1 [email protected]KENDA Siminaria, ARPA-SIM, Bologna, 22 November 2010 PP KENDA : Contents the Priority Project KENDA (K m-Scale En semble-Based D ata A ssimilation): an overview of the new assimilation scheme for the COSMO model Christoph Schraff Deutscher Wetterdienst, Offenbach, Germany • basic theory • overview on PP KENDA / preliminary tasks + technical implementation • results from preliminary experiments • current work and plans within PP KENDA
39
Embed
PP KENDA : Contents the Priority Project KENDA · PP KENDA : Contents the Priority Project KENDA (Km-Scale Ensemble-Based Data Assimilation): ... Lucio Torrisi (CNMCA, Italy) Amalia
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Assumptions: - no systematic errors (no bias)- Gaussian distribution of errors
Generalisations
y vector of N observationsR matrix of observation error covariances x vector of M model grid pts * variables (M>>N)B matrix of background error covariances
H ‘observation operator’ : projectingfrom model space into observation space
provides optimal solution to data assimilation problem if – no systematic errors– Gaussian random errors– system linear (M, H ) – (practical issue:) problem small enough to allow computing matrix inversions
Basic theory :Kalman filter
[ ] [ ]bbbTaba HxyKxHxyRHPxx OO −+=−+= −1
( )[ ] 1-1HRH PP
-1 −+= Tba
… and also compute explicitly analysis error covariances (notation: B → Pb )
and then, forecast to the next analysis time (using the forward model M )
na
nbf Mxxx == +1
( )QMPMPP errormodel+== +T
na
nbf
1
analysis step: compute analysis of system state as before (K : Kalman Gain) …
but also need new analysis ensemble xa(i) in order to cycle the system:
– apply Kalman gain to each xb(i) → obs must be artificially perturbed to get reasonable spread (stochastic filter)
– ensemble square root filters, e.g. (L)ETKF :computes analysis ensemble in a ‘deterministic’ way (deterministic filter)
idea : choose ensemble (k members) of initial conditions (model states) such thatensemble spread around analysis mean characterises analysis error Pa ;
then propagate ensemble to next analysis time using the non-linear model, and use resulting forecast ensemble xb(i) to represent forecast error Pb :
Basic theory :Ensemble Kalman filter (EnKF)
( ) bibbTbbb ik xxXXXP −=−= )(,)1( of column th- where
Pb known → can compute Pa, K , analysis mean
for : – non-linear forecast systems (M → NWP model M ) – large problems (large state vector, many obs)
remaining assumptions : – no systematic errors– Gaussian random errors
→ model error is not accounted for → covariance inflation
Pb = (k – 1) Xb (Xb)T has rank ≤ (k – 1)
→ sampling error for small correlations → localise error covariances→ cannot represent all errors in all degrees of freedom of the global system (model)
• locally, model has much less degrees of freedom → localisation • increase ensemble spread artificially → perturbed obs / covariance inflation
initial (analysis) ensemble spread ‘characterises’ analysis error Pa , but ensemble size is limited, ensemble can only sample but not fully represent errors
Basic theory :Ensemble Kalman filter (EnKF)
ensemble members
area of high probabilitynon-linear
use resulting forecast ensemble xb(i) to represent forecast error Pb :
not represented by ensemble
→ Local ETKF : compute analysis (transform matrix) for each ‘grid point’ separatelyusing only nearby obs
→ ensemble spread is mainly localised over frontal area→ forecast errors assumed in EnKF are mainly localised over frontal area→ observation causes analysis increments over frontal area (see right panel)→ EnKF accounts for errors of the day
Whitaker et al., 2005
Ensemble Kalman filter (EnKF) : a favourable property
use forecast ensemble perturbations Xb(i) to represent forecast error Pb :
→ wanted: ensemble of perturbed analyses, to start next cycle of ensemble forecasts→ analysis mean and each analysis ensemble member is (locally) a linear combination of the forecast ens. members, where those forecast members that match obs well get larger weights; observations reduce spread of ensemble→ explicit solution for minimisation of cost function in ensemble space provides required transform matrix
perturbed forecasts
ensemble mean forecast
(local, inflated) transform
matrixanalysis mean(computed only in ensemble space)
Task 1: General issues in the convective scale (e.g. non-Gaussianity)
• COSMO Newsletter: M. Tsyrulnikov: Is the Local Ensemble Transform Kalman Filter suitable for operational data assimilation ? (difficulty: assimilate non-local satellite data & achieve good resolution in local analysis)
• COSMO Newsletter: K. Stephan and C. Schraff: The importance of small-scale analysis on the forecasts of COSMO-DE(small-scale analysis can be beneficial, depends on situation, daytime of analysis)
• COSMO Technical Report: D. Leuenberger: Statistical Analysis of high-resolution COSMO
Ensemble forecasts, in view of Data Assimilation(no clear indication that Gaussian assumption of LETKF will be detrimental → COSMO Met Services will (continue to) focus on developing LETKF)
T@5400 m over N. GermanyT@10 m over Northern Germany (flat)
+3h / +9h+3h / +9h
Statistical characteristics (COSMO-DE) :background term (EPS)
COSMO-DE ensemble forecast perturbations
• results largely determined by physical perturbation technique
• LETKF ensemble will differ form the current set-up of COSMO-DE EPSe.g. Gaussian spread in initial conditions→ need to re-do such evaluations with LETKF ensemble
• for data assimilation, need perturbations that are more Gaussian → physics perturbations: stochastic instead of fixed parameters ?
