A description of the tangent linear
normal mode constraint (TLNMC) in GSI
David Parrish
I first want to give thanks to the Central Weather Bureau
and the National Central University for inviting us. It
has been especially wonderful to have our jet lag
cured with the extra time you allowed for us. I speak
for Daryl and myself in thanking Wan Shu and her
sister Jean for their hospitality and graciousness as
they gave us a most excellent tour of Southern
Taiwan and provided the medicine for a speedy cure
from the jet lag.
Second, I want to thank Daryl for providing a copy of his
PhD qualifying exam presentation, which I have
made liberal use of. I invite him (and others) to
correct me when I make incorrect statements.
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Improving global variational data assimilation
at NCEP
Daryl T. Kleist
AOSC PhD Qualifying Exam Seminar
18 January 2011
Adviser: Dr. Kayo Ide
Acknowledgements: Dr. John Derber and Dr. David Parrish
In this talk, my goal is for everyone here to have some
understanding of the following, which is the abstract
representation of TLNMC, short for
Tangent Linear Normal Mode Constraint
and why it is useful in the GSI system and also to
mention its limitations.
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Tangent Linear Normal Mode Constraint
   
  



1 ' T T 1 1 '
1 '
' T
J x  x c C B C x c  y o  Hx c R 1 y 'o  Hx 'c  J c
2
2
'
c
x'c  Cx'
•
analysis state vector after incremental NMI
–
C = Correction from incremental normal mode
initialization (NMI)
•
represents correction to analysis increment that filters out the
unwanted projection onto fast modes
•
No change necessary for B in this formulation
•
Based on:
–
Temperton, C., 1989: “Implicit Normal Mode
Initialization for Spectral Models”, MWR, vol 117, 436451.
* Similar idea developed and pursued independently by Fillion et al. (2007)
5
Before getting to the previous expression, I will try to get
us to the basic cost function of variational data
assimilation:
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Variational Data Assimilation
   
1 '
J x  x
2
'
T
  
1 '
B x  y o  Hx '
2
1
'
 R y
T
1
'
o

 Hx '  J c
J : Penalty (Fit to background + Fit to observations + Constraints)
x’ : Analysis increment (xa – xb) ; where xb is a background/first guess
B : Background error covariance
H : Observations (forward) operator
R : Observation error covariance (Instrument + representativeness)
yo’ : Observation innovations
Jc : Constraints (physical quantities, balance/noise, etc.)
7
For a long time now, cost functions like this are always
the first thing and the main thing that is shown in
data assimilation papers and lectures. But many
people, especially those outside the data
assimilation community, I think are immediately
unable to follow what we are talking about , partly
because the notation is abstract and difficult to
understand.
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To give some meaning to this, I like to use zero
dimensional examples. The example I will use here
comes from our last weekend adventures. It was
quite warm in the south of Taiwan. Speaking for
myself being from the U. S., my mental reference for
experiencing the air temperature is in the Fahrenheit
scale, where feeling hot for me right now is when
the temperature is above 86 F. But for most of the
world, including here, feeling hot is when the
temperature is above (5/9)*(86 – 32) = 30 C.
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Yesterday, we were warm and sweaty as we left the
beautiful Kenting National Park. When we got off of
the train in Taipei, it felt like winter had arrived. But
my internal observation of temperature of 59 F
didn’t match with the sign on a tower that said in
bold red 20 C. So I have to do a mental calculation,
which isn’t always accurate.
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So let us call my perceived temperature estimate when
arriving in Taipei
x_b = 59 F (the bold type here and following is for
vectors and matrices, which here are all of
dimension = 1, and in space/time of dimension 0)
It is not that cold, but my perception is not very accurate,
with an estimated standard deviation error of say
sqrt(B) = 5 F
and no bias (I say I am unbiased for simplicity, but that is
certainly not correct—and this is true for NWP
models also!)
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The observed temperature in Taipei is
y_o = 20 C
and the measurement error (also unbiased for simplicity)
is
sqrt(R) = 1 C
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The simulated observation based on my perception x_b
is
y_b = H(x_b) = (5/9) ( x_b – 32 )
H(x) is the called the full forward operator to derive a
simulated observation from the model (in this case,
my perception of temperature) and in general is a
nonlinear function of x.
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The analysis increment is
x’ = x_a – x_b where x_a is the correction to my
perception x_b based on the Taipei observation y_o
The observation innovation is
y_o’ = y_o – H(x_b)
The “tangent linear” of H(x_b) is
H = (5/9)
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So now we have identified all of the components of the
basic cost function (except J_c which will be
discussed briefly later):
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Variational Data Assimilation
   
