Stable downward continuation of potential field data using Thikhoniv regularization for depth estimation of the mining bodies

Semi-automated interpretation methods in applied gravity and magnetic are based on the estimation of source parameters directly from the measured (processed) and/or transformed potential fields data. These methods are built on the introduction of a prior information on the properties of the desired solution, mainly at the mathematical level. In the majority of cases, the priori information is based on the recognition of a predefined source type response in the interpreted data. Transformations of geophysical potential fields (mainly in gravity and magnetic) play an important role in their processing and interpretation. Due to the harmonic property of potential field data derived from Laplace equation, it is possible to realize the operation of analytical continuation of them, upwards and downwards, in the source free domain. In the case, when we continue the data further from the sources, we speak about upward continuation, in the opposite direction (closer to the sources), we deal with downward continuation. This description of the operation is independent of the orientation of the vertical coordinate axis, which can be different depending upon the application (either pointing downwards or upwards). In geophysical data processing, analytical continuation is used in various situations: e.g. to compare airborne and ground geophysical potential surveys data (measured on different height levels). In the interpretation of potential field data, upward continuation is used to enhance the regional components in the original data by suppressing near surface sources manifestation. Conversely, the downward continuation is used to enhance the detection of shallower sources by extracting the local or residual anomalies. Downward continuation is also often used to calculate the depth of the important shallowest sources. A great variety of mathematical treatments of the classical analytical continuation problem, either in space or spectral domain, have been reported. Downward continuation of potential fields is a powerful, but very unstable tool used in the processing and interpretation of geophysical data sets. It has been analytically proved from potential field theory that in downward continuation, we can only continue an interconnected potential field function to the depth of its nearest source (its upper edge). In the very downward continuation process, it is common that even in depths shallower than first source, the noise is amplified. Treatment of the instability problem has been realized by various authors in different ways. The Tikhonov regularization approach is one of the most robust ones. Regularization approach (Tikhonov et al., 1968) gives a straightforward and elegant way to the solution of the problem of achieving stable downward continuation of potential fields. It is based on a low-pass filter, derived in the Fourier spectral domain, by means of a minimization problem solution. We highlight the most important characteristics from its theoretical background and present its realization in the form of a Matlab-based program (REGCONT) written by Pašteka et al (2012). As we have shown in the presented synthetic model studies and practical data transformation, the proposed regularization method gives stable results, which are relatively close to the correct values particularly at shallow continuation depths. In comparison with other approaches to stabilize downward continuation, it shows a relative small dependency on the sampling rate of the data sets to be interpreted. For the selection of the optimum regularization parameter value , the behavior of the constructed norm functions has been used. In the majority of cases, C-norm gave a better developed and necessary local minimum in the function shape, which is connected with the searched optimum value. Positions of local minima for the other Lp-norm functions give in general higher values of , which lead to more smooth solutions. On the other hand, in some cases these norm functions can give a better developed local minimum and so they can be better used for depth estimation purposes (mainly in the case of under-sampled data sets). We demonstrate very good stabilizing properties of this method on several synthetic models and also the real gravity and magnetic datasets respectively form Shavaz Iron ore and Yazd geological quadrangle. In Shavaz Iron ore, the depth to the top of the subsurface body was estimated 60 m which is in agreement with the drilling data. The main output of the proposed solution is the estimation of the depth to the source below the potential field measurement level. In this study, the REGCONT Matlab code was used.