Journal of Applied Geophysics 136 (2017) 315–325
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Source depth estimation of self-potential anomalies by spectral methods Rosa Di Maio ⁎, Ester Piegari, Payal Rani Department of Earth Sciences, Environment and Resources, University of Naples Federico II, Largo San Marcellino, 10, I-80138 Napoli, Italy
a r t i c l e
i n f o
Article history: Received 1 February 2016 Received in revised form 8 November 2016 Accepted 9 November 2016 Available online 22 November 2016 Keywords: Self-potential Data inversion Spectral methods
a b s t r a c t Spectral analysis of the self-potential (SP) field for geometrically simple anomalous bodies is studied. In particular, three spectral techniques, i.e. Periodogram (PM), Multi Taper (MTM) and Maximum Entropy (MEM) methods, are proposed to derive the depth of the anomalous bodies. An extensive numerical analysis at varying the source parameters outlines that MEM is successful in determining the source depth with a percent error less than 5%. The application of the proposed spectral approach to the interpretation of field datasets has provided depth estimations of the SP anomaly sources in very good agreement with those obtained by other numerical methods. © 2016 Elsevier B.V. All rights reserved.
1. Introduction During last few decades, various spectral methods have been successfully applied for depth estimation of potential field sources, like gravity and magnetic anomaly sources (Spector and Grant, 1970; Negi et al., 1986; Maus and Dimri, 1995; Bansal et al., 2006; Bansal and Dimri, 2010). As concerns self-potential (SP) anomaly interpretation, spectral analysis approach has been first proposed by employing the Fourier transform. In particular, the Fourier amplitude and phase spectra have been analyzed to find the parameters of sheet-like sources (Atchuta Rao et al., 1982; Rao and Mohan, 1984) and polarized spherical and cylindrical bodies (Skianis et al., 1991; Asfahani et al., 2001). More recently, the energy spectrum method has been used to SP data analysis (Das and Agarwal, 2012). In this case, energy spectrum (power spectrum) is taken as the square of the Fourier amplitude spectrum and the SP source depth is estimated as half of the slope of the straight line of the spectrum. Lately, high-resolution spectral methods (HRSMs) of power spectrum estimation, such as multi-taper method (MTM) and maximum entropy method (MEM), have been applied by the authors for depth estimation of SP anomaly generated by some simple geometrical bodies (Rani et al., 2015). In particular, in the present work, an extended comparative study has been carried out among the three different spectral methods: Periodogram Method (PM), MTM and MEM to estimate the depth of sources of SP anomalous signals. Once power spectrum is computed, the depth of the anomalous body is estimated as half of the slope of straight line fitted to the log of power spectrum, P(k), versus the wavenumber, k, by following the
⁎ Corresponding author. E-mail address:
[email protected] (R. Di Maio).
http://dx.doi.org/10.1016/j.jappgeo.2016.11.011 0926-9851/© 2016 Elsevier B.V. All rights reserved.
approach of Spector and Grant (1970). In the following, after a brief theoretical introduction, the proposed methods are applied to synthetic SP data generated by geometrically simple anomalous bodies, such as sphere, horizontal and vertical cylinder, and inclined sheet. Then, the application of the three spectral methods to different SP field datasets is presented. In particular, the effectiveness of the proposed HRSMs has been confirmed by the good agreement with the source depth values estimated by other numerical approaches. Furthermore, the reliability of the obtained results has suggested the integration of the proposed spectral approach with other inversion methods for a full characterization of the SP anomaly source parameters (Di Maio et al., 2016a; Di Maio et al., 2016b). 2. Spectral analysis approach The spectral approach is based on selection of an appropriate method for power spectrum estimation. There are different methods for power spectrum evaluation, which can be categorized in parametric and nonparametric methods (Stoica and Moses, 2005). The nonparametric methods apply a band pass filter with a narrow bandwidth to a data sequence and use the filter output power divided by the filter bandwidth as a measure of the spectral content of the input data. The parametric methods select a model, estimate the model parameters for the given data and, then, compute the power spectrum by using the estimated parameters. The most accurate estimates of the power spectrum can be obtained by using parametric or nonparametric methods depending on if the data indeed satisfy or not the model assumed by the parametric methods (Stoica and Moses, 2005). In the present study, the power spectrum of SP data is computed by using the nonparametric methods PM and MTM and the parametric method MEM, which are shortly described in this section.
