Ensemble Adaptive Data Assimilation Techniques Applied to Land-Falling North American Cyclones - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 548

sensitivity-based targeting for improving forecasts, and similar methods are still
in use today with regard to high-impact weather events such as tropical cyclones
( Reynolds et al. 2009 ).
Early efforts were also made to account for analysis uncertainty in addition to
dynamical error growth through SV or adjoint sensitivity techniques. Taking into
account analysis uncertainty is important because if observations are taken and
assimilated in regions based on leading SVs, for example, they would have little
impact if the background uncertainty was very small because the data assimilation
system would essentially ignore the targeted observations. In turn, other locations
with less-amplifying SVs may produce larger forecast impacts if large uncertainty
and larger analysis increments were produced there, even if the dynamical error
growth rates of those perturbations were smaller. Barkmeijer et al. ( 1998 ) addressed
this issue with Hessian SVs, which are calculated with a norm based on analysis
uncertainty provided by a 3DVAR system at initial time. Bishop and Toth ( 1999 )
developed the ensemble transform method, which accounts for uncertainty within
the framework of an ensemble.
A major step forward in observation targeting techniques came with the real-
ization that the characteristics of the data assimilation system used to assimilate
the targeted observations should be considered. Data assimilation systems not
only provide background uncertainty estimates at initial time, but also include
observation error estimates, and contain the exact procedure that would be used
to assimilate targeted observations. In turn, by considering both the assimilation
characteristics and a way to estimate error growth (such as through SVs or adjoint
sensitivity), more appropriate observation targeting techniques can be formulated
that estimate more accurately how hypothetical observations would impact forecasts
in a specific assimilation system. Both Berliner et al. ( 1999 )and Langland ( 2005 )
elaborate on the necessity to include error evolution dynamics, analysis uncertainty,
observation errors, and the specific assimilation system in formulating observation
targeting schemes. This holistic approach to targeted observing laid the groundwork
for modern adaptive data assimilation techniques using variational and ensemble
methods.
Modern adaptive data assimilation was marked by the extension of initial
condition sensitivity into observation sensitivity, and was first described in Baker
and Daley ( 2000 ) in the context of a 3DVAR system. Observation sensitivity
describes not how perturbations to initial conditions would change the forecast
(as adjoint sensitivity does), but how an assimilated observation would change the
forecast, and can be written as:

@ R =@ y o D @ R =@ x o @ x o =@ y o (23.3)

where y o is the observation sensitivity, which is a function of the adjoint

sensitivity and the change to the analysis given observations (

@ R =@

. For data
assimilation systems that assume Gaussian statistics to achieve a most-likely state,
the term

@ x o =@ y o /

y o simply becomes the Kalman gain matrix. Equation ( 23.3 ) is a form
of observation targeting as described in Baker and Daley ( 2000 ) as it allows one to

@ x o =@