Magnetic interpretation utilizing a new inverse algorithm for assessing the parameters of buried inclined dike-like geological structure

Abstract

A new algorithm has been established to interpret magnetic anomaly data due to inclined dike-like structure. This algorithm uses first horizontal derivative anomalies attained from magnetic anomaly data utilizing filters of sequential window lengths. The final estimated parameters are the half-width, the depth, angle of magnetization and amplitude factor of an inclined dike-like geological structure. A minimum variance criterion is used for selecting the most suitable variables. This algorithm has been realized to theoretical data without and with random noise. The effects of interference due to near structures have additionally been studied. The method was then applied to two field examples from Turkey and Peru, which demonstrate its effectiveness and accurateness. Thus, it is a respectable correspondence among the model parameters retrieved from this approach, drilling information, and the outcomes published in the literature. For example, in Turkey, we applied the technique to gauge the source variables and also the results were precise to w = 64.74 m and h = 87.65 m (2% and 3% errors, respectively) based on information from Aydin and Gelişli (Jeofizik 10:41–49, 1996).

Introduction

In magnetic elucidation problems, the anomalies owing to an inclined dike model have a wide use, e.g., oil, mineral, groundwater exploration and others (Smith et al. 2005; Abdelrahman et al. 2007a; Abdelrahman et al. 2012; Abo-Ezz and Essa 2016; Baiyegunhi and Gwavava 2017; Essa et al. 2018; Gadirov et al. 2018). However, in practice, an inclined dike configuration is often approximated by a vertical dike. This simple model may not be geologically accurate, but it is often times utilized in the geophysical interpretation to get the depth, width, and the thickness of a class of dike structures (Mehanee 2015; Mehanee and Essa 2015). Normal and rapid elucidation of such easy geometric models is administrated by hand through graphical approaches involving characteristic curves (Bean 1966; Koulomzine et al. 1970; Rao et al. 1972). However, the precision of the estimated outcomes by these methods depends on the accuracy of the residual field separated from an observed field.

In addition, several numerical-approaches are adjusted to understand the magnetic anomaly caused by an inclined dike-like structure, such as curve matching techniques, which depend on trial and error of fitting the measured and calculated profile (Dondurur and Pamukçu 2003). Werner (1953) designed the deconvolution method (Werner deconvolution) to investigate inclined magnetized dikes by separating the anomaly owing to a selected dike from the interference of neighboring dikes. In spectral analysis approaches (Cassano and Rocca 1974; Sengupta and Das 1975; Bhimasankaram et al. 1978), the source depth is evaluated from the power band of their magnetic anomalies; the drawbacks of this method are that the precision of the outcomes is affected by the precision of the drawn interpolating line and it is difficult to apply this method when there are more than one body positioned at various depths (Cassano and Rocca 1974).

Rao et al. (1981) utilized a fancy gradient technique that depends on a few distinctive points (designated on the amplitude/phase plots) from which the variables of the dike are determined. Keating and Pilkington (1990) devised an automatic numerical routine to infer the vertical magnetic gradient profiles of an inclined dike. Abdelrahman et al. (2007b) suggested a semiautomatic least squares approach that convolves the inclined dike model with a moving average filter. However, the challenge facing this approach in estimating the model parameters is choosing the correct origin location of the dike from the measured magnetic data. Cooper (2012) discussed a new semiautomatic method to interpret the magnetic anomalies of dike by determining the fraction of the derivatives of the whole magnetic field. The disadvantage of this method is that it is applicable to thin dikes with different dips. Another method relies upon the amplitude of the analytic signal, which was devised by Cooper (2015) for the magnetic anomaly from a thin sheet to calculate the depth to the upper surface, amount of dip, and product of susceptibility and thickness of the dike. Al-Garni (2015) used a neural network method to estimate several parameters of dipping dikes. Essa and Elhussein (2017) discussed a semiautomatic method that depends on the second horizontal derivative of the measured magnetic anomaly for delineating dipping dikes sources. The shortcoming of the usage of higher order of derivatives is that sensitivity to noise is emphasized.