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2.1. PM
2.2. MTM
PM is a fast and conventional method to compute the power spectrum of discrete data. It estimates the power spectrum by computing the Discrete Fourier Transform (DFT) and appropriately scaling the magnitude squared of the result. The DFT is evaluated with a Fast Fourier Transform algorithm. In particular, the so called radix-2 FFT procedure is generally used, which is easy to encode and quite computationally efficient (Cooley and Tukey, 1965). The PM method provides reasonably high resolution for sufficiently large number of data, but is a poor spectral estimator because its variance is high and does not decrease with increasing data length. However, PM is considered a relevant basic spectral estimator as many other nonparametric estimators derive from it (Stoica and Moses, 2005).
MTM is a nonparametric method (Thomson, 1982; Percival and Walden, 1993) that reduces the variance of spectral estimates by using a small set of tapers rather than a single taper (or spectral window), like conventional PM does. The data are multiplied by orthogonal tapers and the power spectrum is obtained by averaging over the set of independent computed power spectra. The orthogonal tapers are constructed to minimize the leakage outside of a frequency band with bandwidth equals to 2pf, where f = 1 / (NΔ) is the Rayleigh frequency, N is the number of data points, Δ is the sampling interval and p is a suitably chosen integer related to the number of tapers J. Actually, since only the first 2p-1 tapers provide usefully small spectral leakage (Slepian, 1978; Thomson, 1982; Park et al., 1987), J should be b2p-1.
Fig. 1. (a) SP anomaly due to a sphere characterized by the parameters indicated in Table 1; (b) power spectrum estimated by PM; (c) power spectrum estimated by MTM; (d) order selection for AR process and (e) power spectrum estimated by MEM. The estimated order of the AR process is 35.
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Fig. 2. (a) SP anomaly due to a horizontal cylinder characterized by the parameters indicated in Table 1; (b) power spectrum estimated by PM; (c) power spectrum estimated by MTM; (d) order selection for AR process and (e) power spectrum estimated by MEM. The estimated order of the AR process is 27.
However, the optimal choice of p and J depends on the length and the properties of the data series under study (Ghil et al., 2002). 2.3. MEM MEM consists in approximating the data series under study by a linear autoregressive (AR) process of order M (Burg, 1975; Ulrych and Bishop, 1975; Kay and Marple, 1981; Dimri, 1992). Power spectrum estimation from MEM basically requires: (i) selection of the order M of the AR model, and (ii) evaluation of the AR coefficients. The latter can be estimated by different approaches. In the following, the Burg's approach (Burg, 1975) is applied, which finds a set of AR parameters that minimizes the sum of the squares of the forward and backward prediction
errors. The minimization is performed by using recursion equations and direct estimation of the reflection coefficients (Stoica and Moses, 2005). In selecting the order M of the AR process, several criteria have been proposed. The selection of M is crucial, as if M is too small, the spectral estimation will have poor resolution, on the other hand, if M is too large, power spectrum may display spurious peaks and line splitting. In the following, the Akaike's Final Prediction Error (FPE) criterion is used, which is suggested by Ulrych and Bishop (1975) for selection of M for geophysical data. The FPE is written as: FPE ¼
NþMþ1 2 σ ; N−M−1
ð1Þ
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Fig. 3. (a) SP anomaly due to a vertical cylinder characterized by the parameters indicated in Table 1; (b) power spectrum estimated by PM; (c) power spectrum estimated by MTM; (d) order selection for AR process and (e) power spectrum estimated by MEM. The estimated order of the AR process is 28.
where N is the number of data points in the profile, M is the order of AR process and σ2 is the variance of residual noise after fitting the AR model to the series. FPE is plotted versus M and the order corresponding to the first minimum in the plot is selected as the order of the AR process.
