How does 4D-Var handle Nonlinearity and non-gaussianity?

Transcription

1 How does 4D-Var handle Nonlinearity and non-gaussianity? Mike Fisher Acknowledgements: Christina Tavolato, Elias Holm, Lars Isaksen, Tavolato, Yannick Tremolet Slide 1

2 Outline of Talk Non-Gaussian pdf s in the 4d-Var cost function - Variational quality control - Non-Gaussian background errors for humidity Can we use 4D-Var analysis windows that are longer than the timescale over which non-linear effects dominate? - Long-window, weak constraint 4D-Var Slide 2

3 Non-Gaussian pdf s in the 4D-Var cost function The 3D/4D-Var cost function is simply the log of the pdf: J ( x) log( p( x y, x ) log( p( y x) ) log( p( x x) ) K Non-Gaussian pdf s of observation error and background error result in non-quadratic cost functions. In principle, this has the potential to produce multiple minima and difficulties in minimization. In practice, there are many cases where the ability to specify non-gaussian pdf s is very useful, and does not give rise to significant minimization problems. - Directionally-ambiguous scatterometer winds - Variational quality control b - Bounded variables: humidity, trace gasses, rain rate, etc. b Slide 3

4 Variational quality control and robust estimation Variational quality control has been used in the analysis for the past 10 years. Observation errors are modelled as a combination of a Gaussian and a flat (boxcar) distribution: ( ) p y x = (1 P ) N + P G, where P = p(gross error), and: G G G y H( x) N = exp σ 2 2 o π σ o 1 G = if y H( x) < D/ 2, zero otherwise D With this pdf, observations close to x are treated as if Gaussian, whereas those far from x are effectively ignored. Slide 4

5 Variational quality control and robust estimation An alternative treatment is the Huber metric: ( ) 2 1 a exp aδ if a< δ σ 2 2 o π 1 1 exp δ δ σ 2 2 o π 2 1 b exp bδ if δ > b σ 2 2 o π 2 p y x = a b where δ = y H( x) σ Equivalent to L 1 metric far from x, L 2 metric close to x. With this pdf, observations far from x are given less weight than observations close to x, but can still influence the analysis. Many observations have errors that are well described by the Huber metric. o Slide 5

6 Variational quality control and robust estimation 18 months of conventional data -Feb 2006 Sep 2007 Normalised fit of PDF to data - Best Gaussian fit - Best Huber norm fit Slide 6

7 Variational quality control and robust estimation Gaussian Huber Gaussian + flat Slide 7

8 Comparing optimal observation weights Huber-norm (red) vs. Gaussian+flat (blue) More weight in the middle of the distribution Weight relative to gaussian (no VarQC) case -σ o was retuned More weight on the edges of the distribution More influence of data with large departures 25% -Weights: 0 25% Slide 8

9 Test Configuration Huber norm parameters for - SYNOP, METAR, DRIBU: surface pressure, 10m wind - TEMP, AIREP: temperature, wind - PILOT: wind Relaxation of the fg-check - Relaxed first guess checks when Huber VarQC is done - First Guess rejection limit set to 20σ Retuning of the observation error - Smaller observation errors for Huber VarQC Slide 9

10 French storm Surface pressure: -Model (ERA interim T255): 970hPa -Observations: 963.5hPa -Observation are supported by neighbouring stations! Slide 10

11 Data rejection and VarQC weights Era interim UTC +60min fg rejected used VarQC weight = 50-75% VarQC weight = 25-50% VarQC weight = 0-25% Slide 11

12 Data rejection and VarQC weights Huber exp. VarQC weight = 50-75% VarQC weight = 25-50% VarQC weight = 0-25% Slide 12

13 MSL Analysis differences: Huber Era Diff AN = 5.6 hpa New min 968 hpa Low shifted towards the lowest surface pressure observations Slide 13

14 Humidity control variable b b b b ( ) b P( δrh 2rh1 δrh) P rh rh rh rh = + Joint pdf: ( b b P rh ) 1, rh2 for two members of an ensemble of 4D-Var analyses. δ rh = rh rh b b 2 1 is representative of background error rh 1 (%) rh 2 (%) The pdf of background error is asymmetric when stratified by The pdf of background error is symmetric when stratified by Slide 14 b b ( 2 rh1 ) b b P( δ rh rh1 rh1 ) P rh = + 5.9E E E E E E E E E-06 b rh b 1 rh + δ rh 2

15 Humidity control variable Probability density function Lowest RH+dRH/2 Median RH+dRH/2 Highest RH+dRH/2 Gaussian Standard deviation Bias Standard deviation δrh RH + drh/2 The symmetric pdf b 1 P δrh rh + δrh can be modelled by a normal distribution. 2 b 1 The variance changes with rh + δ rh and the bias is zero. 2 A control variable with an approximately unit normal distribution is obtained by a nonlinear normalization: δ rh δ rh = b 1 σ rh + δrh 2 Slide 15

16 Humidity control variable The background error cost function J b is now nonlinear. Our implementation requires linear inner loops (so that we can use efficient, conjugate-gradient minimization). - Inner loops: use T 1 ( δ ) ( δ ) where ( δ ) Jb = f rh B f rh f rh = b ( ) δrh = δrh σ rh δ rh δ rh b 1 σ rh + δrh 2 - Outer loops: solve for from the nonlinear equation: δ rh = δ rh b 1 σ rh + δrh 2 Slide 16

17 What about Multiple Minima? Example: strong-constraint 4D-Var for the Lorenz threevariable model: from: Roulstone, 1999 Figure 1: The MSE cost function in the Lorenz model as a function of error in the initial value of the Y coordinate. The function becomes increasingly pathological as the assimilation perio d is increased. Slide 17

