isztar zawadzki and bernat puigdomènech treserras mcgill...
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Ensemble Retrievals
Isztar Zawadzki and Bernat Puigdomènech TreserrasMcGill University
Tuesday, 29 May, 12
There is never sufficient information in radar measurement to resolve unambiguously the complexity of microphysics.
Background
Thus, whatever model we use there will be a good deal of uncertainty in the quantitative interpretation of radar measurements.
Radar measurements are not proper measurements without a good assessment of their uncertainties.
When radar measurements are interpreted quantitatively we always use a model microphysics of the backscattering targets.
In radar data assimilation this is particularly fundamental.
Model microphysics is an approximation to a very complex reality
And all this applies to satellite measurements as well
Tuesday, 29 May, 12
What do we call a Retrieval
Inference of atmospheric parameters from observations related but not equal to the parameters of interest.
Simplest example: retrieving LWC, M from Z
Tuesday, 29 May, 12
What do we call a Retrieval
Inference of atmospheric parameters from observations related but not equal to the parameters of interest.
The correspondence between observations and retrieved parameters is established through physical equations. These equations are affected by uncertainties due to the natural
variability of the parameters in these equations. Thus, these are approximate “models” of the physical processes.
Simplest example: retrieving LWC, M from Z
Tuesday, 29 May, 12
What do we call a Retrieval
Inference of atmospheric parameters from observations related but not equal to the parameters of interest.
The correspondence between observations and retrieved parameters is established through physical equations. These equations are affected by uncertainties due to the natural
variability of the parameters in these equations. Thus, these are approximate “models” of the physical processes.
Simplest example: retrieving LWC, M from Z
Simplest example: M = π6ρw D3
0
∞
∫ N(D)dD ; Z = D60
∞
∫ N(D)dD
N(D) has a good deal of natural variability
Tuesday, 29 May, 12
Ensemble Retrievals
Each combination of the possible values of the parameters leads to one set of physical relationships and to one corresponding retrieval.
N(D) = N0 Dµ exp(−λD)
Simplest example: for a given set of possible values of
we get one retrieval of M from ZN0 ,µ, λ
Tuesday, 29 May, 12
Ensemble Retrievals
Each combination of the possible values of the parameters leads to one set of physical relationships and to one corresponding retrieval.
By sweeping through all the possible values of the parameters we obtain the ensemble of possible models leading to the ensemble of retrievals. This ensemble contains all the
uncertainty of the physical model.
N(D) = N0 Dµ exp(−λD)
Simplest example: for a given set of possible values of
we get one retrieval of M from ZN0 ,µ, λ
Tuesday, 29 May, 12
Ensemble Retrievals
Each combination of the possible values of the parameters leads to one set of physical relationships and to one corresponding retrieval.
By sweeping through all the possible values of the parameters we obtain the ensemble of possible models leading to the ensemble of retrievals. This ensemble contains all the
uncertainty of the physical model.
N(D) = N0 Dµ exp(−λD)
Simplest example: for a given set of possible values of
we get one retrieval of M from ZN0 ,µ, λ
In our simplest example, the Z-M relationship, as an alternative we can use a set of observed DSDs to generate the ensemble.
Tuesday, 29 May, 12
A particular precipitation event
From our entire disdrometer data base we select the days which the DSDs records had maximum values of dBZ and durations compatible with this
particular case, namely a mixture of convection and wide spread rain. We have ~200 days of DSD data of this type of events.
Tuesday, 29 May, 12
A particular precipitation event
From our entire disdrometer data base we select the days which the DSDs records had maximum values of dBZ and durations compatible with this
particular case, namely a mixture of convection and wide spread rain. We have ~200 days of DSD data of this type of events.
We treat the 200 days as equiprobable realizations of the same processTuesday, 29 May, 12
Data Base
From the 200 days of distrometric records we have this distribution of logM versus logZ
Typically an average Z-M relationship, obtained by
some kind of regression, is used in retrieving M from Z.
Let us apply ensemble retrieval to this problem.
