Background Error Correlation Modeling with Diffusion Operators - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 189

a b

37.2 37.2

36.8 36.8

1 1

36.4 36.4

0.8 0.8

0.6 0.6

36.0 36.0

0.4 0.4

0.2 0.2

35.6 35.6

−123.2 −122.8 −122.4 −122.0 −121.6 −123.2 −122.8 −122.4 −122.0 −121.6

Fig. 8.6 Diagonal approximation errors under the zeroth-order ( a ), and first-order ( b ) LH methods
for the B g model. The thin black line inside the boundaries shows the domain of error averaging

k 1=2 by its nature, whereas the efficiency of
smoothing (targeted at removing the small-scale error constituents) degrades as the
signal-to-noise ratio of the diagonal estimates increases with the iteration number

HM method converges faster than

.
From the practical point of view, it is not reasonable to do more than several
hundred iterations, as h " i drops to the value of a few per cent (Fig. 8.5 a), which
is much smaller than the accuracy in the determination of the background error
variances. It can therefore be concluded that it is advantageous to use the HM
technique, when making more than a 100 iterations is computationally affordable.

k

8.3.3.3 Asymptotic Expansion Method

Since the principal axes of the diffusion tensor at every point are defined by con-
struction, computation of the zeroth-order approximation ( 8.44 ) to the normalization
factors is not expensive. Near the boundaries, however, the factors described by
( 8.44 ) have to be adjusted by taking into account the geometric constraints imposed
on the diffusion. This adjustment was computed for points located closer than 3 1

from the boundary and it was assumed that the boundary had negligible impact on
the shape of the diffused

-function ( Yaremchuk and Carrier 2012 ).
Figure 8.6 demonstrates the horizontal distribution of the error

ı

obtained by
approximating the diagonal elements of B g with ( 8.44 ) (zeroth-order LH method,
or LH0) and with ( 8.51 ), (the first-order LH method LH1). Despite an apparent
violation of the LH assumption in many regions (e.g.,

". x /

1 changes from 20 ı to the

background value of 3 L 5 6ı < 1 across the shelf break), the

mean approximation error of the diagonal elements appears to be relatively small
(19 %) for the LH0 method, with most of the maxima confined to the regions of
strong inhomogeneity (Fig. 8.6 a). The next approximation (Fig. 8.6 b) reduces h " i to
9 %. Numerical experiments with the B 2 model have shown similar results (16 and
10 % errors).

ı at distances