## 1 Introduction  
Gravity exploration and magnetic exploration are the earliest and most widely used methods of geophysical exploration. They have been applied in mineral exploration, oil and gas surveys, and the geological mapping of different scales. In the above-mentioned applications, the calculation of the depth of the field source is one of the most important interpretation objectives, for which a number of automated methods have been proposed to quickly determine the depth, width, horizontal position and shape index of a geological body. The most widely used among these methods are Werner deconvolution [1; 2; 3], Euler deconvolution [4; 5; 6; 7], analytic signal [8; 9; 10; 11] and DEXP [12; 13]. Among these methods, Euler deconvolution does not require a priori knowledge about anomalous source magnetization as far as it is homogeneous throughout the anomalous body and requires neither high-speed computers nor efficient algorithms. Euler deconvolution is a semiautomatic interpretation technique proposed by [PERSON] et al. [5] which allows the fast processing of large data sets. The technique is based on the Euler equation for homogeneous functions and relates potential field measurements and their gradients. In Euler deconvolution, a sliding data window operator is applied in a piecewise way over the whole data set. Using the measured data inside the window, Euler deconvolution estimates the source positions by solving linear equations. However, because of the interference from adjacent field sources (background field data), the calculated solution is inconsistent with the source position. The background field is one of the main factors affecting the accuracy of inversion results. To reduce its influence, many methods have been developed. These methods can generally be divided into three categories. Firstly, taking into account a constant background, the method that was developed earliest is the most popularly applied way to deal with background. [PERSON] [4] creatively added a constant term to the Euler equation to correct the deviation of the inversion result, and this indeed reduces the influence of background on Euler solutions to a certain degree. [PERSON] et al. [5] implemented [PERSON]'s suggested approach to gridded data. After that, many people adopted this strategy, which treats the background field as a constant. [PERSON] [14] used the high-order vertical derivative of the anomaly to replace the measured anomaly for Euler deconvolution. The Hilbert transform of constant is zero, and [PERSON] and [PERSON] [15] used the Hilbert transform of the anomaly to replace the measured anomaly in Euler deconvolution. [PERSON] [16] considered the product of the structural index (SI) and background as a constant and proposed a finite-difference Euler deconvolution algorithm based on the finite-difference method. To some extent, the accuracy of inversion results was improved by taking the background field as constant. However, the background field data actually varies with the coordinates. Second, taking into account a linear background, [PERSON] [17] proposed a Euler algorithm based on the properties of the differential similarity transformation (DST), and the algorithm linearizes [PERSON]'s equation in the case of a linear background. When the central point of similarity (CPS) coincides with the source singular point, the operator becomes a zero at all observation points. [PERSON] et al. proposed two algorithms: MaG-SoundDST [18] and MaGSOundFDST [19]. MaGSOundDST is also based on the differential similarity transformation, and when the CPS coincides with a source's singular point, the property of DST of a magnetic or a gravity anomaly becomes zero or linear at observation points. In each sliding window, a series of assumed structural indices are used to calculate a 3D function that evaluates the linearity of the DST. MaGSOundFDST is based on the linearity of the finite-difference similarity transformation.