92
P. Courtier
Remark 4
A variant of P~D is the quadratic problem
1
1 N
P~/D : minimize J( 8x( to)) = 28x( to)t B- 1 8x(to)+2 ~)Yb,i+ H;8x( ti)-Yi)tO;1 (Yb,i+ H;8x( til-v;)
1=0
(4.19)
with 8X(ti} = R(ti, to)8x(to) and Yb,i = H[M(ti' to)Xb).
The cost of P~D and P: D are similar but the storage requirement for the background trajectory
is different: in P~D it is the background vertical column at the observation point and in P: D
it is the observation equivalent of the background which have to be stored. In P~D' R is
an approximate linearisation of M, similarly in P: D , HI is an approximate linearisation of Hi.
Approximating the full problem by a quadratic one has theoretical advantages since the solution
involves solving only linear equations in principle, as is guaranteed unique.
The structure functions used in the current T213 optimal interpolation operational at ECMWF
have a cut-off at wave number 63 (Liinnberg, 1988). If we were to use a T106 truncation for
R, this would already be an enhancement in terms of resolution. An adiabatic version for R
with some basic simplified diabatic processes like horizontal and vertical diffusion and surface
friction would produce the same benefits in terms of implicit flow dependent structure functions
as obtained by Thepaut et al. (1993).
The CPU cost of an adiabatic semi-Lagrangian Tl06 L31 model is typically 1/16 of the CPU
cost of the T213 L31 version. The gain of the incremental approach is then of one order of
magnitude.
Remark 5
Another variant of P~D can be obtained by replacing HI8x(ti) in Eq. 4.19 by a finite difference:
P:~ : minimize J(8x(t o)) = ~8x(to}t B- 1 8x(to)+ ~ f,(Yb,i - Yb,i+Yi - ydO;1 (Yb,i -Yb,i+ih - Yilt
1=0
(4.20)
with Yb,i = Hi(M(:h(to)) being the model equivalent of the observation obtained from a simplified (low resolution, adiabatic) background trajectory and Yi being the model equivalent
of the observation obtained from the simplified trajectory issued from Xb(t O) + 8x(to). Xb(tO}
is the background but at lower resolution. Using the Taylor formula, one has to first order
Yi - Yb,i = HIJx(t;). The practical advantage of this formulation over P:D or P~D is that there
is less technical development required once the full 4D-Var problem has already been implemented. It is with this formulation that the numerical experimentations are performed at ECMWF.
These three implementations P~D' P: D and P:~ are equivalent in the quasi-linear context. They
are expected to behave differently in the presence of strong nonlinearities, however we have not
seen any arguments as to why one should be superior to the others.
4.4.4 Further developments
There are two ways of improving the model R. Firstly, one could increase the horizontal
resolution: the main drawback here is the cost involved since the CPU follows a power law
close to 3. In addition the trajectory storage and thus the 10 also follow a cubic law (quadratic
at a given time step but the number of time steps increases linearly).
Secondly, it is necessary to take into account the physics. The experiments performed so
far (Thepaut et aI., 1993a and Rabier and Courtier, 1992) have used only horizontal and/or
vertical diffusion with a simple surface friction. Rabier et al. (1993) showed that large-scale
P. Courtier
Remark 4
A variant of P~D is the quadratic problem
1
1 N
P~/D : minimize J( 8x( to)) = 28x( to)t B- 1 8x(to)+2 ~)Yb,i+ H;8x( ti)-Yi)tO;1 (Yb,i+ H;8x( til-v;)
1=0
(4.19)
with 8X(ti} = R(ti, to)8x(to) and Yb,i = H[M(ti' to)Xb).
The cost of P~D and P: D are similar but the storage requirement for the background trajectory
is different: in P~D it is the background vertical column at the observation point and in P: D
it is the observation equivalent of the background which have to be stored. In P~D' R is
an approximate linearisation of M, similarly in P: D , HI is an approximate linearisation of Hi.
Approximating the full problem by a quadratic one has theoretical advantages since the solution
involves solving only linear equations in principle, as is guaranteed unique.
The structure functions used in the current T213 optimal interpolation operational at ECMWF
have a cut-off at wave number 63 (Liinnberg, 1988). If we were to use a T106 truncation for
R, this would already be an enhancement in terms of resolution. An adiabatic version for R
with some basic simplified diabatic processes like horizontal and vertical diffusion and surface
friction would produce the same benefits in terms of implicit flow dependent structure functions
as obtained by Thepaut et al. (1993).
The CPU cost of an adiabatic semi-Lagrangian Tl06 L31 model is typically 1/16 of the CPU
cost of the T213 L31 version. The gain of the incremental approach is then of one order of
magnitude.
Remark 5
Another variant of P~D can be obtained by replacing HI8x(ti) in Eq. 4.19 by a finite difference:
P:~ : minimize J(8x(t o)) = ~8x(to}t B- 1 8x(to)+ ~ f,(Yb,i - Yb,i+Yi - ydO;1 (Yb,i -Yb,i+ih - Yilt
1=0
(4.20)
with Yb,i = Hi(M(:h(to)) being the model equivalent of the observation obtained from a simplified (low resolution, adiabatic) background trajectory and Yi being the model equivalent
of the observation obtained from the simplified trajectory issued from Xb(t O) + 8x(to). Xb(tO}
is the background but at lower resolution. Using the Taylor formula, one has to first order
Yi - Yb,i = HIJx(t;). The practical advantage of this formulation over P:D or P~D is that there
is less technical development required once the full 4D-Var problem has already been implemented. It is with this formulation that the numerical experimentations are performed at ECMWF.
These three implementations P~D' P: D and P:~ are equivalent in the quasi-linear context. They
are expected to behave differently in the presence of strong nonlinearities, however we have not
seen any arguments as to why one should be superior to the others.
4.4.4 Further developments
There are two ways of improving the model R. Firstly, one could increase the horizontal
resolution: the main drawback here is the cost involved since the CPU follows a power law
close to 3. In addition the trajectory storage and thus the 10 also follow a cubic law (quadratic
at a given time step but the number of time steps increases linearly).
Secondly, it is necessary to take into account the physics. The experiments performed so
far (Thepaut et aI., 1993a and Rabier and Courtier, 1992) have used only horizontal and/or
vertical diffusion with a simple surface friction. Rabier et al. (1993) showed that large-scale