Parametrizations in Data Assimilation

Transcription

1 Parametrizations in Data Assimilation ECMWF Training Course 9- May 05 Philippe Lopez Physical Aspects Section, Research Department, ECMWF (Room 00)

2 Lecture 3 Lecture Lecture Parametrizations in Data Assimilation Introduction An eample of physical initialization A very simple variational assimilation prolem 3D-Var assimilation The concept of adjoint 4D-Var assimilation Tangent-linear and adjoint coding Issues related to physical parametrizations in assimilation Physical parametrizations in ECMWF s current 4D-Var system Eamples of applications involving linearized physical parametrizations Summary and conclusions

3 Why do we need data assimilation? By construction, numerical weather forecasts are imperfect: discrete representation of the atmosphere in space and time (horizontal and vertical grids, spectral truncation, time step) sugrid-scale processes (e.g. turulence, convective activity) need to e parametrized as functions of the resolved-scale variales. errors in the initial conditions. Physical parametrizations used in NWP models are constantly eing improved: more and more prognostic variales (cloud variales, precipitation, aerosols), more and more processes accounted for (e.g. detailed microphysics). However, they remain approimate representations of the true atmospheric ehaviour. Another way to improve forecasts is to improve the initial state. The goal of data assimilation is to periodically constrain the initial conditions of the forecast using a set of accurate oservations that provide our est estimate of the local true atmospheric state.

4 General features of data assimilation Goal: to produce an accurate four dimensional representation of the atmospheric state to initialize numerical weather prediction models. This is achieved y comining in an optimal statistical way all the information on the atmosphere, availale over a selected time window (usually 6 or h): Oservations with their accuracies (error statistics), Short-range model forecast (ackground) with associated error statistics, Atmospheric equiliria (e.g. geostrophic alance), Physical laws (e.g. perfect gas law, condensation) The optimal atmospheric state found is called the analysis.

5 Which oservations are assimilated? Operationally assimilated since many years ago: * Surface measurements (SYNOP, SHIPS, DRIBU, ), * Vertical soundings (TEMP, PILOT, AIREP, wind profilers, ), * Geostationary satellites (METEOSAT, GOES, ) Polar oriting satellites (NOAA, SSM/I, AIRS, AQUA, QuikSCAT, ): - radiances (infrared & passive microwave in clear-sky conditions), - products (motion vectors, total column water vapour, ozone, ). More recently: * Satellite radiances/retrievals in cloudy and rainy regions (SSM/I, TMI, ), * Precipitation measurements from ground-ased radars and rain gauges. Still eperimental: * Satellite cloud/precipitation radar reflectivities/products (TRMM, CloudSat), * Lidar ackscattering/products (wind vectors, water vapour) (CALIPSO), * GPS water vapour retrievals, * Satellite measurements of aerosols, trace gases,... * Lightning data (TRMM-LIS).

6 Why physical parametrizations in data assimilation? In current operational systems, most used oservations are directly or indirectly related to temperature, wind, surface pressure and humidity outside cloudy and precipitation areas (~ 0 million oservations assimilated in ECMWF 4D-Var every hours). Physical parametrizations are used during the assimilation to link the model s prognostic variales (typically: T, u, v, q v and P s ) to the oserved quantities (e.g. radiances, reflectivities, ). Oservations related to clouds and precipitation are starting to e routinely assimilated, ut how to convert such information into proper corrections of the model s initial state (prognostic variales T, u, v, q v and, P s ) is not so straightforward. For instance, prolems in the assimilation can arise from the discontinuous or non-linear nature of moist processes.

7 Improvements are still needed More oservations are needed to improve the analysis and forecasts of: Mesoscale phenomena (convection, frontal regions), Vertical and horizontal distriution of clouds and precipitation, Planetary oundary layer processes (stratocumulus/cumulus clouds), Surface processes (soil moisture), The tropical circulation (monsoons, squall lines, tropical cyclones). Recent developments and improvements have een achieved in: Data assimilation techniques (OI 3D-Var 4D-Var Ensemle DA), Physical parametrizations in NWP models (prognostic schemes, detailed convection and large-scale condensation processes), Radiative transfer models (infrared and microwave frequencies), Horizontal and vertical resolutions of NWP models (currently at ECMWF: T79 ~ 5 km, 37 levels), New satellite instruments (incl. microwave imagers/sounders, precipitation/cloud radars, lidars, ).

