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Reading, UK Parametrizations and data assimilation © ECMWF 2010 Marta JANISKOVÁ ECMWF Parametrizations and data assimilation

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 Why is physics needed in data assimilation ? How the physics is applied in variational data assimilation system ? Which parametrization schemes are used at ECMWF ? What are the problems to be solved before using physics in data assimilation ? What is an impact of including the physical processes in assimilating model ? How the physics is used for assimilation of observations related to the physical processes ? Parametrization = description of physical processes in the model.

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 POSITION OF THE PROBLEM IMPORTANCE OF THE ASSIMILATING MODEL the better the assimilating model (4D-Var consistently using the information coming from the observations and the model) the better the analysis (and the subsequent forecast) the more sophisticated the model (4D-Var containing physical parametrizations) the more difficult the minimization (on-off processes, non-linearities) DEVELOPMENT OF A PHYSICAL PACKAGE FOR DATA ASSIMILATION = FINDING A TRADE-OFF BETWEEN: Simplicity and linearity (using simplified linear model is a basis of incremental variational approach) Realism

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 IMPORTANCE OF INCLUDING PHYSICS IN THE ASSIMILATING MODEL using an adiabatic linear model can be critical especially in the tropics, planetary boundary layer, stratosphere missing physical processes can increase the so-called spin-up/spin-down problem using the adjoint of various physical processes should provide: – initial atmospheric state more consistent with physical processes – better agreement between the model and data inclusion of such processes is a necessary step towards: – initialization of prognostic variables related to physical processes – the use of new (satellite) observations in data assimilation systems (rain, clouds, soil moisture, …)

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 STANDARD FORMULATION OF 4D-VAR the goal of 4D-Var is to define the atmospheric state x (t 0 ) such that the distance between the model trajectory and observations is minimum over a given time period [t 0, t n ] finding the model state at the initial time t 0 which minimizes a cost-function J : x i is the model state at time step t i such as: M is the nonlinear forecast model integrated between t 0 and t i H is the observation operator (model space observation space)

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 WHY AND WHERE ARE PHYSICAL PARAMETRIZATIONS NEEDED IN DATA ASSIMILATION? In 1D-Var, physical parametrizations can be needed in observation operator, H (no time evolution involved). In 4D-Var, physical parametrizations are involved in the observation operator, H, but also in the forecast model, M. Physical parametrizations are needed in data assimilation (DA): – to link the model state to the observed quantities, – to evolve the model state in time during the assimilation (trajectory, tangent-linear (TL) and adjoint (AD) computations in 4D-Var) Example: to assimilate reflectivity profiles, H must perform the conversion: Model state (T, q, u, v, P s ) Cloud and precipitation profiles Simulated reflectivity profile moist physics reflectivity model

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 OPERATIONAL 4D-VAR AT ECMWF – INCREMENTAL FORMULATION 4D-Var can be then approximated to the first order as minimizing: where is the innovation vector In incremental 4D-Var, the cost function is minimized in terms of increments: with the model state defined at any time t i as: tangent linear model Gradient of the cost function to be minimized: computed with the non-linear model at high resolution using full physics M computed with the tangent-linear model at low resolution using simplified physics M computed with a low resolution adjoint model using simplified physics M T Adjoint operators Tangent-linear operators

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 WHY SIMPLIFIED PHYSICS? One of the main assumptions in variational DA is that parametrizations and operators that describe atmospheric processes should be linear. otherwise, the use of the tangent-linear and adjoint approach is inappropriate and the analysis is suboptimal In practise, weak nonlinearities can be handled through successive trajectory updates (e.g., 3 outer loops in ECMWF 4D-Var) Physical parametrizations used in DA (TL and AD) are usually simplified versions of the original parametrizations employed in the forecast models: – to avoid nonlinearities (see further), – to keep the computational cost reasonable, – but they also need to be realistic enough ! ECMWF TL and AD models are coded using a manual line-by-line approach. Automatic AD coding softwares exist (but far from perfect and non-optimized).