In this paper, we establish a new algorithm for calculating the response of an inclined dike model using the first derivative. The suggested method relies on evaluating the horizontal derivative anomalies from measured magnetic data utilizing several filters of successive window lengths (graticule spacings). The advantages of using this methodology are adjusted to estimate the depth and half-width of the covered inclined dike structures and also it is less sensitive to noise. The technique relies on calculating the variance (Var) of depths for every half-width. The tiniest variance is employed as a principle condition for deciding the most effective depth and half-width of the buried structure and does not need any graphical utilities. The accuracy of this methodology is demonstrated at synthetic examples, using simulated data generated from a model with random errors and a statistical distribution. The suggested method is then applied to real field examples from Turkey and Peru. There is an acceptable agreement between the results gained by our approach and those attained from different methods. Thus, the depths estimated from this approach are consistent with those attained from drilling information and published ones.

The method

Hood (1964), McGrath and Hood (1970) and Essa and Elhussein (2017) present an equation for the magnetic anomaly response of an inclined dike (Fig. 1) as follows:

$$\begin{aligned} & H\left( {x_{j } ,h,w, \theta } \right) \\ & \quad = A_{c} \left[ {\sin \left( {\theta \times \frac{\pi }{180}} \right) \left( {\tan^{ - 1} \left( { \frac{{x_{j} + w}}{h} } \right){-}\tan^{ - 1} \left( { \frac{{x_{j} - w}}{h}} \right)} \right)} \right. \\ & \quad \quad \left. { - \frac{{\cos \left( {\theta \times \frac{\pi }{180}} \right)}}{2} \ln \left( {\frac{{\left( {x_{j} + w} \right)^{2} + h^{2} }}{{\left( {x_{j} - w} \right)^{2} + h^{2} }}} \right)} \right], \\ & \quad \quad j = 1, 2, 3, 4, \ldots N \\ \end{aligned}$$

(1)

where h (m) is the depth to the upper surface of an inclined dike, w (m) is the half-width of the inclined dike, θ (°) is the angle of magnetization, x_j (m) are the horizontal coordinates of the observation points, and A_c (nT) is the amplitude factor related to magnetization.

Fig. 1

A sketch diagram of inclined dike source

Using the following three points (x_j − s, x_j, x_j + s) on the anomaly profile, the first horizontal derivative (H_x) of the magnetic anomaly is given by subsequent expression:

$$H_{x} \left( {x_{j } ,h, w, \theta ,s} \right) = \frac{{H \left( {x_{j } + s} \right) - H \left( {x_{j } - s } \right)}}{2 s} ,$$

(2)

where s = 1, 2, …M spacing units which are called window lengths.

Substituting Eq. (1) in Eq. (2), the FHD (first horizontal derivative) of the magnetic anomaly is:

$$\begin{aligned} & H_{x} \left( {x_{j } , h, w, \theta ,s} \right)\\ & \quad = \frac{{ A_{c} }}{2 s}\left[ \sin \left( {\theta\times \frac{\pi }{180}} \right)\left( {\tan^{ - 1} \left({\frac{{x_{j } + s + w}}{h}} \right) + \tan^{ - 1} \left( {\frac{{ -x_{j} - s + w}}{h}} \right) + \tan^{ - 1} \left( {\frac{{ - x_{j} +s - w}}{h}} \right)} \right. \right. \\ &\quad \quad \left. \left.{ +\tan^{ - 1} \left( {\frac{{x_{j } - s - w}}{h}} \right)} \right) +\frac{{\cos \left( {\theta \times \frac{\pi }{180}} \right)}}{2}\ln\left( {\frac{{\left( {\frac{{(x_{j } - s + w)^{2} + h^{2} }}{{(x_{j} - s - w)^{2} + h^{2} }}} \right)}}{{\left( {\frac{{(x_{j } + s +w)^{2} + h^{2} }}{{(x_{j } + s - w)^{2} + h^{2} }}} \right)}}}\right) \right] \\ \end{aligned}$$

(3)

From the above equation and by rearrangement, we can gauge the depth (h_f), the half-width (w), the index angle (θ) and the amplitude factor (A_c) (“Appendix”) as follows:

$$h_{\text{f}} = \frac{3s + w}{{\tan \left( {\frac{P}{q}} \right)}}$$

(4)

where the factors P and q are defined in “Appendix”.

$$\theta_{c} = \tan^{ - 1} \left[ {\frac{{\frac{{H_{x} \left( { + s} \right). \ln \left( { \frac{{\left( {s - w } \right)^{2} + h^{2} }}{{\left( { s + w } \right)^{2} + h^{2} }}} \right)}}{{H_{x} \left( 0 \right)}} - \frac{1}{2} \ln \left( {\frac{{\left( {2s - w} \right)^{2} + h^{2} }}{{\left( {2s + w} \right)^{2} + h^{2} }}} \right)}}{{\tan^{ - 1} \left( {\frac{2s + w}{h} } \right) + \tan^{ - 1} \left( {\frac{w - 2s}{h} } \right) + 2\tan^{ - 1} \left( {\frac{ - w}{h} } \right)}}} \right].$$