1 '
J x  x
2
'
T
  
1 '
B x  y o  Hx '
2
1
'
 R y
T
1
'
o

 Hx '  J c
J : Penalty (Fit to background + Fit to observations + (Constraints) )
x’ : Analysis increment (xa – xb) ; where xb is a background/first guess (my estimate of
T in deg F, and xa is the improvement on my perception by the Taipei data yo )
B : Background error covariance (error variance (square of std deviation error of my
perception = 52 = 25 F2 )
H : Observations (forward) operator (tangent linear = 5/9 in this case)
R : Observation error covariance (Instrument + representativeness) (here just my ges of
1 C2 for the tower display)
yo’ : Observation innovations (yo - H( xb ) )
16
Jc : Constraints (physical quantities, balance/noise, etc.)
Because all vectors and matrices are now just 1 element,
we can trivially solve for the analysis increment. We
first take the derivative of J with respect to x’ and set
to zero:
0 = B-1 x’ – HT R-1 ( yo’ – Hx’ )
or
xa = xb + (B-1 + HT R-1 H)-1 HT R-1 ( yo – H(xb) )
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Because in this example all quantities are really just
scalars, we can rearrange however we want and get
rid of the transpose. Of course for real cases we
work with, only certain orders are valid. One that I
believe is the basis for the EnKF is
xa = xb + B HT (H B HT + R)-1 ( yo – H(xb) )
CWB
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Tangent Linear Normal Mode Constraint
   
  



1 ' T T 1 1 '
1 '
' T
J x  x c C B C x c  y o  Hx c R 1 y 'o  Hx 'c  J c
2
2
'
c
x'c  Cx'
•
analysis state vector after incremental NMI
–
C = Correction from incremental normal mode
initialization (NMI)
•
represents correction to analysis increment that filters out the
unwanted projection onto fast modes
•
No change necessary for B in this formulation
•
Based on:
–
Temperton, C., 1989: “Implicit Normal Mode
Initialization for Spectral Models”, MWR, vol 117, 436451.
* Similar idea developed and pursued independently by Fillion et al. (2007)
19
Variational Data Assimilation
   