• basically for verification purposes, COSMO obs operators incl. quality controlwill be implemented in 3DVAR / LETKF environment→ future: hybrid 3DVAR-EnKF approaches in principle applicable to COSMO
• analysis step (LETKF) outside COSMO code→ ensemble of independent COSMO runs up to next analysis time→ separate analysis step code, LETKF included in 3DVAR package of DWD
• standard experimentation system not yet adapted to perform LETKF (but soon)→ stand-alone scripts allow only preliminary LETKF experiments up to now
• use in-situ obs (TEMP, AIREP, SYNOP) from GME analysis (sparse density)→ near future: use set of in-situ obs with higher density
• 3-hourly cycles (and only up to 2 days: 7 – 8 Aug. 2009: quiet + convective day)→ (near) future: 1-hourly / 30-min / 15-min cycles
• lateral (and upper) boundary conditions (BC)– ensemble BC from COSMO-SREPS (3 * 4 members),– or deterministic BC from COSMO-EU→ future: ens. BC from global LETKF (GME/ICON)
• ensemble size: 32 (→ near future: ~ 40)
• initial ensemble perturbations reflect global 3DVar-B
• ensemble mean verified against nudging analysiswhich used much more obs
• stochastic physics (Palmer et al., 2009) (by Lucio Torrisi, available Jan. 2011)
• estimation and modelling of model-errors(by M. Tsyrulnikov, V. Gorin (Russia), for 2 years, start in June 2010)
1. develop an objective estimation technique for model (tendency) errors(not to be confused with forecast errors), and set upstochastic model (parameterisation) for model error : e = u * M(x) + eadd
involves stochastic physics ( u * M(x) ) and additive components eadd
and includes multi-variate and spatio-temporal aspects
– build statistics of forecast tendencies minus observation tendencies d(using pairs of lagged obs (increments) at nearly the same location)cov(d) = ( cov(u) M(xi) M(xi) + cov(δM(x)) + cov(eadd) ) * ∆t 2 + 2 R
where δM(x) = error in the model tendency due to the analysis error, cov(δM(x)) estimated by sample covariance of the ens. tendencies
– then develop different approximations and test their validity (both looking at the simulated obs and doing assimilation experiments) in order to obtain sufficiently accurate and efficient operators
Particular issues for use in LETKF: obs error variances and correlations,superobbing, thinning,localisation
LETKF, Task 4: inclusion of additional observations
• cloud information based on satellite and conventional data
→ DWD: Eumetsat fellowship (plus resources from regular staff)
– derive incomplete analysis of cloud top + cloud base, using conventional obs (synop, radiosonde, ceilometer) and NWC-SAF cloud products from SEVIRI
– use obs increments of cloud or cloud top / base height or derived humidity
– use SEVIRI brightness temperature directly in LETKF in cloudy (+ cloud-free) conditions, in view of improving the horizontal distribution of cloud and the height of its top
– compare approaches
Particular issues: non-linear observation operators, non-Gaussian distribution of observation increments
LETKF, Task 4: inclusion of additional observations