Download presentation

Presentation is loading. Please wait.

1
**Parametrizations and data assimilation**

Marta JANISKOVÁ ECMWF

2
**PARAMETRIZATIONS IN DATA ASSIMILATION**

Parametrization = description of physical processes in the model. Why is physics needed in data assimilation ? How the physics is applied in variational data assimilation system ? What are the problems to be solved before using physics in data assimilation ? Which parametrization schemes are used at ECMWF ? 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 ?

3
**POSITION OF THE PROBLEM**

GOAL OF DATA ASSIMILATION production of an accurate representation of the atmospheric state to initialize numerical weather prediction models 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 Realism

4
**IMPORTANCE OF INCLUDING PHYSICS IN THE ASSIMILATING MODEL**

MISSING PHYSICAL PROCESSES: can be critical especially in the tropics, planetary boundary layer, stratosphere can increase the so-called spin-up/spin-down problem INCLUDING PHYSICAL PROCESSES: provides an initial atmospheric state more consistent with physical processes creates a better agreement between the model and data 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, …) PARAMETRIZATIONS OF PHYSICAL PROCESSES: constantly being improved however, remain approximate representation of the true atmospheric behaviour

5
**STANDARD FORMULATION OF 4D-VAR**

the goal of 4D-Var is to define the atmospheric state x(t0) such that the “distance” between the model trajectory and observations is minimum over a given time period [t0, tn] finding the model state at the initial time t0 which minimizes a cost-function J : H is the observation operator (model space observation space) xi is the model state at time step ti such as: M is the nonlinear forecast model integrated between t0 and ti

6
**WHY AND WHERE PHYSICAL PARAMETRIZATIONS NEEDED IN DATA ASSIMILATION?**

In 1D-Var, physical parametrizations can be needed in observation operator, H (no time evolution involved). Example: to assimilate reflectivity profiles, H must perform the conversion: Model state (T, q, u, v, Ps ) Cloud and precipitation profiles Simulated reflectivity profile moist physics reflectivity model 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)

7
**OPERATIONAL 4D-VAR AT ECMWF – INCREMENTAL FORMULATION**

In incremental 4D-Var, the cost function is minimized in terms of increments: with the model state defined at any time ti as: tangent linear model Tangent-linear operators 4D-Var can be then approximated to the first order as minimizing: where is the innovation vector Gradient of the cost function to be minimized: Adjoint operators 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 MT

8
**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).

9
**finite difference (FD) TL integration**

LINEARITY ISSUE 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, …) finite difference (FD) TL integration u-wind increments fc t+12, ~700 hPa x 105 Without adequate treatment of most serious threshold processes, the TL approximation can turn to be useless.

10
**TL increments correspond well to finite differences**

IMPORTANCE OF THE REGULARIZATION OF TL MODEL 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 finite difference (FD) TL integration

11
**Potential source of problem (example of precipitation formation)**

dx dyTL dyNL

12
**Possible solution, but …**

dyTL1 dyTL2 dyNL dx

13
**… may just postpone the problem and influence the performance of NL scheme**

dx2 dyTL3 dyTL2 dyNL dx1

14
**However, the better the model the smaller the increments**

dyTL1 dyTL2 dyNL dx

15
**SIMPLIFCATIONS AND REGULARIZATIONS OF “PERTURBATION” MODEL**

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

16
**EXAMPLES OF REGULARIZATIONS (1)**

Regularization of vertical diffusion scheme: Ri number 10.0 0.0 f(Ri) - 10.0 - 20.0 Function of the Richardson number Exchange coefficients K are function of the Richardson number: 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 ) 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

17
**EXAMPLES OF REGULARIZATIONS (2)**

selective regularization of the exchange coefficients K based on the linearization error and a criterion for the numerical stability (Laroche et al., 2002) perturbation of the exchange coefficients neglected, K’ = 0 (Mahfouf, 1999) in the operational ECMWF version used to the middle of 2008 New operational ECMWF version: using reduction factor for perturbation of the exchange coefficients

18
**ECMWF LINEARIZED PHYSICS (as operational in 4D-Var)**

Currently used schemes in ECMWF operational 4D-Var minimizations – main simplifications with respect to the nonlinear versions are highlighted in red (Janisková and Lopez 2012): No evolution of surface variables 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. Dry processes: Radiation: – TL and AD of longwave and shortwave radiation [Janisková et al. 2002], – shortwave: only 2 spectral intervals (instead of 6 in nonlinear version), – longwave: called every 2 hours only. Non-orographic gravity wave drag: – TL and AD of the non-linear scheme for non-orographic gravity waves [Orr et al. 2010], – suppressing increments for momentum flux for the highest phase speed.

19
**ECMWF LINEARIZED PHYSICS (as operational in 4D-Var)**

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. Moist processes: 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. After solving the threshold problems clear advantage of the diabatic TL evolution of errors compared to the adiabatic evolution

20
**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.

