Control of smoothness in non-linear filtering of geomagnetic data by a one-pass method with a piecewise cubic polynomial

A smoother realized by a one-pass method with a piecewise cubic polynomial and a modified Powell criterion is termed a PCP filter. Since its action is data-adaptive, the PCP filter is non-linear. Furthermore, this filler is also non-causal and in practice causes little phase shift in the output and/or residual data as long as the input data has no dominant components with frequencies within its transition band. An average amplitude response of the PCP filter for sinusoidal inputs is determined numerically for each value of the parameter μ, which appears in the modified Powell criterion and controls the smoothness in output of the filter. An objective criterion for the selection of μ in the practical application of the PCP filter is then obtained from this response. Furthermore, the cut-off frequency of the PCP filter is defined as the first frequency at which the attenuation of the average amplitude response becomes 3 dB. It is found that the cut-off frequency is a good measure of the smoothness in output of the PCP filter. This frequency decreases with increasing μ, meaning that the output of the PCP filter becomes smoother and the oscillations with longer periods can be separated as residuals as μ grows large. As μ increases, the average number of sample points between two adjacent knots becomes larger and the computational cost increases for data processing by the PCP filter.