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13 . Dual formulation of 3D/4D-Var (PSAS)
The 3D-Var formulation (A5) can be rewritten into a form
called PSAS (Physical Space Assimilation System1) which is equivalent in the linear
case only. The idea is to notice that the expression
can be split as the following two equalities
where  has the same dimension as  and can be regarded
as a kind of "increment" in observation space2, whereas  is a smoothing
term that maps the increment from observation to model space. The aim is
to solve the analysis problem in terms of  rather than in model space. One way is to solve for  the linear system
which can be regarded as the dual of the OI algorithm. Another way is to
find a cost function that  minimizes,
for instance
which is a quadratic cost function. The practical PSAS analysis algorithm
is as follows:
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1) Calculate the background departures
 |
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2) Minimize  . Some possible
preconditionings are given by the symmetric square root of  or  . |
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3) Multiply the minimum  by  to obtain analysis increments. |
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4) Add the increments to the background
 . |
A 4-D generalization of PSAS is obtained by a suitable
redefinition of the space  to be a concatenation  of all the values  at all observation time steps  . Then  must be replaced
by an operator that uses the tangent linear model  to map the initial model state to the observation space at each
time step  , i.e.  . The factorization
of the cost function evaluation using the adjoint method is applied to the
computation of the term  , so that the
evaluation of the 4D-PSAS cost function  is as follows:
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1) Calculate the departures  for each time step, (this needs only be done once) |
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2) Integrate the adjoint model
from final to initial time, starting with a null model state, adding
the forcing  at each observation timestep, |
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3) Multiply the resulting adjoint
variable at initial time by  , which yields
 , |
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4) Integrate the tangent-linear
model, starting with  as model state,
storing the state times  at each
observation time step. The collection of the stored values is  . |
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5) Add  and  (both obtained by sums of already computed quantities) to obtain
 . |
More comments on the 4D-PSAS algorithm are provided in
Courtier
(1997). The PSAS algorithm is equivalent to the representer method
(Bennett and Thornburn 1992).
As of today it is still unclear whether PSAS is superior
or not to the conventional variational formulations, 3D and 4D-Var. Here
are some pros and cons:
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• PSAS is only equivalent
to 3D/4D-Var if  is linear, which means that it cannot be extended
to weakly non-linear observation operators. |
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• However, most implementations
of 3D/4D-Var are incremental, which means that they do rely on a linearization
of  anyway: they include non-linearity through
incremental updates, which can be used identically in an incremental
version of PSAS. |
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• It is awkward to include
a  term in PSAS for constraints expressed in model space. |
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• Background error models
can be implemented directly in PSAS as the  operator. In
3D/4D-Var they need to be inverted (unless they are factorized and
used as preconditioner). |
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• The size of the PSAS
cost function is determined by the number of observations  instead of the
dimension of the model space  . If  then the PSAS minimization is done in a smaller space
than 3D/4D-Var. In a 4D-Var context,  increases
with the length of the minimization period whereas  is fixed, so that this apparent advantage of PSAS may disappear. |
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• The conditioning of a
PSAS cost function preconditioned by the square root of  is identical
to that of 3D/4D-Var preconditioned by the square root of  . However the comparison may be altered if more sophisticated preconditionings
are used, or if one square root or the other is easier to specify. |
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• Both 3D/4D-Var and PSAS
can be generalized to include model errors. In 3D/4D-Var this means
increasing the size of the control variable, which is not the case
in PSAS, although the final cost of both algorithms looks the same. |
Ref: Bennett
and Thornburn 1992, Courtier
1997.
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1 The misleading name PSAS was introduced for
historical reasons and is widely used, probably because it sounds like the
US slang word pizzazz.
2 Note, though, that it does not have the right
physical dimensions. The actual increment in observation space is  , and a precise physical interpretation of  is difficult.
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