# A Stable and High-Precision Downward Continuation Method of Magnetic Data
## 1 Introduction  
Downward continuation of magnetic data can be used to obtain the magnetic data measured on one surface to the data that would be measured on another arbitrary lower surface, with which the high frequency information induced by the causative sources can be enhanced. Over the years, it has been widely used in many practical applications [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. However, it suffers various disadvantages such as the non-convergence of the continuation operator, amplitude attenuation, and susceptibility to noise. Therefore, the downward continuation method has been the research focus over the past decades.  
Generally, the theories for improving downward continuation can be summarized into three categories. The first category is reforming the downward continuation operator, such as using the Taylor series method to solve the operator non-convergence problem and applying the horizontal derivative to substitute the vertical derivative in the operator to stable the process [1, 2, 3, 4]. The second category is replacing the downward continuation operators with other stable operators, such as adopting a three-order Adams-Bashforth formula method for downward continuation [5]. Using iterative upward continuation replaces the downward continuation process in the wavenumber domain [6, 7, 8] and combines the measured data of the lower interface with the Taylor series iterative method to derive the constrained downward continuation method [9]. The third is employing the equivalent source method to perform downward continuation, which is the main idea of this paper, and it is different from the above two categories. It simulates the forward process of the magnetic source rather than reforming or replacing the continuation operator and hasbeen widely used in potential field data processing, such as magnetic and gravity anomaly transformation, potential filed data interpolation, and noise reduction of the observation data [10; 11; 12; 13]. Because of its noise insensitivity and high continuation accuracy, this method also can be used in field continuation processing. For example, scholars have adopted a magnetic single-layer or dipole equivalent source to conduct continuation of aerial survey data, surface random point magnetic anomaly potential field data, and three component magnetic field data [14; 15; 16]. To further improve the accuracy of results, the wave number response of the regularized equivalent source has been studied, and then the iterative compensation algorithm and the optimization strategy of equivalent field source have been proposed [17; 18; 19]. Although above improved methods could stabilize the process and improve the precision indeed, most of these methods still have disadvantages such as susceptibility to noise and amplitude attenuation, especially when the continuation distance is large.  
In this paper, we focus on reducing the impact of noise and improving the computational accuracy during the long-distance continuation. The measured magnetic data of the lower surface is used innovatively to constrain the construction of the equivalent source. The unbiased predictive risk estimator method (UPRE) is adopted to estimate the regularization parameters for achieving a stable result. The theoretical formulas of the presented method are provided. The feasibility of this method is investigated on synthetic and real data.