Geoscience Reference
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6.5 Angular and radial spectra of the field
Naidu (1969) and Mishra & Naidu (1974) gave the spectrum of a two-dimensional magnetic
survey as
K
1
1
2
(39a)
Suv
,
Xuv
KLL
,
K
k
k
1
xy
Where S (u, v) is the power spectrum of a 2-D aeromagnetic field, X (u, v) is the Fourier
transform of the field, L
x
and L
y
are length dimensions, and u and v are frequency in the x
and y directions respectively given as u = 2π/L
x
and v = 2π/L
y
. A two-dimensional
spectrum can be expressed in a condensed form as one-dimensional spectra; radial and
angular (Spector & Grant, 1970; Mishra & Naidu, 1974; Naidu, 1980; Naidu & Mishra, 1980;
Naidu & Mathew, 1998). The radial spectrum is defined as
2
1
R
f
(s) =
f
Ss
(cos ,sin )
s
d
(39b)
2
0
Where again
=√
+
is the magnitude of the frequency vector and = tan
-1
(v/u) is the
direction of the frequency vector in the spatial frequency plane v and u (in radian/km) in
the x- and y-directions respectively. The angular spectrum is defined as
s
s
1
0
A() =
Ss
(cos ,sin )
s
ds
(40)
f
s
s
0
Where, s is the radial frequency band starting from s
0
to s
0
+ s, over which the averaging
is carried out. It is useful as a rule to look at power spectra in one-dimensional or profile
form rather than in two-dimensional or map form. This is because S is a somewhat bumpy
function of (Fedi et al., 1997) when the width of the model is moderately large, and the
bumpiness imparts a certain irregularity to the contours (Spector & Grant, 1970). The power
spectra in one-dimension also enables ensemble of magnetic block parameters (average
magnetic moment/unit depth, average depths to top, average thickness and average
widths) to be factored out completely for effective analysis.
Usually the angular spectrum is normalized with respect to the radial spectrum so as to free
this spectrum from any radial variation. In this case (Naidu & Mathew, 1998)
s
s
Ss
(cos ,sin )
s
1
0
f
A
norm
()=
(41)
ds
s
R
()
s
f
s
0
The computations of spectra in equations (40) and (41) may require the use of template. The
radial spectrum is computed by averaging the 2D spectrum over a series of annular regions
while the angular spectrum is computed by averaging over angular sectors in the template.
Naidu (1980) and Naidu & Mathew (1998) have shown that the angular spectrum of the total
field of a uniform magnetized layer having an uncorrelated random magnetization is given by
h
eI s
2242
2
2
2
2
S
fT
(u,v) =
(
)cos (
)
(42)
0
0
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