Introduction to Data Assimilation

Transcription

1 Introduction to Data Assimilation Alan O Neill Data Assimilation Research Centre University of Reading What is data assimilation? Data assimilation is the technique whereby observational data are combined with output from a numerical model to produce an optimal estimate of the evolving state of the system. 8 th 9 th August 003

2 Why We Need Data Assimilation range of observations range of techniques different errors data gaps quantities not measured quantities linked 8 th 9 th August 003

3 Some Uses of Data Assimilation Operational weather and ocean forecasting Seasonal weather forecasting Land-surface process Global climate datasets Planning satellite measurements Evaluation of models and observations 3 8 th 9 th August 003

4 c What We Want To Know x( t) s( t) atmos. state vector surface fluxes model parameters X( ) ( x( t), s( t), c) t = What We Also Want To Know Errors in models Errors in observations What observations to make 4 8 th 9 th August 003

5 DATA ASSIMILATION SYSTEM Data Cache A Error Statistics model observations O DAS A Numerical Model F B The Data Assimilation Process observations forecasts compare reject adjust estimates of state & parameters errors in obs. & forecasts 5 8 th 9 th August 003

6 Retrievals and Assimilation Problems with retrievals a priori state and poorly known errors Problems with assimilation of radiance systematic errors and cloud clearing expensive (multi-channels) Need effective interface Information content with error analysis Statistical Approach to Data Assimilation 6 8 th 9 th August 003

7 Minimum Variance Combination of Data Unbiased, Uncorrelated Errors ~ x = α x + βx α + β = Var( x ) = σ Var( x ) = σ 7 8 th 9 th August 003

8 Minimum Variance Combination of Data Unbiased, Uncorrelated Errors ~ ) Var(x ( = α σ + α ) σ α Var( ~ x ) = ασ ( ) α σ = 0 α, β Minimum Variance Combination of Data Unbiased, Uncorrelated Errors ~ x = x σ + σ x σ + σ var( ~ x ) min( σ Best Linear Unbiased Estimate, σ ) 8 8 th 9 th August 003

9 Variational Method J ( x) = ( x σ x ) ( x x) + σ J (x) x~ at min J ( x) x~ x Maximum Likelihood Estimate Obtain or assume probability distributions for the errors The best estimate of the state is chosen to have the greatest probability, or maximum likelihood If errors normally distributed,unbiased and uncorrelated, then states estimated by minimum variance and maximum likelihood are the same 9 8 th 9 th August 003

10 Multivariate Case observation vector state vector y y y m x( t) = x x x n Variance becomes Covariance Matrix Errors in x i are often correlated spatial structure in flow dynamical or chemical relationships Variance for scalar case becomes Covariance Matrix for vector case COV Diagonal elements are the variances of x i Off-diagonal elements are covariances between x i and x j Observation of x i affects estimate of x j 0 8 th 9 th August 003

11 Estimating Covariance Matrix for Observations, O O usually quite simple: diagonal or for nadir-sounding satellites, non-zero values between points in vertical only Calibration against independent measurements Estimating Covariance Matrix for Model, B Use simple analytical functions constructed experimentally, e.g. B = σ σ exp( d ) ij where d ij i j is thehorizontal distancebetween x and i x j Run ensemble of forecasts from slightly different conditions and analyse error growth. ij 8 th 9 th August 003

12 Methods of Data Assimilation Optimal interpolation (or approx. to it) 3D variational method (3DVar) 4D variational method (4DVar) Kalman filter (with approximations) Optimal Interpolation observation observation operator analysis x a = x b background (forecast) + K( y H ( xb)) linearity H H matrix inverse K T T = BH ( HBH + O) limited area 8 th 9 th August 003

13 = y H ( x b ) at obs. point data void x b X observation model trajectory t 3 8 th 9 th August 003

14 Data Assimilation: an analogy Driving with your eyes closed: open eyes every 0 seconds and correct trajectory Variational Data Assimilation J (x) vary x to minimise J(x) x a x 4 8 th 9 th August 003

15 J ( x) ( x ( y Variational Data Assimilation x = b ) T B H ( x)) T ( x O x ( y b ) + H ( x)) nonlinear operator assimilate y directly global analysis Choice of State Variables and Preconditioning Free to choose which variables to use to define state vector, x(t) We d like to make B diagonal may not know covariances very well want to make the minimization of J more efficient by preconditioning : transforming variables to make surfaces of constant J nearly spherical in state space 5 8 th 9 th August 003

16 Cost Function for Correlated Errors x x Cost Function for x Uncorrelated Errors x 6 8 th 9 th August 003

17 x Cost Function for Uncorrelated Errors Scaled Variables x 4D Variational Data Assimilation obs. & errors given X(t o ), the forecast is deterministic t o t vary X(t o ) for best fit to data 7 8 th 9 th August 003

18 4D Variational Data Assimilation Advantages consistent with the governing eqs. implicit links between variables Disadvantages very expensive model is strong constraint Problem model error covariances 8 8 th 9 th August 003

19 Kalman Filter (expensive) Use model equations to propagate B forward in time. B B(t) KAL Analysis step as in OI What are the benefits of data assimilation? Quality control Combination of data Errors in data and in model Filling in data poor regions Designing observational systems Maintaining consistency Estimating unobserved quantities 9 8 th 9 th August 003

20 Some Applications of Data Assimilation Skill Measures: Observation Increment, (O-F) The difference between the forecast from the first guess, F, and the observations, O, also known as observed-minusbackground differences or the innovation vector. This is probably the best measure of forecast skill. 0 8 th 9 th August 003

21 _ + T ECMWF Ozone monthlymean analysis increment. Sept th 9 th August 003

22 Best observations to make to characterize the chemical system Impact on NWP at the Met Office Mar 99. 3D-Var and ATOVS Jul 99. ATOVS over Siberia, sea-ice from SSM/I Oct 99. ATOVS as radiances, SSM/I winds May 00. Retune 3D-Var Feb/Apr 0. nd satellites, ATOVS + SSM/I 8 th 9 th August 003

23 Impact of satellite data in NWP Future Advanced Infrared Sounders 3 8 th 9 th August 003

24 IASI vs HIRS 4 8 th 9 th August 003

25 5 8 th 9 th August 003

26 6 8 th 9 th August 003

27 MERIS ocean colour Regional Scale: Walnut Gulch (Monsoon 90) Model Observation Tombstone, AZ 0% 0% Model with 4DDA Houser et al., th 9 th August 003

28 Land Initialization: Motivation Knowledge of soil moisture has a greater impact on the predictability of summertime precipitation over land at mid-latitudes than Sea Surface Temperature (SST). Conclusions Data assimilation should be essential part of ground-segment of all satellite missions. Aim should be to provide all data in near real time. Need to find optimal ways to assimilate data efficient interfaces. Challenges: errors, resolution, data. 8 8 th 9 th August 003