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 Variational assimilation is based on the strong assumption that the analysis is performed in quasi-linear framework. However, in the case of physical processes, strong nonlinearities or thresholds can occur in the presence of discontinuous/non-differentiable processes (e.g. switches or thresholds in cloud water and precipitation formation, …) Without adequate treatment of most serious threshold processes, the TL approximation can turn to be useless. LINEARITY ISSUE finite difference (FD) TL integration u-wind increments fc t+12, ~700 hPa x 10 5

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 regularizations help to remove the most important threshold processes in physical parametrizations which can effect the range of validity of the tangent linear approximation after solving the threshold problems TL increments correspond well to finite differences u-wind increments fc t+12, ~700 hPa TL integration finite difference (FD) IMPORTANCE OF THE REGULARIZATION OF TL MODEL © ECMWF 2010

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 dy NL dy TL Potential source of problem (example of precipitation formation) dx

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 dx dy NL dy TL1 dy TL2 Possible solution, but …

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 dx1 dy NL dy TL2 … may just postpone the problem and influence the performance of NL scheme dy TL3 dx2

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 dx dy NL dy TL1 dy TL2 However, the better the model the smaller the increments

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 In NWP – a tendency to develop more and more sophisticated physical parametrizations they may contain more discontinuities For the perturbation model – more important to describe basic physical tendencies while avoiding the problem of discontinuities Level of simplifications and/or required complexity depends on: which level of improvement is expected (for different variables, vertical and horizontal resolution, …) which type of observations should be assimilated necessity to remove threshold processes Different ways of simplifications: development of simplified physics from simple parametrizations used in the past selecting only certain important parts of the code to be linearized

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 Regularization of vertical diffusion scheme: reduced perturbation of the exchange coefficients (Janisková et al., 1999): – original computation of Ri modified in order to modify/reduce f(Ri), or – reducing a derivative, f(Ri), by factor 10 in the central part (around the point of singularity ) Ri number f(Ri) Function of the Richardson number EXAMPLES OF REGULARIZATIONS (1) reduction of the time step to 10 seconds to guarantee stable time integrations of the associated TL model (Zhu and Kamachi, 2000) not possible in operational global models Exchange coefficients K are function of the Richardson number:

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 selective regularization of the exchange coefficients K based on the linearization error and a criterion for the numerical stability (Laroche et al., 2002) New operational ECMWF version: using reduction factor for perturbation of the exchange coefficients perturbation of the exchange coefficients neglected, K = 0 (Mahfouf, 1999) in the operational ECMWF version used to the middle of 2008 EXAMPLES OF REGULARIZATIONS (2)

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 ECMWF LINEARIZED PHYSICS (as operational in 4D-Var) Currently used in ECMWF operational 4D-Var minimizations (main simplifications with respect to the nonlinear versions are highlighted in red): Radiation: – TL and AD of longwave and shortwave radiation [ Janisková et al ] – shortwave: only 2 spectral intervals (instead of 6 in nonlinear version) – longwave: called every 2 hours only. Vertical diffusion: – mixing in the surface and planetary boundary layers, – based on K-theory and Blackadar mixing length, – exchange coefficients based on Louis et al. [1982], near surface, – Monin-Obukhov higher up, – mixed layer parameterization and PBL top entrainment recently added. – Perturbations of exchange coefficients are smoothed out. Gravity wave drag: [Mahfouf 1999] – subgrid-scale orographic effects [Lott and Miller 1997], – only low-level blocking part is used. No evolution of surface variables Dry processes:

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 ECMWF LINEARIZED PHYSICS (as operational in 4D-Var) Convection scheme: [Lopez and Moreau 2005] – mass-flux approach [Tiedtke 1989], – deep convection (CAPE closure) and shallow convection (q-convergence) – perturbations of all convective quantities included, – coupling with cloud scheme through detrainment of liquid water from updraught, – some perturbations (buoyancy, initial updraught vertical velocity) are reduced. Large-scale condensation scheme: [Tompkins and Janisková 2004] – based on a uniform PDF to describe subgrid-scale fluctuations of total water, – melting of snow included, – precipitation evaporation included, – reduction of cloud fraction perturbation and in autoconversion of cloud into rain. After solving the threshold problems clear advantage of the diabatic TL evolution of errors compared to the adiabatic evolution Moist processes:

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 VALIDATION OF THE LINEARIZED PARAMETRIZATION SCHEMES Non-linear model: Forecast runs with particular modified/simplified physical parametrization schemes Check that Jacobians (=sensitivities) with respect to input variables look reasonable (not too noisy in space and time) classical validation: TL - Taylor formula, AD - test of adjoint identity Tangent-linear (TL) and adjoint (AD) model: examination of the accuracy of the linearization: comparison between finite differences (FD) and tangent-linear (TL) integration Singular vectors: Computation of singular vectors to find out whether the new schemes do not produce spurious unstable modes.

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 Comparison: finite differences (FD) tangent-linear (TL) integration TANGENT-LINEAR DIAGNOSTICS

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 TL ADIAB – adiabatic TL model TL WSPHYS – TL model with the whole set of simplified physics (Mahfouf 1999) TL WSPHYS TL ADIAB FD Zonal wind increments at model level ~ 1000 hPa [ 24-hour integration]

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 Diagnostics: mean absolute errors: relative error TANGENT-LINEAR DIAGNOSTICS Comparison: finite differences (FD) tangent-linear (TL) integration

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 adiabsvd || vdif EXP - REF EXP relative improvement REF = ADIAB [%] X N 60N 40N 20N 0 20S 40S 60S 80S Temperature Impact of operational vertical diffusion scheme © ECMWF 2010

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 adiabsvd || vdif + gwd + radold + lsp + conv EXP - REF EXP relative improvement REF = ADIAB [%] X N 60N 40N 20N 0 20S 40S 60S 80S Temperature Impact of dry + moist physical processes (1 st used setup) © ECMWF 2010

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Reading, UK Parametrizations and data assimilation © ECMWF adiabsvd || vdif + gwd + rad + cloud+conv current new new new cycle EXP - REF REF = ADIAB EXP Temperature Impact of all physical processes (including improved schemes) relative improvement[%] X X X X 80N 60N 40N 20N 0 20S 40S 60S 80S adiab adiabsvd vdif oper_old oper_2007 © ECMWF 2010

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 Zonal wind EXP - REF Impact of all physical processes relative improvement [%] X X X X conv current new cycle adiab adiabsvd vdif oper_old oper_2007 adiab adiabsvd vdif oper_old oper_2007 EXP - REF Specific humidity X X X conv current new cycle relative improvement[%]

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 IMPACT OF THE LINEARIZED PHYSICAL PROCESSES IN 4D-VAR (1) comparisons of the operational version of 4D-Var against the version without linearized physics included shows: – positive impact on analysis and forecast – reducing precipitation spin-up problem when using simplified physics in 4D-Var minimization Time evolution of total precipitation in the tropical belt [30S, 30N] averaged over 14 forecasts issued from 4D-Var assimilation

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Reading, UK Parametrizations and data assimilation © ECMWF DAY FORECAST ERROR OF 500 hPa GEOPOTENTIAL HEIGHT OPER (very simple radiation) vs. NEWRAD (new linearized radiation) (27/08/ h t+24) A1: FC_OPER – ANAL_OPER A2: FC_NEWRAD – ANAL_OPER A2 – A IMPACT OF THE LINEARIZED PHYSICAL PROCESSES IN 4D-VAR (2) impact of new linearized radiation