(5)

$$A_{c} = \frac{{2s H_{x} \left( 0 \right)}}{{ \cos \theta \ln \left( {\frac{{\left( {s - w} \right)^{2} + h^{2} }}{{\left( {s + w} \right)^{2} + h^{2} }}} \right)}}.$$

(6)

The RMS (root-mean-square error) is estimated at various w values from the following equation:

$${\text{RMS}} = \sqrt {\frac{{\mathop \sum \nolimits_{i = 1}^{N} \left[ {H\left( {x_{j} } \right) - H_{\text{c}} \left( {x_{j} } \right)} \right]^{2} }}{N}} ,$$

(7)

where $H\left( {x_{j} } \right)$ is the measured field and $H_{\text{c}} \left( {x_{j} } \right)$ is the computed field. This is considered the misfit between the observed and calculated anomalies. The procedures and flowchart of applying our technique are summarized in Fig. 2.

Fig. 2

Generalized scheme for parameters estimation by using our method

The limitation of our method is based on: (a) the number of data points (larger than 21 points), (b) the convergence and stability of the minimizer, (c) a priori information accessible from different techniques that can be obtained as underlying model data.

Uncertainty analysis

We examined the uncertainties in estimating the model parameters (h, w, θ, A) of an inclined dike source by our method. For this purpose, each noise-free and noise-corrupted magnetic anomaly has been scrutinized and the impact of interference from neighboring bodies is considered.

Analysis of synthetic example

The total magnetic anomaly (ΔH) for an inclined dike with the parameters: $A_{c}$ = 1500 nT, θ = 40°, h = 9 m, w = 4 m, and profile length = 100 m, is

$$\begin{aligned} & \Delta H\left( {x_{j } } \right) \\ & \quad = A_{c} \left[ {\left( {\sin \left( {40^{o} \times \frac{\pi }{180}} \right)} \right)\left( {\tan^{ - 1} \left( { \frac{{x_{j} + 4}}{9} } \right){-}\tan^{ - 1} \left( { \frac{{x_{j} - 4}}{9}} \right)} \right)} \right. \\ & \quad \quad \left. { - \frac{{\cos \left( {40^{o} \times \frac{\pi }{180}} \right)}}{2} \ln \left( {\frac{{\left( {x_{j} + 4} \right)^{2} + 9^{2} }}{{\left( {x_{j} - 4} \right)^{2} + 9^{2} }}} \right)} \right]. \\ \end{aligned}$$

(8)

Equation (2) is utilized to evaluate the horizontal derivative anomalies from the magnetic anomaly (ΔH) utilizing diverse s values (s = 5, 6, 7, 8 and 9 m). We estimated h values at various w for each s value and afterward calculated the average depth and variance (Table 1) by using our inversion approach. Table 1 demonstrates the outcomes for noise-free data. The estimated parameters from the planned method are in fair agreement with the model of the 2D inclined dike (h = 9 m and w = 4 m). In the end, we can observe that the minimum variance (Var = 0) occurs at the true depth (h = 9 m) and true half-width (w = 4 m).

Table 1 Numerical results for a synthetic example with and without 10% random noise (h = 9 m, θ = 40°, A = 1500 nT, w = 4 m, and profile length = 100 m)

Subsequently, the real data are polluted with random noise; we imposed 10% random noise on the composite anomaly to see the impact of this noise on our algorithm. The FHD anomalies were computed using the same s values mentioned above.

Table 1 also shows the computational results for the noisy magnetic data. The average depth is 9.11 m, which is an error of 1.22% from the true h value and the estimated half-width is w = 4 m. The obtained solution for the different parameters corresponds to a minimum variance with an RMS which equal to 12.587 nT. This demonstrates that our method is valuable when applied to noisy magnetic data.

Analysis of the interference effect

The magnetic anomalies may be distorted due to nearby structures; a synthetic model composed of a 2D inclined dike (with h = 9 m, θ = 40°, A_c = 1500 nT and w = 4 m) and a vertical thin dike (with h = 4 m, α = 45°, K = 2500 nT and x_o = 40 m) was computed using a profile length of 120 m. The analysis by our method was applied to this synthetic data to evaluate and study the response of this neighboring structure on assessing the body parameters (h, w, θ and A_c) by using our new approach.

