Post on 01-Apr-2015
Model error estimation employing ensemble data assimilation
Dusanka Zupanski and Milija ZupanskiCIRA/Colorado State University, Fort Collins, CO, U.S.A.
EGU General Assembly 2005 NP5.01: Quantifying predictability
24-29 April 2005Vienna, Austria
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
OUTLINE
Why do we need to estimate model error?- Data assimilation point of view- General point of view
Methodology- MLEF+State Augmentation
Experimental results- KdVB model (1-d)- CSU-RAMS model (3-d, non-hydrostatic)
Conclusions and future work
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Why do we need to estimate model error?
Goal of CLASSICAL data assimilation methods is to estimate(1) atmospheric state
Goal of ENSEMBLE data assimilation methods is to estimate(1) atmospheric state(2) uncertainty of the estimated state
Data assimilation point of view
Model error influences - adversely - both estimates
ENSEMBLE approaches are more sensitive to model error
Use this opportunity to further improve new methods.
Be happy with the limited benefits of the new methods.
or
Why do we need to estimate model error?
Many additional applications in geophysics would benefit from model error estimation:
Improving current dynamical models Developing new dynamical models Quantifying predictability Quantifying information content of observations Obtaining new knowledge about geophysical processes
General point of view
This presentation is mostly focused on the data assimilation aspect, as a first step towards more general applications.
min]([]([2
1][][
2
1 11 obs
Tobsb
-f
Tb HHJ yxRyxxxxx ))P
2121 )( CIPfbxx
Change of variable (preconditioning)
x
- control vector in ensemble space of dim Nens
Minimize cost function J
- model state vector of dim Nstate >>Nens
ZZC T
)()( 2121 xRpxRz HH fii
C - information matrix of dim Nens Nens
METHODOLOGY: MLEF approach
fip - columns of
21
fP iz - columns of Z
METHODOLOGY: MLEF + State Augmentation
011,0
1
1,
0
0
00
00
01
nnnn
n
n
1-n
1-nn
nn
n
n
n
F
GM
zb
x
I
Ib
x
zn
nΦxx 1-n1-nn,n M
)(bΦΦ Gn0
nn
01-nn
bbbb 01-nn
011, nnnnF zzn
- model state time evolution
- AUGMENTED state time evolution
- serially correlated model error
- model bias
- vector of empirical parameters
Approach applicable to other EnKF methods.
RESULTS: Parameter estimation, KdVB modelESTIMATION OF DIFFUSION COEFFICIENT
(102 ens, 101 obs)
2.00E-02
4.00E-02
6.00E-02
8.00E-02
1.00E-01
1.20E-01
1.40E-01
1.60E-01
1.80E-01
2.00E-01
2.20E-01
2.40E-01
2.60E-01
1 11 21 31 41 51 61 71 81 91
Cycle No.
Dif
fusi
on
co
efic
ien
t va
lue
estim value (0.07)
true value (0.07)
estim value (0.20)
true value (0.20)
Innovation histogram (incorrect diffusion)(neglect_err, 10 ens, 101 obs)
0.00E+00
1.00E-01
2.00E-01
3.00E-01
4.00E-01
5.00E-01
-5 -4 -3 -2 -1 0 1 2 3 4 5
Category bins
PD
F
Innovation histogram (incorrect diffusion)(param_estim, 10 ens, 101 obs)
0.00E+00
1.00E-01
2.00E-01
3.00E-01
4.00E-01
5.00E-01
-5 -4 -3 -2 -1 0 1 2 3 4 5
Category bins
PD
F
Innovation histogram (correct diffusion)(correct_model, 10 ens, 101 obs)
0.00E+00
1.00E-01
2.00E-01
3.00E-01
4.00E-01
5.00E-01
-5 -4 -3 -2 -1 0 1 2 3 4 5
Category bins
PD
F
True parameter recovered. Improved innovation statistics.