2.4. Source depth estimation The first attempt to invert power spectra for finding the depth of potential fields anomaly sources has been made by Spector and Grant (1970), who applied power spectrum analysis for interpretation of
Table 1 Parameters characterizing the sources of the SP anomalies shown in Figs. 1a, 2a and 3a. The last three columns report the depth values estimated by applying PM, MTM and MEM to the synthetic curves. Synthetic source model
Electric dipole moment (K)
Polarization angle (α)
Body center depth (z0)
Sphere Horizontal cylinder Vertical cylinder
−100 mV m −100 mV m −100 mV m
30° 30° 30°
30 m 30 m 30 m
Estimated depth (m) PM
MTM
MEM
25.8 ± 3.5 34.1 ± 4.3 40.5 ± 17.2
28.8 ± 1.1 31.4 ± 1.3 35.5 ± 3.6
29.6 ± 0.7 30.1 ± 0.4 31.4 ± 0.7
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Table 2 Depth values estimated for synthetic source models (sphere, horizontal and vertical cylinder) of SP anomalies by using different polarization angles, for z0 = 30 m. Synthetic source model
Polarization angle (α)
Estimated depth (m) PM
Sphere
Horizontal cylinder
Vertical cylinder
10° 20° 30° 40° 10° 20° 30° 40° 10° 20° 30° 40°
25.3 25.6 25.8 26.0 33.9 34.0 34.1 34.8 39.6 40.1 40.5 40.9
aeromagnetic maps. In particular, they modeled the magnetic anomaly source distribution by a statistical ensemble of rectangular prismatic bodies and found the following linear relationship between the logarithmic power spectrum of aeromagnetic data and the depth of the anomaly source:
MTM ± ± ± ± ± ± ± ± ± ± ± ±
3.8 4.0 3.5 4.1 6.5 6.3 4.3 4.8 20.0 18.5 17.2 17.5
28.7 28.8 28.8 28.9 31.2 31.4 31.4 31.5 35.4 35.4 35.5 35.6
MEM
± ± ± ± ± ± ± ± ± ± ± ±
1.4 1.2 1.1 1.0 1.7 1.5 1.3 1.2 4.6 4.2 3.6 3.0
29.2 29.3 29.6 29.9 29.9 30.0 30.1 30.3 29.9 31.2 31.4 31.5
± ± ± ± ± ± ± ± ± ± ± ±
0.4 0.6 0.7 0.6 0.5 0.5 0.4 0.6 0.5 0.7 0.7 0.6
the above mentioned spectral analysis approaches to the SP field data interpretation. In particular, as in the present work only single synthetic and/or field anomaly sources are considered, we refer to Eq. (2) for the estimation of the SP source depth. 3. Numerical analysis
logP ðkÞ ¼ logA−2jkjz;
ð2Þ
where P(k) is the power spectrum, A is the constant when the source distribution is uncorrelated, k is the wavenumber and z is the depth to the top of the anomalous source. Thus, from Eq. (2), z is estimated as half of the slope of the straight line fitted to the log P(k) as a function of k. If more than one straight line can be fitted in the logP − k plot, the slopes of subsequent line fits will give the shallower source depths for higher wavenumbers (Spector and Grant, 1970). Later, the direct proportionality between the logarithmic power spectrum and the source depth, Eq. (2), has been also retrieved for other different source geometries (Naidu, 1972; Hahn et al., 1976; Pedersen, 1991; Pawlowski, 1994; Garcia-Abdeslem and Ness, 1994). Then, Maus and Dimri (1995) improved the Spector and Grant approach by developing an inversion method that is able to estimate the depth of magnetic and gravity anomaly sources and to take into account the scaling behavior of the corresponding geophysical parameters, i.e. magnetic susceptibility and density. This method is based on scaling source distributions with power spectra proportional to f-β, where f is the wavenumber and β is the scaling exponent of the source distribution. The best values of the source depths and the scaling exponents are then obtained by minimizing the difference between the theoretical power spectrum of the assumed scaling source distribution and the power spectrum of the measured data. Actually, as the equations governing gravitational and electrical fields are the same, Rani et al. (2015) have proposed the application of
To check the efficiency of the spectral methods in SP data interpretation, first PM, MTM and MEM were applied to synthetic examples of SP anomalies generated by geometrically simple polarized bodies, such as sphere, horizontal and vertical cylinder, and inclined sheet. 3.1. SP anomalies due to polarized spherical and cylindrical sources The SP observed at any point P(x, z) on the surface along a profile line perpendicular to the strike over a polarized body whose center is placed at P0(x0, z0) can be expressed as (Yüngül, 1950; Bhattacharya and Roy, 1981; Agarwal and Srivastava, 2009): V ðxÞ ¼ K
ðz−z0 Þ sinα−ðx−x0 Þ cosα h iq ; ðx−x0 Þ2 þ ðz−z0 Þ2
ð3Þ
where K is the electric dipole moment, α is the polarization angle between the polarization and horizontal axes, x0 and z0 are the coordinates of the center axis of the body and q is the shape factor, which is 1.5, 1 or 0.5 in case of sphere, horizontal cylinder or vertical cylinder, respectively. The numerical analysis was performed by varying the shape factor, q, in Eq. (3) for computation of SP anomalies due to different polarized structures. For each structure, the analysis was carried out at varying the depth of the body, the polarization angle and the sampling interval.
