Adjoint Models as Analytical
Tools
Dr. Ronald M. Errico
Goddard Earth Sciences and Technology Center (UMBC)
Global Modeling and Assimilation Office (NASA)
Outline
1. Sensitivity analysis: The basis for adjoint model applications
2. Examples of adjoint-derived sensitivity
3. Code construction: an example
4. Nonlinear validation
5. Efficient solution of optimization problems
6. Adjoint models as paradigm changers
7. Singular vectors
8. Observation sensitivity
9. Other applications
10. Other considerations
11. Summary
Sensitivity Analysis:
The basis for adjoint model applications
Impacts vs. Sensitivity
xi
xj
Adjoint Sensitivity Analysis
Impacts vs. Sensitivities
A single impact study yields exact response measures
(J) for all forecast aspects with respect to the particular
perturbation investigated.
A single adjoint-derived sensitivity yields linearized
estimates of the particular measure (J) investigated
with respect to all possible perturbations.
Examples of Adjoint-Derived Sensitivities
Example Sensitivity Field
Errico and
Vukicevic
1992 MWR
Contour interval 0.02 Pa/m M=0.1 Pa/m
Lewis et al. 2001
J=average surface pressure in
a small box centered at P
From Errico and Vukicevic 1992
J=barotropic component of
vorticity at point P
Development of Adjoint Model From
Line by Line Analysis of Computer Code
Development of Adjoint Model From
Line by Line Analysis of Computer Code
Development of Adjoint Model From
Line by Line Analysis of Computer Code
Parent NLM :
TLM :
Y = X * (W**A)
Z=Y*X
Ytlm = Xtlm * (W**A) + Wtlm *A* X *(W**(A-1))
Ztlm = Ytlm * X + Xtlm * Y
Xadj = Xadj + Zadj * Y
Yadj = Yadj + Zadj * X
Adjoint :
Xadj = Xadj + Yadj * (W**A)
Wadj = Wadj + Yadj * X *(W**(A-1))
Development of Adjoint Model From
Line by Line Analysis of Computer Code
Parent NLM :
TLM :
Y = X * (W**A)
Z=Y*X
Ytlm = Xtlm * (W**A) + Wtlm *A* X *(W**(A-1))
Ztlm = Ytlm * X + Xtlm * Y
Xadj = Xadj + Zadj * Y
Yadj = Yadj + Zadj * X
Adjoint :
Xadj = Xadj + Yadj * (W**A)
Wadj = Wadj + Yadj * X *(W**(A-1))
Development of Adjoint Model From
Line by Line Analysis of Computer Code
Automatic Differentiation
TAMC
Ralf Giering (superceded by TAF)
TAF
FastOpt.com
ADIFOR
Rice University
TAPENADE INRIA, Nice
OPENAD
Argonne
Others
www.autodiff.org
See http://imgi.uibk.ac.at/MEhrendorfer/
work_7/present/session3/giering.pdf
Development of Adjoint Model From
Line by Line Analysis of Computer Code
1.
2.
3.
4.
5.
6.
TLM and Adjoint models are straight-forward to derive from
NLM code, and actually simpler to develop.
Intelligent approximations can be made to improve efficiency.
TLM and (especially) Adjoint codes are simple to test
rigorously.
Some outstanding errors and problems in the NLM are typically
revealed when the TLM and Adjoint are developed from it.
It is best to start from clean NLM code.
The TLM and Adjoint can be formally correct but useless!
Nonlinear Validation
Does the TLM or Adjoint model tell us anything about
the behavior of meaningful perturbations in the nonlinear
model that may be of interest?
Linear vs. Nonlinear Results in Moist Model
24-hour SV1 from case W1
Initialized with T’=1K
Final ps field shown
Contour interval 0.5 hPa
Errico and Raeder
1999 QJRMS
Linear vs. Nonlinear Results in Moist Model
Non-Conv
Precip. ci=0.5mm
Convective
Precip. ci=2mm
Non-Conv
Precip. ci=0.5mm
Convective
Precip. ci=2mm
Comparison of TLM and Nonlinearly Produced Precip Rates
12-Hour Forecasts with SV#1
NonConvective
Convective.
Errico et al.
QJRMS 2004
Linear
Nonlinear
Linear
Nonlinear
Contours: 0.1, 0.3, 1., 3., 10. mm/day
Linear vs. Nonlinear Results
In general, agreement between TLM and NLM results
will depend on:
1.
