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+∫zqv
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

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