Table 3 Depth values estimated for synthetic source models (sphere, horizontal and vertical cylinder) of SP anomalies by using different source depth values, for α = 50°. Synthetic source model
Body center depth (z0)
Sphere
20 30 40 20 30 40 20 30 40
Estimated depth (m) PM
Horizontal cylinder
Vertical cylinder
m m m m m m m m m
16.7 24.9 35.9 21.1 33.5 41.3 26.0 40.1 49.6
MTM ± ± ± ± ± ± ± ± ±
1.2 2.5 3.5 2.6 5.0 4.8 13.8 15.6 22.1
18.8 29.3 39.1 20.3 31.5 41.6 22.0 32.4 42.8
± ± ± ± ± ± ± ± ±
MEM 0.5 1.4 1.6 0.4 1.2 2.2 1.3 2.0 4.1
18.7 28.9 39.0 20.6 30.0 40.0 21.5 31.6 41.3
± ± ± ± ± ± ± ± ±
0.6 0.8 0.8 0.6 0.4 0.7 0.8 0.7 1.4
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Fig. 4. (a) SP anomaly due to an inclined sheet characterized by the parameters indicated in Table 1; (b) power spectrum estimated by PM; (c) power spectrum estimated by MTM; (d) order selection for AR process and (e) power spectrum estimated by MEM. The estimated order of the AR process is 36.
It is worth noting that if the radius of the structure is much lower than the depth of the center, then z0 could be approximated by the depth of the body top.
The results of the numerical analysis on the synthetical examples have shown that, for all the examined cases, MEM is able to provide depth values closer to actual values than those provided by PM and MTM. As examples, Figs. 1, 2 and 3 illustrate the power spectrum
Table 4 Parameters characterizing the inclined sheet source of the SP anomaly shown in Fig. 4a. The last three columns report the depth values estimated by applying PM, MTM and MEM at varying the polarization angle. Synthetic source model
Electric dipole moment (K)
Body center depth (z0)
Half-width (a)
Polarization angle (α)
Inclined sheet
−100 mV m
30 m
15 m
10° 20° 30° 40°
Estimated depth (m) PM 41.3 39.5 35.5 32.6
MTM ± ± ± ±
18.7 16.8 12.1 9.0
32.2 27.6 24.9 23.8
± ± ± ±
MEM 4.3 3.1 2.5 2.1
30.1 28.9 25.8 23.0
± ± ± ±
1.5 1.2 1.5 0.9
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Table 5 Depth values estimated for the inclined sheet source model by using different depth values, for α = 50°. Synthetic source model
Body center depth (z0)
Inclined sheet
20 m 30 m 40 m
Estimated depth (m) PM
MTM
MEM
20.9 ± 8.1 29.5 ± 5.8 34.3 ± 6.7
17.5 ± 1.2 25.8 ± 3.1 35.2 ± 3.0
18.2 ± 1.4 28.9 ± 1.2 38.6 ± 2.0
Table 6 Depth values estimated for synthetic source models (sphere, horizontal cylinder, vertical cylinder and inclined sheet) of SP anomalies due to different polarization angles by varying the number of tapers. Synthetic source model
Body center depth (z0)
Polarization angle (α)
Sphere
30 m
Horizontal cylinder
30 m
Vertical cylinder
30 m
Inclined sheet
30 m
10° 20° 30° 40° 10° 20° 30° 40° 10° 20° 30° 40° 10° 20° 30° 40°
Estimated depth from MTM (m) p=2 27.4 27.4 27.5 27.6 30.9 30.9 30.9 31.0 33.7 33.6 33.4 33.5 31.5 27.4 24.5 23.2
estimated by PM (Figs. 1b, 2b and 3b), MTM (Figs. 1c, 2c and 3c) and MEM (Figs. 1e, 2e and 3e) for SP anomalies generated, respectively, by a polarized sphere, a horizontal cylinder and a vertical cylinder for a profile length of 300 m and a sampling interval of 2 m. In particular, for the MEM analysis the chosen order of the AR processes corresponds to the first minimum in the plots of Figs. 1d, 2d and 3d. Table 1 indicates the source parameters for the examples shown in the figures. As it can be seen, the application of MEM is successful in determining the depth of the causative source with a percent error less than 5%. A numerical analysis has been then performed to investigate whether power spectrum estimation by PM, MTM and MEM is affected by variations in the values of the polarization angle, α. In Table 2, as an example, the results obtained for source models with center depth, z0, of 30 m and electric dipole moment, K, equals to − 100 mV m are shown. As it can be seen, the dependence of the power spectrum on the polarization angle is negligible for spherical and cylindrical sources. Additionally, a numerical analysis has been then performed by varying the depth of the anomalous source keeping fixed the polarization angle. As an example, in Table 3 the results obtained for α = 50° are shown for different source models. The SP values are computed for a profile length of 300 m with a sampling interval of 2 m, and for K = − 100 mV m and x0 = 0. As it can be seen, the application of MEM is successful in determining the depth of the causative source, with a percent error less than 5%. 