[Mitgcm-support] Note: Self-Consistency of the K-filter Assimilation

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Assessing Self-Consistency of the K-filter Assimilation: 
(or figuring out what's wrong with the K-filter) 
-------------------------------------------------------------

This note (and its thread) summarizes pertinent analyses assessing
self-consistency of the K-filter.  Configuration details of the
different K-filter runs can be found in
        guppy:/escher6/medea/btang/kf_bc_runs/kf_run.txt[1]

1) Trend in T/P data
---------------------

Synopsis: There is a trend in model and T/P.  Removing it prior to
          assimilation reduces model degradation by the assimilation.  

A first attempt with the baroclinic filter (kf023a) degraded the
simulation off Taiwan (not Philippine as previously reported).  Skill
is measured by the difference of model-data residual variance;
        Sum_(simulation - T/P)^2  - Sum_(dynup - T/P)^2

Here "dynup" is filtered estimate immediately prior to assimilation in
the K-filter recursion.  The skill of kf023a is shown in this figure[2].
Positive (yellow-red) indicates improvement.  Run kf023a assumed
    - NCEP wind covariance on coarse BAROCLINIC grid as process noise
      Q, tapered away from equator with 10-degrees Gaussian e-folding
      on amplitude.  
    - Q decorrelates every 3-days
The figure is the skill vs space; Summation in the skill above is a
time-average over 1993-1997 on the T/P grid binned into 3-deg squares.  

Sea level in a 12-deg by 6-deg square (129E-141E, 22N-28N) surrounding
the degradation (black square in previous plot) is plotted here[3] for
inspection.  The lower plot is a 10-day running average of the T/P
samples in the upper panel; T/P (black), simulation (blue), kf023a
dynup (red).  The degradation of kf023a is seen to be caused by a
larger trend in the filtered estimate. 

The trend in T/P and simulation sea level is shown here[4]; values are
equivalent sea level (m) change over 5-years.  There is a sizable
trend, especially for the model in the western and eastern tropical
Pacific.  Although these trends are not directly under the degradation
area off Taiwan, it may affect the area because of its inconsistencies
with assumptions in the K-filter. 

Indeed the degradation is largely eliminated by removing these trends
in doing the assimilation.  This figure[5] compares the skill of kf023a
(top) and the same with assimilation of detrended model and data sea
level difference (kf023b3; bottom).

There is not much trend dependence in the barotropic filter.  This
figure[6] compares the skill of a barotropic filter with (kf021a; top)
and without the trend (kf021c; bottom).  Both runs assumed
  - NCEP wind covariance on coarse BAROTROPIC grid as process noise Q 
  - Q decorrelates every day
The detrended assimilation does pick up less baroclinic signal in the
tropics associated with ENSO toward the end of '97.  The detrended
solution also seems to have slightly less degradation south of New
Zealand. 

Addendum
--------

Synopsis: The trend can probably be handled by directly estimating it
          rather than "avoiding" it in the estimation.   

The skills above were based on 
        Sum_(simulation - T/P)^2  - Sum_(dynup - T/P)^2
with no detrending.  The same can be computed with respect to
detrended model and T/P.  These figures compare such measure for the
baroclinic filter[7] (kf023a, kf023b3) and the barotropic filter[8] (kf021a,
kf021c).   In each, the top panels are the same as before (i.e., skill
measure based on raw sea level), whereas the lower panels are measures
based on detrended sea level (trend removed is that of T/P and model
simulation).

There is very little difference in measure for the detrended
assimilations (kf023b3, kf021c) whether the skill measure uses
detrended sea level or not.  A larger difference is found for the non
detrended assimilation (kf023a, kf021a).  Moreover, the measure using
detrended sea level (lower panels) are similar between the two
assimilated estimates especially for the baroclinic filter.

If the trend were independent of the assimilated correction, the skill
should not depend on which measure is used; that is indeed what we see
with the detrended assimilations (kf023b3, kf021c).  By removing the
trend prior to assimilation, the assimilated corrections (i.e.,
improvements) become (nearly) independent of the actual trend.

When a measure using detrended sea level is used, the assimilations
with and without prior trend removal become similar to each other
(lower panel; kf023a vs kf023b3).  That would indicate that the
non-detrended assimilation (kf023a) might be OK after all and only
needs some additional trend correction to be equivalent with a
detrended assimilation (kf023b3).  Indeed there are studies that
suggest bias (and trend) can be corrected by a separate estimation
added on top of the usual assimilation (Dee and da Silva, 1998, QJRMS,
124, 269-295.)  

If we don't want to estimate the trend separately, removing it prior
to assimilation seems most prudent and simple.

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[1] http://escher.jpl.nasa.gov:2000/hosts/escher/escher6/medea/btang/kf_bc_runs/kf_run.txt
[2] http://escher.JPL.NASA.GOV:2000/hosts/escher/escher2/medea/if/forum/t73f5h_d.ps
[3] http://escher.JPL.NASA.GOV:2000/hosts/escher/escher2/medea/if/forum/t73c6d_b.ps
[4] http://escher.JPL.NASA.GOV:2000/hosts/escher/escher2/medea/if/forum/t73f5d_3.ps
[5] http://escher.JPL.NASA.GOV:2000/hosts/escher/escher2/medea/if/forum/t73f5h_e.ps
[6] http://escher.JPL.NASA.GOV:2000/hosts/escher/escher2/medea/if/forum/t73f5h_f.ps
[7] http://escher.JPL.NASA.GOV:2000/hosts/escher/escher2/medea/if/forum/t73f5h_e2.ps
[8] http://escher.JPL.NASA.GOV:2000/hosts/escher/escher2/medea/if/forum/t73f5h_f2.ps