8 To summarize Oservations with errors a priori information from model = ackground state with errors Data assimilation system (e.g. 4D-Var) Analysis NWP model Forecast Physical parametrizations are needed in data assimilation: - to link the model variales to the oserved quantities, - to evolve the model state in time during the assimilation (e.g. 4D-Var).

9 Empirical initialization Eample from Ducrocq et al. (000), Météo-France: - Using the mesoscale research model Méso-NH (prognostic clouds and precipitation). - Particular focus on strong convective events. - Method: Before running the forecast: ) A mesoscale surface analysis is performed (esp. to identify convective cold pools) ) the model humidity, cloud and precipitation fields are empirically adjusted to match ground-ased precipitation radar oservations and METEOSAT infrared rightness temperatures. Radar METEOSAT

10 h FC from modified analysis Rain gauges + Nîmes + Ducrocq et al. (004) 00 km.5-km resolution model Méso-NH h FC from operational analysis Nîmes radar Flash flood over South of France (8-9 Sept 00) + Nîmes+ h accumulated precipitation: 8 Sept UTC 9 Sept UTC

11 o J y J J o o ) ( A very simple eample of variational data assimilation - Short-range forecast (ackground) of m temperature from model: with error. - Simultaneous oservation of m temperature: y o with error o. The est estimate of m temperature ( a =analysis) minimizes the following cost function: In other words: o o a o o a a y y d dj a 0 And the analysis error, a, verifies: ), min( o a o a = quadratic distance to ackground and os (weighted y their errors) The analysis is a linear comination of the model ackground and the oservation weighted y their respective error statistics.

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13 o o y J o T o T H H J y R y B ) ( ) ( ) ( ) ( 0D-Var 3D-Var

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15 Important remarks on variational data assimilation Minimizing the cost function J is equivalent to finding the so-called Best Linear Uniased Estimator (BLUE) if one can assume that: - Model ackground and oservation errors are uniased and uncorrelated, - their statistical distriutions are Gaussian. (then, the final analysis is the maimum likelihood estimator of the true state). The analysis is otained y adding corrections to the ackground which depend linearly on ackground-oservations departures. In this linear contet, the oservation operator (to go from model space to oservation space) must not e too non-linear in the vicinity of the model state, else the result of the analysis procedure is not optimal. The result of the minimization depends on the ackground and oservation error statistics (matrices B and R) ut also on the Jacoian matri (H) of the oservation operator (H).

16 An eample of oservation operator H: input = model state (T,q v ) output = surface convective rainfall rate Jacoians of surface rainfall rate w.r.t. T and q v H T H q v H T H q v Marécal and Mahfouf (00) Betts-Miller (adjustment scheme) Tiedtke (ECMWF s oper mass-flu scheme)

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20 o o y J o o a y o a o T o T H H J y R y B ) ( ) ( ) ( ) ( ) ( ) ( o T T a H y R HBH BH H R H B A T 0D-Var 3D-Var

21 The minimization of the cost function J is usually performed using an iterative minimization procedure cost function J J( ) J( n n n n ) n J( n ) J mini Eample with control vector = (, )

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23 model state 3D-Var y o analysis time t a All oservations y o etween t a -3h and t a +3h are assumed to e valid at analysis time (t a =00 UTC here) a a = final analysis = model first-guess min J ( J 9 5 time ) T B B ( ( ) ) H T T H ( ) y R H ( ) y R o H ( ) y 0 o o

24 model state initial time t 0 4D-Var analysis time t a All oservations y o etween t a -9h and t a +3h are valid at their actual time y o Model trajectory from corrected initial state a min J 3 ( J B ) T B ( 0 9 assimilation window - ) n i0 M T n i0 [ t,t i 5 T H ( M )) y R H ( M 0 ] H Model trajectory from first guess Adjoint of forecast model ( 0 - ) with simplified linearized i physics 0 T i R i H ( M )) y 0 i time oi 0 i Forecast model is involved in minimization i oi 0 )) y oi

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26 (T79 L9) L37) (T55 L37) (T55 L37) minimization needed in each with minimization. 35 and around 30 iterations are in the second and third minimizations).

