Spectral Analysis of Geophysical Data - Advances in Data, Methods, Models and Their Applications in Geoscience - page 47

Where h is depth to the magnetic layer, I 0 is the direction of the current earth's magnetic
field, ,  and  are directional cosines while  0 is the declination of the earth's magnetic
field. The non-exponent component of equation (44) is exactly the same as the square of
Γ(u, v) expressed in equation (34). The angular spectrum, A norm (θ) may now be expressed
as
A norm (θ) = Ω[γ 2 + (α 2 + β 2 )cos 2 (θ - θ 0 )] 2
Where Ω is a constant. Thus the angular spectrum is a maximum in the direction of
polarization vector. Naidu (1970) had earlier shown that the shape of the angular spectrum
is a product of a number of factors such as rock type, strike and polarization vector, but the
latter two factors influence the shape of the spectrum significantly. Thus the presence of
peaks in the angular spectrum gives an indication of linear features in the map.
On the other hand, the radial spectrum gives a measure of the rate of decay with respect to
radial frequency of the spectral power, which may represent a deep-seated phenomenon
(Bhattacharyya, 1966; Naidu, 1970, Spector & Grant, 1970).
6.6 Estimation of radial and angular spectra of aeromagnetic data
We shall apply the concept of radial and angular spectra to synthetic and real data. For the
synthetic data, we have taken the field over a magnetic dipole buried at a depth of 4 km.
Other parameters of the dipole include: inclination and declination of the inducing field
(and remanent field of the dipole) on the dipole are respectively 6 o and 8 o (i.e. at low
magnetic latitudes), where the field strength of the inducing field is 33510 nT and the
magnetization intensity is 0.01 A/m, while its susceptibility is assumed as 9.5 x 10 -5 cgs
(0.0012 SI). For space, the anomaly map is not shown. In computing the angular spectrum of
this dipole anomaly, the 50x50 data grid was padded with sufficient zeros and cosine
tapered to make data matrix amenable for FFT computations. This resulted in a data matrix
size of 64x64. The angular spectrum is then calculated for three frequency sub-bands (1-10,
10-20 and 20-30 frequency numbers) for angular interval of 180 o . The highest frequency in
the data is 32.
Figure 3 shows the curves representing the low (1-10), middle (10-20) and high (20-30)
frequency numbers. Observation of these curves shows that 8 o and 90 o spectral peaks appear
in the three frequency subbands followed by a peak at 156 o in the mid to high frequency
sub-bands though with reduced magnitude. Thus we see the emergence in the three
frequency sub-bands, of 8 o : the declination of both the inducing and remanents fields. The
other angular feature (90 o ) at subdued level is due to the tapered window used in the
analysis. The other peak (156 o ) corresponds to the direction of the polarization in the
direction of the inducing field at this magnetic latitude.
The real data used for the computation of radial and angular spectra were obtained from
aeromagnetic total field intensity of the Middle Benue Trough, Nigeria(MBT) collected from
1974 to 1976. The MBT is the central part of the main Benue Trough of Nigeria. The Benue
trough is linked genetically to the oil/gas bearing rocks of the inland Nigerian Niger Delta
area. With an upbeat in petroleum efforts in the inland basins, attention is now focused on
the Benue trough with more emphasis laid on the structural setting of the basin.
The composite total field intensity data were corrected for the main field using the IGRF
1975 model and this resulted in the residual field map (Fig. 4) used for the present analysis.