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Many scientists used the power spectrum of the gravity anomaly to obtain the average depth of the disturbing surface or equivalently the average depth to the top of the disturbing body. The spectrum of gravity anomaly due to layered source is separated into multiple segments in frequency domain that can be interpreted in terms of mean depth of the interface. The half of the slope of the segments gives the mean depth of the interfaces. This study aims to estimate the average residual depth anomalies of various regions of the northern Logone Birni sedimentary basin of Cameroon using polynomial separation of gravity anomalies, and spectral analysis along different profiles (segments). The profiles were derived from residual anomaly maps obtained by fitting the Bouguer anomalies, the interpretation used polynomial separation and depth average was done using spectral analysis. Positive and negative residual gravity anomalies were highlighted and their interpretation revealed the structural directions of the sedimentary basin (NW-SE and NE-SW), as well as an intimate relationship between the negative anomalies and the northern Logone Birni sedimentary basin. Three distinct residual anomalies were identified over the Goulfey, Tom Merifine and Tourba basin with an average depth varying between 0.24 km and 4.55 km.

The Logone Birni sedimentary Basin is located in the far north region of Cameroon, south of Lake Chad (^{2} and it is part of the West and Central African Rift system (WCARS). The West and Central African Rift system is divided into two Cretaceous subsystems genetically linked, but physically separated [

Significant hydrocarbon deposits were discovered in the sedimentary basins which are part of the WARS and CARS (e.g., the Doba basin in Chad, Muglad and Melut basins in Sudan). Despite the scientific and economic importance of these zones, recent surveys date from the 1960s to the 1990s [

In this study, we calculated the average depths of the basin using spectral analysis of the residual anomaly maps obtained after filtering the Bouguer anomaly map computed by [

The study area (

The region is under a tropical climate, which favours the growth of a shrubby savannah and sparse forest vegetation. The Lake Chad region has the lowest altitude ( 280 m ). The main rivers that drain the region are the Chari and the Logone; the Chari is tributary to the Lake Chad while the Logone is an effluent of the Chari. The geological history of the region is related to the formation of the Chad basin.

Throughout the Lower Cretaceous, the Chad basin was an extension zone between the western and central Africa plates. It was subject to various tectonic activities from the primary era to the Quaternary era marked by relief inversions [

- The first transgression might have started at the end of the Tertiary, resulting to basin infill in the Middle Quaternary from material derived from partial ablation of ancient deposits and border massifs. The process was amplified by subsidence of the basin and, inversely, by massifs uplifting.

- The second transgression dates from about 21.000 years.

- The third transgression linked to the third humid period, dated from 12.000 years ago and marks the constitution of the first trace of the actual drainage pattern.

The northern Logone Birni Basin is a sedimentary plain covered by sandy clayey alluvial deposits of quaternary age. Its lowest altitude makes it vulnerable to regular flooding by the River Chari and the Lake Chad. The sedimentary cover date from the Tertiary to the Quaternary throughout the region. This cover is made by river alluvia, lacustrine, wind sediments, and shows three main series [

- The Bodele series constituted by lightly differentiated sediments of the Upper Tertiary composed of sand and sandstone with some clayey intercalations. This series which date from the Pliocene is mainly fluvial.

- The Soulias series (Middle and upper Pleistocene) constituted by Aeolian sand and lacustrine limestone.

- The Labde series, which includes thin lacustrine deposits date from 2400 years to recent.

The sequence is underlained by granitic, gneissic and migmatitic basement rocks which appear at variable depths.

The polynomial separation method was used to produce the first, second and third degree residual maps. The algorithm by [

The least-square method was used to compute the mathematical surface which gave the best fits to the gravity field within specific limits [

The Bouguer anomaly B(x, y) in the given point M(x, y) of the earth in cartesiane coordinates is governed by the relation:

B ( x i , y i ) = A ( x i , y i ) + R ( x i , y i ) (3.1)

B ( x i , y i ) is the sum of the residual anomaly A ( x i , y i ) and the regional anomaly R ( x i , y i ) .

The surface F ( x i , y i ) which is adapted to the gravity field data g ( x , y ) is given by the following relation (Radhakrishhna and Krishnamacharyulu, 1990):

F ( x i , y i ) = C 1 + C 2 X i + C 3 Y i + C 4 X i 2 + C 5 X i Y i + C 6 Y i 2 + ⋯ + C M − N Y i x i N − 1 + C M − N − 1 Y i X i N − 1 + ⋯ + C M Y i N (3.2)

where N is the order of the polynomial, M = ( N + 1 ) ( N + 2 ) 2 the number of

terms of the polynomial, and C M the coefficients to be determined:

The first order Polynomial is:

F ( x i , y i ) = C 1 + C 2 X i + C 3 Y i (3.3)

The second order polynomial is:

F ( x i , y i ) = C 1 + C 2 X i + C 3 Y i + C 4 X i 2 + C 5 X i Y i + C 6 Y i 2 (3.4)

The third order polynomial is:

F ( x i , y i ) = C 1 + C 2 X i + C 3 Y i + C 4 X i 2 + C 5 X i Y i + C 6 Y i 2 + C 7 X i 3 + C 8 Y i X i 2 + C 9 X i Y i 2 + C 10 Y i 3 (3.5)

We denote by ε i = B ( x i , y i ) − F ( x i , y i ) the difference between the homologous points of the experimental and analytical surfaces respectively and by N_{0} the number of stations P_{i} in which the Bouguer anomaly is known. The adjustment of the surfaces which consists in making the quadratic deviation minimal is expressed by:

E = ∑ i = l N 0 ε i 2 then ∂ E ∂ C k = 0 with 1 ≤ k ≤ ( N + 1 ) ( N + 2 ) 2 (3.6)

E = ∑ i = l N 0 [ B ( x i , y i ) − F ( x i , y i ) ] 2 and ∂ E ∂ C k = 0

We then obtain a system of (M) equations with (M) unknowns. The unknowns are the coefficients C k of the polynomial F ( x i , y i ) of order N. Once the coefficients are determined we determine the analytic regional anomaly R ( x i , y i ) = F ( x i , y i ) and the residual by:

A ( x i , y i ) = B ( x i , y i ) − F ( x i , y i ) (3.7)

The polynomial method is particularly used when the amplitude of the residual anomalies is negligible compared to the regional one. Apart from the polynomial method there are other methods such as the upward continuation method.

The spectral analysis method was used to investigate the wave numbers of gravity residual anomalies and to estimate the depths of the bottom and top of the residual anomalies source bodies.

Many scientists used the calculation of the power spectrum from the Fourier coefficients to obtain the average depth of the disturbing surface or equivalently the average depth to the top of the disturbing body [

The approach is based on the following procedure that allows the determination of different depths [

f ( x j , z ) = ∑ j = 0 n − 1 A k e i 2 π x j e ± 2 π k z (1)

In this function the wavenumber k is defined as k = 1 / λ and A k is the amplitude coefficients of the spectrum is defined as,

A k = ∑ j = 0 n − 1 f ( x j , z ) e − i 2 π k x j e ± 2 π k z (2)

For an altitude of z = 0, the amplitude coefficients of the spectrum can be written as,

( A k ) 0 = ∑ j = 0 n − 1 f ( x j , 0 ) e − i 2 π k x j (3)

Then the amplitude coefficients of the spectrum can be rewritten in terms of (3) as,

A k = ( A k ) 0 e ± 2 π k z (4)

The power spectrum P k is defined as,

P k = ( A k ) 2 = ( P k ) 0 e ± 4 π k z (5)

Taking the logarithm of both sides,

log e P k = log e ( P k ) 0 ± 4 π k z (6)

We can plot the wavenumber, k, against log e P k to attain the average depth to the disturbing interface. The interpretation of this plot requires an estimate of the best line fit of the lowest wavenumbers where a change in gradient is observed. The average depth can be estimated from Equation (6) as,

H ¯ = Δ P 4 π Δ k (7)

where H ¯ is the average depth, Δ P and Δ k are derivative of P and k respectively. The observed residual anomalies depths using power spectral approach as shown in Figures 4-6.

