International Journal of Civil, Structural, Environmental and Infrastructure Engineering Research and Development (IJCSEIERD) ISSN 2249-6866 Vol. 3, Issue 2, Jun 2013, 145-154 © TJPRC Pvt. Ltd.

NUMERICAL INVESTIGATION INTO THE DOUBLE FOURIER ANALYSIS FOR DETERMINING THE QUANTITY OF UNDERGROUND MINERAL DEPOSIT ABUBAKAR T & IDOWU T. O Department of Surveying and Geoinformatics, Modibbo Adama University of Technology, Yola, Adamawa, Nigeria

ABSTRACT Geophysical exploration is a reliable technique for detecting underground mineral resources. Its methods include natural-source method such as gravimetric method and artificial–source method like seismic method. Previous studies have shown that the gravimetric method is a good alternative to the more expensive seismic method. Yet, seismic method is widely preferred in Nigeria and many other countries to the gravimetric method for mineral exploration. This is due, probably, to the inverse problem in the gravimetric method, which has defied satisfactory solution. This inherent problem includes amongst others the determination of the most probable quantity of the underground mineral deposit. In this study, an attempt is made to carry out numerical determination of the estimated quantity of underground mineral deposit. Data used for the study are the rectangular coordinates and residual gravity anomalies of gravity stations obtained along some profiles in Bauchi State of Nigeria. The data distributed in profile pattern was later resolved into square grids using Kriging method of interpolation. Thereafter, the estimated quantity of the underground mineral deposit was determined using the method of double Fourier analysis. It can be inferred, from the analysis of the results obtained, that the computed quantity of the mineral deposit is satisfactory. Therefore, it shows that the solution of the inverse problem in gravimetric technique has been satisfactorily attempted. Also, double Fourier analysis has been shown to serves as effective tool for the determination of the quantity of underground mineral deposit,

KEYWORDS: Underground Mineral, Residual Gravity Anomalies, Fourier Analysis, Gravimetric Technique and Bauchi State of Nigeria

INTRODUCTION Geophysical surveying can be described as a comparatively young science discovered around 1925 when an authoritative English text on the subject was compiled (Kearey and Brooks, 1988). The method is now universally applied and plays important roles in the discovery of mineral deposits in the whole World. Reynolds (1998), defined Geophysical surveying as “making and interpreting measurements of physical properties of the earth to determine sub-surface conditions, usually with economic objectives such as discovery of mineral deposition’’ while its use could lead to the improvement of the recent rate of sub-surface mineral discovery. Gravimetric surveying is a natural sourced method of geophysical Surveying, it was more cost- effective than other methods. Gumert (1992), Robin (1995) and Kearsley et al. (1975), observed that the current level of accuracy, resolution of the measured gravity data and high speed of operation of gravity survey have led to a good understanding of regional geology and economically limited the use of more expensive seismic survey for mineral exploration. Hence, it can be inferred that the natural-source method is a good alternative to the artificial-source method. Artificial-source method, that is the seismic surveying, involves propagation of artificial waves through the earth’s interior to search for local perturbation in the artificial field that may be caused by the concealed geological features.

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Gravimetric method can provide information on sub-surface geological body to a significantly greater depth and is logistically simpler to carry out than the artificial-source method (Kearey and Brooks 1988). Gravimetric method is used to provide data for a sort of reconnaissance survey for the real exploratory work. One may tend to wonder why the gravimetric method is not used for investigation of underground mineral resources for the real exploratory work despite its financial advantage over the widely used method (seismic method). Much work has been done in order to show the efficacy of the gravimetric method in the investigation of underground mineral resources. The approaches reveal a good result that can be relied upon. However skepticism about the quality of result obtained might be the factor debilitating against the total acceptance of the gravimetric method. More inputs in the data used for the investigation will certainly improve the result and give more reliability and acceptance by mineral explorers. In general, both natural-source and artificial-source methods are used for investigation of underground minerals resources. In all cases, the natural-source method serves as reconnaissance or confirmatory survey to the artificial-source method. Determination of quantity of mineral deposit forms part of the analysis of gravity anomalies. Various authors have tried different techniques to accomplish this task. For instance, in the mathematical formulations derived from the works of Kearey and Brooks (1988), Telfort et al (1990) and Reynolds (1998), the residual gravity anomalies were used in the determination of the quantity of mineral deposit. The methods so far used for the determination of quantity of mineral deposit are the direct application without sufficient mathematical back-up theory and formulations this created room for further investigation. Idowu (2005) uses both the Newtonâ€™s method and one dimensional Fourier analysis with elaborate mathematical back-up and formulations to determine the quantity of mineral deposit and the result was seen to be accepted and reliable. Therefore it is the objective of this study to use an interpolation technique to interpolate residual gravity anomalies at grid points and apply a mathematical procedure, that is, a two dimensional Fourier analysis to determine the quantity of the underground mineral deposit.

