Accounting for non-gaussian observation error

Transcription

1 Accounting for non-gaussian observation error Lars Isaksen & Christina Tavolato ECMWF Royal Meteorological Society Meeting Comparing observations with models 5 May 2009, ECMWF Acknowledgements to Erik Andersson and Elias Holm, ECMWF RMetS Comparing observations with models ECMWF 5 May

2 Outline Does departures follow a Gaussian distribution? Accounting for non-gaussian observation error Justifying the use of Huber norm based QC Use of Huber norm for diagnosing observations Implementation of Huber norm in ECMWF s assimilation system and case studies - North America data rejections 6 th of June French/German storm 27 th of December 1999 Summary and perspectives RMetS Comparing observations with models ECMWF 5 May

3 Even good-quality data show significant deviations from the pure Gaussian form Actual distribution Gaussian Tail QC-rejection or good data? obs-bg departures The real data distribution has fatter tails than the Gaussian Aircraft temperature observations shown here RMetS Comparing observations with models ECMWF 5 May K

4 The Normal observation cost function Jo (1) The general expression for the observation cost function is based on the probability density function (the pdf) of the observation error distribution (see Lorenc 1986): J o = ln p + const p is the probability density function of observation error arbitrary constant, chosen such that Jo=0 when y=hx RMetS Comparing observations with models ECMWF 5 May

5 The Normal observation cost function Jo (2) When for p we insert the normal (Gaussian) distribution (N): J N 0 = N = ln σ N 1 exp 2π we obtain the usual expression + const y: observation x: represents the model/analysis variables H: observation operators σ o : observation error standard deviation o = y Hx σ o y Hx σ o In VarQC a non-gaussian pdf will be used, resulting in a non-quadratic expression for Jo. 2 2 Normalized departure RMetS Comparing observations with models ECMWF 5 May

6 Accounting for non-gaussian effects In an attempt to better describe the tails of the observed distributions, Ingleby and Lorenc (1993) suggested a modified pdf (probability density function), written as a sum of two distinct distributions: QC p = (1 A) N + G Ap Normal distribution (pdf), as appropriate for good data pdf for data affected by gross errors A is the prior probability of gross error RMetS Comparing observations with models ECMWF 5 May

7 Variational quality control Thus, a pdf for the data affected by gross errors (p G ) needs to be specified. Several different forms could be considered. In the ECMWF implementation (Andersson and Järvinen 1999, QJRMS) a flat distribution was chosen. 1 p G = D D is the width of the distribution The consequence of this choice will become clear in the following RMetS Comparing observations with models ECMWF 5 May

8 Gross errors of that type occur occasionally Positive observed temperatures ( C) reported with wrong sign. (Chinese aircraft data 1-21 May 2007) Innovation Statistics Sample=429,000 All data OBS - FG OBS TEMPERATURE (C) Rejected data OBS - FG RMetS Comparing observations with models ECMWF 5 May

9 Gaussian + flat PDF Sum of 2 Gaussians Gradient Gradient QC Weight QC Weight RMetS Comparing observations with models ECMWF 5 May

10 Huber-norm an alternative A compromise between the l 2 and l 1 norms Gaussian Huber norm (New) Gaussian + flat (operational) Huber norm: Robust method: a few erroneous observations does not ruin analysis Adds some weight on observations with large departures A set of observations with consistent large departures will influence the analysis Concave cost function RMetS Comparing observations with models ECMWF 5 May

11 Huber norm variational quality control The pdf for the Huber norm is: ( ) 2 1 a exp aδ if a < δ σ 2 2 o π 1 1 δ δ σ 2 2 o π 2 1 b exp bδ if δ > b σ 2 2 o π 2 p y x = exp 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 norm. o RMetS Comparing observations with models ECMWF 5 May

12 Comparing observation weights: Huber-norm (red) versus Gaussian+flat (blue) More weight in the middle of the distribution More weight on the edges of the distribution More influence of data with large departures 25% -Weights: 0 25% RMetS Comparing observations with models ECMWF 5 May

13 Departure statistics for radiosonde temperatures is well described by a Huber-norm distribution Based on 18 months of data Feb 2006 Sep 2007 Normalised fit of pdf to data - Best Gaussian fit - Best Huber norm fit Used data Data counts (log scaling) All data Normalized departures RMetS Comparing observations with models ECMWF 5 May

14 METAR surface pressure data (Tropics) Blacklisting data may well contain gross errors Data counts (log scaling) Blacklisted data included What is left after removing blacklisted data Normalized departures After removing the blacklisted data the departures are well described by a Huber norm (black crosses & red line) RMetS Comparing observations with models ECMWF 5 May

15 Aircraft temperature and winds N.Hemis Huber norm distributed with some deviation for cold departures Data counts (log scaling) Normalized departures Normalized departures RMetS Comparing observations with models ECMWF 5 May