Figure 3 shows the composite magnetic anomaly due to the two structures with and without 10% random noise. The FHD technique was utilized to the magnetic anomaly without random noise using four graticule spacings (s = 5, 6, 7 and 8 m).

Fig. 3

A composite magnetic field anomaly of a buried inclined dike with h = 9 m, θ = 40°, A_c = 1500 nT and w = 4 m, and a vertical thin dike with h = 4 m, α = 45°, K = 2500 nT and x_o = 40 m, and profile length = 120 m with and without 10% random noise and a sketch diagram of the 2 sources

The values of h at several w for each s value were appraised by our method, and then the average and Var values of the different factors were calculated at various w for each s value (Table 2). From Table 2, the average depth of 8.33 m, w of 4.5 m, θ of 40.04° and A_c of 1208.03 nT correspond to the minimum RMS (RMS = 136.337 nT). The percentage error in depth is 7.44%, in half-width is 12.5%, in magnetization angle is 0.1% and in amplitude factor is 19.46%.

Table 2 Numerical results for a synthetic example composed of a 2D inclined dike (with h = 9 m, θ = 40°, A_c = 1500 nT and w = 4 m), and a vertical dike (with h = 4 m, α = 45°, K = 2500 nT and x_o = 40 m) with and without 10% random noise

The FHD technique was repeated with 10% random noise for the same previous spacings (s = 5, 6, 7 and 8 m). From Table 2, the average depth h_c is 8.19 m, w is 4.5 m, θ_c is 40.99° and A_c is 1176.46 nT; the corresponding minimum (RMS = 134.398 nT). The error of computed depth is 9% from the true depth (9 m), half-width is 12.5% from the true value (4 m), magnetization angle is 2.475% from the true value (40°); finally, the amplitude factor is 21.57% from the actual value (1500 nT).

Analysis of the effect of choosing origin

The selection of an incorrect origin of the magnetic profile from real data leads to errors in determining the inclined dike structure parameters. To determine the implication of choosing an incorrect origin, we used a synthetic model of A_c = 1500 nT, θ = 50°, h = 9 m, w = 4 m, and profile length = 100 m) with an error (1.5 m) in the source location (see Eq. 9) as follows:

$$\begin{aligned} & \Delta H\left( {x_{j } } \right) \\ & \quad = A_{c} \left[ {\left( {\sin \left( {50^{^\circ } \times \frac{\pi }{180}} \right)} \right)\left( {\tan^{ - 1} \left( { \frac{{(x_{j} - 1.5) + 4}}{9} } \right){-}\tan^{ - 1} \left( { \frac{{(x_{j} - 1.5) - 4}}{9}} \right)} \right)} \right. \\ & \quad \quad \left. { - \frac{{\cos \left( {50^{^\circ } \times \frac{\pi }{180}} \right)}}{2} \ln \left( {\frac{{\left( {(x_{j} - 1.5) + 4} \right)^{2} + 9^{2} }}{{\left( {(x_{j} - 1.5) - 4} \right)^{2} + 9^{2} }}} \right)} \right]. \\ \end{aligned}$$

(9)

Our investigation begins by utilizing Eq. (2) for the FHD separation anomalies from the magnetic anomaly (ΔH) using available s values (s = 5, 6, 7 and 8 m).

Table 3 displays the results after applying the error in the horizontal coordinate. The estimated parameters (h, w, θ and A_c) are 8.04 m, 4 m, 55.15° and 1392.76 nT, respectively, and correspond to the minimum RMS of 83.692 nT. The error in computed depth is 10.66%, in half-width is 0%, in index parameter is 10.3% while the error of estimated amplitude factor is 7.15%.

Table 3 Numerical results for a synthetic example composed of a 2D inclined dike (with h = 9 m, θ = 50°, A_c = 1500 nT and w = 4 m), with incorrect origin

Field examples

To test the validity and the rationality of the anticipated approach, we have explored two real field cases with increasing complexity of the geological sources collected from the available published literature.

Magnetic anomaly of the Bayburt–Sarihan skarn zone, Turkey

A vertical component magnetic field profile was taken over the Bayburt–Sarihan (northeast of Turkey) skarn dike by Dondurur and Pamukçu (2003) (Fig. 4). The length of this profile is 714.5 m, and it was digitized at 8.93 m sample interval. We have applied our new technique to determine the body parameters (z, d) using five successive windows (s = 62.51, 71.44, 80.37, 89.3, and 98.23 m) for the horizontal derivative anomalies. Table 4 shows that the parameters with the best-fit parameters are: w = 64.74 m, h = 87.65 m, θ = − 71.37° and A_c = − 409.25 nT (RMS = 69.353). Table 5 shows the comparison with other interpretation methods from the published literature.

