Variational assimilation: method

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Variational assimilation method technicalities

• Claude Fischer Patrick Moll,
• CNRM/GMAP, Météo-France

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Overview of some principles of VAR assimilation
as applied in our NWP systems
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Principle of VAR (in as short as possible )

• 4D variational assimilation
• Search for the model trajectory approaching best
  the available observations gt iterative
  minimization process
• 3D variational assimilation a  reduced 
  version of 4D-VAR gt no time integration, all obs
  are considered valid for time t1

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Picture view of 4D-VAR assimilation
obs
Jo
previous forecast
analysis
Jo
obs
xb
background
corrected forecast
Jb
Jo
xa
obs
t09h
t112h
t215h
Assimilation window
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General formulation of 4D-VAR in NWP systems

• Full (Nonlinear) cost function

• Linearized cost function (kouter loop)

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Incremental formulation (I)
Cost function
is the  simplified  (low resolution) increment
at outer loop
and
with
Trajectory (analysis) updates
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Incremental formulation (II)
To pre-condition the problem, an additional
change of variable is performed
CHAVAR
New cost function
CHAVARIN
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Incremental 4D-Var algorithm
(ECMWF documentation)
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3D-VAR LAM - (I)
Cost function gt no model integration gt
There is no time window (1 timeslot) and only one
outer loop
and
Analysis updates gt no  simplification 
operator ! gt the increment is computed on the
same grid than the background
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3D-VAR LAM - (II)
To pre-condition the problem, an additional
change of variables is Performed (4D-VAR
3D-VAR)
CHAVAR
New cost function
CHAVARIN
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Computation of Jo in 4D-VAR
where
are the  high resolution  departures.
These high resolution departures are computed
during high the resolution integrations and
stored in the ODB files. In practice, one
computes
Low Resol. depart. - Obs.
High Resol. depart.
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Computation of Jo in LAM 3D-VAR
With k1 and i0 only and
are the  high resolution  departures.
The departures are computed only once, during
high the screening step and stored in the ODB
files. In the code, one computes like for
incremental 4D-VAR
 Low Resol.  depart.- Obs.
 High Resol.  depart.
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Chain of operations in a VAR minimisation loop
(SIM4D)

• compute trajectory (take Xb in 3D-VAR)
• read (stored in ODB)
• () compute Jb and its gradient
• compute TL of Jo
• And store in the ODB
• compute AD of Jo
• compute J
• call minimizer go to () until convergence

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Major messages to keep in mind

• 3D-VAR is a reduced ( simpler ) version of
  4D-VAR
• Aladin 3D-VAR is a code installed inside the
  global IFS/Arpège code (as we will see in the
  last slides)

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Back to application with pictures !
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SURFACE OBSERVATIONS FOR ALADIN-France
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Assimilation of Reflectivity Observation
operator implemented in the 3DVar ALADIN/AROME
model level
N
r
j
q
z
h

• Bi-linear interpolation of the simulated
  hydrometeors (T,q, qr, qs, qg)
• Compute  radar reflectivity  on each model
  level

Diameter of particules
Resolution volum, ray path standard refraction
(4/3 Earthradius)
Backscattering cross section Rayleigh
(attenuation neglected)
Microphysic Scheme in AROME
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The VAR assimilation in Ald/Aroscripting
sequence

• Observations are taken /- 3 h around base time
  in Aladin /- 1.5 h in Arome
• Analysis time background time and no  high  /
   low  resolution
• Full analysis sequence (altitude fields)
• Full ODB database enters Screening gt computes
  departures and performs QC and first-guess check
• ODB base is compressed ( keep only active data
  in the output ODB)
• Variational analysis minimization of cost
  function
• Write out analysis file

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Technical aspects about the LAM VAR code inside
the IFS
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The observation operators in the variational cost
function

