Background Error Correlation Modeling with Diffusion Operators - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 184

is inversely proportional to the ensemble size . In other words, one may expect to

achieve 10 % accuracy at the expense of approximately 100 multiplications by B if
the first ensemble member gives a 100 % error. This estimate may seem acceptable
since in geophysical applications the BE variances are usually known with limited
precision and approximating the diagonal with 5-10 % error seems satisfactory.
The above described Monte-Carlo (MC) technique was developed further by
Bekas et al. ( 2007 ), who noticed that the method may converge to d in the finite
number of iterations that equals to the matrix dimension

K

if the ensemble vectors
are mutually orthogonal. An easy way to construct such an ensemble is to draw the
vectors s k from the columns of the

N

N N Hadamard matrix (HM), which span the

model's state space (see Appendix 3).
In the numerical experiments below we use MC and HM techniques as testbeds
for the diagonal estimation methods which can be derived from analytical consider-
ations and take into account prior knowledge of the structure of B .

8.3.2 Locally Homogeneous Approximations

a 2 D 1=2

Consider homogeneous ( and assume

that the coordinate axes are aligned along the eigenvectors of the diffusion tensor,
whose (positive) eigenvalues are

D const ) operators ( 8.2 ) with

i ;i D 1;::;n . Then the matrix elements of B g;m

can be written down explicitly as

exp 2
2

B g . x ; y / D exp . D =2/ D d (8.42)

s

N K s . N /

=2m/ m D d

B m . x ; y / D . I D (8.43)

2

s 1 .

s /

where

q

D . x y /

T

1 . x y /

is the distance between the correlated points (measured in terms of the smoothing
scales

d D .2/ n=2 ˝ 1 are the (constant) diagonal elem ents of B g;m ,

i ), ˝ D

˘ i D p det

N D p

2m

is the diffusion volume element, and .

varies in space, ( 8.42 )and( 8.43 ) are no longer valid, and the diagonal
elements d depend on x and the type of the B operator. However, if we assume that

When

L

is locally homogeneous (LH), i.e. varies in space on a typical scale which is

i , the diagonal elements d . /

much larger than x can be expanded in t he powers of

is the mean eigenvalue of p

D N =L

,where N

the small parameter . The zeroth-

order LH approximation term (LH0) is apparently

d 0 . / D .2/ n=2 ˝. / 1 x x (8.44)