3.2. SP anomaly due to inclined sheet The SP anomaly at any point P(x) on a profile perpendicular to the strike of a 2D inclined sheet can be written as (Murty and Haricharan, 1985; Sundararajan et al., 1998): (
) ½ðx−x0 Þ−a cosα 2 þ ðz0 −a sinα Þ2 V ðxÞ ¼ K ln 2 ; ½ðx−x0 Þ þ a cosα 2 þ z0 þa sinα
ð4Þ
± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±
p=3 1.5 1.4 1.3 1.4 3.2 2.9 3.5 2.2 5.5 5.1 4.9 4.3 4.6 4.2 3.6 2.5
28.7 28.8 28.8 28.9 31.2 31.4 31.4 31.5 35.4 35.4 35.5 35.6 32.2 27.6 24.9 23.8
± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±
p=4 1.4 1.2 1.1 1.0 1.7 1.5 1.3 1.2 4.6 4.2 3.6 3.0 4.3 3.1 2.5 2.1
30.5 30.6 30.7 30.7 32.0 32.2 32.1 32.3 35.2 35.2 35.2 35.1 35.0 30.8 27.8 25.2
± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±
1.6 1.5 1.4 1.7 2.2 1.8 1.9 1.3 2.3 2.2 2.1 2.0 2.4 1.9 1.7 1.5
where K is the electric dipole moment, x0 and z0 are, respectively, the position along the profile axis and the depth of the center of the sheet, α is the inclination with respect to the horizontal plane, and a is the half-width of the sheet. From Eq. (4), the SP anomaly of an inclined sheet is a function of five parameters, one more than in the case of spherical and cylindrical bodies, therefore more extended analysis is required for its numerical characterization. In Fig. 4, as an example, the power spectrum estimated by PM, MTM and MEM is reported for the SP anomaly computed over an inclined sheet for a profile length of 300 m and a sampling interval of 2 m. The source parameters for the example shown in the figure are reported in Table 4. As it can be seen, the application of MEM is successful in determining the depth of the causative source, even if with a percentage error larger than that found for the case of the sphere, horizontal and vertical
Table 7 Depth values estimated for synthetic source models (sphere, horizontal cylinder, vertical cylinder and inclined sheet) of SP anomalies by varying the order of AR process. The numbers in bold mark selected values of the orders for successful applications of MEM. Synthetic source model
Body center depth (z0)
Order
Estimated depth from MEM (m)
Sphere
30 m
Horizontal cylinder
30 m
Vertical cylinder
30 m
Inclined sheet
30 m
25 35 45 20 27 35 20 28 35 30 36 45
29.2 29.6 30.1 29.9 30.1 30.6 31.5 31.4 32.0 24.9 25.8 27.0
± ± ± ± ± ± ± ± ± ± ± ±
1.2 0.7 1.0 1.8 0.4 0.5 2.0 0.7 1.1 1.2 1.5 1.1
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Table 8 Depth values estimated for synthetic SP data generated by polarized structure like horizontal cylinder and inclined sheet with addition of different level of random noise. Synthetic source model
Body center depth (z0)
Random noise
Estimated depth (m) PM
Horizontal cylinder
30 m
Inclined sheet
30 m
5% 10% 15% 5% 10% 15%
cylinder. The loss of precision is due to the dependence of the power spectrum shape on the value of the half-width of the sheet, a. The performed numerical analysis has shown that the best agreement between
28.9 29.9 31.3 27.3 28.9 29.6
MTM ± ± ± ± ± ±
5.1 5.0 5.3 4.2 3.7 4.5
32.1 32.3 32.6 26.1 24.4 23.5
± ± ± ± ± ±
MEM 2.7 3.1 4.3 2.2 2.8 4.7
31.2 31.7 32.0 30.6 30.5 31.3
± ± ± ± ± ±
2.8 3.0 4.1 4.2 5.0 5.6
predicted and actual values of the source depth are found, keeping fixed α, when a is about z0 / 2. A numerical analysis has been then performed to study the behavior of the power spectrum estimated by PM, MTM and MEM for different
Fig. 5. (a) Sulleymonkoy SP anomaly, Ergani, Turkey (after Bhattacharya and Roy, 1981); (b) power spectrum estimated by PM; (c) power spectrum estimated by MTM; (d) order selection for AR process and (e) power spectrum estimated by MEM.
