In all signals there will exist a certain amount of ‘noise’ or data in which we have no interest. In certain circumstances this data may have a characteristic frequency which we can remove if we know it. In other circumstances we may know the frequency range of the signal that we are interested and we can consequently remove any frequencies within the signal that fall outside this frequency band. We can perform this operation by filtering our original signal to remove any unwanted frequencies and leave us with the data in which we are interested.

In the case of volcano seismology we are primarily interested in time series which represent the motion of the ground as recorded by seismometers. We can filter time series using any of the four filters described below. The filter that will be chosen depends on the type of signal that you are trying to isolate in your original time series.

Low-Pass Filter – A low-pass filter removes all frequencies above a given cut-off frequency, effectively letting the low frequency components of the signal ‘pass through’ unaltered.

High-Pass Filter – The same principle as a low-pass filter but frequencies below a given cut-off are removed: the high frequency components are allowed to pass through.

Notch Filter – A notch filter removes all the frequencies present in a given frequency band. Again, the frequencies defining the band are termed cut-off frequencies.

Band-Pass Filter – This filter removes all the frequencies outside a given frequency band, letting the frequencies within the defined band pass through unaltered.  

Signal Processing

                                                                  a _* f = n                                                             (1)

In the frequency domain this convolution takes place by multiplication of the amplitude spectra such that:

                                                                A . F = N                                                            (2)

The characteristic of any filter is often best viewed in the frequency domain and when equation 2 is considered the effect of filtering a time series should be clear. However, the filter characteristic can be seen in both the time and frequency domain. An example of a filter characteristic in the time domain is shown in figure 1(a). In this case a Butterworth low-pass filter has been used. Any filter characteristic can be seen by convolving a single spike (a delta pulse) with the filter since the result of this convolution will simply return the time series of the filter. Figure 1(b) shows the amplitude spectra of the same filter, which is high for low frequencies and decays away to zero at higher frequencies. When we perform convolution with this filter in the frequency domain, we can see from equation 2 that this will result in the frequencies outside the cut-off frequencies being lost as multiplication by zero is taken place.

Figure 1 - (a) the filter characteristic of the Butterworth low-pass filter in the time domain and (b) in the frequency domain.

However, you will note that the filter characteristic in figure 1(b) does not have a sharp cut-off between the desired and undesired frequency ranges: the filter characteristic is tapered. Filters need to be tapered in order to prevent the ringing that will occur in the resultant time series as a result of the Gibbs Phenomenon (which occurs whenever there is a sharp cut-off in the frequency domain). However, we can adjust the steepness of the taper. The equation for the Butterworth low-pass filter is given below (from Gubbins, 2004):

                                                    |F_l(ω)|² = 1/ (1+ (ω)/ ω_c²ⁿ)                                               (3)

where ω is the frequency to be filtered, ω_c is the cut-off frequency of the low-pass filter and n is the number of poles of the amplitude spectrum. In equation 3 it is the number of poles, n, which controls the steepness of the filter cut-off. When choosing which value to use for n a trade off must be made between keeping unwanted frequencies beneath the taper and introducing time delays in the resulting time series: for high values of n you will have a narrower band of frequencies retained but the onset of the time series will be shifted forwards in time and greater ringing may occur. An example of increasing the number of poles for the low-pass filter affects the resultant time series is given in figure 2. Notice how both the onset of the signal and the location of the first peak are shifted forwards in time.

Figure 2 - The effect of increasing the number of poles (n) of the Butterworth low-pass filter characteristic in the time domain. Note the delayed onset of the signal and the movement of the peak of the signal in time as the number of poles is increased where for (a) n=1, (b) n=3, (c) n=5.

The increasing number of poles can also be seen to increase the sharpness of the cut-off in the frequency domain as seen in figure 3:

Figure 3 - The effect of increasing the number of poles (n) of the Butterworth low-pass filter. Note the steepening cut-off as the number of poles is inceasesd. (a) n=1, (b) n=3, (c) n=5.

One way of by passing the problems which can occur in the time domain as a result of filtering is to perform the convolution operation twice using the same filter, once forwards and once backwards, which is known as two-pass filtering. This creates a filter that is zero-phase and the problem of shifting the time series is generally solved. As can be seen in figure 4, the position of the peak is constant in time. However, it can be seen that the problem of ringing can still occur with increasing the number of poles. There is also the added problem that energy is introduced into the signal before the event has taken place. This can lead to confusion over the onset of sharp events in the filtered signal. The amplitude spectra of the two-pass filter can be seen in figure 5. It can be seen that the cut-offs of the two-pass spectra are slightly steeper than those of the one-pass filter for the same number of poles.

Figure 4 - The filter characteristic of the two-pass Butterworth low-pass filter in the time domain. The number of poles used in the low-pass filter are increased in each time series: (a) n=1, (b) n=3, (c) n=5. Note that although the location of the peak remains at the correct time, some ringing is still introduced into the time series with increasing numbers of poles.

Figure 5 - The amplitude spectra of the two-pass Butterworth low-pass filter in the frequency domain. Note how the cut-offs steepen with increasing number of poles: (a) n=1, (b) n=3, (c) n=5. Not also that the cut-offs of the two-pass filter are slightly steeper than those of the one-pass filter.