Geoscience Reference
In-Depth Information
6.1.1
Tangent Linear and Adjoint of the Forecast Model
The tangent linear and adjoint of an NWP model were introduced by Dimet and
Talag r an d ( 1986 ) for data assimilation experiments. An overview of tangent linear
and adjoint models in meteorology and references for seminal works can be found
in paper by Errico ( 1997 ). A tangent linear model M calculates the first order
response of the output to a perturbation of the input of a nonlinear model
M
is quasi-linear, the perturbation forecast by M will be similar to the difference in
output between perturbed and non-perturbed runs of
M
.If
. The tangent linear model
is constructed by differentiating the nonlinear model with respect to the model's
forecast state vector x .
M
M D @M.
/
x
x :
(6.1)
@
J
Adjoint models provide the gradient of some scalar function
of the forecast state
vector within a numerical weather prediction NWP model with respect to the initial
state vector (analysis). The forecast state vector depends on the initial conditions in
the following way,
J.
/ D J.M.
x a //;
x
(6.2)
where x a is initial state. The gradient of
J
with respect to the initial model state is
@J
@
x a D M T @J
x ;
(6.3)
@
where M T is the adjoint model. The adjoint model maps the gradient of
J
with
respect to the forecast state to the initial time, resulting in the sensitivity of
J
with respect to the analysis. The input to the adjoint model, @ @ x
is calculated by
J
differentiating
with respect to the forecast state. Since the adjoint model is
derived from the tangent linear model, it's ability to calculate meaningful gradients
is dependent on the quasi-linearity of the forecast model. Therefore, adjoint model
fields in the atmosphere are most accurate over short time scales and for dry
processes.
6.1.2
Observation Sensitivity
More recently, the adjoint of a DA system was formulated to compute gradients with
respect to observations ( Baker and Daley 2000 ). In DA, to obtain the analysis x a ,
the linear three dimensional analysis equations is written as
x a D x b C K
.
y Hx b /:
(6.4)
The vector of observations y is length
, while the model space analysis x a
and background vectors x b are both length
N
L
. The linear forward operator H
 
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