2.
3.
4.
5.
6.
Amplitude of perturbations
Stability properties of the reference state
Structure of perturbations
Physics involved
Time period over which perturbation evolves
Measure of agreement
The agreement of the TLM and NLM is exactly
that of the Adjoint and NLM if the Adjoint is exact
with respect to the TLM.
Efficient solution of optimization problems
Gradient
at point P
Contours of J
in phase (x) space
P
M
Adjoint models as paradigm changers
100
hPa
Sensitivity field for J=ps with respect to T for an idealized cyclone
500
hPa
1000
hPa
From Langland and Errico 1996 MWR
Bao and Errico: MWR 1997
Adjoint of Nudging Fields
Contour 0.1 unit
Contour 1.0 unit
t=0
0.4 units
t=48 hours
5.0 units
Errico et al.
Tellus 1993
What is the modeled precipitation rate sensitive to?
Errico, Raeder, Fillion: 2003 Tellus
Non-Convective Forecast Precipitation Rate
Contour interval 2 cm/day
Vort optimized
Errico et al.
2003 Tellus
Rc optimized
Impacts for adjointderived optimal
perturbations for
forecasts starting
indicated hours in
the past.
Rn Optimized
Perturbations in Different Fields Can Produce the Same Result
12-hour v TLM forecasts
Initial u, v, T, ps Perturbation
Errico et al.
QJRMS 2004
Initial q Perturbation
Singular Vectors
Gelaro et al.
MWR 2000
Bred Modes (LVs)
And SVs
Results for
Leading 10 SVs
Gelaro et al.
QJRMS 2002
Balance of Singular Vectors
Vertical modes of the 10-level MAMS model
(from Errico, 2000 QJRMS)
H1  10216m
H2  2060m
H 7  14m
Balance of Singular Vectors
E=E_t, K=KE, A=APE, R=R mode E, G=G mode E
A,G
t=0
E,K,R
t=24 hours
Errico 2000
The Balance of Singular Vectors
v(  0.55)
T(  0.55)
Initial
R mode
Contour
2 units
Initial
G mode
Contour
1 unit
Errico 2000
How Many SVs are Growing Ones?
EM
ED
TM
TD
E-norm Moist Model
E-norm Dry Model
R-norm Moist Model
R-norm Dry Model
Singular
Value
Squared
Errico et al.
Tellus 2001
Mode Index
From Novakovskaia et al. 2007 and Errico et al. 2007
Sensitivity to Observations
The use of an adjoint of a data analysis algorithm
Sensitivity to analyzed potential temperature at 500 hPa
J = mean squared
24 hour forecast
error using E-norm
Sensitivity to raob temperatures at 500 hPa
From Gelaro
and Zhu 2006
Baker 2000 PhD. Thesis
Impacts of various observing systems Totals
GEOS-5 July 2005
15
NH observations
10
Observation
Count (millions)
5
tw
ind
s
sp
ss
mi
air
cra
ft
su
rfa
ce
qk
sw
nd
bs
sa
rao
u
ms
s
hir
air
s
b
su
am
-10
go
es
eo
s_
am
su
a
(J/Kg)
am
e
-5
su
a
0
e  0
-15
-20
15
SH observations
10
Observation
Count (millions)
…all observing
systems provide
total monthly
benefit
5
From R. Gelaro
-15
-20
s
sp
ss
mi
air
cra
ft
su
rfa
ce
qk
sw
nd
sa
tw
i
nd
bs
rao
u
ms
go
es
eo
s_
am
su
a
s
hir
air
s
su
b
-10
am
(J/Kg)
am
e
-5
su
a
0
e  0
Diagnosing impact of hyper-spectral observing systems
GEOS-5 July 2005 00z Totals
AMSU-A (15 ch)
AIRS (153 ch)
Forecast
degradation
-7.0
Channel
Channel
H2O Channels
e
(J/Kg)
0
-0.6
 eError
Forecast
(J/Kg)
Reduction (J/Kg)0
…some AIRS water vapor channels currently degrade the 24h forecast
in GEOS-5…
From R. Gelaro
Other Applications
1.
2.
3.
4.