R. Di Maio et al. / Journal of Applied Geophysics 136 (2017) 315–325 Table 9 Estimated depth values for Sulleymonkoy anomaly source by different analysis methods. Sulleymonkoy SP anomaly: source depth estimated by different methods Method
Depth (m)
Characteristic curves (Yüngül, 1950) Nomograms (Bhattacharya and Roy, 1981) Analytic signal approach (Agarwal, 1984) Enhanced local wavenumber (Srivastava and Agarwal, 2009) Euler's deconvolution (Agarwal and Srivastava, 2009) PM MTM MEM
38.8 40.0 30.1 28.9 27.0 30.5 27.4 30.0
323
values of the polarization angle, α. As it can be seen from Table 4 for the inclined sheet the dependence of the power spectrum on the polarization angle is not negligible as for the case of sphere, horizontal and vertical cylinder. In particular, the depth of the anomaly source estimated from the slope of the log P(k) is less accurate as α grows. As for the previous analyzed structures, a numerical analysis has been then performed by varying the depth of the anomalous source keeping fixed the polarization angle. In Table 5 the results obtained for α = 50° are shown. As it can be seen, MEM is the most effective method in providing the actual values of the source depth with a percent error of about 5%.
Fig. 6. (a) Neem-Ka-Thana SP anomaly, Rajasthan Copper belt, India (after Srivastava and Agarwal, 2009); (b) power spectrum estimated by PM; (c) power spectrum estimated by MTM; (d) order selection for AR process and (e) power spectrum estimated by MEM.
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Table 10 Estimated depth values for Neem-Ka-Thana anomaly source by different analysis methods. Neem-Ka-Thana SP anomaly: source depth estimated by different methods Method
Depth (m)
Enhanced local wavenumber (Srivastava and Agarwal, 2009) Global optimization (Göktürkler and Balkaya, 2012) Genetic algorithm Particle swarm optimization Simulated annealing PM MTM MEM
10.8 18.6 17.6 16.3 22.1 17.2 16.8
4. Sensitivity analysis As it has been mentioned in Section 2, power spectrum estimation from MTM and MEM depends on number of tapers, J, and suitable selection of the order of the AR process, respectively. Therefore, to investigate how such parameters affect the results, a detailed analysis has been performed to check the appropriateness of the choice made. As concerns the number of tapers, the estimation of the power spectrum has been performed by using three different values of J. Specifically, we set J = 2p-1, and used p = 2, 3 and 4. As it can be seen from the results summarized in Table 6, it has been found that p = 3 produces less variance in spectral estimation, while for p = 2 and 4, a higher error is observed between actual and estimated depth values. Thus, the value J = 5 has been used for all the numerical analyses discussed in the present study. As concerns the order selection of the AR process required for spectral estimation by MEM, the order corresponding to the first minimum in the plot FPE vs order has been selected as the order of the AR process to consider for successful application of the method. To check the appropriateness of such a choice, an analysis has been performed by varying the order of the AR process, for fixed values of depth and polarization angle. As it can be seen from Table 7, the estimated depths of the anomaly source obtained by considering lower and higher orders with respect to the order corresponding to the first minimum in the plot FPE vs order (indicated with numbers in bold) show higher error. Thus, the adopted criterion for the choice of the order has been validated. Finally, in order to check the stability of the proposed inversion methods to noisy data, a numerical analysis has been performed by adding different levels of Gaussian random noise to original synthetic datasets, at varying depth and polarization angle of the anomalous body. For all the examined cases, it has been found that MEM is able to provide the actual depths with the lowest percent error (b12%). As an example, Table 8 summarizes the results obtained for the SP anomaly curves generated by a polarized horizontal cylinder and an inclined sheet with the addition of 5%, 10% and 15% of random noise. The SP values are computed for a profile length of 300 m with a sampling interval of 2 m, and for K = −100 mV m, z0 = 30 m and α = 50°. 5. Applications to field data The spectral methods PM, MTM and MEM have been applied to the analysis of different examples of SP field data taken from the published literature. 5.1. Sulleymonkoy SP anomaly, Ergani, Turkey As a first example, the Sulleymonkoy SP anomaly, from Ergani (Turkey), has been analyzed. This anomaly has been previously studied by many authors with different analysis techniques (Yüngül, 1950; Bhattacharya and Roy, 1981; Agarwal, 1984; Sundararajan and Srinivas, 1996; Agarwal and Srivastava, 2009; Srivastava and Agarwal, 2009). For our study, the anomaly has been digitized after