10 20 30 40 50 60Z [dBZ]
0.001
0.01
0.1
1.0
10
100
M [g
/m3 ]
Tuesday, 29 May, 12
ExampleHere we formulate the retrieval of M as
M = m p(m | [Z ±δ ]) dm∫
p(m | Z )and for each dBZ interval determine
δ = .5 dB if 10<35;1dBZ if 35<Z<42; 2.5dB if Z>42dBZLet us take
Tuesday, 29 May, 12
Example
M = 0.15 = m p(m | [29 − 30dBZ ]) dm∫
29.0 29.2 29.4 29.6 29.8 30.0Z [dBZ]
-1.5 -1.0 -0.5 0.0log10 (M)
0
50
100
150
200Pmean -9.16e-01
1 2 3 4 5D [mm]
10-4
10-2
100
102
104
N(D
)
Num
ber o
f DSD
s
0.1
1.0
M [k
g/m
3 ]
mean 0.151 [g/kg3]X2 =1.75e-02
Here we formulate the retrieval of M as
M = m p(m | [Z ±δ ]) dm∫
p(m | Z )and for each dBZ interval determine
δ = .5 dB if 10<35;1dBZ if 35<Z<42; 2.5dB if Z>42dBZLet us take
Tuesday, 29 May, 12
Data BaseFrom our sample of 200 days of disdrometric records we have this
expected value of logM versus logZ
10 20 30 40 50Z [dBZ]
0.1
1.0
10.0
M [g
/m3 ]
Z=12355M1.55
Tuesday, 29 May, 12
Data BaseFrom our sample of 200 days of disdrometric records we have this
expected value of logM versus logZ
10 20 30 40 50Z [dBZ]
0.1
1.0
10.0
M [g
/m3 ]
Z=12355M1.55
Tuesday, 29 May, 12
Data BaseFrom our sample of 200 days of disdrometric records we have this
expected value of logM versus logZ
10 20 30 40 50Z [dBZ]
0.1
1.0
10.0
M [g
/m3 ]
Z=12355M1.55Some small progress here!
We get a more complex relationship that in fact has
some physical sense:it is consistent with the
tendency to equilibrium DSDs at Z>40dBZ and the expected
behaviour at very low rates where cloud collection is the
prevailing mechanism od precipitation growth
Tuesday, 29 May, 12
Data BaseFrom our sample of 200 days of disdrometric records we have this
expected value of logM versus logZ
10 20 30 40 50Z [dBZ]
0.1
1.0
10.0
M [g
/m3 ]
Z=12355M1.55Some small progress here!
We get a more complex relationship that in fact has
some physical sense:it is consistent with the
tendency to equilibrium DSDs at Z>40dBZ and the expected
behaviour at very low rates where cloud collection is the
prevailing mechanism od precipitation growth
10-2
100
102
104
N [
m-3
mm
-1]
0 2 4 6D [mm]
45-50dBZ(68 min)
40-45 dBZ(208 min)35-40 dBZ(697 min)
Tuesday, 29 May, 12
Data BaseFrom our sample of 200 days of disdrometric records we have this
expected value of logM versus logZ
10 20 30 40 50Z [dBZ]
0.1
1.0
10.0
M [g
/m3 ]
Z=12355M1.55Some small progress here!
We get a more complex relationship that in fact has
some physical sense:it is consistent with the
tendency to equilibrium DSDs at Z>40dBZ and the expected
behaviour at very low rates where cloud collection is the
prevailing mechanism od precipitation growth
0 1D [mm]
100
102
104
N(D
) [m
-3 m
m-1
] 1 June 200413:15-14:04
105 15Z [dBZ]
0.1
0.5
R [m
m/h
r]
Z= 36 R1.04
10-2
100
102
104
N [
m-3
mm
-1]
0 2 4 6D [mm]
45-50dBZ(68 min)
40-45 dBZ(208 min)35-40 dBZ(697 min)
Tuesday, 29 May, 12
Variance is a function of intensity
10 20 30 40Z [dBZ]
0.18
0.20
0.22
0.24
0.26
0.28
0.30ST
DEV
(log 1
0(M))
10 20 30 40Z [dBZ]
0.01
0.10
1.00
STD
EV(M
)
10.013.116.219.422.525.628.831.935.038.141.244.447.550.653.856.960.0
dBZ
-200 -100 0 100 200[km]
-200
-100
0
100
200
[km
]
21/07/2010 18:24:53
0.010.010.020.030.060.100.170.280.480.821.402.404.096.99
11.9320.3834.80
STD
EV o
f M
-200 -100 0 100 200[km]
-200
-100
0
100
200
[km
]
Tuesday, 29 May, 12
Variance is a function of intensity
10 20 30 40Z [dBZ]
0.18
0.20
0.22
0.24
0.26
0.28
0.30ST
DEV
(log 1
0(M))
10 20 30 40Z [dBZ]
0.01
0.10
1.00
STD
EV(M
)
10.013.116.219.422.525.628.831.935.038.141.244.447.550.653.856.960.0
dBZ
-200 -100 0 100 200[km]
-200
-100
0
100
200
[km
]
21/07/2010 18:24:53
0.010.010.020.030.060.100.170.280.480.821.402.404.096.99
11.9320.3834.80
STD
EV o
f M
-200 -100 0 100 200[km]
-200
-100
0
100
200
[km
]
Here we have the diagonal terms of
the error covariance matrix
Tuesday, 29 May, 12
Extremes (shown here for logR)
29.5 30.5Z [dBZ]
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
log
R
e2
-0.2 0.0 0.2 0.4 0.6 0.8 1.0log R
