A simple method for depth determination from self-potential anomalies due to two superimposed structures

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1 CSIRO PUBLISHING Exploration Geophysics, 16, 47, A simple method for depth determination from self-potential anomalies due to two superimposed structures El-Sayed M. Abdelrahman 1, Eid R. Abo-Ezz 1 Tarek M. El-Araby 1 Khalid S. Essa 1 1 Geophysics Department, Faculty of Science, Cairo University, Giza 1613, Egypt. Corresponding author. sayed55@yahoo.com Abstract. In this paper, we develop a method to determine the depth to two superimposed sources from a self-potential anomaly profile. The method is based on finding a relationship between the depths to the two superimposed structures from a combination of observations at symmetric points with respect to the coordinate of the sources centre. Formulae have been derived for two classes of geometric sources: spheres and cylinders. Equations are also formulated to estimate the other model parameters, including the polarisation angle and electric dipole moment of both structures. The proposed method is tested both on synthetic data with and without random noise, as well as real self-potential data from asetoffield data collected in Turkey. In all cases, the model parameters obtained are in good agreement with the actual ones. Key words: depth-curves method, interpretation, noise, SP data, superimposed structures. Received 4 July 14, accepted 6 June 15, published online July 15 Introduction The self-potential (SP) technique has a wide range of applications in locating metallic sulfides (Yüngül, 195), geothermal exploration (Corwin and Hoover, 1979; Anderson, 1984), cavity detection (Schiavone and Quarto, 199) and engineering and geophysical investigations (Markiewicz et al., 1984). Many methods of evaluation have been developed to determine the depth from residual SP anomalies due to simple geometries, including those given by Yüngül (195), Banerjee (1971), Fitterman (1979), Bhattacharya and Roy (1981), Abdelrahman and Sharafeldin (1997), El-Araby (4), Abdelrahman et al. (1997, 6, 8, 9) and many others. However, these methods cannot be used to interpret residual anomalies due to two superimposed structures such as superposed folds and other geological features (Weiss, 1959; Spencer, 1977). Abdelrahman et al. () developed a depth-curves method to determine the depth to shallow and deep-seated structures from a magnetic anomaly profile. This method involves using a relationship between the depths to two superimposed sources obtained by combining observations at symmetric points with respect to the coordinate of the sources centre. In this paper, we apply the same methodology given by Abdelrahman et al. () to determine the depths to two superimposed sources from an SP anomaly profile. Procedures are also formulated to estimate the other model parameters including the polarisation angle and electric dipole moment of both structures. Also, this method can be used to distinguish whether the interpreted anomaly is due to a single source or two superimposed sources. The validity of the method is tested on synthetic data and on a field example from Turkey. Formulation of the problem The SP response of a buried structure can be expressed as (Yüngül, 195) Journal compilation ASEG 16 V ðx i ; Z; qþ ¼ AX i þ B ðx i þ Z Þ q i ¼ 1; ; :::L; A ¼ K cos and B ¼ KZ sin : In Equation 1, X i is a horizontal position coordinate, Z is the depth, y is the polarisation angle, K is the electric dipole moment, L is the number of data points and q is the shape factor. For example, the