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29 As an alternative to the matri method, adjoint coding can e carried out using a line-y-line approach (what we do at ECMWF). Automatic adjoint code generators do eist, ut the output code is not optimized and not ug-free.

30 Tangent linear code δ = 0 δ * = 0 δ = A δy + B δz δy * = δy * + A δ * δz * = δz * + B δ * δ * = 0 δ = A δ + B δz δz * = δz * + B δ * do k =, N δ(k) = A δ(k) + B δy(k) end do Basic rules for line-y-line adjoint coding () Adjoint statements are derived from tangent linear ones in a reversed order if (condition) tangent linear code δ * = A δ * Adjoint code do k = N,, (Reverse the loop!) δ * (k ) = δ * (k) + A δ * (k) δy * (k ) = δy * (k) + B δ * (k) δ * (k) = 0 end do if (condition) adjoint code And do not forget to initialize local adjoint variales to zero!

31 Basic rules for line-y-line adjoint coding () To save memory, the trajectory can e recomputed just efore the adjoint calculations. The most common sources of error in adjoint coding are: ) Pure coding errors (often: confusion trajectory/perturation variales), ) Forgotten initialization of local adjoint variales to zero, 3) Mismatching trajectories in tangent linear and adjoint (even slightly), 4) Bad identification of trajectory updates: if ( > 0) then δ = A δ / = A Log() end if Tangent linear code Trajectory and adjoint code Trajectory store = (storage for use in adjoint) if ( > 0) then = A Log() end if Adjoint if ( store > 0) then δ * = A δ * / store end if

32 Perturation scaling factor machine precision reached

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34 Linearity assumption Variational assimilation is ased on the strong assumption that the analysis is performed in a (quasi-)linear framework. However, in the case of physical processes, strong non-linearities can occur in the presence of discontinuous/non-differentiale processes (e.g. switches or thresholds in cloud water and precipitation formation). Regularization needs to e applied: smoothing of functions, reduction of some perturations. y original tangent in 0 Precipitation formation rate Dy (tangent-linear) 0 Dy (non-linear) 0 D (finite size perturation) Cloud water amount

35 Linearity issue y Precipitation formation rate threshold 0 0 Dy (non-linear) tangent in 0 D (finite size perturation) Dy (tangent-linear) = 0 Cloud water amount

36 Illustration of discontinuity effect on cost function shape: Model ackground = {T, q }; Oservation = RR os Simple parametrization of rain rate: RR = {q q sat (T)} if q > q sat (T), 0 otherwise J T T T q q q [ q qsat( T)] RR RR os os q Saturated ackground q Dry ackground {T, q } J min OK No convergence! Single minimum of cost function T Several local minima of cost function T

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38 Janisková et al. 999

39 Importance of regularization to prevent instailities in tangent-linear model Evolution of temperature increments (4-hour forecast) with the tangent linear model using different approaches for the echange coefficient K in the vertical diffusion scheme. Perturations of K included in TL Perturations of K set to zero in TL

40 Importance of regularization to prevent instailities in tangent-linear model -hour ECMWF model integration (T59 L60) Temperature on level 48 (appro. 850 hpa) Finite difference etween two non-linear model integrations Corresponding perturations evolved with tangent-linear model No regularization in convection scheme

41 Importance of regularization to prevent instailities in tangent-linear model -hour ECMWF model integration (T59 L60) Temperature on level 48 (appro. 850 hpa) Finite difference etween two non-linear model integrations Corresponding perturations evolved with tangent-linear model Regularization in convection scheme (uoyancy & updraught velocity reduced pertur.)