0
10
20
30
40
50
Num
ber o
f day
s
Pmean:logR=0.43
Pmax:logR=0.42
1 2 3 4 5D [mm]
10-2
100
102
104
N(D
)
X
logR = log r p(log r | dBZ ±δ ) dr∫
Tuesday, 29 May, 12
Extremes (shown here for logR)
29.5 30.5Z [dBZ]
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
log
R
e2
-0.2 0.0 0.2 0.4 0.6 0.8 1.0log R
0
10
20
30
40
50
Num
ber o
f day
s
Pmean:logR=0.43
Pmax:logR=0.42
1 2 3 4 5D [mm]
10-2
100
102
104
N(D
)
X
logR = log r p(log r | dBZ ±δ ) dr∫
Tuesday, 29 May, 12
Extremes (shown here for logR)
Here are some extreme members of the ensemble. The last two are averages of all possibilities outside the SD of the entire population
0.050.080.140.230.400.691.182.023.455.92
10.1417.3429.7450.8187.09
149.18254.65
R [m
mh
-1]
-200 -100 0 100 200[km]
-200
-100
0
100
200
[km
]
Z-R
0.050.080.140.230.400.691.182.023.455.92
10.1417.3429.7450.8187.09
149.18254.65
-200 -100 0 100 200[km]
-200
-100
0
100
200
[km
]
Z-Rupper mean
0.050.080.140.230.400.691.182.023.455.92
10.1417.3429.7450.8187.09
149.18254.65
-200 -100 0 100 200[km]
-200
-100
0
100
200
[km
]
Z-Rlower mean
R [m
mh
-1]
R [m
mh
-1]
mean
29.5 30.5Z [dBZ]
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
log
R
e2
-0.2 0.0 0.2 0.4 0.6 0.8 1.0log R
0
10
20
30
40
50
Num
ber o
f day
s
Pmean:logR=0.43
Pmax:logR=0.42
1 2 3 4 5D [mm]
10-2
100
102
104
N(D
)
X
logR = log r p(log r | dBZ ±δ ) dr∫
Tuesday, 29 May, 12
Extremes (shown here for R)
0.070.130.220.370.621.051.793.045.158.77
14.8725.2242.8772.63
123.41209.56354.66
Z = 199 R1.24
0.070.130.220.370.621.051.793.045.158.77
14.8725.2242.8772.63
123.41209.56354.66
Z = 686 R1.63
R [m
mh
-1]
-200 -100 0 100 200[km]
-200
-100
0
100
200
[km
]
-200 -100 0 100 200[km]
-200
-100
0
100
200
[km
]
R [m
mh
-1]
The second pair of extremes comes from taking the DSDs that give the maximum and minimum amount of precipitation
Tuesday, 29 May, 12
Non-diagonal terms of the ECM
We can only estimate the time correlations of the perturbations around the Z-R relationship and invoke the Taylor Hypothesis for estimation of the space correlation.
10 20 30 400
10
20
30
40
50
Nº o
f day
s
7.5241.548
Y of {log(R)-log(RZR)} [min]0 10 20 30 40Y of {log(R)-log(RZR)} [min]
0
10
20
30
40
Y (1
/e) o
f dBZ
[min
]
Y=decorrelationtime to 1/e
Tuesday, 29 May, 12
Non-diagonal terms of the ECMFor R in linear scale we do our best to measure decorrelation
time as a function of reflectivity:
700 800 900 1000 1100 1200time [min]
15
20
25
30
35
40Z
[dBZ
]
700 800 900 1000 1100 1200time [min]
-6
-4
-2
0
2
4
R -
RZ-
R
0 10 20 30 40time [min]
0.0
0.2
0.4
0.6
0.8
1.0
Cor
rela
tion
10-20 dBZ20-30 dBZ30-40 dBZ >40 dBZ
10 20 30 40 50Z [dBZ]
0.1
1.0
10
100
R [m
mh-1
]
Tuesday, 29 May, 12
Non-diagonal terms of the ECM
For R in linear scale we get a decorrelation time varying with reflectivity:
10 20 30 40 50Z [dBZ]
5
10
15
20
25Y(
1/e)
of {
R -
RZ-
R}
Tuesday, 29 May, 12
Conclusions
Ensemble retrievals reduce biases
Tuesday, 29 May, 12
Conclusions
Ensemble retrievals reduce biases
Give information on the error covariance matrix
Tuesday, 29 May, 12
Conclusions
Lead to new insight
Ensemble retrievals reduce biases
Give information on the error covariance matrix
Tuesday, 29 May, 12
Are as simple to perform as common retrievals
Conclusions
Lead to new insight
Ensemble retrievals reduce biases
Give information on the error covariance matrix
Tuesday, 29 May, 12
Are as simple to perform as common retrievals
Conclusions
Lead to new insight
Ensemble retrievals reduce biases
Give information on the error covariance matrix
Help to establish a common language between observational and numerical teams of researchers
Tuesday, 29 May, 12
Are as simple to perform as common retrievals
Conclusions
Lead to new insight
There is no reason for not doing things this way
Ensemble retrievals reduce biases
Give information on the error covariance matrix
Help to establish a common language between observational and numerical teams of researchers
Tuesday, 29 May, 12