16 SYNOP ship Surface Pressure and buoy winds Both follow a Huber norm distribution fairly well Data counts (log scaling) Normalized departures Normalized departures RMetS Comparing observations with models ECMWF 5 May

17 SYNOP land surface pressure N. and S. Hemisphere S.Hemisphere follow a Huber norm distribution nicely. N.Hemisphere bump linked to incorrect station height for some Alpine stations. Data counts (log scaling) Normalized departures Normalized departures RMetS Comparing observations with models ECMWF 5 May

18 All data Tropics NOAA-18 AMSU-A ch. 11 and 6 Channel 11 almost Gaussian. Channel 6 cloud and surface contaminated Channel 11 Peak 25hPa Channel 6 Peak 350hPa Data counts (log scaling) Normalized departures Normalized departures RMetS Comparing observations with models ECMWF 5 May

19 All and used S.Hem METOP AMSU-A channel 7 data Channel 7 has complex contamination. The used data cloud QC well. Channel 7 Peak 250hPa All data Data counts (log scaling) Channel 7 Peak 250hPa Used data Normalized departures Normalized departures RMetS Comparing observations with models ECMWF 5 May

20 All and used S.Hem METOP AMSU-A channel 14 Channel 14 is free of contamination. The fg QC applied is too strict! Channel 14 Peak 2hPa Channel 14 Peak 2hPa Data counts (log scaling) Normalized departures Normalized departures RMetS Comparing observations with models ECMWF 5 May

21 Implementation of Huber norm variational QC in ECMWF s assimilation system Huber norm implemented and tested for - SYNOP, METAR, DRIBU: surface pressure, 10m wind - TEMP, AIREP: temperature, wind - PILOT: wind Relaxation of the first guess check - Relaxed first guess checks introduced, because Huber VarQC is a robust method - Relaxation out to ~ 20 Sigma (20 St.Dev.) Retuning of the observation error - Smaller observation errors for Huber VarQC introduced, representing the good data centre of the distribution RMetS Comparing observations with models ECMWF 5 May

22 North America 6 th of June 2008 case Time series curves 500hPa Geopotential Anomaly correlation forecast Europe Lat 35.0 to 75.0 Lon to 42.5 oper T UTC,12UTC Large departures over the US: - Wind data - ~200hPa - TEMP, AIREP Very bad 120h forecast scores for Europe JUNE 2008 RMetS Comparing observations with models ECMWF 5 May

23 First guess rejections. Operations ~200hPa, wind UTC RMetS Comparing observations with models ECMWF 5 May Departures of fg-rejections VarQC weights < 25%

24 Huber assimilation experiment Rejections ~200hPa UTC Very few first guess rejections! What are the VarQC weights? RMetS Comparing observations with models ECMWF 5 May Departures of first guess rejections VarQC weights < 25%

25 VarQC weights ~200hPa 00UTC +10min VarQC weight = 50-75% oper VarQC weight = 25-50% VarQC weight = 0-25% Huber fg rejected used -Weights > 75% RMetS Comparing observations with models ECMWF 5 May

26 6 th of June day forecast scores Time series curves 500hPa Geopotential Anomaly correlation forecast Europe Lat 35.0 to 75.0 Lon to 42.5 T+120 oper 00UTC,12UTC f1ww 00UTC,12UTC Huber JUNE 2008 RMetS Comparing observations with models ECMWF 5 May Oper

27 27 Dec 1999 French storm 18UTC Era interim analysis produced a low with min 970 hpa Lowest pressure observation (SYNOP: red circle) hpa (supported by neighbouring stations) At this station the analysis shows 977 hpa Analysis wrong by 16.5 hpa! High density of good quality surface data for this case RMetS Comparing observations with models ECMWF 5 May

28 Data rejection and VarQC weights Era interim UTC +60min fg rejected used RMetS Comparing observations with models ECMWF 5 May

29 Data rejection and VarQC weights Huber exp. VarQC weight = 50-75% VarQC weight = 25-50% VarQC weight = 0-25% RMetS Comparing observations with models ECMWF 5 May

30 MSL Analysis differences: Huber v. Era DiffAN = 5.6 hpa New min 968 hpa Low correctly shifted towards west and intensified in better agreement with surface pressure observations RMetS Comparing observations with models ECMWF 5 May

31 Summary and perspectives Robust estimation is essential for operational data assimilation The Huber-norm fits the observed pdf better than the Gaussian does, and better than Gaussian&flat pdf The associated cost-function is guaranteed concave no secondary minima Huber norm allows to relax the first guess check and adds analysis weight to observations with large departures Huber norm QC has a positive impact on analyses, especially for severe weather events Opens many new avenues for diagnosing observations Thank you! RMetS Comparing observations with models ECMWF 5 May

32 Radiosonde humidity observations Data counts (log scaling) Normalized departures Normalized departures RMetS Comparing observations with models ECMWF 5 May