Bhattacharya and Roy (1981) at a sampling interval of 6 m for a profile length of 264 m. Fig. 5 shows the digitized SP anomaly and the power spectrum computed by PM, MTM and MEM. The obtained depth value from the spectral approach is in the range of 27 m to 30 m, which is in good agreement with the depth values provided by other authors as summarized in Table 9. 5.2. Neem-Ka-Thana SP anomaly, Rajasthan Copper belt, India The second example of field data interpretation concerns a SP anomaly observed in the Ahirwala deposit of the Neem-Ka-Thana Copper belt, Rajasthan, India (Reddi et al., 1982). The Neem-Ka-Thana SP anomaly has been previously studied by using different sophisticated interpretation methods (e.g. Srivastava and Agarwal, 2009; Göktürkler and Balkaya, 2012). For our analysis, the anomaly has been digitized after Srivastava and Agarwal (2009) at a sampling interval of 2 m for a profile length of 250 m. Fig. 6 illustrates the power spectrum estimated by PM, MTM and MEM for the digitized SP curve. As it can be seen, both depth values provided by MEM and MTM are in good agreement with the depth values estimated by global optimization approach, as summarized in Table 10. Anyway, as in the previous case, the percentage error associated with the depth value obtained by using MEM is the smallest one. 6. Conclusions Many applications of spectral analysis to inversion processes of gravimetric and magnetic data have demonstrated that the spectral approach is a useful tool for characterizing anomaly sources of potential field data. In the present paper, three high-resolution spectral methods, i.e. Periodogram (PM), Multi Taper (MTM) and Maximum Entropy (MEM) methods, have been proposed for the interpretation of natural electric potential fields, in particular for estimating the depth of the causative sources. An extended comparative study among the three proposed methods has been performed by applying the spectral methods to SP synthetic data generated by simple geometrical bodies, such as sphere, horizontal and vertical cylinder and inclined sheet. As a result of this numerical analysis, it has been found that generally MEM is able to give depth values closer to actual values than those provided by PM and MTM. The effectiveness of the proposed spectral approach has also emerged from the analysis of SP experimental signals. Indeed, a very good correlation of the obtained depth values with those provided by other numerical methods has been found. Although the spectral approach is able to characterize the SP anomaly only providing depth values of the source, the effectiveness of the proposed HRSMs suggests the use of such approach in combination with other inversion methods for a full characterization of the SP anomaly source parameters (Di Maio et al., 2016a; Di Maio et al., 2016b). Acknowledgments The authors are very grateful to the Reviewers, who made useful suggestions and criticisms for improving the manuscript. References Agarwal, B.N.P., 1984. Quantitative interpretation of self-potential anomalies. 54th Annual International Meeting, SEG. 154-157 Expanded Abstract. Agarwal, B.N.P., Srivastava, S., 2009. Analyses of self-potential anomalies by conventional and extended Euler deconvolution techniques. Comput. Geosci. 35, 2231–2238. Asfahani, J., Tlas, M.J., Hammadi, M., 2001. Fourier analysis for quantitative interpretation of self-potential anomalies caused by horizontal cylinder and sphere. Journal of King Abdulaziz University, Earth Sciences 13, 41–54. Atchuta Rao, D., Ram Babu, H.V., Sivakumar Sinha, G.D.J., 1982. A Fourier transform method for the interpretation of self-potential anomalies due to two-dimensional inclined sheets of finite depth extent. Pure and App. Geophys. 120, 365–374. Bansal, A.R., Dimri, V.P., 2010. Scaling spectral analysis: a new tool for interpretation of gravity and magnetic data. Earth Sci. India e-Journal 3 (I), 54–68.
