The Application of
Observation Adjoint
Sensitivity to Satellite
Assimilation Problems
Nancy L. Baker
Naval Research Laboratory
Monterey, CA
Outline
• Introduction and motivation
• Data assimilation adjoint theory – what is
observation sensitivity?
• Examples of observation sensitivity with
NAVDAS adjoint
– 00 UTC 7 February 1999 for TOVS
– 00 UTC 10 February 2002 for ATOVS/AMSU-A
• Summary and conclusions
Introduction to Observation
Adjoint Sensitivity
• A significant component of the medium-range
forecast error is due to errors in the initial conditions
– particularly true for relatively poorly sampled regions
• Classical adjoint sensitivity computes the sensitivity
of a cost function J (e.g., 72-h forecast error) to the
initial conditions for that forecast.
– highlights regions that are very sensitive to small errors in
the initial conditions (e.g., temperatures and winds).
• The complete NWP adjoint sensitivity to J must
include the adjoint of the data assimilation system.
– sensitivity of J to the observations and background
Motivation
• Research was originally motivated by preparations
for FASTEX in late 1996
– targeting methods were not able to take into account how
the data assimilation system would use the additional
observations
– the presence of other observations in the area
– as such, could not provides guidance on where to place the
adaptive observations in the sensitive region
• Observation sensitivity has applicability far beyond
targeting applications
– help us to understand how observations are used by the
assimilation system
– help us identify potential sources of forecast errors due to
errors in the initial conditions
What is observation sensitivity?
forward problem
Observations
(y)
Data
Assimilation
System
Analysis
(xa)
Forecast
Model
Forecast
(xf)
Background
(xb)
adjoint problem
Observation
Sensitivity
(J/ y)
Background
Sensitivity
(J/ xb)
Adjoint of
the Data
Assimilation
System
Sensitivity
to the
Analysis
(J/ xa)
Adjoint of the
Forecast Model
Tangent
Propagator
Gradient of
Cost Function J
(J/ xf)
NAVDAS* ADJOINT
*NRL Atmospheric Variational Data Assimilation System
• NAVDAS adjoint computes the sensitivity of the
forecast aspect J (such as forecast error) to the
observations and background.
• The sensitivity of J to the observations is given by
J
J
T J
T
1
K
 (HPbH  R ) HPb
,
y
x a
x a
J/xa - sensitivity of J to the initial conditions.
Compare to linear analysis equation
xa  xb  K (y  Hxb )
Sensitivity of the 72-h energy-weighted NOGAPS forecast error to the initial
wind and height fields at 00 UTC 07Feb 1999. Verification area over west coast
of U.S.
t
Magnitude of the sensitivity of the 72-h NOGAPS energyweighted forecast error to the MSU-2 brightness temperatures.
The “t” indicates time-window discontinuities.
Sensitivity to MSU-2
Brightness Temperatures
• Maximum MSU-2 sensitivity occurs when
– the observations are relatively isolated, or
– the observation density abrupt changes,
– coincident with large amplitude and spatial scale
sensitivity to the initial fields.
• Large observation sensitivities near 45oN, 175oE
– occur in the middle of a large-scale sensitivity to the
initial 700-hPa temperature field,
– and are associated with a time-window data
discontinuity
Sensitivity to TOVS Brightness
Temperatures
• Other abrupt changes in TOVS brightness
temperature density may be associated with larger
observation errors.
– TOVS brightness temperatures over land and ice are more
difficult to assimilate properly and are eliminated in the
present NAVDAS configuration
– This creates an abrupt change in the observation density
along the coastlines and ice-edge boundaries where the
observations are less accurate (mixed field-of-view).
– Abrupt changes in the data density also occur for the less
accurate observations along the edges of the satellite scan.
Implications
• Sensitivity to the relatively inaccurate observations
along the data discontinuity is larger than the
sensitivity to the more accurate (data dense)
observations.
– assuming identical observation errors are assigned
• This implies that the less accurate observations have
greater potential to change the forecast aspect, and
to influence the analysis.
• Increasing the assumed observation error variance
will decrease both the observation sensitivity and the
influence of the observation on the analysis.
Estimate of the potential forecast
impact due to the observations
• Define the change in J as the projection of the
analysis error (a) onto the analysis sensitivity
gradient
• J  T J
a
x a
• The reduction in the expected variance of the change
in J due to the observations is
 J 
T
2
o
 J 
J
T
   (HPbH  R )
y
 y 
Sensitivity of NOGAPS 72-h forecast
error to the initial T,u,v,p fields
10 UTC 09 February 2002
 J 
2
o
for different observation
types
Discussion
• Are very large values of observation sensitivity
desirable?
– implies that the analysis depends upon a few observations in
highly sensitive regions
– even small errors in the observation may contribute to the
forecast error
– the corresponding background sensitivity is large
– as observations are added, the analysis becomes less
dependent upon individual observations and the
background, and the observation and background
sensitivities decrease
– intermediate values of observation sensitivity are desirable
Conclusions and Future Work
• The observation sensitivity gives an estimate of the
potential for an observation to make changes to the
analysis with the amplitude and structure suggested
by the analysis sensitivity.
• Actual impact cannot be determined until the
observations have been taken and the forecast
computed.
• Develop observation sensitivity techniques to
diagnose sources of forecast error
• Investigate alternate impact functions
T
 J 
S    ( y  Hx b )
 y 
(Doerenbecher and Bergot, 2001)
Supplemental Slides
Summary
• The observation sensitivity is largest when
– the analysis sensitivity is large in amplitude
– the spatial scales of the analysis sensitivity and background error
covariances are similar (i.e., large targets)
– the observation is assumed to be more accurate than the
background
– the observations are relatively isolated or associated with an abrupt
change in the observation density (e.g, along coastlines or edges of
satellite swaths)
• The observation sensitivity is weak when
– the analysis sensitivity is weak amplitude or small-scale
– the length scales of the analysis sensitivity is shorter than the
background error covariance length scales
– the observation is assumed to be less accurate than the
background
– the observation density is high
Understanding Observation Sensitivity
J y  K T J x a
xa  xb  K (y  Hxb )
• For relatively isolated observations, K and KT
are large in amplitude and spatial scale.
– When KT projects strongly onto the analysis
sensitivity, both J y and the potential change to J
will be large
– the observation has more independent information
– the observation will be given more weight in the
analysis
– potential changes to the analysis due to the
observation are large in amplitude and spatial scale
Understanding Observation Sensitivity
J y  K T J x a
xa  xb  K (y  Hxb )
• For high density observations, KT is small in
amplitude and spatial scale.
– The projection of KT onto the analysis sensitivity
will be weaker, and both J y and the potential
change to J are small.
• Similarly, if the observations are more
accurate than the background, the
observations will be given more weight and
the potential changes to J are larger.
Observation Sensitivity for a Hypothetical Flight Path
Understanding Observation Sensitivity