shape factor, q, is.5, 1. and for a semi-infinite vertical cylinder, horizontal cylinder and a sphere, respectively (Abdelrahman et al., 1997). Figure 1 shows a crosssectional view of sphere, horizontal cylinder and vertical cylinder models. Using Equation 1, the anomaly expression produced by two superimposed sources having depths Z 1 and Z, and shape factors q 1 and q, respectively, is then given as V ðx i ; Z 1 ; Z Þ¼ A 1X i þ B 1 ðx i þ Z 1 Þq 1 i ¼ 1 ; ; 3; :::; L; þ A X i þ B ðx i þ Z Þq where A 1 = K 1 cosy 1, A = K cosy, B 1 = K 1 Z 1 siny 1 and B = K Z siny. Equation gives the following values at X i =,X i = N and X i = M sampling units, respectively V ðþ ¼ B 1 Z q1 1 ð1þ ðþ þ B ; X i ¼ ð3þ Z q V ðnþ ¼ A 1N þ B 1 ðn þ Z 1 Þq 1 þ A N þ B ðn þ Z Þq ; X i ¼ N Vð NÞ ¼ A 1N þ B 1 ðn þ Z 1 Þq 1 A N B ðn þ Z Þq ; X i ¼ N V ðmþ ¼ A 1M þ B 1 ðm þ Z 1 Þq1 þ A M þ B ðm þ Z Þq ; X i ¼ M ð4þ ð5þ ð6þ

2 Depth determination using SP anomalies Exploration Geophysics 39 V ð MÞ ¼ A 1M þ B 1 ðm þ Z1 A M B Þq 1 ðm þ Z ; Xi ¼ --M: ð7þ Þq Equations 3 7, when solved simultaneously, yield A 1 ¼ R 1R 3 ½NPR 4 MDR Š NM½R 1 R 4 R R 3 Š ; Z 1 ¼ 6 4 Sphere (q = ) x +x x +x x +x Z P (x i, z) Z P (x i, z) Z P (x i, z) O q Z q r Horizontal cylinder (q = 1.) X O r X O X q q x x x A ¼ R R 4 ½NPR MDR 1 Š NM½R R 3 R 1 R 4 Š ; ð8þ ð9þ B 1 ¼ R 1R 3 ½WR 4 FR Š ; ð1þ ½R 1 R 4 R 3 R Š B ¼ R R 4 ½WR 3 FR 1 Š ; ð11þ ½R R 3 R 1 R 4 Š R 1 R 3 ½WR 4 FR Š þ Z q1 1 ½R 1 R 4 R 3 R Š VðÞZ q 31 R R 4 ½WR 3 FR 1 Š = q 1 ½R R 3 R 1 R 4 Š 7 5 ; ð1þ where F ¼ V ðnþþvð NÞ; W ¼ VðMÞþVð MÞ; D ¼ VðNÞ Vð NÞ; P ¼ VðMÞ Vð MÞ; R 1 ¼ N þ Z1 Þq 1 ; R ¼ðN þ Z Þq ; R 3 ¼ðM þ Z1 Þq 1 ; R 4 ¼ðM þ Z Þq : The depth Z 1 can be obtained by solving Equation 1 using a simple iteration method (Demidovich, and Maron, 1973) ifz is known. The iterative form of Equation 1 is given as Z 1f ¼ f ðz 1 jþ; Vertical cylinder (q =.5) Fig. 1. Cross-sectional view of sphere, horizontal cylinder and semi-infinite vertical cylinder models. ð13þ where Z 1j is the initial depth and Z 1f is the revised depth; Z 1f will be used as the Z 1j for the next iteration. The iteration stops when Z 1f Z 1j e, where e is a small predetermined real number close to zero. The e value has units of length and acts as an estimate of model depth resolution. However, Equation 1 can be used to simultaneously estimate the depth Z 1 and Z by using the depth-curves method developed by Abdelrahman et al. (). Equation 1 is applied to the input data yielding depth solutions Z 1 for a representative range of all possible Z values for fixed N and M values. The computed Z 1 values are plotted against Z values representing a depth-curve. The depth-curves should intersect at a single point, i.e. the value of Z 1 at the point of intersection is the depth to the first structure, and the value of Z gives the depth to the second structure. Theoretically, any two curves associated with two different values of M are sufficient to simultaneously determine Z 1 and Z. In practice, more than two values of M might be necessary because of the presence of noise in the data. Knowing Z 1 and Z from the depth-curves, A 1, A, B 1 and B can be determined from Equations 8, 9, 1 and 11, respectively. Because Z 1, Z, A 1, A, B 1 and B are known, the polarisation angles y 1 and y can be determined from the following equations using the relationships given in Equation 1 ¼ tan 1 B 1 ; ð14þ Z 1 A 1 ¼ tan 1 B Z A : ð15þ Once y 1 and y are known, the electric dipole moments K 1 and K can be determined from the following equations using also the relationships given in Equation K 1 ¼ A 1 cos 1 ; ð16þ K ¼ A : ð17þ cos It should be mentioned here that it is not necessary for all of the Z 1 solutions to be shallow and all of the Z solutions to be deep. This means that our inversion process is reversible. Theoretical examples In the following synthetic examples, the shapes of the two superimposed structures are assumed to be known a priori. Two superimposed structures Figure shows a composite SP anomaly consisting of the combined effect of a sphere with q 1 =, Z 1 = 3 m and K 1 = 1 mv, and a semi-infinite vertical cylinder with q =.5, Z = 1 m and K = 5 mv from the following expression DV 1 ðx i Þ¼ 1 X i cosð3 Þþ3 sinð3 Þ ðx i þ 9Þ 1:5 5 X i cosð6 Þþ1 sinð6 Þ ðxi þ 1Þ :5 : sphere þ semi-infinite vertical cylinder ð18þ In Figure, anomaly 1 and anomaly are due to the sphere and the semi-infinite vertical cylinder, respectively. Assuming q 1 = (sphere) and q =.5 (semi-infinite vertical cylinder), Equation 1 is then applied to the composite SP anomaly yielding depth solutions Z 1 for different Z values for N =5m and M = 1,, 3, 4 and 5 m. The e value used in this case is.1 m. The computed Z 1 values are plotted against Z values. The results are shown in Figure 3. It is verified that the solution for Z 1 and Z occurs at the common intersection of the depth-curves. Figure 3 shows the intersection at the correct depths Z 1 = 3 m and Z =1m. Contaminated with noise In this example, the composite SP anomaly, DV 1 (X i ), is contaminated with 5% random error using the following equation DV rnd1 ðx i Þ¼DV 1 ðx i Þ½1 þðrndðiþ :5Þ:5Š; ð19þ where DV rnd1 (X i ) is the contaminated anomaly value at X i, and RND(i) is a pseudorandom number whose range is (, 1).

3 31 Exploration Geophysics E.-S. M. Abdelrahman et al. Self-potential anomaly (mv) K 1 = 1 mv Model parameters K = 5 mv q 1 = 3 q = 6 Z 1 = 3 m Z = 1 m q 1 = q =.5 Sphere Semi-infinite vertical cylinder Anomaly 1 1 Anomaly Composite anomaly Horizontal position X (m) Fig.. Composite SP anomaly of a buried sphere and semi-infinite vertical cylinder Z 1 = 3 m Z = 1 m q 1 = (Sphere) q =.5 (Semi-infinite vertical cylinder) Z 1 = 3.1 m Z = 1.4 m q 1 = (Sphere) q =.5 (Semi-infinite vertical cylinder) Intersectionpoint Fig. 3. Interpretation of the data in Figure using the depth-curves method. Fig. 4. Data interpretation of Figure after adding 5% random errors using the present depth-curves method. The interval of the pseudorandom number is an open interval, i.e. it does not include the extremes and 1. Applying the depth-curves method to the noisy composite SP anomaly, the results are shown in Figure 4. The curves nearly intersect within a narrow region defined by.8 m < Z 1 < 3.4 m and 9.5 m < Z < 1 m. Averaging over the three intersections yields the solution Z 1 = 3.1 and Z = 1.4 m (Figure 4). The results are in good agreement with the depth parameters given in model Equation 18. This demonstrates that the present depth-curves method will give reliable results when applied to noisy SP data. Laterally offset structures A composite SP anomaly consisting of the combined effect of a horizontal cylinder and a semi-infinite vertical cylinder are computed using several offsets (D) between the horizontal cylinder (Z 1 = 5 m) and a semi-infinite vertical cylinder (Z = 1 m). The model equation is DV ðx i Þ¼ X i ðcos 5 Þþ5ðsin 5 Þ ðxi þ 5Þ 5 ðx i DÞðcos 75 Þþ1ðsinð75 Þ ððx i DÞ þ 144Þ :5 ; horizontal cylinder þ semi-infinite vertical cylinder ðþ where D =1,,3,4,5and6m. For each offset value D, the depth-curves method is applied to the data assuming q 1 = 1 and q =.5 and using an e value of.1 m. The results are shown in Figure 5.