R. Di Maio et al. / Journal of Applied Geophysics 136 (2017) 315–325 Bansal, A.R., Dimri, V.P., Vidya Sagar, G., 2006. Depth estimation from gravity data using the maximum entropy method (MEM) and multi taper method (MTM). Pure and App. Geophys. 163, 1417–1434. Bhattacharya, B.B., Roy, N., 1981. A note on the use of nomograms for self-potential anomalies. Geophys. Prospect. 29, 102–107. Burg, J.P., 1975. Maximum Entropy Spectral Analysis. Ph.D. Thesis. Stanford University, California . Cooley, J.W., Tukey, J.W., 1965. An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297–301. Das, B., Agarwal, B.N.P., 2012. Depth determination of 2-D SP anomaly source using Energy Spectrum method and its advantages. 9th Biennial International Conference & Exposition on Petroleum Geophysics, Hyderabad 2012. P-187 Expanded Abstract. Di Maio, R., Piegari, E., Rani, P., Avella, A., 2016a. Self-potential data inversion through the integration of spectral analysis and tomographic approaches. Geophys. J. Int. 206, 1204–1220. Di Maio, R., Piegari, E., Rani, P., Avella, A., 2016b. A comparative analysis of SP data inversion by spectral, tomographic and global optimization approaches. 78th EAGE Conference and Exhibition 2016. http://dx.doi.org/10.3997/2214-4609.201600560 . Dimri, V.P., 1992. Deconvolution and Inverse Theory: Application to Geophysical Problems. Elsevier Science, Amsterdam . Garcia-Abdeslem, J., Ness, G.E., 1994. Inversion of the power spectrum from magnetic anomalies. Geophysics 59, 391–401. Ghil, M., Allen, M.R., Dettinger, M.D., Ide, K., Kondrashov, D., Mann, M.E., Robertson, A.W., Saunders, A., Tian, Y., Varadi, F., Yiou, P., 2002. Advanced spectral methods for climatic time series. Rev. Geophys. 40, 3.1–3.41. Göktürkler, G., Balkaya, Ç., 2012. Inversion of self-potential anomalies caused by simplegeometry bodies using global optimization algorithms. J. Geophys. Eng. 9, 498–507. Hahn, A., Kind, E.G., Mishra, D.C., 1976. Depth estimation of magnetic sources by means of Fourier amplitude spectra. Geophys. Prospect. 24, 287–308. Kay, S.M., Marple, S.L., 1981. Spectrum analysis - a modern perspective. Proceedings of the IEEE 69, 1381–1419. Maus, S., Dimri, V.P., 1995. Potential field power spectrum inversion for scaling geology. J. Geophys. Res. 100, 12605–12616. Murty, B.V.S., Haricharan, P., 1985. Nomograms for the complete interpretation of spontaneous potential profiles over sheet-like and cylindrical two-dimensional sources. Geophys. 50, 1127–1135. Naidu, P.S., 1972. Maximum likelihood (ML) estimation of depth from the spectrum of aeromagnetic fields. Pure Appl. Geophys. 95, 141–149. Negi, J.G., Dimri, V.P., Agarwal, P.K., Pandey, O.P., 1986. A spectral analysis of the profiles for thickness estimation of flood basalt of India. Expl. Geophys. 17, 105–111.
325
Park, J., Lindberg, C.R., Vernon III, F.L., 1987. Multitaper spectral analysis of high-frequency seismograms. J. Geophys. Res. 92, 12,675–12,684. Pawlowski, R.S., 1994. Green's equivalent-layer concept in gravity band-pass filter design. Geophysics 59, 69–76. Pedersen, L.B., 1991. Relations between potential fields and some equivalent sources. Geophysics 56, 961–971. Percival, D.B., Walden, A.T., 1993. Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques. Cambridge University Press, Cambridge . Rani, P., Di Maio, R., Piegari, E., 2015. High-resolution spectral analysis methods for selfpotential data inversion. SEG Technical Program Expanded Abstracts. 2015, pp. 1596–1601. Rao, S.V.S., Mohan, N.L., 1984. Spectral interpretation of self-potential anomaly due to an inclined sheet. Curr. Sci. 53, 474–477. Reddi, A.G.B., Madhusudan, I.C., Sarkar, B., and Sharma, J.K. (1982), An album of geophysical responses from base metal belts of Rajasthan and Gujarat Geological Survey of India Miscellaneous publication no. 51. Skianis, G.E., Papadopoulos, T.D., Vaiopoulos, D.A., 1991. 1-D and 2-D spatial frequency analysis of self-potential field anomalies, produced by a polarized sphere. Pure and App. Geophys. 137, 251–260. Slepian, D., 1978. Prolate spheroidal wave functions, Fourier analysis, and uncertainty - V: the discrete case. Bell System Technical Journal 57, 1371–1430. Spector, A., Grant, F.S., 1970. Statistical models for interpreting aeromagnetic data. Geophysics 35, 293–302. Srivastava, S., Agarwal, B.N.P., 2009. Interpretation of self-potential anomalies by enhanced local wavenumber technique. J. Appl. Geophys. 68, 259–268. Stoica, P., Moses, R.L., 2005. Spectral Analysis of Signals. Prentice Hall, New Jersey . Sundararajan, N., Srinivas, Y., 1996. A modified Hilbert transform and its application to self-potential interpretation. J. Appl. Geophys. 36, 137–143. Sundararajan, N., Srinivasa Rao, P., Sunitha, V., 1998. An analytical method to interpret self-potential anomalies caused by 2D inclined sheets. Geophysics 63, 1551–1555. Thomson, D.J., 1982. Spectrum estimation and harmonic analysis. Proc. IEEE 70, 1055–1096. Ulrych, T.J., Bishop, T.N., 1975. Maximum entropy spectral analysis and autoregressive decomposition. Rev. Geophys. Space Phys. 13, 183–200. Yüngül, S., 1950. Interpretation of spontaneous-polarization anomalies caused by spherical ore bodies. Geophysics 15, 237–246.