Fig. 4

A vertical component magnetic anomaly Bayburt–Sarihan (northeast of Turkey) skarn dike by Dondurur and Pamukçu (2003)

Table 4 Numerical results for Bayburt–Sarihan (northeast of Turkey) skarn zone field example

Table 5 Comparison between numerical results of different methods for Bayburt–Sarihan (northeast of Turkey) skarn zone field example

Magnetic anomaly in the Marcona district, Peru

The magnetic field profile taken by Gay (1963) near the magnetic equator in the Marcona district, Peru, is shown in Fig. 5. The profile length is 1125 m and was digitized at 14.06 m. Following the previous outlined process, the horizontal derivative anomalies were calculated from Eq. (2) by using seven s value (s = 126.54, 140.6, 154.66, 168.72, 182.78, 196.84 and 210.9 m). For our algorithm, the retrieved parameters are as follows: h = 138.28 m, w = 196.84 m, θ = 40.49° and A_c = 1862.75 nT which are the best-fit model parameters (RMS = 48.5567 nT) (Table 6). The estimated results from our process are in sensible agreement with those published in the literature (Table 7).

Fig. 5

Observed magnetic anomaly near the magnetic equator at Marcona district, Peru (Gay 1963)

Table 6 Numerical results for Marcona district field example, Peru

Table 7 Comparison between numerical results of different methods for Marcona district field example, Peru

Conclusions

The estimation of the parameters of an inclined dike are very important in geophysical exploration. We have devised an algorithm, which is based on the first horizontal derivative technique to gauge the body parameters. Our new approach is easy, semiautomatic and it does not necessitate any graphical supports. To verify the accurateness and pertinence of our method, the approach has been applied to synthetic data without and with random noise, and applied it also to a more complicated synthetic model including interference effect, again with and without random noise.

From our outcomes, we show that the best-fit parameters for the inclined dike can be determined with reasonable accuracy from our new algorithm even if the observed data are tainted with noise. Furthermore, the method has been relevantly applied to two field data from Turkey and Peru, obtained over mineral rich deposits. The limitation of our approach concerns the convergence of s-values, i.e., not all s-values can give results. The good agreement of the results acquired from our method with those published in the literature shows that the inclined dike parameters can be reasonably and efficiently determined.

Appendix

Hood (1964), McGrath and Hood (1970) and Essa and Elhussein (2017) represent an equation for the total magnetic anomaly of an inclined dike (Fig. 1) as follows:

$$\begin{aligned} & H\left( {x_{j } ,h,w, \theta } \right) \\ & \quad = A_{c} \left[ {\sin \left( {\theta \times \frac{\pi }{180}} \right) \left( {\tan^{ - 1} \left( { \frac{{x_{j} + w}}{h} } \right){-}\tan^{ - 1} \left( { \frac{{x_{j} - w}}{h}} \right)} \right)} \right. \\ & \quad \quad \left. { - \frac{{\cos \left( {\theta \times \frac{\pi }{180}} \right)}}{2} \ln \left( {\frac{{\left( {x_{j} + w} \right)^{2} + h^{2} }}{{\left( {x_{j} - w} \right)^{2} + h^{2} }}} \right)} \right], \\ & \quad \quad j = 1, 2, 3, 4, \ldots N \\ \end{aligned}$$

(10)

where h (m) is the depth to the top of the inclined dike, w (m) is the half-width of the inclined dike, θ (°) is the angle of magnetization (index parameter), x_j (m) are the horizontal coordinates, and A (nT) is the amplitude factor.

Using three observation points (x_j − s, x_j, x_j + s) along the anomaly profile; the first horizontal derivative (H_x) of the total magnetic anomaly is given by the following expression:

$$H_{x} \left( {x_{j } ,h, w, \theta ,s} \right) = \frac{{H\left( {x_{j } + s} \right) - H\left( {x_{j } - s } \right)}}{2 s} ,$$

(11)

where s = 1, 2, …M spacing units which is called the graticule spacing or window length.

