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).

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)