**Authors**

**Abstract**

The analytic signal method is a semiautomatic method for estimating the location of causative bodies in magnetic and gravity methods. The application of analytic signal for interpretation of two dimensional (2D) structures was introduced by Nabighian (1972). The analytic signal is defined as a complex function in which the real and imaginary parts are a pair of Hilbert transforms. In other words, the analytic signal is a combination of horizontal and vertical gradients of potential field.

Analytic signal is a symmetric function with amplitude sensitive to parameters of the source. In case of 2D structures, the amplitude of the analytic signal is independent of the directional parameters such as inclination, declination and strike angle (Nabighian, 1972; Atchuta et al., 1981; Roest et al., 1992).

The depth of 2D structures can be estimated using the width of the analytic signal or the ratio of the analytic signal to its higher derivatives (Hsu et al., 1996; Roest et al., 1992). Source’s parameters of a dyke such as width, dip, strike, magnetization and depth can be estimated by analytic signal method (Bastani & Pedersen, 2001). The nth-order enhanced analytic signal is defined as the analytic signal of the nth-order vertical derivative of the potential field.

An automated method for estimating the depth, horizontal location and shape of 2D magnetic structures is the horizontal gradient of analytic signal method. This method is capable of interpreting low quality data because of using the first and second order derivatives of potential field in the main equations. The method of analytic signal estimates the horizontal location of the source by approximating the maximum amplitude of the signal; hence noise can affect the estimations. On the other hand, by using the horizontal gradient of analytic signal expressions, all of the source’s parameters could be approximated simultaneously.

In this method, equations of the analytic signal, Euler enhanced analytic signal and horizontal gradient of analytic signal are combined to derive a linear equation. Using the first order analytic signal, horizontal gradient of analytic signal and linear inversion method, the depth and horizontal location of 2D magnetic bodies are obtained. The location estimation is independent of the shape of the causative bodies. The causative body’s geometry is estimated as a structural index by applying the least squares method.

Data selection for solving the equations or width of windows is based on data quality. The optimum size is defined somehow to detect a signal specific anomaly and also variations of the anomaly in one window. In this study, in order to solve the equations of the horizontal gradient of analytic signal method, the data greater than twenty percent of maximum amplitude of the analytic signal were used.

The analytic signal-Euler deconvolution combined method is an automated method to estimate depth and shape of the sources. This method is used to interpret 2D & 3D magnetic and gravity data. After substituting the appropriate derivatives of the Euler’s homogeneous equation in the equation of the analytic signal, major independent equations which are used to estimate the depth and shape of causative bodies, are derived. The horizontal location of causative bodies is estimated by Euler method or locating the maximum amplitude of the analytic signal.

In this study, the accuracy and efficiency of each of the mentioned methods in interpretation of magnetic anomalies are evaluated. Methods were tested for different synthetic datasets provided by forward modeling. 2D magnetic models placed at different depths and random noise added for some models. Derivatives were calculated in frequency domain by using Fourier transform techniques. In this technique, bell-shapedness effect appears at the edges of the profiles. This effect could be corrected by linearly expanding the profiles. Upward continuation filter was applied on some synthetic data to decrease the noise level.

In this paper, the applicability of the horizontal gradient of analytic signal method and the analytic signal-Euler combined method were tested. Both methods estimate the parameters of the causative bodies without any prior information. In both methods, there is not any explicit dependence on directional parameters (e.g. magnetization) in the main equations; hence, as the results show, estimations were not affected by remanent magnetization. The results also show accurate estimations of the horizontal gradient of analytic signal method for shape and horizontal location and efficient estimations of the analytic signal-Euler deconvolution combined method for depth.

Analytic signal is a symmetric function with amplitude sensitive to parameters of the source. In case of 2D structures, the amplitude of the analytic signal is independent of the directional parameters such as inclination, declination and strike angle (Nabighian, 1972; Atchuta et al., 1981; Roest et al., 1992).

The depth of 2D structures can be estimated using the width of the analytic signal or the ratio of the analytic signal to its higher derivatives (Hsu et al., 1996; Roest et al., 1992). Source’s parameters of a dyke such as width, dip, strike, magnetization and depth can be estimated by analytic signal method (Bastani & Pedersen, 2001). The nth-order enhanced analytic signal is defined as the analytic signal of the nth-order vertical derivative of the potential field.

An automated method for estimating the depth, horizontal location and shape of 2D magnetic structures is the horizontal gradient of analytic signal method. This method is capable of interpreting low quality data because of using the first and second order derivatives of potential field in the main equations. The method of analytic signal estimates the horizontal location of the source by approximating the maximum amplitude of the signal; hence noise can affect the estimations. On the other hand, by using the horizontal gradient of analytic signal expressions, all of the source’s parameters could be approximated simultaneously.

In this method, equations of the analytic signal, Euler enhanced analytic signal and horizontal gradient of analytic signal are combined to derive a linear equation. Using the first order analytic signal, horizontal gradient of analytic signal and linear inversion method, the depth and horizontal location of 2D magnetic bodies are obtained. The location estimation is independent of the shape of the causative bodies. The causative body’s geometry is estimated as a structural index by applying the least squares method.

Data selection for solving the equations or width of windows is based on data quality. The optimum size is defined somehow to detect a signal specific anomaly and also variations of the anomaly in one window. In this study, in order to solve the equations of the horizontal gradient of analytic signal method, the data greater than twenty percent of maximum amplitude of the analytic signal were used.

The analytic signal-Euler deconvolution combined method is an automated method to estimate depth and shape of the sources. This method is used to interpret 2D & 3D magnetic and gravity data. After substituting the appropriate derivatives of the Euler’s homogeneous equation in the equation of the analytic signal, major independent equations which are used to estimate the depth and shape of causative bodies, are derived. The horizontal location of causative bodies is estimated by Euler method or locating the maximum amplitude of the analytic signal.

In this study, the accuracy and efficiency of each of the mentioned methods in interpretation of magnetic anomalies are evaluated. Methods were tested for different synthetic datasets provided by forward modeling. 2D magnetic models placed at different depths and random noise added for some models. Derivatives were calculated in frequency domain by using Fourier transform techniques. In this technique, bell-shapedness effect appears at the edges of the profiles. This effect could be corrected by linearly expanding the profiles. Upward continuation filter was applied on some synthetic data to decrease the noise level.

In this paper, the applicability of the horizontal gradient of analytic signal method and the analytic signal-Euler combined method were tested. Both methods estimate the parameters of the causative bodies without any prior information. In both methods, there is not any explicit dependence on directional parameters (e.g. magnetization) in the main equations; hence, as the results show, estimations were not affected by remanent magnetization. The results also show accurate estimations of the horizontal gradient of analytic signal method for shape and horizontal location and efficient estimations of the analytic signal-Euler deconvolution combined method for depth.

**Keywords**