### Appendix

### Magnetic method

The geomagnetic field of the Earth is complex with changes in time, space, direction, and intensity. The geomagnetic field at the surface of the Earth is considered to result from two processes. The first is from the Earth's interior (internal magnetic field), and the second from beyond the Earth (external magnetic field) (Le Mouël 1969).

The main or dipole field (which constitutes 99% of the measured field at the surface represents the internal field. Its origin is due to complex deep-seated magneto-fluid dynamic processes and the crustal field generated by magnetized rocks. The external field is generated by electric currents in the ionosphere and in the magnetosphere.

The magnetic field measured at the surface (*T*_{obs}) is the sum of three components:

*B*_{n}: The main or regular magnetic field, which corresponds to large wavelength anomalies (over thousands of kilometers) with intensity of several thousand nT.

*B*_{a}: A magnetic anomaly field that corresponds to short wavelength magnetic anomalies (over a few thousands of meters) and generally with maximum amplitude of the order of hundreds nT.

*B*_{t}: Transient or temporary field, with much lower and variable intensity with the time. It originates mainly from external sources.

In a surface magnetic survey, only the magnetic anomaly field is of practical interest, and can be obtained according to the following equation:

$$\left| {B_{a} } \right| \, = \, \left| {T_{{{\text{obs}}}} } \right| \, - \, \left| {B_{n} } \right| \, + \, |{\text{DB}}_{t} |.$$

### Elements of the geomagnetic field

The Earth's magnetic field is a vector quantity, \(\overrightarrow{B}\) with properties of direction and intensity. In the spherical coordinate system, the components of the field (*B*_{r}, *B*_{ɵ}, and *B*_{φ}) are given by complex relationships.

The full determination of the magnetic field at a point in space necessitates measurement of three independent elements that can be chosen from among the seven following (Fig. 22):

1. The North component *X* = − B_{ϴ}.

2. The East component *Y* = B_{ϴ}.

3. The vertical component *Z* = − B_{r}.

4. The declination *D*: angle between the Magnetic Meridian and the Geographic Meridian, is positive or ″East″ when the magnetic meridian is to the East of the Geographic Meridian, and is given by one of the following formulae:

$$D = {\text{arctan}}\left( {Y/X} \right){\text{ or }}D = {\text{arcsin}}\left( {Y/H} \right).$$

5. The inclination *I*, angle between the field vector and the horizontal plane, is positive when the field vector points towards the center of the earth (Northern Hemisphere). It is expressed by:

$$I = {\text{ arctan}}\left( {Z/H} \right).$$

6. The total *F* (or sometimes *T*) is the intensity of the magnetic field. It is given by the following formula:

$$F=\sqrt{{X}^{2}+{Y}^{2}+{Z}^{2}.}$$

The intensity of the horizontal component H is expressed by the Cartesian components X and Y (North and East, respectively) and is given by the following formula:

$$H=\sqrt{{X}^{2}+{Y}^{2}.}$$

### International magnetic reference field formula (IGRF)

The mathematical representation models of the geomagnetic field was adopted according to an internationally agreed protocol under the auspices of the International Association of Geomagnetism and Astronomy (IAGA), which is a branch of the International Union of Geodesy and Geophysics (IUGG). These models known as the IGRF (International Geomagnetic Reference Field) are models of global reference, calculated every 5 years. The models are rounded off to the tenth of a degree for the geomagnetic field and to the eighth of a degree for the temporal variations, which correspond to a certain time.

These IGRFs, based on different models, are constructed from data available at a given time then extended by a model of secular variation at the time of the reference model. Due to the fact that it is impossible to predict exactly the variations in the time of the magnetic field, certain discrepancies are noted between the true geomagnetic field and the IGRF model. In the long term, it is possible to correct and improve the former IGRF with the input of data, including satellite collected data.

### Variations of the magnetic field (*B*
_{t})

When field measurements are being taken at a fixed location, it will be noticed that these measurements vary on the time scale of milliseconds to millions of years. Changes in short periods are essentially caused by electric currents in the ionosphere. These temporal variations are known as the diurnal variations. The variations in the long periods have their origin in the Earth's nucleus and are called secular variations. These are often represented on maps indicating the rate of change in nT/year for the intensity of the magnetic field, and in degree/year for the inclination or the magnetic declination.

### Magnetic field of the crust (*T*
_{nor})

The spectral analysis of the energy allows the separation of the main field (which is a complex magnetic fluid dynamic phenomena associated with the liquid outer core) from the crustal field, which is associated with the magnet rocks of the Earth's crust.

### Total magnetic field anomalies (*T*
_{anom})

The modern precession magnetometer (proton magnetometer) is used to measure the total magnetic field strength. Total field magnetic anomalies are obtained by subtraction of a regional component (for example the IGRF) from the measured field.

Referring to the anomalous field (magnetic anomalies) due to a certain source, as *T*_{anom}, this quantity can be represented by the following relationship (Fig. 23):

$$T_{{{\text{anom}}}} = \, T_{{{\text{obs}}}} - \, T_{{{\text{nor}}}} .$$

The intensity of the magnetic anomaly field is very small compared to that of the normal field (\(\left|{T}_{\mathrm{nor}}\right|\gg \left|{T}_{\mathrm{anom}}\right|\)). This condition is usually true for crustal anomalies.

The magnetic survey of the study area was acquired using a Geometrics G-856 AX Proton magnetometer. The total number of magnetic stations is 603 with a spacing interval ranging from 10 to 1500 m depending on the accessibility, which was sometimes limited due to the presence of sand dunes or zone restrictions (private and military).