• We have seen that the Jo part of the cost
  function can be finally written
  Jo(x) ? ? (yik Hik xb)TRik-1(yi Hik
  xb)
• Where Hik is the observation operator for obs n
  k of type n i.
• The operator Hik is subdivided into a sequence of
  operators, each one of which performs part of the
  transformation from control variable to observed
  parameters
• Conversion from control variable to model
  variables
• Inverse spectral transforms put the model
  variables on the gridpoint space
• A horizontal interpolation provides vertical
  profiles of model variables at observation
  locations
• Vertical integration if necessary (hydrostatic
  equation for geopotential, radiative transfer
  equation for radiances)
• Vertical interpolation to the level of the
  observations

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The observation operators in the variational cost
function

• Horizontal interpolations
• A 12-point bi-cubic or 4-point bi-linear
  horizontal interpolation gives vertical profiles
  of model variables at observation locations. The
  surface fields are interpolated bilinearly to
  avoid spurious maxima and minima
• Vertical interpolations it depends on the
  variable
• Linear in pressure for temperature and specific
  humidity
• Linear in logarithm of pressure for wind
• Linear in logarithm of pressure for geopotential
  (performed in terms of departures from the ICAO
  standard atmosphere). New operator until a few
  years, specific to Météo-France, better
  consistent with hydrostatism.
• Vertical interpolations for surface observations
  (T2m, V10m, Hu2m) are done consistently with the
  physics of the model

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The observations in the variational cost function
Adjoints of the observation operators The
solution of the minimisation problem is given by
?J(xa)0 At each step of the descent algorithm,
?J has to be computed ?J(x) 2B-1(x-xb)
2HTR-1(y Hxb) Where HT is the the adjoint
(?transpose) of the the observation operator (or
the tangent linear of the observation operator
when it is non-linear). It means that the
adjoints of the obs operators have to be
computed, which is easy for the interpolations,
but more difficult for the vertical integrations,
in particular for the radiative transfer equation
or other operators containing non differentiable
processes.
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DIRECT AND ADJOINT OBSERVATION OPERATORS
ymod
X (model var.)
Hn
H2
.
H1
..
Jo
Chain of direct operators
y
Hn
Jo
H1
H2
.
ymod
X
Chain of adjoint operators
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Chain of operations in a VAR minimisation loop
(SIM4D)

• compute trajectory (take Xb in 3D-VAR)
• read (stored in ODB)
• () compute Jb and its gradient
• compute TL of Jo
• And store in the ODB
• compute AD of Jo
• compute J
• call minimizer go to () until convergence

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Global // LAM code interfacing for VAR

• Simulator and model

Jb?? ?Jb2?
obshortl
Minimisation
dxB1/2.?
cpgtl
obsvtl
Var inner loop (SIM4D)
JJbJo ?J
H.dx
einv_trans
ecoupl1
H.dx
edir_trans
?JoB1/2. GradJo
GradJoH.R-1. (y-H(x)-H.dx)
espcmtl
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Global // LAM code interfacing for VAR

• Architecture

obshortl
Slightly different code LELAM key
coupling
Completely different code
Fully shared dataflow between IFS and Aladin
(especially in gridpoint space), but quite
separate dimensioning and addressing in spectral
buffers (spherical versus bi-Fourier). Coupling
code is of course only LAM.
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Global // LAM code interfacing for VAR

• Change of variable

cv2spa
chavarin
cvar2in
sqrtbin
cvaru3i
jgcori
ejgvcori
ejghcori
jgnrsi
etransinv_jb
cvargptl
etransdir_jb
ebalstat
ebalvert
spa1ylvazxlamcvtmeanuver
addbgs
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Phasing of LAM 3D-VAR

• ODB bator need to work (almost same as Arpège,
  plus LAMFLAG)
• At every new cycle with IFS, a careful inspection
  of B-matrix setup and change of variable
  routines, SPECTRAL_FIELDS and CONTROL_VECTOR
  structures, specific observation-related code is
  needed ( 1 week of work).
• Additionally, TL and AD LAM code may need to be
  checked.
• LAM 3D-VAR does not use multi-incremental I/O
  prepared for IFS-Arpège no WRMLPPADM/RDFPINC, no
  SAVMINI/GETMINI, NUPTRAgt1,  traj  recomputation
  of innovations

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Thank you for your attention