The analytic signal and derivatives of the fractional orders for potential fields (applications in processing and interpretation)

Horizontal and vertical gradients of the potential fields are used routinely to enhance the edge of the magnetic and gravity sources; furthermore, they are used as useful tools in interpreting and processing of magnetic and gravity data. In general, the derivatives of the potential fields are divided into horizontal and vertical derivatives, and they have always been significant tools in interpreting and processing of potential data. The derivatives can be determined in two procedures, direct measuring when the data are recorded, and calculation using mathematical and numerical methods. Many interpreting methods, that estimate the depth, location and the shape of a potential source, are based on using the gradients of potential fields. For example, both analytic signals and Euler Deconvolution methods, that have been widely applied, basically use the potential field derivatives. In these methods, different kinds of first order derivatives or derivatives of other positive integer orders are commonly used. In the basic equations of these methods, it is possible to use the derivatives of fractional orders in place of derivatives of other positive integer orders. Derivatives are high pass filters. They intrinsically amplify any noise and shallow anomalies present in the data. Therefore, using high order derivatives would be less common. Instead of using high order derivatives, one should use fractional order derivatives of the field. Besides, negative order derivatives are applicable in these kinds of methods and equations, and they can be considered as an interesting property of negative order derivation that acts as a low pass filter. In addition, horizontal fractional derivatives can be used instead of reduction to the pole at low latitudes to eliminate the instability of the reduced data. In this paper, the methods of the field gradient calculation, their alternation, and the application of the fractional order derivatives in analytic signals and reduction to the pole were inquired. To study the effects of the derivatives of different orders, the method was applied to synthetic data generated by various magnetic models such as a thin dike, and a horizontal cylinder. In the next stage, to simulate the real cases, the data was contaminated by random noise. To produce the synthetic data, the forward modeling was used. Finally, the method was applied to an aeromagnetic data set acquired over an area in Sweden. According to the geological studies in this region, there exists a granite intrusive body with certain fractures in which Diabase veins have penetrated. The results show that the fractional order derivatives as well as negative order ones are useful in data processing, and they can be considered as the principle of some of interpreting methods. All of the processing steps in this paper have been performed by using the code that we have written in Matlab.
Horizontal and vertical gradients of the potential fields are used routinely to enhance the edge of the magnetic and gravity sources; furthermore, they are used as useful tools in interpreting and processing of magnetic and gravity data. In general, the derivatives of the potential fields are divided into horizontal and vertical derivatives, and they have always been significant tools in interpreting and processing of potential data. The derivatives can be determined in two procedures, direct measuring when the data are recorded, and calculation using mathematical and numerical methods. Many interpreting methods, that estimate the depth, location and the shape of a potential source, are based on using the gradients of potential fields. For example, both analytic signals and Euler Deconvolution methods, that have been widely applied, basically use the potential field derivatives. In these methods, different kinds of first order derivatives or derivatives of other positive integer orders are commonly used. In the basic equations of these methods, it is possible to use the derivatives of fractional orders in place of derivatives of other positive integer orders. Derivatives are high pass filters. They intrinsically amplify any noise and shallow anomalies present in the data. Therefore, using high order derivatives would be less common. Instead of using high order derivatives, one should use fractional order derivatives of the field. Besides, negative order derivatives are applicable in these kinds of methods and equations, and they can be considered as an interesting property of negative order derivation that acts as a low pass filter. In addition, horizontal fractional derivatives can be used instead of reduction to the pole at low latitudes to eliminate the instability of the reduced data. In this paper, the methods of the field gradient calculation, their alternation, and the application of the fractional order derivatives in analytic signals and reduction to the pole were inquired. To study the effects of the derivatives of different orders, the method was applied to synthetic data generated by various magnetic models such as a thin dike, and a horizontal cylinder. In the next stage, to simulate the real cases, the data was contaminated by random noise. To produce the synthetic data, the forward modeling was used. Finally, the method was applied to an aeromagnetic data set acquired over an area in Sweden. According to the geological studies in this region, there exists a granite intrusive body with certain fractures in which Diabase veins have penetrated. The results show that the fractional order derivatives as well as negative order ones are useful in data processing, and they can be considered as the principle of some of interpreting methods. All of the processing steps in this paper have been performed by using the code that we have written in Matlab.
 
Key words: Fractional derivatives,vertical derivatives, horizontal derivatives, analytic signal, Euler deconvolution, potential fields