Substituting Eq. (10) in Eq. (11), the first horizontal derivative (FHD) of the total magnetic anomaly is given by:

$$\begin{aligned} & H_{x} \left( {x_{j } , h, w, \theta ,s} \right) \\ & \quad = \frac{{ A_{c} }}{2 s}\left[ {\sin \left( {\theta \times \frac{\pi }{180}} \right)\left( {\tan^{ - 1} \left( {\frac{{x_{j } + s + w}}{h}} \right) + \tan^{ - 1} \left( {\frac{{ - x_{j} - s + w}}{h}} \right)} \right.} \right. \\ & \quad \quad \left. { + \tan^{ - 1} \left( {\frac{{ - x_{j} + s - w}}{h}} \right) + \tan^{ - 1} \left( {\frac{{x_{j } - s - w}}{h}} \right)} \right) \\ & \quad \quad \left. { + \frac{{\cos \left( {\theta \times \frac{\pi }{180}} \right)}}{2}\ln \left( {\frac{{\left( {\frac{{(x_{j } - s + w)^{2} + h^{2} }}{{(x_{j } - s - w)^{2} + h^{2} }}} \right)}}{{\left( {\frac{{(x_{j } + s + w)^{2} + h^{2} }}{{(x_{j } + s - w)^{2} + h^{2} }}} \right)}}} \right)} \right]. \\ \end{aligned}$$

(12)

By substituting x_j = 0 in Eq. (12), the amplitude factor (A_c) can be estimated as follows when s > w:

$$A_{c} = \frac{{2s H_{x} \left( 0 \right)}}{{ \cos \left( {\theta \times \frac{\pi }{180}} \right) \ln \left( {\frac{{\left( {s - w} \right)^{2} + h^{2} }}{{\left( {s + w} \right)^{2} + h^{2} }}} \right)}}.$$

(13)

Using Eq. (13), Eq. (12) can be written as the follows:

$$\begin{aligned} & H_{x} \left( {x_{j } ,h, w, \theta ,s} \right) \\ & \quad = \frac{{H_{x} \left( 0 \right) }}{{\cos \left( {\theta \times \frac{\pi }{180}} \right) \ln \left( { \frac{{\left( {s - w } \right)^{2} + h^{2} }}{{\left( { s + w } \right)^{2} + h^{2} }}} \right)}} \\ & \quad \quad \times\, \left[ {\sin \left( {\theta \times \frac{\pi }{180}} \right)\left( {\tan^{ - 1} \left( {\frac{{x_{j } + s + w}}{h}} \right) + \tan^{ - 1} \left( {\frac{{ - x_{j} - s + w}}{h}} \right)} \right.} \right. \\ & \quad \quad \left. { + \tan^{ - 1} \left( {\frac{{ - x_{j} + s - w}}{h}} \right) + \tan^{ - 1} \left( {\frac{{x_{j } - s - w}}{h}} \right)} \right) \\ & \quad \quad \left. { + \frac{\cos \theta }{2}\ln \left( {\frac{{\left( {\frac{{(x_{j } - s + w)^{2} + h^{2} }}{{(x_{j } - s - w)^{2} + h^{2} }}} \right)}}{{\left( {\frac{{(x_{j } + s + w)^{2} + h^{2} }}{{(x_{j } + s - w)^{2} + h^{2} }}} \right)}}} \right)} \right] \\ \end{aligned}$$

(14)

Putting x_j = + s, x_j = − s, x_j = + 2s, x_j = − 2s, we get the following four equations, respectively:

$$\begin{aligned} & H_{x} \left( { + s} \right) \\ & \quad = \frac{{H_{x} \left( 0 \right) }}{{\cos \left( {\theta \times \frac{\pi }{180}} \right) \ln \left( { \frac{{\left( {s - w } \right)^{2} + h^{2} }}{{\left( { s + w } \right)^{2} + h^{2} }}} \right)}} \\ & \quad \quad \times\, \left[ {\sin \left( {\left( {\theta \times \frac{\pi }{180}} \right) \times \frac{\pi }{180}} \right)\left( { \tan^{ - 1} \left( {\frac{2s + w}{h} } \right)} \right.} \right. \\ & \quad \quad \left. { + \tan^{ - 1} \left( {\frac{w - 2s}{h} } \right) + 2 \tan^{ - 1} \left( { - \frac{w}{h} } \right)} \right) \\ & \quad \quad \left. { + \frac{{\cos \left( {\theta \times \frac{\pi }{180}} \right)}}{2} \ln \left( {\frac{{\left( {2s - w} \right)^{2} + h^{2} }}{{\left( { 2s + w} \right)^{2} + h^{2} }}} \right)} \right] \\ \end{aligned}$$

