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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
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.
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
MULTI-SENSOR BAYESIAN COMBINATIONSMULTI-SENSOR BAYESIAN COMBINATIONS
RAINGAUGES + RADAR
RAINGAUGES + SATELLITE
RAINGAUGES + RADAR + SATELLITE
RADAR + SATELLITE
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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)
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ 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
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
llu
u
Cov
0
Covariance of the estimation errors:
llT
T
blT
Tbl
T
T
T
u
uu
u
Cov
*
*
*
*
*
**
0
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
KALMAN FILTERKALMAN FILTER
Time Update(“Predict”)
Measurement Update(“Correct”)
Initial estimates for:
1kx
1kP
An optimal recursive data processing algorithm.
(Maybeck, 1979)
(Gelb, 1974)
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
Time Update(“Predict”)
Measurement Update(“Correct”)
A priori estimate Radar
Measurement BK raingauges
A posteriori estimate Raingauges and RadarBayesian combination
(Gelb, 1974)
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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
Compute the Kalman gain:
Update estimate with measurement zk:
Update the error covariance:
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
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
ERROR FREE GAUGES
NormalDistribution
Bias Variance
Log-NormalDistribution
Bias Variance
Raw dataCombined data
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
ERROR PRONE GAUGES
NormalDistribution
Bias Variance
Log-NormalDistribution
Bias Variance
Raw dataCombined data
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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 ~~
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
KALMAN FILTER at the SATELLITE SCALEKALMAN 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
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ 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
y yP~
yUx T ˆˆ UPUP yT
x ~~
xxP~~
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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)
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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~~~~
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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 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
WE GENERATED A NOISE FIELD FOR THE GAUGES AND WE ADDED IT TO THE TRUE RAINFALL FIELD TO GET SATELLITE LIKE ESTIMATES
Distributions used in the data generation: Normal Distribution Log-Normal Distribution
Variograms used in data generation:- Gaussian- Exponential- Modified spherical
TEST WITH SYNTHETIC DATATEST WITH SYNTHETIC DATA
CARPE DIEM Meeting – Helsinki, 24th June 2004
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
)
GAUGES:Bias = 0.00Exp. Variance = 0.90
RADAR:Bias = 5.00Exp. Variance = 0.75
SATELLITE:Bias = 10.00Exp. Variance = 0.70
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
TRUE BK RADAR BK+RADAR
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
BK+RADAR
SATELLITE BK+RAD+SAT
BK+RAD+SAT
TRUEBK+RADAR
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ 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
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
O O
O
O O
O
O
O
O
O
WE GENERATED THE “TRUE” RAINFALL FIELD ON THE GRID AND ON THE GAUGES
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
Distributions used in the data generation: Normal Distribution Log-Normal Distribution
Variograms used in data generation:- Gaussian- Exponential- Modified spherical
TEST WITH SYNTHETIC DATATEST WITH SYNTHETIC DATA
Grid: 20x20 Km = 400 Km2
Satellite pixels 5x5 Km
9 raingauges
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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 aggr 0.50 0.92 0.85 0.89 0.67 0.89
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 aggr 0.44 0.92 0.70 0.89 0.66 0.88
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 aggr 0.39 0.93 0.75 0.89 0.63 0.88
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)
GAUGES:Bias = 0.00Exp. Variance = 0.90
SATELLITE:Bias = 10.00Exp. Variance = 0.70
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
TRUE BK BK aggr. SATELLITE
BK+SATELLITE
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
BK+SATELLITE aggr.
BK+SATELLITE
TRUE
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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
CARPE DIEM Meeting – Helsinki, 24th June 2004
15th April 1998 – ore 15 BLOCK KRIGING
BK + RADAR BK + SATELLITE BK+RADAR+SATELLITE
RADAR SATELLITE2.62.6
0.00.0
CARPE DIEM Meeting – Helsinki, 24th June 2004
4.44.4
0.00.0
BLOCK KRIGING RADAR SATELLITE
BK + RADAR BK + SATELLITE BK+RADAR+SATELLITE
15th April 1998 – ore 16
CARPE DIEM Meeting – Helsinki, 24th June 2004
8.78.7
0.00.0
BLOCK KRIGING RADAR SATELLITE
BK + RADAR BK + SATELLITE BK+RADAR+SATELLITE
15th April 1998 – ore 18
CARPE DIEM Meeting – Helsinki, 24th June 2004
2.82.8
0.00.0
BLOCK KRIGING RADAR SATELLITE
BK + RADAR BK + SATELLITE BK+RADAR+SATELLITE
15th April 1998 – ore 19
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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
Discharge at CASALECCHIO
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
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
THANK YOU
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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.
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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.
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
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
Compute the Kalman gain:
Update estimate with measurement zk:
Update the error covariance:
Measurement equation:
A priori estimate:
Compute the Kalman gain:
Update estimate with measurement zk:
Update the error covariance:
Measurement equation:
CARPE DIEM Meeting – Helsinki, 24th June 2004
ALMA MATER STUDIORUMUNIVERSITA’ DI BOLOGNA MUSIC Project
Discharge at CASALECCHIO
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