Geoscience Reference
In-Depth Information
ii
.k;l/
D
C
exp
.k
2
C
l
2
/
L
2
(16.26)
where
are non-dimensional wave numbers (associated with the indexing of
grid points in the FFT routine),
k
and
l
(a non-dimensional length scale) controls the
horizontal correlation length scale in spectral space. Decreasing L increases the
spatial scale of the random fields by (
16.24
). The scale
L
C
is an amplitude factor
E
T
are equal to unity and
that is used to ensure that the diagonal elements of
E
ƒ
E
T
is a valid correlation matrix. The values of
hence that
E
used in
our experiments are 0.5, 10, and 0.5 h, respectively. With these parameters, (
16.24
)
produces a spatially correlated field of time shifts with a standard deviation of
˛
D
0:5
ƒ
C;L
and
˛
h.
1
To create a time shift vector
that varies in time as well as space, we used
(
16.24
) to create two entirely independent time-shift vector shifts
ı
t
.t/
ı
t
.t
i
/
and
ı
t
.t
i
C
1
/
corresponding to the discrete times
t
i
C
1
. These two times might be 24 or 72 h
apart depending on the perceived decorrelation time of atmospheric forcing errors.
(In our study independent fields were generated every 24 h). To ensure that the time
shift vector varied smoothly between these two times, we set
t
i
and
cos
2
t
t
i
t
i
C
1
t
i
sin
2
t
t
i
t
i
C
1
t
i
ı
t
.t/
D
ı
t
.t
i
/
C
ı
t
.t
i
C
1
/
(16.27)
Equation
16.27
implies that the evolution of the covariance of time shifts is given by
ƒ
i
E
T
D
i
cos
2
2
t
t
i
t
i
C
1
t
i
D
.t/
T
E
ı
t
.t/ı
t
D
D
i
E
ƒ
i
C
1
E
T
D
i
C
1
sin
2
2
t
t
i
t
i
C
1
t
i
C
D
i
C
1
E
(16.28)
This formulation allows both the scale and magnitude of the deformations to be
a function of time. Note also that in the special case that
D
i
C
1
E
ƒ
i
C
1
E
T
D
i
C
1
D
ƒ
i
E
T
D
i
, the trigonometric rule cos
2
C sin
2
D
1
D
i
E
ensures that the covariance
of the time shifts given by (
16.27
)and(
16.28
) is constant even though each
individual time shift is smoothly evolving through time.
For the experiments reported in this Chapter, the eigenvector matrix
E
was
comprised by the set of sinusoidal basis functions spanning a two dimensional plane.
By making the domain on which the time shifts
t
were generated larger than that
of the regional ocean model, it was possible to produce aperiodic time-shifts.
The temporally shifted fields include surface wind, air temperature, relative
humidity, precipitation, sea-level pressure, and short- and long-wave radiation. Each
ı
1
This technique has been used to perturb an initial best-guess unperturbed state of sea surface
temperature (SST) to provide an ensemble of ocean-surface lower boundary conditions for
atmospheric ensemble forecast (
Hong et al. 2011
).
Search WWH ::
Custom Search