42 M Forecasts Climate runs No OK? Timing: Yes No Pure coding M M(+)M() M OK? M* <M,y> = <,M*y> Yes No OK? Yes Deugging and testing (incl. regularization) No OK? Singular Vectors (EPS) No OK? 4D-Var (minim) NL TL AD APPL

43 ECMWF operational LP package (operational 4D-Var) Currently used in ECMWF operational 4D-Var minimizations (main simplifications with respect to the full non-linear versions are highlighted in red): Large-scale condensation scheme: [Tompkins and Janisková 004] - ased on a uniform PDF to descrie sugrid-scale fluctuations of total water, - melting of snow included, - precipitation evaporation included, - reduction of cloud fraction perturation and in autoconversion of cloud into rain. Convection scheme: [Lopez and Moreau 005] - mass-flu approach [Tiedtke 989], - deep convection (CAPE closure) and shallow convection (q-convergence) are treated, - perturations of all convective quantities are included, - coupling with cloud scheme through detrainment of liquid water from updraught, - some perturations (uoyancy, initial updraught vertical velocity) are reduced. Radiation: TL and AD of longwave and shortwave radiation availale [Janisková et al. 00] - shortwave: ased on Morcrette (99), only spectral intervals (instead of 6 in non-linear version), - longwave: ased on Morcrette (989), called every hours only.

44 ECMWF operational LP package (operational 4D-Var) Vertical diffusion: - miing in the surface and planetary oundary layers, - ased on K-theory and Blackadar miing length, - echange coefficients ased on Louis et al. [98], near surface, - Monin-Oukhov higher up, - mied layer parametrization and PBL top entrainment recently added. - Perturations of echange coefficients are smoothed (esp. near the surface). Orographic gravity wave drag: [Mahfouf 999] - sugrid-scale orographic effects [Lott and Miller 997], - only low-level locking part is used. Non-orographic gravity wave drag: [Oor et al. 00] - isotropic spectrum of non-orographic gravity waves [Scinocca 003], - Perturations of output wind tendencies elow 00 hpa reset to zero. RTTOV is employed to simulate radiances at individual frequencies (infrared, longwave and microwave, with cloud and precipitation effects included) to compute model satellite departures in oservation space.

45 Impact of linearized physics on tangent-linear approimation Comparison: Finite difference of two NL integrations TL evolution of initial perturations Eamination of the accuracy of the linearization for typical analysis increments: M( an) M( g) M( an g) typical size of 4D-Var analysis increments Diagnostics: mean asolute errors: M M M an g an g EXP REF relative error change: 00% (improvement if < 0) REF here: REF = adiaatic run (i.e. no physical parametrizations in tangent-linear)

46 Error difference: SPHYSvdif - ADIAB: -5.8 % Temperature Temperature - 5/03/00 h t+ Impact of operational vertical diffusion scheme EXP - REF REF = ADIAB N 60N 40N 0N 0 0S 40S 60S 80S 80N 60N 40N 0N 0 0S 40S 60S 80S relative improvement [%] hour T59 L60 integration X EXP Adia simp vdif vdif

47 Error Temperature difference: WSPHYSold_oper - ADIAB: -9.7 % Temperature - 5/03/00 h t+ Impact of dry + moist physical processes EXP - REF REF = ADIAB N 60N 40N 0N 0 0S 40S 60S 80S 80N 60N 40N 0N 0 0S 40S 60S 80S relative improvement [%] hour T59 L60 integration X EXP Adia simp vdif vdif + gwd + radold + lsp + conv

48 Error Temperature difference: WSPHYSnew_oper - ADIAB: % Temperature - 5/03/00 h t+ Impact of all physical processes (including new moist physics & radiation) EXP - REF REF = ADIAB N 60N 40N 0N 0 0S 40S 60S 80S 80N 60N 40N 0N 0 0S 40S 60S 80S relative improvement [%] hour T59 L60 integration X X X X EXP Adia simp vdif vdif + gwd + radnew + cl_new + conv_new

49 Applications

50 D-Var with radar reflectivity profiles Background =(T,q, ) J T B J B Initialize = =(T,q, ) no J 0? MINIM J J o Moist Physics Reflectivity Model yes Analysis a = Moist Physics (ADJ) Reflectivity Model (ADJ) Z mod J o K k Z Z k k mod os k os Z J o Z k mod Z k os k os k, K Reflectivity oservations Z os with errors os K = numer of model vertical levels

51 D-Var with TRMM/Precipitation Radar data Tropical Cyclone Zoe (6 Decemer UTC; Southwest Pacific) AQUA MODIS image TRMM Precipitation Radar Cross-section TRMM-PR swath

52 D-Var with TRMM/Precipitation Radar data A5 Rain Background Rain D-Var Analysed Rain A5 Reflectivity Background Reflect. D-Var Analysed Reflect. Tropical Cyclone Zoe (6 Decemer UTC) Vertical cross-section of rain rates (top, mm h - ) and reflectivities (ottom, dbz): oserved (left), ackground (middle), and analyzed (right). Black isolines on right panels = D-Var specific humidity increments.