INNOVATION 2 TEST (biased model)(neglect_err, 10 ens, 10 obs)
0.00E+002.00E+004.00E+006.00E+008.00E+001.00E+011.20E+01
1 11 21 31 41 51 61 71 81 91
Analysis cycle
INNOVATION 2 TEST (biased model)(bias_estim, 10 ens, 10 obs, bias dim = 101)
0.00E+002.00E+004.00E+006.00E+008.00E+001.00E+011.20E+01
1 11 21 31 41 51 61 71 81 91
Analysis cycle
INNOVATION 2 TEST (biased model)(bias_estim, 10 ens, 10 obs, bias dim = 10)
0.00E+002.00E+004.00E+006.00E+008.00E+001.00E+011.20E+01
1 11 21 31 41 51 61 71 81 91
Analysis cycle
INNOVATION 2 TEST (non-biased model)(correct_model, 10 ens, 10 obs)
0.00E+002.00E+004.00E+006.00E+008.00E+001.00E+011.20E+01
1 11 21 31 41 51 61 71 81 91
Analysis cycle
NEGLECT BIAS BIAS ESTIMATION (vector size=101)
BIAS ESTIMATION (vector size=10) NON-BIASED MODEL
RESULTS: Bias estimation, KdVB model
It is beneficial to reduce degrees of freedom of the model error.
RESULTS: Bias estimation, KdVB model
Augmented analysis error covariance matrix is updated in each data assimilation cycle. It includes cross-covariance between the initial conditions (IC) error and model error (ME).
An experiment with a simple state dependent model error
Estimate state dependent model error . Define model error components for u, v, T,…,q as:
nun uΦ
nvn vΦ
nqn qΦ
Estimate single parameter
In real atmospheric applications, model errors are commonly more complex, but ARE often STATE DEPENDENT.
EXPERIMENTAL DESIGN
Non-hydrostatic atmospheric model (CSU-RAMS)
- 3d model
- simplified microphysics (level 2)Hurricane Lili case25 1-h DA cycles: 13UTC 1 Oct 2002 – 14 UTC 2 Oct30x20x21 grid points, 15 km grid distance (in the Gulf of Mexico)
- model domain 450km X 300kmControl variable:
- u,v,w,theta,Exner, r_total (initial conditions, dim=54000)
- (dim=1)Model simulated observations with random noise
(7200 obs per DA cycle)Nens=50Iterative minimization of J (1 iteration only)
RESULTS: Parameter estimation, RAMS model
Analysis RMS errors for u-wind(analysis-truth)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1 6 11 16 21
Analysis cycle
RM
S u-
win
d (m
s-1)
correct_modelparam_estimneglect_errno_assim
Analysis RMS errors for w-wind(analysis-truth)
0
0.005
0.01
0.015
0.02
0.025
1 6 11 16 21
Analysis cycle
RM
S w
-win
d (m
s-1)
correct_modelparam_estimneglect_errno_assim
U-WIND ANALYSIS ERRORS
W-WIND ANALYSIS ERRORS
Parameter estimation is almost as good as the perfect (correct) model data assimilation experiment.
Both the initial conditions and the parameter are adjusted. Control variable size is 54001.
RESULTS: Parameter estimation, RAMS model
Analysis RMS errors for Exner(analysis-truth)
0
0.5
1
1.5
2
2.5
3
1 6 11 16 21
Analysis cycle
RM
S E
xner
correct_modelparam_estimneglect_errno_assim
Analysis RMS errors for Total Mixing Ratio(analysis-truth)
0.00E+00
5.00E-04
1.00E-03
1.50E-03
2.00E-03
1 6 11 16 21
Analysis cycle
RM
S r
correct_modelparam_estimneglect_errno_assim
Both the initial conditions and the parameter are adjusted. Control variable size is 54001.
EXNER FUNCTION ANALYSIS ERRORS
TOTAL WATER MIXING RATIO ANALYSIS ERRORS
Neglecting model error reduces the benefits of data assimilation.
RESULTS: Parameter estimation, RAMS model
Estimated parameter value is close to the true parameter value.
Empirical parameter
-1.50E-05
-1.00E-05
-5.00E-06
0.00E+00
5.00E-06
1.00E-05
1.50E-05
1 6 11 16 21
Analysis cycle
estimatedtrue
Neglect_err Param_estim
No_assim Correct_model
True
Theta_il
Differences of the order of1.0K-3.0K.
Differences of the order of0.1K.
CONCLUSIONS
Dusanka Zupanski, CIRA/CSUZupanski@CIRA.colostate.edu
Ensemble based data assimilation methods, if coupled with state augmentation approach, can be effectively used to estimate empirical parameters.
Estimation of model errors can also be effective if number of degrees of freedom of the model error is reduced.
Neglecting model errors leads to degraded data assimilation results.
Capability to update augmented forecast error covariance is an advantage of ensemble based data assimilation approaches.
Sensitivity of ensemble data assimilation approaches to model errors is an OPPORTUNITY for further improvements. This will be further explored in the future.