4 Depth determination using SP anomalies Exploration Geophysics Z 1 = 4.95 m Z =1.15 m q 1 = 1. (H. cylinder) q =.5 (V. cylinder) Offset (D) = 1 m Z 1 = 4.84 m Z = 1.14 m q 1 = 1. (H. cylinder) q =.5 (V. cylinder) Offset (D) = m Z 1 = 4.64 m Z = 1. m q 1 = 1. (H. cylinder) q =.5 (V. cylinder) Intersectionpoint Z 1 = 4. m Z = 11.6 m q 1 = 1. (H. cylinder) q =.5 (V. cylinder) Offset (D) = 3 m Offset (D) = 5 m Z 1 = 4.4 m Z = 11.9 m q 1 = 1. (H. cylinder) q =.5 (V. cylinder) Offset (D) = 4 m Offset (D) = 6 m 1. Z 1 = m Z = 1 m.5 q 1 = 1. (H. cylinder). q =.5 (V. cylinder) Fig. 5. Data interpretation of the composite SP anomaly of a buried horizontal cylinder and semi-infinite vertical cylinder in which a lateral offset of 1,, 3, 4, 5 and 6 m is introduced into the horizontal position X i of Equation 19. Table 1 summarises the results of the depth curves shown in Figure 5. It is numerically verified that as the offset between the two sources increases, the error in the depths increases, particularly in Z 1. However, the depth to the semi-infinite vertical cylinder (Z ) is generally in good agreement with the model depth shown in Equation. These results suggest that the present method produces reliable depth estimates when the sources are offset laterally by an amount less than the depth of the buried horizontal cylinder (Table 1). Application to a single source anomaly Figure 6 shows a SP anomaly profile over a sphere (q =, Z =7m, y =3, K = 1 mv). The depth-curves method is applied to the SP anomaly assuming q 1 = (sphere) and q =.5 (semi-infinite cylinder) and using an e value of.1 m. The result is shown in Figure 7. It is numerically verified that when our method is applied to a residual SP anomaly due to a single buried structure, the depth Z 1 computed from Equation 1 will remain constant for all Z values. This criterion can be used to distinguish whether the

5 31 Exploration Geophysics E.-S. M. Abdelrahman et al. interpreted anomaly is due to a single source or two superimposed sources. Effect of using different N and M values In the above examples, we have used a fixed N value and different M values to construct the depth curves. In this subsection, we test a large range of N and M values to investigate whether or not our method would give consistent results. The depth-curves method is applied to the same composite anomaly shown previously in Figure using a large range of N and M values. The results are shown in Figure 8. It is verified that Table 1. Offset (L) (m) Computed depths from depth curves for each offset (L) using theoretical data. Computed depth % error in Z 1 Computed depth % error in Z the depth curves intersect at the correct depths Z 1 = 3 m and Z = 1 m. The depth curves shown in Figure 3 are similar to the depth curves shown in Figure 8, but they are not identical because of using different N and M values. This demonstrates that our method will give consistent results when using a large range of N and M values. Field example Figure 9 shows the Suleymnkoy SP anomaly profile from the Ergani Copper District, Turkey (Yüngül, 195; Bhattacharya and Roy, 1981). The origin of the anomaly profile is determined using a method described by Stanley (1977) to locate the origin for magnetic anomalies. This anomaly profile of 11 m length was digitised at an interval of.75 m. The depth-curves method (Equation 1) is applied to the SP anomaly profile using six possible combinations of geometric sources (sphere, horizontal cylinder and semi-infinite vertical cylinder) to determine Z 1 and Z using an e value of.1 m. To limit the number of solutions tested, Z 1 was constrained to being either a sphere or a horizontal cylinder. In cases where we assume sphere (q 1 = ) and sphere (q = ), or horizontal cylinder (q 1 = 1) and semi-infinite vertical cylinder (q =.5) combinations, Equation 1 Self-potential anomaly (mv) Model parameters K = 1 mv q = 3 Z = 7 m q = (sphere) Residual anomaly Horizontal position X (m) Fig. 6. A typical SP profile over a single sphere Z 1 = 3 m Z = 1 m q 1 = (sphere) q =.5 (semi-infinite vertical cylinder) N = 44 m, M = m N = 1 m, M = 36 m N = 1 m, M = 5 m N = 7 m, M = 17 m N = 3 m, Fig. 8. Interpretation of the data in Figure using the depth-curves method with a large range of N and M values Z = 7 m q = (sphere) N = 5 m Self-potential anomaly (mv) 1 1 Observed anomaly Horizontal position X (m) Fig. 7. Data interpretation of Figure 6 using the depth-curves method. Fig. 9. Sulemankoy SP anomaly, Ergani Copper District, Turkey.

