Sophie RICCI

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The diabatic errors in the formulation of the
data assimilation Kalman Filter/Smoother system
at JPL Extending the control space
Sophie RICCI CALTECH/JPL Post-doc Advisor
Ichiro Fukumori
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The diabatic errors in the formulation of data
assimilation system
Reduced space Temperature along the vertical
Control space errors on heat flux Qnet and
vertical mixing coefficient Kz
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Formulation of the Kalman filter under these
assumptions

• Steps to the doubling algorithm
• (Anderson et Moore, 1979)
• Transition matrix A (simplified linear model)
• Projection matrix for the errors onto the
  reduced space G
• Error covariance matrix for the heat flux and
  vertical mixing coefficient Q

Transition matrix A at 214W, 29N
Identification of the state error covariance
matrix P (constant in time)
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What we learn from a single point twin experiment

• 1D experiment at 214W, 29N
• Assimilate daily SST observations for 1 year

(obs-Tassim)² lt (obs-Truth) ²
The filter has skills at the surface
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What we learn from a single point twin experiment
Kz profils at 214W, 29N
Dynamic up-date for T
The assumption made on the error profil for Kz in
Q is crucial. It can lead to an irrealist
correction below the mixed layer depth
Need to apply a filter with statistics coherent
with the vertical mixing struture at the instant
of the assimilation. Need to derive several
filters translating the variability of the
vertical mixing coefficient profile.
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Building a set of filters based on the
variability of kz profile
First try was to build 12 filters based on
monthly profiles for kz at a given location
(214W, 29N) The fit between the instantaneaous
and monthly profile is computed with a least
square procedure. The skill of the assimilation
at sub-surface improves when a criteria on the
depth of the mixed layer is added and when
additional profiles are taken into account.
Skill at sub-surface

• The bigger the set is, the better the skill gets
  at depth
• Matching the depth of the mixed layer is the
  major key
• to avoid irrealistic correction
• The monthly set at 214W, 29N is not
  representative of all latitudes

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Building a set of filters based on the
variability of kz profile
Our new set is a function of mixed-layer depth
and the maximum Kz value within the mixed-layer.
This set of 411 profiles is computed binning and
averaging profiles of Kz over a 8 years period
for a simulation run.

• 10 profils per Mixed Layer Depth (MLD)
• MLD varies from surface to level 41
• 1 level for extremely low mixing (lt5.10-4)

Set of profiles for MLD7 and MLD20
Depth
Depth
Kz
Kz
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Building a set of filters based on the
variability of kz profile
Fitting for a summer time at 214W, 56.5S
Perspectives Compute the associated filters to
this large set Perform a global assimilation
experiment with real SST observations
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Associated set of 411 filters P for the
assimilation of SST
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Associated set of 411 filters P for the
assimilation of SST
P matrice for MLD41
Diagonal terms of the 411 filters

• The P matrices are computing with the doubling
  algorithm described earlier
• The observation error is assumed constant in
  time and space (0.16 C²)
• The heat flux error is assumed constant in space
  and time (5.10 7 ²)
• The control space is reduced to the heat flux
  error. The treatment of the vertical mixing
  coefficient error seems to require additional
  work.

C²
Depth
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Assimilation of SST

• The SST data to assimilated are extracted from
  Reynolds SST
• The assimilation occurs every 30 days
• The data are located on a coarse grid at the
  surface, covering the entire globe (963 points)
• The SST and SSS relaxation are turned on
• The assimilation aims at controlling the
  variability of the SST but not an eventual bias.
• The model mean is computed over 8 years of a
  control run (1993-2000)
• The SST mean is computed over the same period
  using the Reynolds product.

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Assimilation of SST
Examples of time series of temperature at 2
different locations where data are assimilated
over 1993
dynup dynamic up-date measup measurement up-date
The assimilation is dragging the control towards
the observations.
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Skill of the assimilation of SST
Skill at the surface for 1993
If the assimilation works correctly, the skill
must be negative
The measurement up-date and the dynamic up-date
lead to a negative skill for more the 90 of the
location where data are assimilated for the
1993-1995 period.
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Independent in-situ data set
Skill of the assimilation versus independent
in-situ data (mostly XBT)
If the assimilation works correctly, the skill
must be negative
The equivalent of the control and assimilation
are computed at the locations in space and time
of the in-situ data. These in-situ data are NOT
assimilated and are used as an independent set of
data to validate the impact of the assimilation
of SST at the surface AND at the sub-surface.
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Independent in-situ data set validation at the
surface
T anomalies for the North Atlantic in 1995
The assimilation is closer to the observation
then the control. The skill is almost every where
negative (improvement).
Skill versus XBT for 1993
XBT Assimilation Control
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Independent in-situ data set validation at the
sub-surface
T anomalies for the North Atlantic 1995
Skill versus XBT for 1993 at level 7 (75 m)
XBT Assimilation Control
degradation
The assimilation is further to the observation
then the control. The skill shows some
degradation areas.
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Why is there a degradation at the sub-surface
T profiles of anomalies for the North Atlantic
1995 at level 7
T anomalie
level 7
In this region, even though the skill is good at
the surface, there is a degradation at
sub-surface.
XBT Assimilation Control
Depth
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Why is there a degradation at the sub-surface
Schematic behavior of the model for this region
If the control space is reduced to an error on
heat forcing, the T increment from the
assimilation will be one sign through out the
water column. The assimilation wont be able to
improve the control in such situation.
The anomalies of the control at level 7 are
always positive (same for 1993 and 1994), then
change sign for 1996-1999. There seems to be a
low frequency signal in this region that the
assimilation is not enable to correct.
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Conclusion and perspectives

• The assimilation scheme works fine and gives
  satisfying results at the sub-surface.
• At sub-surface, in some regions, the model
  presents a low frequency variability that
• the assimilation is enable to correct under the
  currents assumptions.
• Solutions
• Increase the observation error in these regions
  so that the assimilation is inactive
• Use a reference for the model that includes this
  low-frequency signal, such as a 1993-1995 mean.
• Once the Filter gives satisfying results at every
  depth, the ultimate step of this study is to run
  the Smoother to get an new estimate of the heat
  forcing. This new estimate should lead to a
  better representation of the ocean.