METHODOLOGY The method used includes the application of kriging method of interpolation to convert the residual gravity anomalies obtained in profile to grid. Fourier analysis in two dimensions as discussed in Tsuboi (1983) was used to determine the quantity of the mineral deposit.

DATA ACQUISITION The data used is the residual gravity anomalies presented in profile form for 300 stations and their X and Y coordinates. The data is abstracted from Idowu (2005), this covers an area in the Gongola basin, Bauchi state Nigeria, an area suspected to have mineral accumulation. Idowu (2005) in his research work uses the third order polynomial function of the indirect analytical function to extract the residual gravity anomalies from gravity observation carried out by Shell Nigeria Exploration and production Company (SNEPCO). The residual gravity anomalies obtained were satisfactorily extracted from the observed gravity anomalies at one percent level of significance. This indicates that the quality of the data used presented in table 1 is believed to be satisfactory. Columns 1, 2, 3 and 4 of the table show the serial number, Xcoordinates, Y-coordinates and residual gravity anomalies of the Table 1: Extract of Data (Residual Gravity Anomalies) Used for the Research S/No 1. 2. 3. 4.

X(m) 658742.200 658730.900 659104.000 659539.000

Y(m) 1122860.600 1122362.300 1122152.800 1121931.800

R(mgal) -0.094731 00.377338 01.665155 01.441542

147

Numerical Investigation into the Double Fourier Analysis for Determining the Quantity of Underground Mineral Deposit

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Table 1: Contd., 659972.300 1121694.100 660276.400 1121309.800 660496.900 1120872.000 660872.400 1120544.500 661165.200 1120164.800 661634.900 1120015.800 662037.200 1119720.900 662376.200 1119355.800 662698.600 1118979.600 662865.600 1118525.800 663045.100 1118080.900 663246.400 1117626.000 663460.200 1117175.600 663561.500 1116695.800 663800.800 1116320.300

01.906113 02.528534 02.924877 02.783191 02.627027 02.009606 01.223088 00.688359 00.009444 -0.049030 -0.415414 -0.757984 -1.288014 -1.986381 -2.745139

DATA PROCESSING Data processing includes the mathematical procedure followed in interpolation of residual gravity anomalies at grid points. Interpolation by Kriging method is hereby discussed. Also, the computational procedure for the solution of inverse problem in the gravity method of geophysical exploration is explained. This means, the mathematical formulation for the determination of the quantity of the underground mineral deposit by two dimensional Fourier analysis. Gridding by Kriging Method of Interpolation Kriging is a group of geostatistical techniques to interpolate the value Z(x0) of a random field Z(x) (e. g. the elevation Z of the landscape as a function of the geographic location x) at unobserved location x0 from observations =

of the random field at nearby location

unbiased estimator

Kriging computes the best linear

of Z(x0) based on a stochastic model of the spatial dependence quantified either by the

variogram y(x,y) or by expectation µ(x) = E [Z(x)] and the covariance function c(x,y) of the random field. The kriging estimator is given by a linear combination as shown below: of the observed values

with weights

i = 1, ……,n

chosen such that the variance (also called kriging variance or kriging error)

The kriging variance must not be confused with the variance

of the kriging predictor Chiles and Delfiner (1999).

itself. A more detailed explanation of the method is shown in Groverts (1997) and

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Abubakar T & Idowu T. O

Determination of Quantity of the Mineral Deposit by Fourier Analysis Determination of the quantity rather known as actual mass was achieved by a two dimensional Fourier analysis as discussed in Tsuboi (1983): cos f ( xy ) m n mn cos sin mx sin ny