6 Depth determination using SP anomalies Exploration Geophysics M = 19.5 m 1 M = 4.75 m M = 3.5 m N = 7.5 m 5 M = m M = 41.5 m Z 1 = 43.7 m Z = 4 m q 1 = (sphere) q =.5 (semi-infinite vertical cylinder) Fig. 1. Data interpretation of Figure 9 using the depth-curves method when q 1 = and q =, i.e. both anomalies are caused by spheres. N = 7.5 m M = 19.5 m M = 4.75 m M = 3.5 m M = m M = 41.5 m Fig. 11. Data interpretation of Figure 9 using the depth-curves method when q 1 = 1. and q =.5, i.e. when Z 1 is a horizontal cylinder and Z is a semi-infinite vertical cylinder. N = 7.5 m M = 19.5 m M = 4.75 m M = 3.5 m M = m M = 41.5 m Fig. 1. Data interpretation of Figure 9 using the depth-curves method when q 1 = and q =.5, i.e. when Z 1 is a sphere and Z is a semi-infinite vertical cylinder. converges to a depth solution, but the depth curves (Figures 1 and 11) do not intersect each other at a single point. This indicates that these two source combinations are not acceptable in interpreting the observed data. However, in cases where we assume sphere (q 1 = ) and horizontal cylinder (q = 1), horizontal cylinder (q 1 = 1) and horizontal cylinder (q = 1), or horizontal cylinder (q 1 = 1) and sphere (q = ) combinations, Equation 1 does not converge to a depth solution. This result also indicates that the previously mentioned three combinations of geometrical sources are not acceptable. The only successful combination of geometrical sources is obtained with sphere (q 1 = ) and semi-infinite vertical cylinder (q =.5). The depth curves in this particular case are shown in Figure 1. Figure 1 suggests that the shape of the buried deeper structure resembles a sphere model buried at a depth of 43.6 m and the shape of the buried shallow structure resembles a semi-infinite vertical cylinder model buried at depth of 3.9 m. The estimated shape and depth of the deeper structure are found to agree well with results obtained by Yüngül (195), Bhattacharya and Roy (1981) and Abdelrahman and Sharafeldin (1997) who assumed a sphere model. Moreover, it is emphasised that the depth of the sphere model is larger than the depth of the semi-infinite vertical cylinder model, which indicates that the vertical cylinder is passing through the sphere. This suggests that the two structures are not completely separated from each other. Conclusions The problem of determining the depths to two superimposed structures from a composite SP residual anomaly profile can be solved using the depth-curves method. The proposed method can be used to determine not only the depths of two superimposed structures from their composite SP anomaly, but also the depth of a single source from its residual anomaly. It can also be used to determine the shape of the superimposed sources. Theoretical and field examples have illustrated the validity of the method presented. Finally, we envisage the newly introduced formula, which represents the SP effect of two superimposed sources, will result in the future development of SP data interpretation methods. Acknowledgements The authors thank Dr Mark Lackie, the Associate Editor, Dr Kent Inverarity, and two capable reviewers for their keen interest and excellent and valuable comments that improved the original manuscript. The authors would also like to thank Prof. Dr M. M. Gobashy, Head and Chairman of the Geophysics Department, Cairo University, for his constant encouragement. References Abdelrahman, E. M., and Sharafeldin, S. M., 1997, A least squares approach to depth determination from residual self-potential anomalies caused by horizontal cylinders and spheres: Geophysics, 6, doi:1.119/ Abdelrahman, E. M., Ammar, A. A., Sharafeldin, S. M., and Hassanein, H. I., 1997, Shape and depth solutions from numerical horizontal self-potential gradients: Journal of Applied Geophysics, 37, doi:1.116/s (96)58-4 Abdelrahman, E. M., El-Araby, H. M., El-Araby, T. M., and Essa, K. S.,, A new approach to depth determination from magnetic anomalies: Geophysics, 67, doi:1.119/ Abdelrahman, E. M., Essa, K. S., Abo-Ezz, E. R., and Soliman, K. S., 6, Self-potential data interpretation using standard deviations of depths computed from moving-average residual anomalies: Geophysical Prospecting, 54, doi:1.1111/j x

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