Linearized Physics for Data Assimilation at ECMWF - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 269

sensitivity of all outputs by modifying in turn each input variables. More generally,
an adjoint can be applied to analyze the sensitivity of a forecast aspect to initial
conditions as proposed, for instance, by Errico and Vukicevic ( 1992 )or Rabier
et al. ( 1996 ). The adjoint method can also be used to measure the sensitivity
with respect to any parameter of importance of the data assimilation system. In
recent years, adjoint-based observation sensitivity techniques have been used as
a diagnostic tool to monitor the observation impact on short-range forecasts (e.g.
Langland and Baker 2004 ; Cardinali and Buizza 2004 ; Zhu and Gelaro 2008 ;
Cardinali 2009 ). Such technique is restricted by the tangent-linear assumption and
its validity. The better the tangent-linear approximation, the more realistic and useful
the sensitivity patterns. Results obtained through the adjoint integration when using
a too simplified adjoint model with large inaccuracies or adjoint models without a
proper treatment of nonlinearities and discontinuities, can be incorrect.
The adjoint( F T ) of the linear operator F can provide the gradient of any objective
function, J , with respect to x (input variables) given the gradient of J with respect
to y (output variables) as:

@ J

x D F T : @

J

(11.36)

@ @ y

As an example, Fig. 11.6 displays the adjoint sensitivity of the 24-h forecast error
to the initial conditions, i.e. to the analysis @ J
@

x ,where J is a measure of the forecast
error ( Rabier et al. 1996 ; Cardinali 2009 ). The sensitivity with respect to specific
humidity and temperature at the lowest model level are shown for the situation on
28 August 2010 at 21:00 UTC from the run at T255L91 resolution. The results
are presented for two different experiments, the first one run with only the dry
parametrization schemes (i.e. vertical diffusion, gravity wave drag, non-orographic
gravity wave and radiation) included in the adjoint model (Fig. 11.6 a, b) and the
second one with moist processes also taken into account (Fig. 11.6 c, d). With only
dry parametrization schemes, sensitivity to specific humidity is quite small and
localized in areas of strongest dynamical activity. Even for temperature, it is obvious
that some sensitivities are quite weak, especially in convective regions. Adding
moist processes in the adjoint model brings additional structures to the sensitivity in
areas affected by large-scale condensation/evaporation and convection. Therefore,
using a more sophisticated adjoint model also provides more flow-dependent and
more realistic sensitivities.
Another example of adjoint sensitivity computations using the adjoint version of
the linearized physics package is given here, where the cost function was defined
as the 3-h precipitation averaged over the core of a mid-latitude winter storm over
northwestern Europe. One should emphasize that this kind of computation is only
possible if the adjoint of moist physics parameterizations is available. Figure 11.7
shows the field of 3-h precipitation accumulation used for the evaluation of the
precipitation cost function inside the black box at 0000 UTC 10 February 2009.
As an illustration, Fig. 11.8 displays the adjoint sensitivities of the precipitation cost
function with respect to 500 hPa temperature at 0000 UTC 9 February 2009 (i.e.
24 h beforehand and computed at T159L91 resolution). In other words, Fig. 11.8