(1)

Suppose a distribution of a certain quantity f (xy) is given within a square, 0 ≤ x ≤ 2, 0 ≤ y ≤ 2. The distribution of the value of f (xy) in the direction of x along a certain value of y can be expressed by a single Fourier series of x such as:

f ( xy ) m n ( y ) cos sin mx . The coefficient

m

(2)

changes according to , so that it can be expressed by a single Fourier series of

m ( y ) n n cos sin ny.

such as: (3)

Then as a whole, f (xy) is given as: cos f ( xy ) m n n n cos sin mx . sin ny.

(4)

If n n is written as mn then cos f ( xy ) m n mn cos sin mx sin ny

(5)

Therefore, using the double Fourier series, if a distribution of gravity anomaly ( R(xy) ) is given by: cos R ( xy ) m n Bmn cos sin mx sin ny

(6)

Then, the underground mass (M) at a depth (d) that will produce this R(xy) can be given by:

M ( xy) (1 / 2K ) m n Bmn exp (m 2 n 2 )d

cos sin

mx cos sin ny

(7)

In this process the data were observed along x and y directions at a define area. The process involved:

Evaluation of Fourier series to obtain the Fourier coefficients which is equal to the number of points where the residual gravity anomalies were obtained along the x and y direction. I.e. 2m within within

and 2n

. The gravity stations intervals are equal in both x and y direction which is one of the

requirements for achieving satisfactory result. The Fourier coefficients for the residual gravity anomalies are obtained as

a=

1 1 2m 2 n / Ri cos(2 (i 1) j / 2m) Ri cos(2 (i 1) j / 2n) m n i 1 i 1

b=

1 1 2m 2n / Ri cos(2 (i 1) j / 2m) Ri sin(2 (i 1) j / 2n) m n i 1 i 1

149

Numerical Investigation into the Double Fourier Analysis for Determining the Quantity of Underground Mineral Deposit

c=

1 1 2m 2n / Ri sin(2 (i 1) j / 2m) Ri cos(2 (i 1) j / 2n) m n i 1 i 1

l=

1 1 2m 2n / Ri sin(2 (i 1) j / 2m) Ri sin(2 (i 1) j / 2n) m n i 1 i 1

Computing amn exp (m 2 n 2 )d , bmn exp (m 2 n 2 )d , cmn exp (m 2 n 2 )d and l mn exp (m 2 n 2 )d for m = 0, 1, 2, 3, ……., m and n = 0, 1, 2, 3,…., n with an assumed initial value of depth (d). that is d = 0.01 Computation of Quantity of mineral deposit using equation (7) that is,

c exp (m n )d sin mx cos ny l exp (m n )d sin mx sin ny The next values of a exp (m n )d , b exp (m n )d , c exp (m n )d and l exp (m n )d are computed by increasing the initial value of d. This is an iterative process which M ( xy) (1 / 2K ) m n amn exp (m 2 n 2 )d cos mx cos ny bmn exp (m 2 n 2 )d cos mx sin ny 2

2

2

mn

2

mn

2

2

2

2

2

mn

mn

2

2

mn

2

mn

continues until the value of M(xy) decreases smoothly and converges to the minimum value.

PRESENTATION OF RESULT The result of the residual gravity anomalies interpolated at grid notes using kriging are extracted and presented in table 2. Column one is the grid number, columns two and three are the coordinates X and Y respectively while column four is the residual gravity anomalies. Residual gravity anomalies used for the computation of the underground mineral deposit are extracted and presented on Table 3, Column one is the grid number, columns two and three are the X and Y coordinates respectively while column four is the residual gravity anomalies. Figure 1A is the contour map of the study area shown on the gridded area. The demarcated area is the area where the quantity of the mineral deposit was computed that is area with positive anomalies. Figure 1B is the Digital elevation model of the study area and the demarcated area on it shows where the quantity of the mineral deposit was computed. Table 4 below shows the values of computed quantities M(xy) and where the M(xy) converges suggesting the quantity of the underground mineral deposit as computed by two dimensional Fourier analysis. Column 1of table 4 is the serial number, column 2 is the quantity and column 3 is the depth factor. Table 5 is the statistical report of the gridding processes extracted from the process of interpolation by Kriging and table 6 shows the table and computed statistics used in the analysis of the variance. Table 2: Extract of Interpolated Residual Gravity Anomalies at Grid Points Grid No. 001 002 003 004 005 006 007 008 009 010 011 012