1 '
J x  x
2
'
T
  
1 '
B x  y o  Hx '
2
1
'
 R y
T
1
'
o

 Hx '  J c
J : Penalty (Fit to background + Fit to observations + Constraints)
x’ : Analysis increment (xa – xb) ; where xb is a background/first guess
B : Background error covariance
H : Observations (forward) operator
R : Observation error covariance (Instrument + representativeness)
yo’ : Observation innovations
Jc : Constraints (physical quantities, balance/noise, etc.)
20
“Strong” Constraint Procedure
C=[I-DFT]x’
T
nxn
F
mxn
dx
dt
d x'g
Dry, adiabatic
tendency model
Projection onto m
gravity modes
dt
m-2d shallow water
problems
D
nxm
 Δx
Correction matrix to
reduce gravity mode
Tendencies
Spherical harmonics
used for period cutoff
• Practical Considerations:
• C is operating on x’ only, and is the tangent linear of NNMI operator
• Only need one iteration in practice for good results
• Adjoint of each procedure needed as part of minimization/variational
procedure
21
Tangent Linear Normal Mode Constraint
• Performs correction to increment to reduce gravity mode
tendencies
• Applied during minimization to increment, not as postprocessing of analysis fields
• Little impact on speed of minimization algorithm
T
• CBC becomes effective background error covariances for
balanced increment
– Not necessary to change variable definition/B (unless
desired)
– Adds implicit flow dependence
• Requires time tendencies of increment
– Implemented dry, adiabatic, generalized coordinate
tendency model (TL and AD)
22
Surface Pressure Tendency Revisited
Minimal increase
with TLNMC
Zonal-average surface pressure tendency for background (green), unconstrained
GSI analysis (red), and GSI analysis with TLNMC (purple)
23
Vertical Modes
• Gravity Wave Phase Speed:
– 1 (313.89 ms-1)
– 2 (232.91 ms-1)
– 3 (165.45 ms-1)
– 4 (120.07 ms-1)
– 5 (91.19ms-1)
• Global mean temperature and
pressure for each level used
as reference
• First 8 vertical modes are
used in deriving incremental
correction in global
implementation
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Single observation test (T observation)
• Magnitude of TLNMC
correction is small
• TLNMC adds flow
dependence even when
using same isotropic B
Isotropic
response
Flow dependence
added
500 hPa temperature increment (right) and analysis difference (left, along with background geopotential height) valid25at 12Z
09 October 2007 for a single 500 hPa temperature observation (1K O-F and observation error)
Single observation test (T observation)
U wind
Ageostrophic U wind
From
multivariate B
TLNMC
corrects
Smaller
ageostrophic
component
Cross section of zonal wind increment (and analysis difference) valid at 12Z 0926
October 2007 for a single 500 hPa temperature observation (1K O-F and
Analysis Difference and Background
500 hPa zonal wind analysis difference (TLNMC-No Constraint ; left) after
assimilating all observations and zonal wind background (right) valid at 12Z 09
October 2007
27
Little Impact on Minimization
Norm of gradient (left) and total penalty (right) for each iteration for analysis at
12Z 09 October 2007 [Jump at iteration 100 from outer loop update]
No Constraint (orange) versus TLNMC (green)
28
Surface Pressure Tendency Revisited
Minimal increase
with TLNMC
Zonal-average surface pressure tendency for background (green), unconstrained
GSI analysis (red), and GSI analysis with TLNMC (purple)
29
Vertical Velocity Difference
Zonal Mean Difference of the RMS (TLNMC-No Constraint; Pa s-1) of the
derived vertical velocity increment for the analysis valid at 12Z 09 October
2007. Negative values are shaded blue and positive values shaded red.
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“Balance”/Noise Diagnostic
• Compute RMS sum of incremental tendencies in spectral
space (for vertical modes kept in TLNMC) for final analysis
increment
– Unfiltered: Suf (all) and Suf_g (projected onto gravity modes)
– Filtered: Sf (all) and Sf_g (projected onto gravity modes)
– Normalized Ratio:
• Rf = Sf_g / (Sf - Sf_g)
• Ruf = Suf_g / (Suf - Suf_g)
Suf
Suf_g
Ruf
Sf
Sf_g
Rf
NoJC
1.45x10-7
1.34x10-7
12.03
1.41x10-7
1.31x10-7
12.96
TLNMC
2.04x10-8
6.02x10-9
0.419
1.70x10-8
3.85x10-9
0.291
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Fits of Surface Pressure Data in Parallel Tests
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Impact of TLNMC on 500 hPa AC Scores
No Constraint (control, black) versus TLNMC (red)
500 hPa Geo. Height AC Scores for period 01 Dec. 2006 to 14 Jan. 2007
33
Precipitation Verification
Precipitation Equitable Threat and Bias Scores for period from 01 Dec. 2006 to 12
34
Jan. 2007 (No Constraint-Black ; TLNMC-Red)
Tropical Wind Vector RMS Error
No Constraint (control, black) versus TLNMC (red)
200 hPa & 850 hPa Tropical Vector Wind RMS Error for period 01 Dec. 2006 to 14
35
Jan. 2007
TLNMC Summary
• A scale-selective dynamic constraint has been developed
based upon the ideas of NNMI
– Successful implementation of TLNMC into global version of
GSI at NCEP and GMAO
– Incremental: does not force analysis (much) away from the
observations compared to an unconstrained analysis
– Improved analyses and subsequent forecast skill, particularly in
extratropical mass fields
• Work is on-going to apply TLNMC to regional
applications & domains (Dave Parrish – NCEP)
– Initial attempts based upon Briere, S., 1982: “Nonlinear Normal Mode
Initialization of a Limited Area Model”, MWR, vol 110, 1166-1186.
• Adequate for small domains: success with assimilation of radar radial velocities
• Apparent issues with larger domains: variation of map factor/Coriolis
36
Thanks!
Questions?
37