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《Chinese Journal of Sinica》 1977-02
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WANG JI-LUN(The Metallurgical Geophysical Prospecting Company)  
In order to discern slowly varrying weak anomalies on a background of noise field and to deal with problem like limiting the instability of the solution due to high-frequency magnification of errors in the series of calculations such as the downward-continuation of potential field, this paper discusses the designing principles of an optimum linear digital filter in the least square sense. This may be transformed into the mathematical problem, namely how to select the optimum filtering function in the space of the L2 linear normed function. It would be very complicated and difficult, if the problem is to be solved directly in the space domain. We found that it may be mathematically much simplier and more rigorous, if we should directly select the trnasfer function or the wave-number response of the optimum linear filter in the wave-number domain by using the method of isoperimetric problem in calculus of variations. In this way, the expression of the transfer function of the selected optimum linear filter is quite simple, namely,L(f,k)=|Si(f, k)|2/{|Si(f,k)|2+λ|Ni(f,k)|2}where|Si(f,k)|2 and |Ni(f,k)|2 express the energy spectra (or the power spectra) of the filter input signal and noise respectively; f, k are wave numbers on the x and y directions.In regard to the above-mentioned two types of the problem and the two related optimum linear filters, the expressions of L(f,k) are the same. They differ only in the conditions of selecting the parameter (λ).After setting up the theoretical expression of the transfer function L(f,k) of the optimum linear filter, we should be able in the least square sense to examine various linear filtering methods, so far published in foreign and domestic literatures in solving the above-mentioned two types of the problems and to show that the optimum results of the linear filtering can be achieved theoretically for different signal and noise conditions. Thus, it provides theorectical criterion for designing two-dimensional linear digital filters.For the observed results of the harmonic functions of potential fields, the above theory can be applied easily to the designing of optimum linear digital filters, but only in the approximate manner.
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