53 Impact of ECMWF linearized physics on forecast scores Comparison of two T5 L9 4D-Var 3-month eperiments with & without full linearized physics: Relative change in forecast anomaly correlation. > 0 =

54 Own impact of NCEP Stage IV hourly precipitation data over the U.S.A. (comined ground-ased radar & rain gauge oservations) Three 4D-Var assimilation eperiments (0 May - 5 June 005): CTRL = all standard oservations. CTRL_noqUS = all os ecept no moisture os over US (surface & satellite). NEW_noqUS = CTRL_noqUS + NEXRAD hourly rain rates over US ( D+4D-Var ). Mean differences of TCWV analyses at 00UTC CTRL_noqUS CTRL NEW_noqUS CTRL_noqUS Lopez and Bauer (Monthly Weather Review, 007)

55 Adjoint sensitivities Idea: The time integration of the adjoint model allows the computation of adjoint sensitivities of any physical aspect (J) inside a target geographical domain to the model control variales several hours earlier. Here: J = 3h total surface precipitation averaged over a selected domain (N points ). J J R i, t N N steps N points N points steps Si t i i N steps S i N points i S i S i R i, t Adjoint model incl. physics J = sensitivity of rain criterion (J) to model input variales ( j ) several hours earlier j

56 Adjoint sensitivities for a European winter storm: J = mean 3h precipitation accumulation inside lack o.

57 J/ after 4 hours of ackward adjoint integration J/T level 64 (500 hpa)

58 Tropical singular vectors in EPS [Leutecher and Van Der Grijn 003] Proaility of tropical cyclone passing within 0 km radius during net 0 hrs: % 0 S 0 S 00 0 o S UTC ep: ef9h Proaility that KALUNDE will pass within 0km radius during the net 0 hours tracks: lack=oper, 4 60 E green=ctrl, lue=eps numers: oserved positions E at t+..h o S 0 o S E E o S numers real 70 0 S position of the cyclone at the 60certain hour green line control T55 forecast (unpertured memer of ensemle) Tropical singular vectors 0 S VDIF only in TL/AD 0 S /03/ UTC Tropical cyclone Kalunde 5 50 Tropical singular 40 vectors Full physics in TL/AD 0 S o E 80 o E 0

59 Correlation etween opposite twins Influence of time and resolution on linearity assumption in physics Results from ensemle runs with the MC model (3 km resolution) over the Alps, from Walser et al. (004). Comparison of a pair of opposite twin eperiments. 3-km scale Linearity 9-km scale Forecast time (hours) The validity of the linear assumption for precipitation quickly drops in the first hours of the forecast, especially for smaller scales.

60 General conclusions Physical parametrizations have ecome important components in recent variational data assimilation systems. However, their linearized versions (tangent-linear and adjoint) require some special attention (regularizations/simplifications) in order to eliminate possile discontinuities and non-differentiaility of the physical processes they represent. This is particularly true for the assimilation of oservations related to precipitation, clouds and soil moisture, to which a lot of efforts are currently devoted. Developing new simplified parametrizations for data assimilation requires: Compromise etween realism, linearity and computational cost, Evaluation in terms of Jacoians (not to noisy in space and time), Validation of forward simplified code against oservations, Comparison to full non-linear code used in forecast mode (4D-Var trajectory), Numerical tests of tangent-linear and adjoint codes for small perturations, Validity of the linear hypothesis for perturations with larger size (typical of analysis increments). Successful convergence of 4D-Var minimizations.

61 Thank you!

62 3 ch ch ch n n Rad Rad Rad q q T T H y Eample of oservation operator H (radiative transfer model): n ch ch n ch ch n ch ch n ch ch n ch ch n ch ch q Rad q Rad T Rad T Rad q Rad q Rad T Rad T Rad q Rad q Rad T Rad T Rad H and its tangent-linear operator H: