Research Article, Geoinfor Geostat An Overview Vol: 6 Issue: 2

Gravity Data Transformation by Polynomial Fitting and Kriging Data Analysis of the Mamfe Basin (Cameroon)

Nguimbous-Kouoh JJ^1*, Ndougsa-Mbarga T² and Manguelle-Dicoum E³

¹Department of gas and petroleum exploration, Faculty of Mines and Petroleum Industries, University of Maroua, Cameroon

²Department of Physics, Advanced Teacher’s Training College, University of Yaounde, Yaounde, Cameroon

³Department of Physics, Faculty of Science, University of Yaounde, Yaounde, Cameroon

*Corresponding Author : Nguimbous-Kouoh JJ
Department of gas and petroleum exploration, Faculty of Mines and Petroleum Industries, University of Maroua, Cameroon
Tel: (+237) 696 383 408
E-mail: nguimbouskouoh@yahoo.fr

Received: April 06, 2018 Accepted: May 16, 2018 Published: May 23, 2018

Citation: Nguimbous-Kouoh JJ, Ndougsa-Mbarga T, Manguelle-Dicoum E (2018) Gravity Data Transformation by Polynomial Fitting and Kriging Data Analysis of the Mamfe Basin (Cameroon). Geoinfor Geostat: An Overview 6:2. doi: 10.4172/2327-4581.1000181

Abstract

Keywords: Mamfe basin; Polynomial fitting; Boxplot analysis; Kriging analysis

Introduction

Geostatistics includes a number of methods and techniques for analyzing the spatial variability of spatially distributed or spatially structured variables [1]. The techniques developed by Krigeet al. [2] and Matheronet al. [3] to evaluate mineralized bodies have been disseminated in many fields using spatial data. These various disciplines include petroleum geology [4], hydrogeology Hatvan et al. [5], hydrology Kovács et al. [6], meteorology Kohán et al. [7], oceanography Monestiez et al. [8], geochemistry Kern et al. [9], metallurgy Deutsch et al. [10], geography Herzfeldet al. [11], Hatvani et al. [12], environmental control Kint et al. [13], landscapeecology Webster et al. [14], soil science, and agriculture Fortin et al. [15] and Zhao et al. [16]. In the petroleum industries, geostatistics is successfully applied to characterize oil reservoirs on the basis of interpretations from sparsely localized data such as reservoir thickness, porosity, permeability, seismic data, gravity field or magnetic field data [17-20]. One of the most used geostatistical techniques for interpolation of spatial data is the kriging technique. Kriging is a complex interpolation method and is considered as a method of estimating a regionalised variable at selected grid points. It can predict unbiased interpolation values with minimal variance [20,21].

In any kriging analysis, there are several main steps. These steps consist of creating an experimental semi-variogram, linking a theoretical model to the experimental semi-variogram and using the latter’s information to perform kriging. More generally, this may be Determine appropriate theoretical models of semi-variogram; check the possibility of anisotropy of the experimental semi-variogram; generate kriging estimates and estimation errors, i.e., Kriging errors, for a point, area or volume; mapping the spatial distributions of kriging estimates and kriging errors. In a kriging analysis, all these components must be systematically taken into account [22-27] . What we can notice is that the analysis of spatial continuity or roughness of geospatial data in different directions and tolerances takes time [28-31].

The aims of this study are: to carry out the polynomial filtering of the Bouguer data of the mamfe basin, to plot the boxplots to understand the influence of the outliers in the data and to compare the distributions of the Bouguer gravity fields, regional and residual then to realize a kriging analysis of gravity data of the basin and their cartographies.

Location of the Study Area

The Mamfe sedimentary basin (Figure 1) is an intracratonique rift basin formed in response to the dislocation of the Gondwana supercontinent and following the separation of South American and African plates. The basin is a small extension of the Benue sedimentary basin. The basin is favorable to the exploration and exploitation of salt springs, minerals, precious stones and hydrocarbons. It has an area of approximately 2400 km² and is located between latitudes 5°30’N and 6°00’N and longitudes 8°40’ E and 9°50’E (Figure 1). It has the form of a plain with an average altitude that varies between 90 and 300 m above the sea level.

Figure 1 shows the available geological map of the basin. Some geological details were extrapolated or removed. The geological map is preliminary and has jointly been updated following several studies [32-41]. The geomorphology of the area is characterized by a succession of horst and grabens. Overall, the Mamfe sedimentary basin has a NW-SE structural trend with a length of 130 Km and a width of approximately 60 km. It is bordered by faults, lineaments and rivers such as Manyu, Munaya and extends from Cameroon to Nigeria [32-41].

Data Acquisition and Processing

Figure 2 shows the distribution of gravity data collected through the Mamfe Basin. It is deduced from the Cameroon gravity database that manages the RID (Research Institute for Development). The Lacoste-Romberg gravimeter (1975-1976) was used for data recording. The determination of the altitudes was made by barometric leveling with a Wallace-Tiernan altimeter. The RID gravity network established in Douala served as a reference base for measurements. The spatial reference coordinate system used is WGS84 and the data projection system is UTM.

The Bouguer anomaly was calculated at each station using the expression by Poudjom et al. [42]: B=G-(G_o-C_z-T) Where G is the observed value of the gravitational field and its expression is: G=G_r+ΔG with Gr the value of the gravity field in a adopted station as reference and ΔG measuring the gravity field difference between the reference station and a given station. G_o is the theoretical value of the gravitational field at the point of the reference ellipsoid corresponding to the station. G_o has been defined in the IGSN71 reference system whose formula is:

G_o=978031.8 (1+0.053024sin Ï�?²)-0.0000022sin2Ï�?²

The error on the latitude of a station is considered between 0.2 mgal and 0.5 mgal with Ï�?=3°.

• C_z: is the Bouguer correction

• It represents the sum of the correction of the free air C₁(mgal)=0.3086 Z and the plateau correction C₂(mgal)=- 0.0419dZ Where d is the density of the crust and Z the altitude of the station expressed in meters. For the reasons of homogeneity we adopted d = 2.67 for all the stations from where C_z(mgal)=0.1967Z. The imprecision of the barometric leveling has led to an error on C_z generally less than 1 mgal but which has reached 2 or 3 mgal in unfavorable cases.

• T: relief correction. It takes into account the relief around the station. The anomaly value at each point was tainted with a maximum error of 5 mgal in the worst conditions, in most cases the error remained below 3 mgal.

Materials and Methods

In linear geostatistics the choice of software and an interpolation method are very important for spatially analyzing a variable of interest [1-4]. We used R software Ihaka et al. [43] and Surfer Golden Software [44]. Surfer is a 2D and 3D surface mapping program that transforms random data into contours of continuous curved faces. Surfer uses twelve different methods to interpolate data; R is statistical software created by Ihaka et al. [43]. It is both a computer language and a work environment: the commands are executed by means of instructions coded in a relatively simple language, the results are displayed in text form and the graphics are displayed directly in a window of their own. This software is used for manipulating data, graphing and statistical analysis of this data.

In this part, we expose the polynomial fitting method and the different steps of the kriging method in the univaried framework.

The polynomial separation method

The polynomial separation method was used to produce the third degree regional and residual maps. The algorithm by Gupta and Murthy et al. [45,46] was used to adjust the polynomial surfaces to the Bouguer anomaly map. This method is based on the analytical least square method and the polynomial decomposition series.

The least-square method is used to compute the mathematical surface which gave the best fits to the gravity field within specific limits [45-47]. This surface is considered to be the regional gravity anomaly. The residual is obtained by subtracting the regional field from the observed gravity field. In practice, the regional surface is considered as a two-dimensional polynomial. The order of this polynomial depends on the complexity of the geology in the study area. The third-order polynomial surfaces of the regional anomaly obtained in this work is presented and the corresponding residual anomaly.

Mathematic formulation of the method

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

B(x_i, y_i)= A(x_i, y_i)+ R(x_i, y_i) (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 [45-49]:

(2)

Where N is the order of the polynomial, 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)=C₁+C₂X_i+C₃Y_i (3)

The second order polynomial is:

(4)

The third order polynomial is:

(5)

We denote byε_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 N₀ 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:

(6)

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) (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.

Variographic formalism and kriging

In a classical framework, kriging analysis generally involves two stages: the variogram estimation and the use of the model coefficients for Kriging [17-20].

The formalism of variographic estimation: The variogram is the basic tool in kriging analysis and is required for geostatistical spatial prediction. The variogram is estimated by the experimental variogram γ*(h), which is calculated from the discrete data as follows [17-20]:

where N is the number of pairs of points distant from h.

Once the experimental variogram has been calculated, it is necessary to replace it with a theoretical variogram whose coefficients are estimated by a least squares procedure [20].

There are several types of theoretical variogram models used for kriging [17-20]:

The spherical variogram is characterized by :

With a the range, h the interpolation step and C the constant sill

The Gaussian variogram is characterized by:

The exponential variogram is characterized by:

The power variogram is characterized by:

For b = 1,the variogram is linear; b = 2,the variogram is a parabola

The formalism of variographic anisotropy: There are different types of anisotropies which enable to know the different orientations that spatialized data can take [17-20]. Geometric anisotropy is the simplest case. A geometric anisotropy is reflected in the experimental variograms by a range which varies according to the direction. Geometric anisotropy is observed when the variogram respects the following mathematical formulation:

With a_g the maximum range, a_p the minimum range and a_θ the range in the anisotropy direction θ

Anisotropy is called zonal when it cannot be solved by a simple geometric or linear coordinate’s transformation. It causes for the bearing variogram which varies according to the space direction.

The kriging formalism

Kriging is an interpolation method that gives the best unbiased linear estimate of the point value of a regionalised variable [2,3]. The kriging interest is that it enable to calculate the estimation error i.e the kriging variance [2,3]. It allows the taking into account of: the data configuration; the distance between data and targets; spatial correlations and external information. Kriging is an exact interpolator, it has a screen effect, it is almost without conditional bias. It is transitive, generally performs a smoothing and takes into account the field size to estimate and the position of the points between them [2,3].

The mathematic formulation of kriging imposes the minimization of the following variance [1-4]:

Where Z_v represents the known values at different points and:

is the estimator

λ_i is the weight and Z_i is the random variable corresponding to each point.

Since kriging is an unbiased interpolator then:

To minimize the kriging variance we use the Lagrangian [1-4]:

With μ the Lagrange multiplier

By calculating: and we obtain the system of equation:

The variance of kriging is given by:

In order to solve the system numerically, it is convenient to write it in matrix form AX=B We obtain:

The system allows finding the set of weights λ_i and the Lagrange multiplier μ that minimize the variance and make the estimator unbiased. The variance of kriging is then obtained by the following formula:

With the X^T transpose of X=A^-1B

The system of equations can also be expressed in terms of covariance under the assumption that the random process is stationary.

Results

In this part, we present the results of polynomial separation by boxplots, variographic analysis and data mapping.

Polynomial separation boxplots analysis

It is interesting to use box-plots when you want to visualize concepts such as the symmetry, the dispersion or the centrality of the values distribution associated with a variable. They are also very interesting for comparing variables based on similar scales. The graphs (Figure 3) represent Bouguer’s gravity fields and interpolated regional and residual data. They show different shapes and positions of box-plots. These graphs enable to summarize the gravity data in a simple and visual way, to identify the extreme values and to understand the different anomaly values distribution.

It can be noted that overall 50% of gravity anomaly values are inside each box. The values outside the box plots are represented by dots. It cannot be considered that if an observed value is outside the box-plots then it is an outlier. However, this indicates that this observed value needs to be studied in more detail. Outliers are data values that are far removed from other data values and can have a significant impact on the results.

These boxes have variable widths, it is not a simple aesthetic transformation, and the width is proportional to the size of the anomaly values. The widths of the two parts of the box-plot reflect the dispersion of the values located at the center of the series (the box contains 50% (approximately) of all anomalies: 25% at the top of the median and 25% at the bottom); in general, the boxplots will be all the more extensive as the dispersion of the series of values is great.

The median is represented by the line in the box. The median is a common measure of data centering. Half of the values are lower or equal and half of the values are greater or equal.

When the data is asymmetrical, the majority of them are located on the upper or lower side of the graph. Asymmetry indicates that the data may not be normally distributed (Table 1).

Table 1: Bouguer anomaly values as well as regional and residual anomalies obtained from polynomial separation.

Variographic analysis of the data

Variographic analysis plays a key role in the process of assessing the correlation level of spatialized data and their modeling. It indicates problems related to data dispersion and allows kriging or mapping in a particular area [17-20].

Anisotropy analysis of the field data

Figure 4 shows the variograms of all Bouguer gravity data, regional and residual. Each variogram is represented by the cloud, the experimental variogram and the three theoretical models selected for the anisotropy tests. The experimental variogram is a graph that shows the contributions of each pair of points to the final variogram. The scatter plot gives a visual impression of the dispersion of values at different offsets. He has no precise direction. We chose a direction and a tolerance each time to obtain an unidirectional variogram which enable to neglect the lag h.

Anisotropy is the parameter that allows us to better know the spatial continuity of the values of Bouguer, regional and residual gravity fields. Figure 4 shows the simulation of theoretical Gaussian variograms across the experimental variograms cloud. These figures prove that bouguer, regional and residual gravity data present a geometric anisotropy. Spatial continuity is not necessarily the same in all directions. The Gaussian model fit the dataset quite well (Table 2).

Table 2: Characteristics of different models of anisotropy.

Variographic analysis of grid data

A variogram expresses the semi-variance of a regionalized variable and its variation as a function of the lag distance h. This variogram shows that, beyond a certain limit called range, no spatial correlation exists between the points. A theoretical variogram model is first computed to fit the experimental variogram obtained with the data, and is then used from Kriging [17-20].

The gridded data variograms are shown in Figure 5. The different experimental variograms clouds were fitted by the Gaussian theoretical variogram. The tolerances and directions chosen were a compromise between the need to have a sufficient number of pairs of points for calculating the variogram and the risk of too much smoothing of the curve. Overall the three variograms are unidirectional. The tolerance direction varies between 4° and 71° and the direction between -52° and 159°. These values were each time set so that the direction of any particular lag distance h, becomes unimportant (Table 3).

Table 3: Characteristics of fitted experimental variogram.

Interpretation of 3D wireframe maps

3D Wireframe map models are three-dimensional cartographic forms of the superficial and deep geological formations commonly used in geology and mining. The wireframe model maps a set of points with known triaxial Cartesian (x, y, z) coordinates. Before the plot, a grid routine is used to place the randomly located field data into a regular grid with selected spacing. The wireframe plot results in an open grid (x, y) with the anomaly of each grid node corresponding to the z coordinate at that point. The wireframe structures that we present in this part are three-dimensional representations of the gravity data of the Mamfe basin. These wireframe structures cannot be larger or smaller than the extent of the grid data file.

Bouguer 3D wireframemap

The Bouguer 3D wireframe map gives an idea of the structural distribution of the superficial and deep geological formations (Figure 6). It reflects the regular and continuous variations of the Bouguer gravity field of the Mamfe basin. The Bouguer 3D wireframe map indicates areas of existence of slopes and density discontinuities such as faults.

Regional 3D wireframe map

The Regional 3D wireframe map shows the overall gravity regional trends affecting the basement of the Mamfe Basin (Figure 7). These trends are N-S, E-W. This graph shows areas where the basement seems to have undergone a dilation confirming the hypothesis of magmatic upwelling in the basin; while the platform zone can be associated with a consolidation of the basement.

Residual 3D wireframemap

The Residual 3D wireframe map gives a three-dimensional representation of the sedimentary layer in the Mamfe basin (Figure 8). It is characterized by two negative gravity anomalies that can be associated with sedimentary infilling zones.

Conclusion

In this study, thirty six gravity Bouguer data were collected in the Mamfe sedimentary basin. Third-order polynomial filtering was applied to the data. Third-order regional and residual gravity data were obtained. The dataset was plotted as boxplots to understand the influence of the outliers in the data and then to compare Bouguer gravity field distributions, regional and residual. The kriging analysis procedure was initiated to: describe the spatial variability of the different gravity data; identify spatial correlations and different trends and directions of experimental semi-variograms; check the possibility of the experimental semi-variograms anisotropy ; fit interpolated semi-variogram models using mathematical functions; Create the 3D wireframe maps of the different gravity fields from the new data grids to evaluate the interpolated surfaces.

Acknowledgment

This study was conducted as part of the research activities conducted in the Department of Mines, Petroleum, Gas and Water Resources Exploration. Thank to all the colleagues involved in the application of geostatistics in the oil and mining field.

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