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 June 2005 – March 2009 – 1D+4D-Var assimilation of SSM/I brightness temperatures (TBs) in regions affected by rain and clouds. (Bauer et al a, b) Since March 2009 – active all-sky 4D-Var assimilation of microwave imagers AIRS – Advanced Infrared Sounder ARM – Atmospheric Radiation Measurement programme GPS – Global Positioning System SSM/I – Special Infrared Sounder Operational: Experimental: 1D-Var assimilation of cloud-related ARM observations. (Janisková et al. 2002) surface downward LW radiation, total column water vapour, cloud liquid water path Investigation of the capability of 4D-Var systems to assimilate cloud-affected satellite infrared radiances – using cloudy AIRS TBs. (Chevallier et al. 2004) 1D-Var assimilation of precipitation radar data. (Benedetti and Lopez 2003) 1D-Var assimilation of cloud radar reflectivity – retrieved from 35 GHz radar at ARM site. (Benedetti and Janisková 2004) 2D-Var assimilation of ARM observations affected by clouds & precipitation – using microwave TBs, cloud radar reflectivity, rain-gauge and GPS TCWV. (Lopez et al. 2006) 4D-Var assimilation of cloud optical depth from MODIS. (Benedetti & Janisková 2007) 1D+4D-Var assimilation of NCEP Stage IV hourly precipitation data over USA – combined radar + rain gauge observations. (Lopez and Bauer 2007) 1D-Var of cloud radar reflectivity from CloudSat (QuARL project) Reading, UK Assimilation of rain and cloud related observations at ECMWF

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 Impact of the direct 4D-Var assimilation of SSM/I all-skies TBs on the relative change in 5-day forecast RMS errors (zonal means). Period: 22 August 2007 – 30 September 2007 Wind Speed 4D-Var assimilation of SSM/I rainy brightness temperatures (Geer, Bauer et al.) forecast is better forecast is worse Relative Humidity

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 B = background error covariance matrix R = observation and representativeness error covariance matrix H = nonlinear observation operator (model space observation space) (physical parametrization schemes, microwave radiative transfer model, reflectivity model, …) For a given observation y o, 1D-Var searches for the model state x=(T,q v ) that minimizes the cost function: Background term Observation term 1D-Var assimilation of observations related to the physical processes The minimization requires an estimation of the gradient of the cost function: The operator H T can be obtained: – explicitly (Jacobian matrix) – using the adjoint technique

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Reading, UK Parametrizations and data assimilation © ECMWF D-Var 1D-Var moist physics + radiative transfer background T,q v Observed rainfall rates Retrieval algorithm (2A12,2A25) 1D-Var on TBs or reflectivities1D-Var on TMI or PR rain rates Observations interpolated on models T511 Gaussian grid TMI TBs or TRMM-PR reflectivities TCWV obs =TCWV bg + z q v 1D-Var+4D-Var assimilation of observations related to precipitation TRMM – Tropical Rainfall Measuring Mission TMI – TRMM Microwave Imager PR – Precipitation Radar

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 Background PATER obs 1D-Var/RR PATER 1D-Var/TB Tropical Cyclone Zoe (26 December UTC) 1D-Var on TMI Rain Rates / Brightness Temperatures Surface rainfall rates (mm h -1 ) 1D-Var on TMI data (Lopez and Moreau, 2003)

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Reading, UK Parametrizations and data assimilation © ECMWF A25 RainBackground Rain1D-Var Analysed Rain 2A25 Reflect.Background Reflect.1D-Var Analysed Reflect. 1D-Var on TRMM/ Precipitation Radar data (Benedetti and Lopez, 2003) Tropical Cyclone Zoe (26 December UTC) Vertical cross-section of rain rates (top, mm h -1 ) and reflectivities (bottom, dBZ): observed (left), background (middle), and analysed (right). Black isolines on right panels = 1D-Var specific humidity increments.

