Simple time‐variant, band‐pass filtering by operator scaling

A convolutional method of time‐variant, band‐pass filtering presented shows that a change of filter cutoff frequencies with time is achieved by frequency scaling the amplitude spectrum of a reference operator. According to the scaling property of the Fourier transform, this frequency scaling is actually accomplished by a simple time‐domain scaling of the reference operator in which the filter operator at a sample point on a seismogram is obtained by compressing the reference operator after multiplication by a constant value. Therefore, the length of filter operator changes as the cutoff frequencies and the pass band change with time; the higher the cutoff frequencies and the broader the passband, the shorter the operator length. The algorithm does not involve any complex‐valued arithmetic that may significantly reduce the computational efficiency if a small computer is used. Because the time‐variant convolution formula is exact, the change of cutoff frequencies is not limited to slowly varying or monotonic variations used in other algorithms. The way of changing cutoff frequencies restricts the passband of the filter to a constant value in terms of octaves. However, this restriction can be relaxed significantly in practical usage by a cascaded implementation if the Nyquist frequency is well above the passband of the filter. Computational efficiency of the method is quite comparable to that of the time‐invariant, band‐pass filtering. Tests of the method on both real and synthetic data sets confirm the effectiveness of the filter.