21
**TANGENT-LINEAR DIAGNOSTICS**

Comparison: finite differences (FD) tangent-linear (TL) integration

22
**Zonal wind increments at model level ~ 1000 hPa [ 24-hour integration]**

FD TLWSPHYS TLADIAB TLADIAB – adiabatic TL model TLWSPHYS – TL model with the whole set of simplified physics (Mahfouf 1999)

23
**TANGENT-LINEAR DIAGNOSTICS**

Comparison: finite differences (FD) tangent-linear (TL) integration Diagnostics: • mean absolute errors: relative error

24
**Impact of operational vertical diffusion scheme**

Temperature Impact of operational vertical diffusion scheme EXP - REF 10 20 30 40 50 60 REF = ADIAB 80N N N N S S S S relative improvement [%] X EXP adiabsvd || vdif © ECMWF 2011

25
**X EXP - REF Temperature**

Impact of dry + moist physical processes (1st used setup) EXP - REF 10 20 30 40 50 60 REF = ADIAB 80N N N N S S S S relative improvement [%] X EXP adiabsvd || vdif + gwd + radold + lsp + conv © ECMWF 2011

26
**X EXP - REF Temperature**

Impact of all physical processes (including improved schemes) EXP - REF 10 20 30 40 50 60 REF = ADIAB adiab adiabsvd vdif oper_old oper_2007 16 relative improvement [%] X 80N N N N S S S S EXP adiabsvd || vdif + gwd rad + cloud+conv cycle new new new © ECMWF 2011

27
**EXP - REF Temperature**

Impact of all physical processes (versions before and after July 2009) EXP - REF REF = ADIAB EXP = cycle before July 2009 EXP = cycle after July 2009 Changes in linearized radiation schemes

28
**Impact of all physical processes**

adiab adiabsvd vdif oper_old oper_2007 EXP - REF Zonal wind relative improvement [%] X conv cycle new EXP - REF 20 18 Specific humidity X conv cycle new relative improvement [%] adiab adiabsvd vdif oper_old oper_2007

29
**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 N.HEM : 500 hPa geopotential N.HEM : 700 hPa rel. humidity Tropics : 700 hPa rel.humidity Anomaly correlation: grey bars indicate significance at 95% confidence level July – September 2011

30
**– – A2 – A1 IMPACT OF THE LINEARIZED PHYSICAL PROCESSES IN 4D-VAR (2)**

1-DAY FORECAST ERROR OF 500 hPa GEOPOTENTIAL HEIGHT OPER (very simple radiation) vs. NEWRAD (new linearized radiation) (27/08/ h t+24) 75.4 -30.3 63.8 -23.3 A2: FC_NEWRAD – ANAL_OPER A1: FC_OPER – ANAL_OPER -17.9 -10.0 A2 – A1 impact of new linearized radiation

31
**Assimilation of rain and cloud related observations at ECMWF**

Operational: 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. (Bauer et al. 2010, Geer et al. 2010) Since November direct 4D-Var assimilation of NCEP Stage IV radar and gauge hourly precipitation data (Lopez 2011) 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+4D-Var of cloud radar reflectivity from CloudSat (Janisková el al. 2011) AIRS – Advanced Infrared Sounder ARM – Atmospheric Radiation Measurement programme GPS – Global Positioning System SSM/I – Special Infrared Sounder Reading, UK

32
**Relative Humidity Wind Speed -0.1 0.05 0.05 0.1 forecast is better**

4D-Var assimilation of SSM/I rainy brightness temperatures (Geer, Bauer et al. 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 Relative Humidity Wind Speed -0.1 0.05 0.05 0.1 forecast is better forecast is worse

33
**1D-Var assimilation of observations related to the physical processes**

For a given observation yo, 1D-Var searches for the model state x=(T,qv) that minimizes the cost function: Background term Observation term 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, …) The minimization requires an estimation of the gradient of the cost function: The operator HT can be obtained: explicitly (Jacobian matrix) using the adjoint technique

34
**“1D-Var+4D-Var” assimilation of observations related to precipitation**

1D-Var on TBs or reflectivities 1D-Var on TMI or PR rain rates Observations interpolated on model’s T511 Gaussian grid TMI TBs or TRMM-PR reflectivities “Observed” rainfall rates Retrieval algorithm (2A12,2A25) 1D-Var moist physics + radiative transfer background T,qv “TCWVobs”=TCWVbg+∫zqv 4D-Var TRMM – Tropical Rainfall Measuring Mission TMI – TRMM Microwave Imager PR – Precipitation Radar

35
**1D-Var on TMI data (Lopez and Moreau, 2003)**

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)

36
**1D-Var assimilation of cloud related observations (1)**

For a given observation yo, 1D-Var searches for the model state x=(T,qv) that minimizes the cost function: Background term Observation term 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) backscatter cross-sections, reflectivities (level-1 products)

37
**1D-Var assimilation of CloudSat cloud radar reflectivity (1) (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

38
**1D-Var assimilation of CloudSat cloud radar reflectivity (2) (QuARL project)**

Bias and standard deviation of first-guess FG vs.analysis AN departures for reflectivity STD.DEV. - reflectivity BIAS - reflectivity FG departure AN-refl departure height [km] height [km] FG departure AN-refl departure Comparison of FG and AN against cloud optical depth (i.e. independent observations) optical depth profile number

