1. 3.11 Adaptive Data Assimilation to Include Spatially Variable Observation Error Statistics Rod Frehlich University of Colorado, Boulder and RAL/NCAR Funded by NSF (Lydia Gates)

2. Turbulence Small scale turbulence defines the observation sampling error or “error of representativeness” Critical component of the total observation error Turbulence and observation sampling error have large spatial variations Optimal data assimilation must include these variations

3. Spatial Spectra Robust description in troposphere Power law scaling Spectral level defines turbulence (ε 2/3 and C T 2 )

4. Structure Functions Alternate spatial statistic Also has power law scaling Structure functions (and spectra) from model output are filtered Corrections are possible by comparisons with in situ data Produce local estimates of turbulence defined by ε or C T 2 for adaptive data assimilation

5. RUC20 Analysis RUC20 model structure function In situ “truth” from GASP data Effective spatial filter (3.5Δ) determined by agreement with theory

6. DEFINITION OF TRUTH For forecast error, truth is defined by the spatial filter of the model numerics For the initial field (analysis) truth should have the same definition for consistency

7. Observation Sampling Error Truth is the average of the variable x over the LxL effective grid cell Total observation error Instrument error =  x Sampling error =  x The observation sampling pattern and the local turbulence determines the sampling error

8. Sampling Error for Velocity and Temperature in Troposphere Rawinsonde in center of square effective grid cell of length L

9. Sampling Error for Global Model Rawinsonde in center of grid cell Large variations in sampling error Dominant component of total observation error in many regions Most accurate observations in low turbulence regions

10. Optimal Data Assimilation Optimal assimilation requires estimation of total observation error covariance Requires calculation of instrument error which may depend on local turbulence (profiler, Doppler lidar) Requires calculation of sampling error Calculation of analysis error

11. Adaptive Data Assimilation Assume locally homogeneous turbulence around the analysis point r forecast (first-guess) N observations

12. Measurement Geometry Single observation at the analysis coordinate Multiple observations around the analysis coordinate Aircraft track

13. Analysis Error for u Velocity Instrument error is 0.5 m/s ( …... ) sampling ( ___ ) all rawin ( _ _ _ ) one rawin ( _. _.) lidar Turbulence is important > 5%

14. SUMMARY Turbulence produces large spatial variations in total observation error Optimal data assimilation using local estimates of turbulence reduces the analysis error

15. NCAR Collaborators Bob Sharman Francois Vandenberghe Yubao Liu Josh Hacker

16. Optimal Analysis Error Analysis error depends on forecast error and effective observation error forecast error effective observation error (local turbulence)

17. Analysis Error for Temperature Instrument error is 0.5 K (.... ) sampling error ( ___ ) all data ( - - - ) single obs. (. -. - ) aircraft Turbulence is important > 50%

18. UKMO 0.5 o Global Model Effective spatial filter (5Δ) larger than RUC The s 2 scaling implies only linear spatial variations of the field (smooth)

19. GFS 0.5 o Global Model Effective spatial filter (5Δ) is the same as UKMO

20. Estimates of Small Scale Turbulence Calculate structure functions locally over LxL square Determine best-fit to empirical model Estimate in situ turbulence level and ε ε 1/3

21. Climatology of Small Scale Turbulence Probability Density Function (PDF) of ε Good fit to the Log Normal model Parameters of Log Normal model depends on domain size L Consistent with large Reynolds number turbulence

22. Scaling Laws for Log Normal Parameters Power law scaling for the mean and standard deviation of log ε Consistent with high Reynolds number turbulence