X(m) 656649.700 656649.700 656649.700 656649.700 656649.700 656649.700 656649.700 656649.700 656649.700 656649.700 656649.700 656649.700

Y(m) 1104887.100 1105450.059 1106013.018 1106575.977 1107138.936 1107701.895 1108264.854 1108827.813 1109390.772 1109953.731 1110516.691 1111079.650

R(mgal) 00.842308 00.836793 00.376764 00.448133 00.517463 00.576390 00.658272 00.744601 00.066292 -0.166433 00.501225 01.408544

150

Abubakar T & Idowu T. O

013 014 015 016 017 018 019 020

Table 2: Contd., 656649.700 1111642.609 656649.700 1112205.568 656649.700 1112768.527 656649.700 1113331.486 656649.700 1113894.445 656649.700 1114457.405 656649.700 1115020.364 656649.700 1115583.323

02.389070 02.581767 02.253498 01.986430 02.187140 02.707387 02.718768 03.584070

Table 3: Extract of Residual Gravity Anomalies at the Suspected Area of Underground Mineral Accumulation S/No 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

X(m) 673546.9500 673546.9500 673546.9500 673546.9500 673546.9500 673546.9500 673546.9500 673546.9500 673546.9500 681432.3333 681995.5750 681995.5750 681995.5750 681995.5750 681995.5750 681995.5750 681995.5750 681995.5750 681995.5750 681995.5750

Y(m) 1114457.4045 1115020.3600 1115583.3200 1116146.2800 1116709.2400 1117272.2000 1117835.1590 1118398.1181 1118961.0772 1122901.7909 1114457.4045 1115020.3600 1115583.3200 1116146.2800 1116709.2400 1117272.2000 1117835.1590 1118398.1181 1118961.0772 1119524.0363

R(mgal) 0.129999900 0.242259900 0.146722020 0.525574800 0.788450800 0.991356000 1.253334000 1.414522000 1.411378900 0.291538000 0.715512600 0.012709700 1.016062004 0.926954030 0.924087980 1.097611990 1.454454050 1.824401100 2.208005100 2.420137400

Figure 1A: Contour Map of Study Area Showing Area of Mineral Accumulation

151

Numerical Investigation into the Double Fourier Analysis for Determining the Quantity of Underground Mineral Deposit

Figure 1B: Digital Elevation Model of Study Area Showing Area of Mineral Accumulation Table 4: Computed Quantity of Mineral Deposit by Fourier Analysis S/No 1. 2. 3. 4.

Quantity(Tones) 719383934.077313000 719383934.077312000 719383934.077309000 719383934.077308000

5.

719383934.077452000

6. 7. 8. 9. 10.

719383934.079283000 719383934.096645000 719383934.241657000 719383935.366876000 719383943.656713000

Depth (Rad) 0.001 0.011 0.021 0.031 0.041 convergence point (computed quantity) 0.051 0.061 0.071 0.081 0.091

Table 5: Gridding Statistics Report X Minimum: X Maximum: X Spacing: Y Minimum: Y Maximum: Y Spacing: Z Minimum: Z 25%-tile: Z Median: Z 75%-tile: Z Maximum: Z Midrange: Z Range: Z Interquartile Range: Z Median Abs. Deviation: Z Standard Deviation: Z Variance: Z Coef. of Variation: Z Coef. of Skewness: Z Root Mean Square: Z Mean Square:

656649.7 697203.1 563 1104677.8 1129994 563 -4.5398373317825 -1.1449143016791 -0.2459705573493 0.50041784329279 4.7021539791339 0.0811583236757 9.2419913109165 1.6453321449719 0.81898167711211 1.3797224567466 1.9036340576508 -1 -0.07112905338202 1.4205461947554 2.0179514914342

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Abubakar T & Idowu T. O

Table 6: Computed and Table Statistics Degree of freedom

(n-1)