39
**1D+4D-Var assimilation of CloudSat cloud radar reflectivity (Janisková et al. 2011)**

Impact on the subsequent forecast: Difference of 200-hPa wind rms errors for FC_exp– AN_ref T+12 T+24 RMS: ref 1.304 exp 1.259 RMS: ref 2.025 exp 1.978 T+36 T+48 RMS: ref 2.510 exp 2.416 RMS: ref 3.438 exp 3.361

40
Experimental 4D-Var assimilation of cloud optical depth from MODIS (1) (Benedetti and Janisková, 2008) 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 = RMS = 0.57 corr. coef. = 0.480 BIAS = RMS = 0.51 corr. coef. = 0.672 Period: 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: © ECMWF 2012 MODIS – Moderate Resolution Imaging Spectroradiometer

41
Experimental 4D-Var assimilation of cloud optical depth from MODIS (2) Comparison with independent cloud observations MLS retrievals: Ice Water Content at 215 hPa MLS = Microwave Limb Sounder IWC [mg/m3] ECMWF model – CNTRL run ECMWF model – EXP run CNTRL run – MLS obs EXP run – MLS obs Courtesy of F. Li, Jet Propulsory Laboratory, CA, USA

42
**data assimilation systems as they are needed: **

GENERAL CONCLUSIONS 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) © ECMWF 2012

43
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. Q. J. R. Meteorol. 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. Q. J. R. Meteorol. Soc., 132, Bauer, P., Geer A.J., Lopez P. and Salmond D., 2010: Direct 4D-Var assimilation of all-sky radiances. Part I: Implementation. Q. J. R. Meteorol. Soc., 136, 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. Weather Rev., 125, Fillion L. and Mahfouf J.-F., 2000: Coupling of moist-convective and stratiform precipitation processes for variational data assimilation. Mon. Weather. Rev., 128, Geer A.J., Bauer P. and Lopez P., 2010: Direct 4D-Var assimilation of all-sky radiances. Part II: Assessment. Q. J. R. Meteorol. Soc., 136, 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. Q J. R. Meteorol. 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. Q. J. R. Meteorol. 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. Q. J. R. Meteorol. Soc., 130, Janisková M., Thépaut J.-N. and Geleyn J.-F., 1999: Simplified and regular physical parametrizations for incremental four-dimensional variational assimilation. Mon. Weather Rev., 127, 26-45 Janisková M., Mahfouf J.-F., Morcrette J.-J. and Chevallier F., 2002: Linearized radiation and cloud schemes in the ECMWF model: Development and evaluation. Q. J. R. Meteorol. Soc.,128, Janisková, M., Mahfouf, J.-F. and Morcrette, J.-J., 2002: Preliminary studies on the variational assimilation of cloud-radiation observations. Q. J. R. Meteorol. Soc., 128, © ECMWF 2012

44
Janisková M., 2004: Impact of EarthCARE products on Numerical Weather Prediction. ESA Contract Report, 59 pp. Janisková M., Lopez P. and Bauer P., 2011: Experimental 1D+4D-Var assimilation of CloudSat observations. Q. J. R. Meteorol. Soc., DOI: /qj.988 Janisková M. and Lopez P., 2012: Linearized physics for data assimilation at ECMWF. ECMWF Technical Memorandum 666, 26 pp. Laroche S., Tanguay M. and Delage Y., 2002: Linearization of a simplified planetary boundary layer parametrization. Mon. Weather Rev., 130, Lopez P. and Moreau E., 2005: A convection scheme for data assimilation purposes: Description and initial test. Q. J. R. Meteorol. Soc., 131, Lopez. P., Benedetti A., Bauer P., Janisková M. and Köhler M.., 2006: Experimental 2D-Var assimilation of ARM cloud and precipitation observations. Q. J. R. Meteorol. 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. Weather Rev., 135, Lopez P., 2011: Direct 4D-Var assimilation of NCEP Stage IV radar and gauge precipitation data at ECMWF. Mon. Weather Rev., 139, 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. Weather Rev., 128, Marécal V. and Mahfouf J.-F., 2002: Four-dimensional variational assimilation of total column water vapour in rainy areas. Mon. Weather Rev., 130, 43-58 Marécal V. and Mahfouf J.-F., 2003: Experiments on 4D-Var assimilation of rainfall data using an incremental formulation. Q J. R. Meteorol. 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. Q. J. R. Meteorol. Soc., 130, Tompkins A.M. and Janisková M. , 2004: A cloud scheme for data assimilation: Description and initial tests. Q. J. R. Meteorol. Soc., 130, Zhu J. and Kamachi M., 2000: The role of the time step size in numerical stability of tangent linear models. Mon. Weather. Rev., 128,

Similar presentations

OK

Polly Smith, Alison Fowler, Amos Lawless School of Mathematical and Physical Sciences, University of Reading Exploring coupled data assimilation using.

Polly Smith, Alison Fowler, Amos Lawless School of Mathematical and Physical Sciences, University of Reading Exploring coupled data assimilation using.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google