(15)

$$\begin{aligned} & H_{x} \left( { - s} \right) \\ & \quad = \frac{{H_{x} \left( 0 \right) }}{{\cos \left( {\theta \times \frac{\pi }{180}} \right) \ln \left( { \frac{{\left( {s - w } \right)^{2} + h^{2} }}{{\left( { s + w } \right)^{2} + h^{2} }}} \right)}} \\ & \quad \quad \left[ {\sin \left( {\theta \times \frac{\pi }{180}} \right) \left( {2 \tan^{ - 1} \left( {\frac{w}{h} } \right) + \tan^{ - 1} \left( {\frac{2s - w}{h} } \right) + \tan^{ - 1} \left( {\frac{ - 2s - w}{h} } \right)} \right)} \right. \\ & \quad \quad \left. { + \frac{{\cos \left( {\theta \times \frac{\pi }{180}} \right)}}{2} \ln \left( {\frac{{\left( {2s - w} \right)^{2} + h^{2} }}{{\left( {2s + w} \right)^{2} + h^{2} }}} \right)} \right] \\ \end{aligned}$$

(16)

$$\begin{aligned} & H_{x} \left( { + 2s} \right) \\ & \quad = \frac{{H_{x} \left( 0 \right) }}{{\cos \left( {\theta \times \frac{\pi }{180}} \right) \ln \left( { \frac{{\left( {s - w } \right)^{2} + h^{2} }}{{\left( { s + w } \right)^{2} + h^{2} }}} \right)}} \\ & \quad \quad \left[ {\sin \left( {\theta \times \frac{\pi }{180}} \right) \left( {\tan^{ - 1} \left( {\frac{3s + w}{h} } \right) + \tan^{ - 1} \left( {\frac{w - 3s}{h} } \right) + \tan^{ - 1} \left( {\frac{ - s - w}{h} } \right) + \tan^{ - 1} \left( {\frac{s - w}{h} } \right)} \right) } \right. \\ & \quad \quad \left. { + \frac{{\cos \left( {\theta \times \frac{\pi }{180}} \right)}}{2} \ln \left( {\frac{{\left( {\frac{{\left( {s + w} \right)^{2} + h^{2} }}{{\left( { s - w} \right)^{2} + h^{2} }}} \right)}}{{\left( {\frac{{\left( { 3s + w} \right)^{2} + h^{2} }}{{\left( { 3s - w} \right)^{2} + h^{2} }}} \right)}}} \right)} \right] \\ \end{aligned}$$

(17)

$$\begin{aligned} & H_{x} \left( { - 2s} \right) \\ & \quad = \frac{{H_{x} \left( 0 \right) }}{{\cos \left( {\theta \times \frac{\pi }{180}} \right) \ln \left( { \frac{{\left( {s - w } \right)^{2} + h^{2} }}{{\left( { s + w } \right)^{2} + h^{2} }}} \right)}} \\ & \quad \quad \left[ {\sin \left( {\theta \times \frac{\pi }{180}} \right)\left( {\tan^{ - 1} \left( {\frac{w - s}{h} } \right) + \tan^{ - 1} \left( {\frac{s + w}{h} } \right)} \right. } \right. \\ & \quad \quad \left. { + \tan^{ - 1} \left( {\frac{3s - w}{h} } \right) + \tan^{ - 1} \left( {\frac{ - 3s - w}{h} } \right)} \right) \\ & \quad \quad \left. { + \frac{{\cos \left( {\theta \times \frac{\pi }{180}} \right)}}{2} \ln \left( {\frac{{\left( {\frac{{\left( {3s - w} \right)^{2} + h^{2} }}{{\left( {3s + w} \right)^{2} + h^{2} }}} \right)}}{{\left( {\frac{{\left( {s - w} \right)^{2} + h^{2} }}{{\left( { s + w} \right)^{2} + h^{2} }}} \right)}}} \right)} \right] \\ \end{aligned}$$

(18)

By subtracting Eq. (16) from Eq. (15) and Eq. (18) from Eq. (17), we get the following two equations, respectively:

$$\begin{aligned} & H_{x} \left( { + s) - H_{x} ( - s} \right) \\ & \quad = \frac{{H_{x} \left( 0 \right) \sin \left( {\theta \times \frac{\pi }{180}} \right)}}{{ \cos \left( {\theta \times \frac{\pi }{180}} \right) \ln \left( { \frac{{\left( {s - w } \right)^{2} + h^{2} }}{{\left( { s + w } \right)^{2} + h^{2} }}} \right)}} \\ & \quad \quad \times\, \left[ {2 \tan^{ - 1} \left( {\frac{2s + w}{h} } \right) + 2 \tan^{ - 1} \left( {\frac{w - 2s}{h} } \right) + 4 \tan^{ - 1} \left( { - \frac{w}{h} } \right)} \right] \\ \end{aligned}$$