18 What about Multiple Minima? In strong-constraint 4D-Var, the control variable is x 0. K T 1 J = y H M ( x ) R y H M x We rely on the model to propagate the state from initial time to observation time. For long windows, this results in a highly nonlinear J o. In weak-constraint 4D-Var, the control variable is (x 0,x 1,,x K ), and (for linear observation operators) J o is quadratic. K T 1 J = y H ( x ) R y H ( x ) J q is close to quadratic if the TL approximation is accurate over the sub-interval [t k-1, t k ]. K ( 0 ) ( ( 0) ) 0 k 0 k o k k t t k k t t k = 0 o k k k k k k k = 0 T 1 ( ) ( ) k 1 k 1 k 1 k 1 J = x M x R x M x q k t t k k t t k k = 1 Slide 18

19 Cross-section of the cost function for a random perturbation to the control vector Cost Lorenz 1995 model. 20-day analysis window. J o + J q (weak constraint) J o (weak constraint) J o (strong constraint) ε / σ o Slide 19

20 Long-window, weak-constraint 4D-Var 0.4 RMS Analysis Error RMS error for 4dVar dx dt i Lorenz 95 model = x x + x x x + F i 2 i 1 i 1 i+ 1 i with i = 1, 2K40 x x x 0 40 x 1 39 x = = = x 41 1 F = 8 RMS error for OI 0.1 RMS error for EKF Length of Analysis Window (days) Slide 20

21 What about multiple minima? 20 GRD: Estimate dotted=truth solid=analysis diamonds=obs time t From: Evensen (MWR 1997 pp : Advanced Data Assimilation for Strongly Nonlinear Dynamics). - Weak constraint 4dVar for the Lorenz 3-variable system. - ~50 orbits of the lobes of the attractor, and 15 lobe transitions. Slide 21

22 What about multiple minima? The abstract from Evensen s 1997 paper is interesting: - This paper examines the properties of three advanced data assimilation methods when used with the highly nonlinear Lorenz equations. The ensemble Kalman filter is used for sequential data assimilation and the recently proposed ensemble smoother method and a gradient descent method * are used to minimize two different weak constraint formulations. - The problems associated with the use of an approximate tangent linear model when solving the Euler-Lagrange equations, or when the extended Kalman filter is used, are eliminated when using these methods. All three methods give reasonable consistent results with the data coverage and quality of measurements that are used here and overcome the traditional problems reported in many of the previous papers involving data assimilation with highly nonlinear dynamics. *i.e. weak-constraint 4D-Var Slide 22

23 Weak Constraint 4D-Var in a QG model The model: - Two-level quasi-geosptrophic model on a cyclic channel - Solved on a domain with Δx=Δy=300km - Layer depths D 1 =6000m, D 2 =4000m - R o = 0.1 Dq i Dt = 0 (for i = 1, 2) ( ) q = ψ F ψ ψ + βy q2 = ψ2 F2( ψ2 ψ1) + βy + Rs - Very simple numerics: first order semi-lagrangian advection with cubic interpolation, and 5-point stencil for the Laplacian. Slide 23

24 Weak Constraint 4D-Var in a QG model dt = 3600s dx = dy = 300km f = 10-4 s -1 β = s -1 m -1 D1 = 6000m D2 = 4000m Orography: - Gaussian hill m high, 1000km wide at i=0, j=15 Domain: 12000km 6000km Perturbation doubling time is ~30 hours Slide 24

25 Weak Constraint 4D-Var in a QG model One analysis is produced every 6 hours, irrespective of window length: Analysis Forecast Linearisation Trajectory Analysis The analysis is incremental, weak-constraint 4D-Var, with a linear inner-loop, and a single iteration of the outer loop. Inner and outer loop resolutions are identical. Forecast Linearisation Trajectory Analysis Background Slide 25

26 Weak Constraint 4D-Var in a QG model Observations: observing points, randomly distributed between levels, and at randomly chosen gridpoints. - For each observing point, an observation of streamfunction is made every 3 hours. - Observation error: σ o =1.0 (in units of non-dimensional streamfunction) Obs at level 1 Obs at level 2 Slide 26

27 Weak Constraint 4D-Var in a QG model Initial perturbation drawn from N(0,Q) Nonlinearity Index model T159 L31 Θ () t = where: + δψ () t + δψ () t + ( δψ t + δψ t ) () () /2 + δψ ( t) = difference from control of integration with positive initial perturbation. δψ ( t) = difference from control of Nonlinearity dominates for Θ>0.7 (Gilmour et al., 2001) integration with negative initial perturbation Time (hours) Slide 27

28 Weak Constraint 4D-Var in a QG model Long-Window 4D-Var in a Two-Level QG Model Mean Analysis and First-Guess Error for Different Window Lengths 1.2 RMS Error for Non-dimensional Streamfunction According to Gilmour Thin lines et = al. s first guess criterion, nonlinearity Thick lines dominates, = analysis for windows longer than 60 hours. Weak constraint 4D-Var allows windows that are much longer than the timescale for nonlinearity Time Within Analysis Window (hours) Slide 28

29 Summary The relationship: J=-log(pdf) makes it straightforward to include a wide range of non-gaussian effects. - VarQC - Non-gaussian bakground errors for humidity,etc. - nonlinear balances - nonlinear observation operators (e.g. scatterometer) - etc. In weak-constraint 4D-Var, the tangent-linear approximation applies over sub-windows, not over the full analysis window. - The model appears in J q as t k 1 t k Window lengths >> nonlinearity time scale are possible. M Slide 29

30 How does 4D-Var handle Nonlinearity and non-gaussianity? Surprisingly Well! Thank you for your attention. Acknowledgements: Christina Tavolato, Elias Holm, Lars Isaksen, Tavolato, Yannick Tremolet Slide 30