite altimetry. IEEE Trans. Geosci. Remote Sensing, 33(2):280-292.

Kalman, R.E. 1960. A New Approach to Linear Filtering and Prediction Problems. Transaction of the ASME—Journal of Basic Engineering, 35-45.

Koistinen, J., Puhakka, T., 1981. An improved spatial gauge-RADAR adjustment technique, 20th Conference on RADAR Meteorology, AMS Boston USA, 179-186.

Krajewski, W.F., 1987. Cokriging Radar-Rainfall and Rain Gage Data. Journal of Geophysical Research, 92(D8):9571-9580.

Matheron, G., 1970. La théorie des variables regionaliséés et ses applications, Cah. Cent. Morphol. Math., 5.

Gelb, A. 1974. Applied Optimal Estimation, MIT Press, Cambridge, MA.


Maybeck, P.S., 1979. Stochastic Models, Estimation, and Control, Volume 1, Academic Press, Inc.

Rauch, H., Tung, F., Striebel, C., 1965. Maximum likelihood estimates of linear dynamic systems. AIAA J. 3(8):1445-1450.

Todini, E., 2001. Bayesian conditioning of radar to rain-gauges, Hydrol. Earth System Sci., 5:225-232. Todini, E., 2001 (Part 1). Influence of parameter estimation uncertainty in Kriging. Part 1. Theoretical development, Hydrol. Earth System Sci., 5(2):215-223.



Gtt

VV BK RAINGAUGES

Gtt

VV 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.

1 Gttt

VVVK t



kkkkk xHzKxx ˆˆˆ

kkk PHKIP

kkk Hxz RNp ,0

kx

A priori estimate:

kP

xz R Satellite

yUx T ˆˆ xP~

xzKxx xx ˆˆˆ

xxxPKIP ~~~

1~~

RPPK xxx 1 RHHPHPK Tk

Tkk

Measurement Update (“Correct”) Block Kriging + SATELLITE




Measurement equation:

A priori estimate:




Measurement equation:




0

50

100

150

200

250

300

1 5 9

13

17

21

25

29

33

37

41

45

49

53

57

61

65

69

73

77

81

85

89

93

97

10

1

10

5

10

9

11

3

time (h)

Dis

ch

arg

e (

m3

/s)

OBSERVED

BLOCK KRIGING

RADAR

BK + RADAR

BK + SATELLITE

BK + RADAR + SATELLITE

RADAR UNDERESTIMATIONSATELLITE UNDERESTIMATION

BAYESIAN COMBINATIONS & TOPKAPI

13-18 April 1998

THE MULTI-SENSOR BAYESIAN COMBINATIONS Cinzia Mazzetti ALMA MATER STUDIORUM UNIVERSITA’ DI BOLOGNA...

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