(19)

$$\begin{aligned} & H_{x} \left( { + 2s) - H_{x} ( - 2s} \right) \\ & = \frac{{H_{x} \left( 0 \right) \sin \left( {\theta \times \frac{\pi }{180}} \right)}}{{ \cos \left( {\theta \times \frac{\pi }{180}} \right)\ln \left( {\frac{{\left( {s - w } \right)^{2} + h^{2} }}{{\left( { s + w } \right)^{2} + h^{2} }}} \right)}} \\ & \quad \quad \left[ {2 \tan^{ - 1} \left( {\frac{3s + w}{h} } \right) + 2 \tan^{ - 1} \left( {\frac{w - 3s}{h} } \right) } \right. \\ & \quad \quad \left. { + 2 \tan^{ - 1} \left( {\frac{ - s - w}{h} } \right) + 2 \tan^{ - 1} \left( {\frac{s - w}{h} } \right)} \right] \\ \end{aligned}$$

(20)

Then by dividing Eq. (19) by Eq. (20), we get the following:

$$q = \frac{{\tan^{ - 1} \left( {\frac{2s + w}{h} } \right) + \tan^{ - 1} \left( {\frac{w - 2s}{h} } \right) + 2 \tan^{ - 1} \left( { - \frac{w}{h} } \right)}}{{\tan^{ - 1} \left( {\frac{3s + w}{h} } \right) + \tan^{ - 1} \left( {\frac{w - 3s}{h} } \right) + \tan^{ - 1} \left( {\frac{ - s - w}{h} } \right) + \tan^{ - 1} \left( {\frac{s - w}{h} } \right)}},$$

(21)

where

$$q = \frac{{H_{x} \left( { + s) - H_{x} ( - s} \right)}}{{H_{x} \left( { + 2s) - H_{x} ( - 2s} \right)}} .$$

From Eq. (21) and by rearrangement, we can calculate $h_{\text{f}}$ from the following equation:

$$h_{\text{f}} = \frac{3s + w}{{\tan \left( {\frac{P}{q}} \right)}} ,$$

(22)

where

$$\begin{aligned} P & = q \tan^{ - 1} \left( {\frac{3s - w}{{h_{\text{i}} }} } \right) + q \tan^{ - 1} \left( {\frac{s + w}{{h_{\text{i}} }} } \right) + q \tan^{ - 1} \left( {\frac{w - s}{{h_{\text{i}} }} } \right) + \tan^{ - 1} \left( {\frac{2s + w}{{h_{\text{i}} }} } \right) \\ & + \tan^{ - 1} \left( {\frac{w - 2s}{{h_{\text{i}} }} } \right) + 2\tan^{ - 1} \left( { - \frac{w}{{h_{\text{i}} }} } \right). \\ \end{aligned}$$

Equation (22) can be deciphered for h using the standard methods for solving nonlinear equations (Press et al. 1986), and its iteration form can be expressed as:

$$h_{\text{f}} = f\left( {h_{\text{i}} } \right) ,$$

(23)

whereh_i is the initial depth estimate and h_f is the revised depth, for the next iteration h_f will be used as h_i. The iteration stops when $\left| {h_{\text{f}} - h_{\text{i}} } \right| \le e$, where e is a small predetermined real number close to zero.

From Eq. (15), we can calculate the magnetization angle as follows:

$$\theta c = \tan^{ - 1} \left[ {\frac{{\frac{{H_{x} \left( { + s} \right). \ln \left( { \frac{{\left( {s - w } \right)^{2} + h^{2} }}{{\left( { s + w } \right)^{2} + h^{2} }}} \right)}}{{M_{x} \left( 0 \right)}} - \frac{1}{2} \ln \left( {\frac{{\left( {2s - w} \right)^{2} + h^{2} }}{{\left( {2s + w} \right)^{2} + h^{2} }}} \right)}}{{\tan^{ - 1} \left( {\frac{2s + w}{h} } \right) + \tan^{ - 1} \left( {\frac{w - 2s}{h} } \right) + 2\tan^{ - 1} \left( {\frac{ - w}{h} } \right)}}} \right].$$

(24)