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Parameter estimation:
To what extent can data assimilation techniques correctly uncover
stochasticity?
Jim HansenMIT, EAPS
(with lots of help from Cecile Penland and Greg Lawson)
Indistinguishable?
Accounting for vs. reducing model inadequacy
• Accounting for model inadequacy– “If you can show me how I can make better forecasts
using chicken bones and voodoo dolls, then I’m going to use them!”
» Harold Brooks, NSSL
– initial conditions (Q-term)– forecasts (MM, stochastic, MOS, forecast 4d-Var)
• Reducing model inadequacy– Making changes to our model so that it becomes a
better representation of the true system– parametric error– structural error
Accounting for vs. reducing model inadequacy
• Accounting for model inadequacy– “If you can show me how I can make better forecasts
using chicken bones and voodoo dolls, then I’m going to use them!”
» Harold Brooks, NSSL
– initial conditions (Q-term)– forecasts (MM, stochastic, MOS, forecast 4d-Var)
• Reducing model inadequacy– Making changes to our model so that it becomes a
better representation of the true system– parametric error– structural error
Reducing model inadequacy
• Reducing model inadequacy is best framed as an off-line, or “reanalysis” activity
– The process of attempting to identify model inadequacy tends to make both initial conditions and forecasts worse
– The aim is to quantify how the model is wrong, fix it, and then worry about data assimilation and forecasting
A proposed approach
• Use data assimilation tools to alter model parameters to better fit observations
• Identify relationships between fit parameters and prognostic variables (a parametric MOS)
• Change model to reflect relationships• Repeat
When all relationships have been uncovered, the history of fit parameter values provides a distributionfrom which to (carefully) draw for the purpose ofstochastic parameterizations
Use data assimilation tools to alter model parameters to better fit observations
• Augment control vector with unknown parameters
• Augmentation removes the nonlinearity from the observation operator and inserts it into the specification of the control vector
xx
α
Txx
Tαα
Txα
Tαx
Txx
Augmented control vector sample covariance
Parametric error example: L63
• System equations • Model equations
( )x x y
y rx xz y
z xy bz
( )
0
x x y
y rx xz y
z xy bz
[ ]Tx y z x
Importance of a state-dependent background error covariance
• 4d-Var, static covariance • Ensemble 4d-Var
time time
para
met
er
para
met
er
Structural error example: Lorenz ‘96
22 1 1 1i i i i i ix x x x x x F
2 1 1 1i i i i i ix x x x x x F
System:
Model:
[ ]Tx x
ix
x(1)
par
amet
er
Regressing parametervs. prognostic variablegives:
Alter model equations with new information
22 1 1 1i i i i i ix x x x x x F
2 1 1 1i i i i i ix x x x x x F
System:
Original model:
2 1 1 1 ( )i i i i i i ix x x x x x x F New model:
Example: Lorenz ’96 Model II
2 1 1 1 ,1
Jx
i i i i i i j ij
h cx x x x x x F y
b
, 1, 2, 1, 1, ,y
j i j i j i j i j i j i i
h cy cby y cby y cy x
b
2 1 1 1 ,1
Jx
i i i i i i j ij
h cx x x x x x F y
b
, 1, 2, 1, 1, ,y
j i j i j i j i j i j i i
h cy cby y cby y cy x
b
System:
Model:
2 1 1 1 ( )i i i i i i ix x x x x x F t
[ ]Tx x F
par
amet
er
x(1)
i ixF G
Regressing parametervs. prognostic variablegives:
2 1 1 1 ,1
Jx
i i i i i i j ij
h cx x x x x x F y
b
, 1, 2, 1, 1, ,y
j i j i j i j i j i j i i
h cy cby y cby y cy x
b
System:
Original model:
2 1 1 1 ( )i i i i i i ix x x x x x F t
New model:
2 1 1 1 ( 1)i i i i i ii ix x x x x x Gx
SDE crash course
• The type of calculus used to integrate the stochastic bits of SDEs matters– Stratonovich calculus
• noise process is continuous (typical assumption for geophysical fluid flows)
– Ito calculus• noise process is discrete (like DA!)
• SDEs can be tricky (and expensive) to integrate– used stochastic RK4 (Hansen and Penland, 2005)
What if the system really is stochastic?
• System is an SDE • Model is an ODE
0 ( )
( )
( )
( )
s
dx x y dt
x y dW
dy rx xz y dt
dz xy bz dt
( )
0
x x y
y rx xz y
z xy bz
[ ]Tx y z x
Can DA uncover the correct form of the stochasticity? - NO
• Ensemble 4d-Var • EnKF
para
met
er
para
met
er
time time
10.08, ( ) 0.36std 10.02, ( ) 0.32std
0 10, 0.1s
Why can’t DA uncover the correct form of the stochasticity?
• Stochasticity operating at different time-scales – SDE has infinitesimal time-scale– ODE with DA has 6-hourly time-scale
• System is using Stratonovich calculus, DA is using Ito calculus
• All leads to a danger of misinterpretation
Model error!
1. Deterministic model using constant, tuned parameter value ( )
2. Stochastic model using mean and standard deviation of tuned parameter value ( )
3. Deterministic, multi-model ensemble with parameters drawn from ( )
4. Deterministic model where parameter varies in the same manner as it was estimated ( )
How should we use this information for forecasting?