### Reduction to magnetic pole technique

Reduction to the pole (RTP) is a correction that must be applied to the magnetic data before interpretation. If a body is located at the Earth’s magnetic poles, it would be geophysically characterized by a magnetic high situated directly above the body. If the body lies anywhere else, the Earth’s field produces a magnetic high to the South of a magnetic body in the northern hemisphere, and to the north of a body located in the southern hemisphere. This technique was first developed by Baranov (1957), to correct the position of the magnetic anomalies. The requisite mathematical equations, together with the coefficient of RTP in space domain, have been provided by Baranov and Naudy (1964) and Baranov (1975). The calculations for the RTP are carried out on the total field intensity magnetic data in the space domain by convolving the gridded data with a set of coefficients, the space domain operator has magnetic inclination of 37.90° (the inclination of the study area), and 1.79° declination angle. Equation for the reduction process (Sudhir 1988) is:

$${g}^{^{\prime}}=-\mu T\left(O\right)-\frac{1}{2\pi }\iint T\left(\rho ,\omega \right){\Omega }_{3}\left(\omega \right)\frac{\mathrm{d}\rho }{\rho }d\omega ,$$

where following Baranov’s notation:

$${\Omega }_{3}\left(\omega \right)=2\sum_{\lambda =1}{\left(-\eta \right)}^{\lambda }k\left(k+\mu \right)cosk\omega ,$$

*g′* = total magnetic field at point *o,* reduced to the pole; *μ* = *Sin I*, I being the inclination of the observed magnetic field; *T* (ρ,ω) = total field at the magnetic observation point (ρ,ω) (polar coordinates) with reference to the calculation point at the origin where the observed field is T (*O*), and

$$\eta =\text{ }(\text{I-sin}\left( \text{I} \right))/\text{cos}\left( \text{I} \right).$$

### Residual magnetic anomaly (least square polynomial technique)

The RTP magnetic map includes both regional and residual components. The regional anomalies mask some of the residual anomalies. Therefore, it is necessary to treat each anomaly in a separate map. The magnetic anomalies are separated into regional and residual components, using different techniques. The least square polynomial technique: Nettleton 1976; discussed in this method, consists of subtracting a polynomial surface that approximates the regional component. Agocs (1951) represented the regional field by a uniform plane surface, whereas Fajklewicz (1959) approximated the regional field be a second order polynomial and concluded that using higher order polynomials might result in the inclusion of part of the residual field in the regional picture. The regional component can be represented by each of the following polynomial surfaces (orders 1 to 3):

$${Z}_{1}\left(x,y\right)=\sum_{n=0}^{1}\sum_{s=0}^{n}{a}_{n-s,s}{x}^{n-s}{y}^{s}\quad {\text{plane surface}},$$

$${Z}_{2} (x,y)=\sum_{n=0}^{2}{\sum_{s=0}^{n}}{{a}_{n-s,s}}{x}^{n-s}{y^{s}s}\quad {\text{second-order surface}},$$

$${Z}_{3}(x,y)=\sum_{n=0}^{3}{\sum_{s=0}^{n}}{{a}_{n-s,s}}{x}^{n-s}{{y}^{s}}\quad {\text{third-order surface}},$$

where *Z* is the regional component*, **a*_{n-s,s} are ½ (*P*+1) (*P*+2) coefficients; *P* = 1, 2 or 3 is the order of the two-dimensional polynomials, x and y are the coordinates.

Also, the residual component can be separated by subtracting the previously obtained IGRF from the RTP map.

### The improved normalized horizontal tilt angle (INH)

The improved normalized horizontal tilt angle (INH) method helps in locating geological boundaries and magnetic/gravity source edges. The equation is given Li et al. (2014):

$${\text{INH}}=\tan^{-1} \left\{ \sqrt{{{\left( \frac{\partial P}{\partial x} \right)}^{2}}+ {{{ \left( \frac{\partial P}{\partial y} \right) }^{2}}}}\left/{a+\left| \frac{\partial P}{\partial z} \right|} \right.\right\},$$

where *P* is the potential field (gravity field and magnetic field) and *a* is a positive constant value and is decided by the interpreter.

### Horizontal derivative (HD)

The HD was developed by Cordell and Grauch (1985), where:

HD = \(\sqrt{{\left(\frac{\partial P}{\partial x}\right)}^{2}+{\left(\frac{\partial P}{\partial y}\right)}^{2},}\) and *P* is the potential field (gravity field and magnetic field).

### Gravity corrections and Bouguer anomaly

Bouguer anomaly at each station is given by:

$$g={G}_{abs}-\gamma +F-2\pi G\rho H+{T}_{c}\rho ={G}_{abs}-\gamma +\left(\beta -2\pi G\rho \right)H+{T}_{c}\rho .$$

*g* is Bouguer anomaly, *G*_{abs} is absolute gravity at the station, *γ* is normal gravity, *F* = *βH* is free-air correction, *2πGρH* is Bouguer correction, *T*_{c}*ρ* is terrain correction.

### Gravity terrain correction

Terrain correction considers topography up to 60 km from each station. The effect of terrain is greater at closer area. Therefore, we use more precise terrain model at closer area. For terrain correction up to 20 m from the station, the correction is calculated assuming that terrain around the gravity station is two dimensional. For terrain correction from 20 m to 60 km from the station, the correction is calculated based on Hammer method using DEM (SRTM). We have used a computer code developed by Komazawa (1988) for gravity terrain corrections. More details about gravity corrections can be found at Saibi et al. (2019a, b).