Convolutions

The "spikey" optimal solution is smoothed using disrete analogues of the convolutions described in the page link required and page link required.

In the continuous case, the forward equation (8c) is :

(1)

and the initial condition on the forward variable (8d) is:

(2)

Of course, these convolutions might be computed directly, by (1) computing and saving the covariance vectors and then (2) computing the integrals indicated above by numerical integration. Although tractable in this small one-dimensional example, the task becomes computationally overwhelming for data assimilation problems of reasonable size and dimension (see Bennett 2002 , Exercise 3.1.1).

Alternatively, Bennett (2002) and Egbert et all. (1994) describe much more efficient algorithms for computations of the convolutions. There are two "shortcuts," one for space convolutions and one for temporal convolutions, and both solve simple systems of PDE's whose solutions have the form of the convolution desired if:

(3)

and:

 

Space Convolutions

Consider first the space convolutions of the general form:

(4)

where:

(5)

The quantity in (4) can be computed by solving a "Pseudo-heat" equation:

(6)

with initial condition:

(7)

Then, it can be shown that:

 

This shortcut for the spatial convolution can be applied to the initial condition, in (2), and to the forcing, in (1) at every time step.

Time Convolutions

Consider next the time convolutions of the form:

(8)

 

These are computed by solving the "forward" differential equation:

(9)

with:

at (10)

followed by:

(11)

with:

at .

Then, it can be shown that:

 

if:

(12)This shortcut for the time convolution can be applied to the forcing, in (1) at every space step.