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 Background term Observation term For a given observation y o, 1D-Var searches for the model state x=(T,q v ) that minimizes the cost function: 1D-Var assimilation of cloud related observations (1) H(x): moist physics or + radar/lidar radiative model (+ radiation scheme) x _b : Background T,q 1D-Var (analyzed T, q) Y: retrieved cloud parameters (level-2 products) or backscatter cross-sections, reflectivities (level-1 products)

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Reading, UK Parametrizations and data assimilation © ECMWF D-Var assimilation of cloud related observations (2) (QuARL project) OBS – CloudSat (94 GHz radar) FG AN – 1D-Var of cloud reflectivity Cloud reflectivity [dBZ] – 23/01/2007 over Pacific QuARL – Quantitative Assessment of Operation.Value of Space-Borne Radar and Lidar Measurements of Cloud and Aerosol Profiles

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Reading, UK Parametrizations and data assimilation © ECMWF D-Var assimilation of cloud related observations (3) (QuARL project) Comparison of the first guess and analysis against cloud optical depth FG departure AN-refl departure AN-reflopt departure BIAS - reflectivity FG departure AN-refl departure AN-reflopt departure STD.DEV. - reflectivity Reading, UK Bias and standard deviation of first-guess vs. analysis departures for reflectivity height [km] profile number

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 Experimental 4D-Var assimilation of cloud optical depth from MODIS (1) (Benedetti and Janisková, 2008) Period: Assimilation of cloud optical depth at 0.55 m from MODIS – fit to observations Scatter-plot of OBS versus FG Scatter-plot of OBS versus ANA BIAS = 0.22 RMS = 0.57 corr. coef. = BIAS = 0.21 RMS = 0.51 corr. coef. = Positive impact on the distribution of the ice water content, particularly in the Tropics. Impact on 10-day forecast positive for upper level temperature in the Tropics and neutral for the model wind. ECMWF 4D-Var is approaching a level of the technical maturity necessary for global assimilation of cloud related observations. Conclusions: MODIS – Moderate Resolution Imaging Spectroradiometer © ECMWF 2010

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 MLS retrievals: Ice Water Content at 215 hPa ECMWF model – CNTRL run CNTRL run – MLS obs ECMWF model – EXP run EXP run – MLS obs MLS = Microwave Limb Sounder Courtesy of F. Li, Jet Propulsory Laboratory, CA, USA IWC [mg/m 3 ] Experimental 4D-Var assimilation of cloud optical depth from MODIS (2) Comparison with independent cloud observations

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 Physical parametrizations become important components in current variational data assimilation systems as they are needed: to link the model state to the observation quantities to evolve the model state in time during the assimilation Positive impact from including linearized physical parametrization schemes into the assimilating model has been demonstrated. However, there are several problems with including physics in adjoint models: – development and thorough validation require substantial resources – computational cost may be very high – non-linearities and discontinuities in physical processes must be treated with care Constraints and requirements when developing new simplified parametrizations for data assimilation: – find a compromise between realism, linearity and computational cost – evaluation in terms of Jacobians (not too noisy in space and time) – systematic validation against observations – comparison to the non-linear version used in forecast mode (trajectory) – numerical tests of tangent-linear and adjoint codes for small perturbations – validity of the linear hypothesis for perturbations with larger size (typical of analysis increments) GENERAL CONCLUSIONS © ECMWF 2010

