Ocean Ensemble Forecasting and Adaptive Sampling - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 391

16.3 Atmospheric Forcing Ensemble Generation

Based on the theory that model forecast errors are often well described in terms
of shifting and timing errors ( Hoffman et al. 1995 ), the uncertainty of atmo-
spheric forcing can be represented by adding perturbations to surface fields from
a single deterministic atmospheric forecast through spatial and temporal deforma-
tion. The amplitude of the perturbations is chosen to be small enough to ensure that
the perturbed field lies within the error bounds of the forecast. To control the
amplitude and horizontal correlation length scale of the random perturbations, the
covariance matrix of the shift-vector

ı t of shifts at a certain time is given by:

˝ ı

t T ˛ D DE

E T D t ı ƒ (16.23)

where D is a diagonal matrix of the variances we wish to assign to the random
process at each grid point and E

E T defines a correlation matrix whose diagonal
values are all equal to 1. For simplicity, we chose the columns of E to be
the two-dimensional sinusoids and cosinusoids that define a basis for the two-
dimensional domain upon which the ocean state is defined. Let a be a random

ƒ

normal vector with zero mean and covariance ˝ aa T ˛ D ƒ

. Now consider random
vectors y obtained using y D Ea . Note that since the columns of E are the sinusoidal
basis used in inverse Fourier transform, the operation Ea is simply an inverse Fourier
transform. To ensure that the random perturbations satisfy ( 16.1 ), we generate each
perturbation using

and ˝ aa T ˛ D ƒ

ı t D DEa ; where h a i D 0 (16.24)

In other words, a random perturbation is created by

1. Creating a vector b of normally independently identically distributed numbers

each of which has a mean of zero and a variance of 1.
2. Letting a D ƒ 1=2 b .
3. Performing the inverse Fourier transform implied by Ea .
4. Performing the operation

n

ı t D DEa .

To see that this process creates random perturbations that satisfy ( 16.1 ) note that
˝ ı

t T ˛ D ˝ DEaa T E T D ˛
D DE ˝ aa T ˛ E T D because E and D are constant

t ı

because ˝ aa T ˛ D ƒ E T D

ƒ ;

D DE (16.25)

The scales and magnitudes of the random perturbations are thus determined by the
user's specification of D and

ƒ

. Here, we chose D D ˛

˛

I so that the constant gives

ii of ƒ

the variance at each point and let the diagonal elements be given by the

Gaussian function of the total wavenumber to which they pertain that is given by