Geoscience Reference
In-Depth Information
Now we choose for study the Gaussian-shaped form of the correlation function.
Since the random fields are anisotropically distributed on the ground surface, the
function ‰
.
w
/
lp
.
L
;!/ may depend on two correlation radii, so it can be chosen in the
form
lp
.
L
;!/
D
F
lp
.!/ exp
!
:
L
y
y
.!/
L
x
x
.!/
‰
.
w
/
(6.86)
The radii
x
.!/ and
y
.!/ characterize the correlations of the wind-current
fluctuation in the x and y-directions, respectively. The functions F
lp
.!/ depend
on how the height-integrated current I
.
w
/
l
is correlated with the current I
.
w
p
.
In particular, if these currents are statistically independent of each other, then
F
lp
.!/
D
0. In the subsequent discussion we define both the specific form of the
correlation radius and the factor F
lp
.!/.
Substituting Eq. (
6.86
)for‰
.
w
/
lp
into Eq. (
6.85
), performing integration over L
x
and L
y
, yields
exp
!
:
k
x
x
C
k
y
y
4
F
lp
x
y
4
G
lp
.!;
k
/
D
(6.87)
In situ measurements the horizontal magnetic field variations are greater than the
vertical one. As one example, we consider now the correlation function ‰
.B/
xx
.
L
;!/.
In fact, this correlation function describes the spatial correlation of the spectral com-
ponents ıB
x
.
r
;!/ and ıB
x
.
r
0
;!/ taken at different points
r
and
r
0
at frequency !.
For practical purposes, it is interesting to study the spectral density/power spectrum,
which is based on a single-stationed three-component magnetometer recording.
If the data for the power spectra is gathered in a single point, only autocorrelation
function is available. In this notation, substituting Eqs. (
6.76
), (
6.84
), (
6.87
)into
Eq. (
6.81
), applying an inverse Bessel transform, and setting
L
D
R
R
0
D
0,
yields
Z
Z
0
x
y
16g
3
ǚ
F
xx
f
2
C
F
xy
C
F
yx
f
f
C
C
F
yy
f
2
‰
.B/
xx
.0;!/
D
C
1
1
exp
2kd
!
k
x
k
x
x
C
k
y
y
4
k
2
dk
x
dk
y
;
(6.88)
In the extreme case of small correlation radii, i.e., 2d
x
;
y
, one can find that
0
x
.!/
y
.!/‚.!/
256d
2
‰
.B/
xx
.!/
D
‰
.B/
xx
.0;!/
;
(6.89)
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