linear filter for upward- down ward continuation by using wavelet transform and its application in magnetic data processing

linear filters are used for a wide range of magnetic science including noise attenuation, spatial derivatives, upward and downward continuation and reduction to the pole. the majority of these filters are spatially-invariant, meaning that the filter has a constant wave number response over the whole signal. in contrast, many signals which encountered real problems such as magnetic signals, typically exhibit a spatially-varying wave number content which motivates us to design filters with spatially-varying wave number responses. this leads to better preservation of anomaly gradients in the calculated derivatives than is possible using conventional fourier or space domain smoothing techniques. fourier transform because of its stability, the simple physical interpretation of the transform coefficients and the diagonalisation of spatially invariant linear operators in the fourier domain play an important role in magnetic processing. however, fourier filters cannot be designed to adapt to local properties of the signal or to generate spatially-varying filters. one of the methods for generating spatially-varying filters is based on the continuous wavelet transform (cwt) which provides new powerful tools in magnetic data processing. the wavelet transform is ideal for analysing signals such as magnetic signals that contain short duration, transient features. wavelet techniques can be used to provide solutions to problems that are difficult or impossible to solve using conventional global techniques such as fourier-based methods. the wavelet transform preserves both spatial and wave number information about a signal allowing us to design a range of spatially-varying filters that act on the wavelet coefficients. this method provides robust and efficient new frameworks for designing filters that are impractical to implement using conventional space or wave number domain techniques. this method is compared with other techniques in upward-downward continuation. we demonstrate the application of spatially-varying scale filters to the problem of upward and downward continuation from a level observation surface to a new irregular-height surface. downward continuation is the most difficult of these operations as it is highly numerically unstable and is very sensitive to high wave number noise. for comparison, conventional methods of downward continuation, such as the taylor-series and chessboard methods are used which for stabilizing a global low-pass filter are applied to the data to attenuate any high wave number noise that may create difficulties in the continuation procedure. the wavelet implementation produces a superior result compared with conventional techniques such as taylor-series and chessboard algorithms. in this study, the wavelet approach combined with the exponential smoothing filter produces sharper images than either the chessboard or taylor-series methods that are clearly evident in the case study and synthetic examples. in contrast, the taylor-series cosine roll-off filter is designed to ensure that the downward continuation is stable over the largest continuation distance. the chessboard method should theoretically be able to behave like the wavelet method by adjusting the amount of smoothing applied to each downward continuation slice. however, the sliding-rule filter does not appear to be as effective as the wavelet exponential filter. the large differences in the performance of each of these downward continuation methods highlight the significance in the choice of smoothing method. in addition, the wavelet exponential filter has the advantage of being locally adapted to the signal, which means that we do not need to oversmooth the signal when the local downward continuation distances are small but the chessboard method suffers from oversmoothing which is needed to prevent artifacts from continuing downward below the shallowest magnetic sources. one advantage of the wavelet and chessboard methods over the taylor-series method is simplicity to automatically generate the parameters needed to design the smoothing filters for the wavelet and chessboard methods. in contrast, the taylor-series method requires some trial-and-error intervention by the user. with a careful choice of smoothing parameters, the taylor-series method can be designed to perform equally as well as the wavelet method. however, the choice of these parameters is often difficult when the downward continuation distances are large.