, ( )std
, ( )std
( )x x y
y rx xz y
z xy bz
Tuned deterministic
, ( )std
1. Deterministic model using constant, tuned parameter value ( )
2. Stochastic model using mean and standard deviation of tuned parameter value ( )
3. Deterministic, multi-model ensemble with parameters drawn from ( )
4. Deterministic model where parameter varies in the same manner as it was estimated ( )
How should we use this information for forecasting?
, ( )std
, ( )std
Incorrect SDE
, ( )std
( )
( )( )
( )
( )
dx x y dt
std x y dW
dy rx xz y dt
dz xy bz dt
1. Deterministic model using constant, tuned parameter value ( )
2. Stochastic model using mean and standard deviation of tuned parameter value ( )
3. Deterministic, multi-model ensemble with parameters drawn from ( )
4. Deterministic model where parameter varies in the same manner as it was estimated ( )
How should we use this information for forecasting?
, ( )std
, ( )std
Multi-model
, ( )std
N( )( ), var( )x x y
y rx xz y
z xy bz
where the draw from is held constant over the entireforecast period.
, va )N( r( )
1. Deterministic model using constant, tuned parameter value ( )
2. Stochastic model using mean and standard deviation of tuned parameter value ( )
3. Deterministic, multi-model ensemble with parameters drawn from ( )
4. Deterministic model where parameter varies in the same manner as it was estimated ( )
How should we use this information for forecasting?
, ( )std
, ( )std
Hybrid
, ( )std
N( )( ), var( )x x y
y rx xz y
z xy bz
where the draw from is made every 6 model hours.
, va )N( r( )
Median of ensemble mean forecast distributions
Tuned deterministicIncorrect SDEMulti-modelHybridPerfect
Nor
mal
ized
RM
SE
Forecast lead (model days)
std(
err/
ens_
std)
Must assess probabilistically!
Tuned deterministicIncorrect SDEMulti-modelHybridPerfect
Forecast lead (model days)
rela
tive
entr
opy
Relative (to perfect) entropy
Multi-modelHybrid
Forecast lead (model days)
What if we use a stochastic model?
• System is an SDE • Model is an SDE
0 ( )
( )
( )
( )
s
dx x y dt
x y dW
dy rx xz y dt
dz xy bz dt
0[ ]Tsx y z x
0 ( )
( )
( )
( )
s
dx x y dt
x y dW
dy rx xz y dt
dz xy bz dt
Now can DA uncover the correct form of the stochasticity? - NO
para
met
er s
tdtime
0 10.11 0.26s
0 10, 0.1s
para
met
er m
ean
time
0 and are not uniques
What’s the problem this time?
• Wrong trajectory of random numbers
Model error!
SDE forecast errors
s
0
What does it all mean?• Deterministic model DA approaches alone are not
enough to uncover the correct form of stochasticity– Implies that we cannot attach physical significance to
tuned parameter values or distributions
• Our efforts to reduce model inadequacy ultimately lead to a sensible way to account for model inadequacy
• Synoptic time-scale, Ito-like stochasticity via parameter estimation does a great job accounting for model inadequacy during forecasting
The future(?) of data assimilation
• Model error issues• Nonlinearity• New disciplines: e.g. paleo, climate• Improved image• DA is part of a larger problem
The future(?) of data assimilation
• Nonlinearity– Implementing nonlinear approaches– Extend minimum error variance approaches a bit
more into the nonlinear regime• Feature-based non-Gaussianity
The future(?) of data assimilation
• Improved image– DA has a bad/boring reputation– Ensemble methods bringing DA to the masses
• University research can be quasi-operational• Reasonable DA now where none before
ATMOSATMOS
COLLEGECOLLEGE
The future(?) of data assimilation
• Improved image– DA has a bad/boring reputation– Ensemble methods bringing DA to the masses
• University research can be quasi-operational• Reasonable DA now where none before
The future(?) of data assimilation
• DA is part of a larger problem– The future of DA is not independent of the future of
observations, ensemble forecasting, verification, calibration, etc..
• Ensemble forecasting• Targeting• Increasing ensemble forecast size at low cost• Ensemble synoptic analysis
Transformed Lag Ensemble Forecasting (TLEF)
• Ensemble size is increased by using ensemble-based data assimilation techniques to transform (scale and rotate) old forecasts using new observations.
Time
The future(?) of data assimilation
• DA is part of a larger problem– The future of DA is not independent of the future of
observations, ensemble forecasting, verification, calibration, etc..
• Ensemble forecasting• Targeting• Increasing ensemble size at low cost• Ensemble synoptic analysis
HakimandTorn
WRF,100
ensemble members,
surface pressure
obs
HakimandTorn
Ertelaa
Ertel pvAxAXPV 1
Ensembles make PV inversion fun and easy!
Note, no worries about balance assumptions or boundary conditions
Approach defined by
HakimandTorn
The future(?) of data assimilation
• Model error issues• Nonlinearity• New disciplines: e.g. paleo, climate• Sales/Marketing• DA is part of a larger problem