(n 1)

Computed statistics Level of significance

Table statistics

= 143 2

02

= 15.842

(α)

= 0.5

2 1 / 2

at (n-1)

= 177.998

ANALYSIS OF RESULT The gridding Statistical report indicated that the spacing of gridding at 563m was done within the range that will allow for a good interpretation of gravity data in especially places with suspected oil reserve. The variance of the population from the statistics report is 1.9036340576508 and the variance of the sample population that was used in the computation of the mineral deposit was computed as 0.2108308. It is important to investigate whether the data set used in the computation of quantity of mineral deposit actual belongs to the normal population (residual gravity anomalies representing the study area). This is achieved by analysis of the variance where test of hypothesis on the variance is carried out as stated in Ayeni (2001). Here we are concern with the problem of testing whether the variance of a normally distributed population 2 is equal to a given variance 02 i.e Null Hypothesis:

Ho: 2 = 02

Alternative Hypothesis: H1: 2 02 Where the computed statistic (2) =

(n 1) 2

02

This is a two-tail test where the Null hypothesis: Ho is rejected if 2 > 1 / 2 at (n-1) or 2< / 2 at (n-1) 2

2

From table 6 above, it can be seen that value of computed statistic at 0.05 levels of significance is less than the table statistics

12 / 2 at (n-1), therefore we accept the null hypothesis (Ho) that 2 = 02. This suggests that the null

hypothesis, that the residual gravity anomalies used for the computation of the quantity of mineral deposit are extracted from the normal population (residual gravity anomalies representing the study area) is accepted.

CONCLUSIONS This paper has attempted to solve the inverse problem in the gravity method of geophysical exploration. Specifically, it interpolated residual gravity anomalies at grid points and tried to utilize the residual gravity anomalies to determine the quantity of the underground mineral deposit using two dimensional Fourier analysis. From the statistical analysis however, the gridded residual gravity anomalies used in the computation of the quantity of underground mineral deposit are truly extracted from residual gravity anomalies representing the study area (normal population), this fact is accepted at 0.05 level of significance. This further suggests that the computed quantity is a true reflection of the area. It can be inferred therefore that the solution of inverse problem in the gravity method of geophysical exploration has been satisfactorily achieved.

153

REFERENCES 1.

Ayeni, O. O. (2001): Statistical Adjustment and Analysis of Data. Lecture note series. Department of Surveying and Geoinformatics, University of Lagos.

2.

Chiles T. P. and Dufiner P. (1999) Geostatistics, modeling special uncertainty, wiley series in proberbility and statistics

3.

Goovaerts P. (1997) Geostatistics for Natural Resour evaluative. Oxford Unversity Press, New York

4.

Gumert, W. R. (1992): Airborne gravity measurements, in CRC. Handbook of Geophysical Exploration at Sea, 2nd edition, CRC Press.

5.

Idowu, T. O. (2005): Determination and Utilization of optimum Residual Gravity Anomalies for Mineral Exploration. PhD thesis, Department Surveying and Geoinformatics, University of Lagos.

6.

Kearey, P. and Brooks, M. (1988) : An Introduction to Geophysical Exploration. The Garden City, Blackwell Scientific Press, Letchworth, Herts.

7.

Kearsley, A. H. W., Forsberg, R., Olesen, A., Bastor, L., Hehl, K., Meyer U. and Reynolds, J. M. (1998): An Introduction to Applied and Environmental Geophysics. Published by John Wiley & Sons ltd. West Sussex, England.

8.

Robin, E. B. (1995): Advances in Aerogeophysics and Precise Positioning: Gravity, Topography, and High Resolution Applications. U. S. National Report (1991-1994) to IUGG. Lamont-Doherty Earth Observatory, Palisades, New York.

9.

Telford, W. M., Geldart, L. P., Sheriff, R. E. and Keys, D. A. (1990) : Applied Geophysics. 2nd Edition, Cambridge University Press.

10. Tsuboi, C. (1983): Gravity. George Allen & Unwin (Publisher) Ltd. English edition. 40 Museum Street, London WCIA ILU, UK

17 numerical investigation full

Published on Jun 30, 2013

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