A map of the Bouguer anomalies represented with dotted lines is shown in ^{2}. This map with a 5 km grid size was obtained by kriging, with a variogram model that has an angle of 0 degree and an anisotropy coefficient of [

with data uniformly sampled. This Bouguer gravity map is shown in

Qualitative interpretation is based on the geophysical information that can be extracted from the residual anomaly maps and their relationship with the geology of the region. The main features of the residual anomaly map (

The Bouguer anomaly map (

The residual anomaly map (

The results of the structural trends of different anomalies are NW-SE and NE-SW. Tables 1-3, are in agreement with the results of [

The structural directions observed here are similar to those of several major oil fields established in Sudan, for example, Muglad and Melut basins [

The spectral analysis curves from profiles P1, P2 and P3 of Figures 4-6 shows the variation of the logarithms of the power spectrum with respect to the wavenumbers, and the slopes (H) selected to estimate the depths. The shape of the power spectrum depends mainly on the values of the wavenumber along the profile. These values can be qualitatively estimated by the shape of the profile which depends on the geology of the area. The high wavenumbers (0.3; 0.6) are related to the shallow anomalies, and the low frequencies (0.0; 0.3) are related to the deepest anomalies ( [

The profiles (P1, P2 and P3) have been selected along the large residual anomaly observed on the three residual maps, which correlate with the northern Logone Birni Basin (

Profile | Profile direction | Anomaly direction | Depths of shallow sources H1 (km) | Depths of deep sources H2 (km) |
---|---|---|---|---|

Profile 1 | NW-SE | NW-SE | 0.81 | 4.55 |

Profile 2 | NW-SE | NE-SW | 0.34 | 2.10 |

Profile 3 | NW-SE | NW-SE | 0.73 | 3.26 |

Profile | Profile direction | Anomaly direction | Depths of shallow sources H1 (km) | Depth of deep sources H2 (km) |
---|---|---|---|---|

Profile 1 | NW-SE | NW-SE | 0.47 | 4.00 |

Profile 2 | NW-SE | NE-SW | 0.34 | 2.09 |

Profile 3 | NW-SE | NW-SE | 0.49 | 2.96 |

Profile | Profile direction | Anomaly direction | Depths of shallow sources H1 (km) | Depth of deep sources H2 (km) |
---|---|---|---|---|

Profile 1 | NW-SE | NW-SE | 0.25 | 1.63 |

Profile 2 | NW-SE | NE-SW | 0.24 | 1.67 |

Profile 3 | NW-SE | NW-SE | 0.49 | 3.84 |

depth of about 4.55, 4.00 and 1.63 km , while the depth of the shallowest anomalies is at 0.81, 0.47, and 0.25 km .

The estimated depths are quite close to those reported by [

We applied gravity polynomial fitting and spectral analysis to observed residual anomalies from the northern Logone Birni sedimentary basin of the far north region of Cameroon, to estimate the structure of the basin as well the depth of residual anomalies. The residual anomaly maps were obtained by fitting the Bouguer anomalies using polynomial surfaces of first, second and third orders

and 3 profiles were extracted to estimate the residual anomaly depths. The shape of the residual anomalies were found to be almost elliptical and trend N-S, NW-SE, NE-SW and are accompanied by a steep dipping gradient. Interpretations of the results show that the structural style of the Logone Birni Basin is mainly NE-SW.

Three distinct residual anomalies were identified in the basin. Over the Goulfey, the deepest anomaly is located at a depth of about 4.55, 4.00 and 1.63 km , and the shallowest anomaly at 0.81, 0.47, 0.25 km . Over the Tom Merifine, the deepest anomaly is located at a depth of about 2.10, 2.09 and 1.67 km , and the shallowest anomaly at 0.34, 0.34, 0.24 km . Over the Tourba, the deepest anomaly is at a depth of about 3.26, 2.96 and 3.84 km , and the shallowest anomaly at 0.73, 0.49, 0.49 km . The average residual depth along the basin varies between 0.24 km and 4.55 km with the minimum at Tom Merifine and the maximum at Goulfey.

Nguimbous-Kouoh, J.J., Ngos III, S., Mbarga, T.N. and Manguelle-Dicoum, E. (2017) Use of the Polynomial Separation and the Gravity Spectral Analysis to Estimate the Depth of the Northern Logone Birni Sedimentary Basin (CAMEROON). International Journal of Geosciences, 8, 1442-1456. https://doi.org/10.4236/ijg.2017.812085