4DVAR (P. Courtier)
Ensemble Forecasting (R. Buizza, T. Palmer)
Key analysis errors (F. Rabier, L. Isaksen)
Targeting (R. Langland, R. Gelaro)
Problems with Physics
Problems with Physics
Consider Parameterization of Stratiform Precipitation
NLM
R
Modified
NLM
TLM
0
TLM
qs
q
Example of a failed adjoint model development
Sensitivity of forecast J with respect to earlier T in lowest model model level
Time= -3 hours;
contour int.=0.00025
Time= -9 hours;
contour int.=10000.
From R. Errico, unpublished MAMS2 development
Tangent linear vs. nonlinear model solutions
TLM
NLM
60x small pert NLM
Errico and
Raeder 1999
QJRMS
Jacobians of Precipitation
RAS scheme
ECMWF scheme
BM scheme
Fillion and Mahfouf 1999 MWR
Problems with Physics
1. The model may be non-differentiable.
2. Unrealistic discontinuities should be smoothed after
reconsideration of the physics being parameterized.
3. Perhaps worse than discontinuities are numerical instabilities that can be created from physics linearization.
4. It is possible to test the suitability of physics components
for adjoint development before constructing the adjoint.
5. Development of an adjoint provides a fresh and
complementary look at parameterization schemes.
Other Considerations
Physically-based norms and the interpretations of
sensitivity fields
Sensitivity of J with respect to u 5 days earlier at 45ON,
where J is the zonal mean of zonal wind within a narrow
band centered on 10 hPa and 60ON. (From E. Novakovskaia)
0.1 hPa
1 hPa
10 hPa
100 hPa
1000 hPa
- 180
0 Longitude
+ 180
Continuous vs. grid-point representations of sensitivity
Rescaling options for a vertical grid
Delta log p
1 hPa
10 hPa
100 hPa
500 hPa
Delta p
850 hPa
2 Re-scalings of the adjoint results
Mass weighting
Volume weighting
0.1
hPa
10
hPa
1000
hPa
From E. Novakovskaia
Summary
Unforeseen Results Leading to New Paradigms
1. Atmospheric flows are very sensitive to low-level T perturbations.
2. Evolution of a barotropic flow can be very sensitive to perturbations
having small vertical scale.
3. Error structures can propagate and amplify rapidly.
4. Forecast barotropic vorticity can be sensitive to initial water vapor.
5. When significant R is present, moist enthalpy appears a key field.
6. Nudging has strange properties.
7. Relatively few perturbation structures are initially growing ones.
8. Sensitivities to observations differ from sensitivities to analyses.
9. A TLM including physics can be useful.
Misunderstanding #1
False: Adjoint models are difficult to understand.
True: Understanding of adjoints of numerical models
primarily requires concepts taught in early
college mathematics.
Misunderstanding #2
False: Adjoint models are difficult to develop.
True: Adjoint models of dynamical cores are simpler
to develop than their parent models, and almost
trivial to check, but adjoints of model physics
can pose difficult problems.
Misunderstanding #3
False: Automatic adjoint generators easily generate
perfect and useful adjoint models.
True: Problems can be encountered with automatically
generated adjoint codes that are inherent in the
parent model. Do these problems also have a
bad effect in the parent model?
Misunderstanding #4
False: An adjoint model is demonstrated useful and
correct if it reproduces nonlinear results for
ranges of very small perturbations.
True: To be truly useful, adjoint results must yield
good approximations to sensitivities with
respect to meaningfully large perturbations.
This must be part of the validation process.
Misunderstanding #5
False: Adjoints are not needed because the EnKF is
better than 4DVAR and adjoint results disagree
with our notions of atmospheric behavior.
True: Adjoint models are more useful than just for
4DVAR. Their results are sometimes profound,
but usually confirmable, thereby requiring new
theories of atmospheric behavior. It is rare that we
have a tool that can answer such important questions
so directly!
What is happening and where are we headed?
1. There are several adjoint models now, with varying
portions of physics and validation.
2. Utilization and development of adjoint models has been
slow to expand, for a variety of reasons.
3. Adjoint models are powerful tools that are under-utilized.
4. Adjoint models are like gold veins waiting to be mined.
Recommendations
1. Develop adjoint models.
2. Include more physics in adjoint models.
3. Develop parameterization schemes suitable
for linearized applications.
4. Always validate adjoint results (linearity).
4. Consider applications wherever sensitivities
would be useful.
Adjoint Workshop
The 8th will be in fall
2008 or spring 2009.
Contact Dr. R. Errico
rerrico@gmao.gsfc.nasa.gov
to be put on mailing list