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 REFERENCES Bauer, P., Lopez, P., Benedetti, A., Salmond, D. and Moreau, E., 2006a: Implementation of 1D+4D-Var assimilation of precipitation affected microwave radiances at ECMWF. Part I: 1D-Va. Quart. J. Roy. Meteor. Soc., 132, Bauer, P., Lopez, P., Salmond, D., Benedetti, A., Saarinen, S. and Bonazzola, M., 2006b: Implementation of 1D+4D-Var assimilation of precipitation affected microwave radiances at ECMWF. Part II: 4D-Var. Quart. J. Roy. Soc., 132, Benedetti, A. and Janisková, M., 2004: Advances in cloud assimilation at ECMWF using ARM radar data. Extended abstract for ICCP, Bologna 2004 Benedetti, A. and Janisková, M., 2008: Assimilation of MODIS cloud optical depths in the ECMWF model. Mon. Wea. Rev., 136, Errico, R.M., 1997: What is an adjoint model. Bulletin of American Met. Soc., 78, Fillion, L. and Errico, R., 1997: Variational assimilation of precipitation data using moist convective parametrization schemes: A 1D-Var study. Mon. Wea. Rev., 125, Fillion, L. and Mahfouf, J.-F., 2000: Coupling of moist-convective and stratiform precipitation processes for variational data assimilation. Mon. Wea. Rev., 128, Klinker, E., Rabier, F., Kelly, G. and Mahfouf, J.-F., 2000: The ECMWF operational implementation of four-dimensional variational assimilation. Part III: Experimental results and diagnostics with operational configuration. Quart. J. Roy. Meteor. Soc., 126, Chevallier, F., Bauer, P., Mahfouf, J.-F. and Morcrette, J.-J., 2002: Variational retrieval of cloud cover and cloud condensate from ATOVS data. Quart. J. Roy. Meteor. Soc., 128, Chevallier, F., Lopez, P., Tompkins, A.M., Janisková, M. and Moreau, E., 2004: The capability of 4D-Var systems to assimilate cloud_affected satellite infrared radiances. Quart. J. Roy. Meteor. Soc., 130, Janisková, M., Thépaut, J.-N. and Geleyn, J.-F., 1999: Simplified and regular physical parametrizations for incremental four- dimensional variational assimilation. Mon. Wea. Rev., 127, Janisková, M., Mahfouf, J.-F., Morcrette, J.-J. and Chevallier, F., 2002: Linearized radiation and cloud schemes in the ECMWF model: Development and evaluation. Quart. J. Roy. Meteor. Soc.,128,

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Reading, UK Parametrizations and data assimilation © ECMWF 2010 Janisková, M., Mahfouf, J.-F. and Morcrette, J.-J., 2002: Preliminary studies on the variational assimilation of cloud-radiation observations. Quart. J. Roy. Meteor. Soc., 128, Janisková, M., 2004: Impact of EarthCARE products on Numerical Weather Prediction. ESA Contract Report, 59 pp. Laroche, S., Tanguay, M. and Delage, Y., 2002: Linearization of a simplified planetary boundary layer parametrization. Mon. Wea. Rev., 130, Lopez, P. and Moreau, E., 2005: A convection scheme for data assimilation purposes: Description and initial test. Quart. J. Roy.. Meteor. Soc., 131, Lopez., P., Benedetti, A., Bauer, P., Janisková, M. and Köhler, M.., 2006: Experimental 2D-Var assimilation of ARM cloud and precipitation observations. Quart. J. Roy. Meteor. Soc., 132, Lopez, P., 2007: Cloud and precipitation parametrizations in modeling and in variational data assimilation. A review. J. Atmos. Sc., 64, Lopez, P. and Bauer, P., 2007: 1D+4D-Var assimilation of NCEP Stage IV radar and gauge hourly precipitation data at ECMWF. Mon. Wea. Rev., 135, Mahfouf, J.-F., 1999: Influence of physical processes on the tangent-linear approximation. Tellus, 51A, Marécal, V. and Mahfouf, J.-F., 2000: Variational retrieval of temperature and humidity profiles from TRMM precipitation data. Mon. Wea. Rev., 128, Marécal, V. and Mahfouf, J.-F., 2002: Four-dimensional variational assimilation of total column water vapour in rainy areas. Mon. Wea. Rev., 130, Marécal, V. and Mahfouf, J.-F., 2003: Experiments on 4D-Var assimilation of rainfall data using an incremental formulation. Quart. J. Roy. Meteor. Soc., 129, Moreau, E., Lopez, P., Bauer, P., Tompkins, A.M., Janisková, M. And Chevallier, F., 2004: Variational retrieval of temperature and humidity profiles using rain rates versus microwave brightness temperatures. Quart. J. Roy. Meteor. Soc., 130, Tompkins, A.M. and Janisková, M., 2004: A cloud scheme for data assimilation: Description and initial tests. Quart. J. Roy. Meteor. Soc., 130, Zhu, J. and Kamachi, M., 2000: The role of the time step size in numerical stability of tangent linear models. Mon. Wea. Rev., 128,

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