Hybrid Noise Filters

Chapter 4 Error Reduction by Optimal Methods 168

4.2 Noise Filters

4.2.2 Hybrid Noise Filters

Noise filters which decrease the amplitude of the spectrum at points where the signal-to-noise ratio is low can be constructed using a formula analogous to the results of Wiener's approach. However, these filters are no longer optimal in the statistical sense. To avoid phase changes in the signal, such filters should have positive real transfer function. Also, these must have amplitudes in the range of zero to one which increase as a function of the signal-to-noise ratio. Several possibilities are suggested below, and tested on the noise-contaminated synthetic signals for their correction effectiveness.

A modified version of Wiener's optimal noise filter is defined by the transfer function:

mE[O,z], N (4.2.8) where the frequency range is d.c. to the Nyquist frequency corresponding to the discrete time signal, and where )Zrn) is the Fourier amplitude spectrum of the mea- sured data assuming instrument correction is not required, similar to Eq. 4.2.5. If the measured data needs to be instrument-corrected, the noise filter should have a transfer function similar to Eq. 4.2.7, where Hrn is the instrument transfer function.

The power spectral density of the noise, E[l~rn)²], is that of the digitization and recording error. As has been previously discussed in Chs. 2 and 3, such noise can be assumed to be Gaussian-distributed white noise with most probable value zero and standard deviation a, usually of the order of 0.001 g for analog accelerographs.

It can be shown that the power spectral density of the noise is then constant over the whole frequency range of the spectrum and is defined by:

(4.2.9) where N is the number of data points in the discrete time signal.

Alternatively, the power spectral density of the noise can be extracted from the spectrum of the measured data. Typically, accelerograms are discretized at intervals of 0.01 sec, giving a Nyquist frequency of 50 Hz, but analog accelerographs have natural frequencies of about 25 Hz beyond which their response decreases sharply.

Also, ground motion spectra are usually very small above 25 Hz. Hence, it can be argued that between 25 Hz and 50 Hz the spectrum of the measured data primarily reflects the digitizing and recording noise. Since Gaussian white noise has a constant power spectral density, the average value of the power spectrum between 25 Hz and 50 Hz could be defined to be the proper estimate for E[)Llrn)²] through out the whole frequency range of the data between d. c. and 50 Hz. Thus,

1 p

N/2 p=N/2-P+l

I:

where ^Wp is approximately 25Hz.

Vm

E

[l,N/2],

(4.2.10)

The correction effectiveness of the modified Wiener noise filter is tested on signal Q11 CNS, which is the noise-corrupted version of synthetic signal Q11 C scaled as a small earthquake (i.e., the level of noise is large with respect to the signal). No other sources of error exist in the signal to be processed. This signal is composed of 200 modulated frequencies between 0.05 Hz and 25 Hz, and hence all frequency content in the spectrum of the noise-contaminated signal between 25 Hz and 50 Hz is due primarily to the added Gaussian white noise. Eq. 4.2.10 is used to estimate the power spectral density function of the noise. The value of the noise variance a² inferred by equating Eq. 4.2.9 and Eq. 4.2.10 is almost identical to the one used to generate the noise in the uncorrupted synthetic signal. Hence, under the assumption that the noise is Gaussian and white, then Eq. 4.2.10 can provide good estimates of the power spectral density function of the noise.

The normalized measure-of-error J is used to examine how well the modified Wiener filter reduces the noise in the signal, where:

(4.2.11) In this equation, Zi is the noise-contaminated and filtered acceleration, velocity or displacement, and Yi is the exact counterpart. Although in the following discus- sion J is computed using the time-domain results, because of Parseval's identity, Eq. 4.2.11 could be equally viewed as comparing the change in the noise level in the frequency domain.

The values of J for the error in the acceleration, velocity and displacement, in the cases where no filter or the modified Wiener filter are implemented, are summarized in Table 4.2.1. These results indicate that the modified Wiener filter is capable of reducing the error in the acceleration by 35%, in the velocity by 60%, and in the displacement also by 60%. The differences in the time histories between the unfiltered and the filtered cases can be observed in Figs. 3.4 and 4.2.1 respectively.

The plots of the error in the acceleration between the exact and the processed signals (top-right figure), show that the Wiener filter reduced the level of the white noise throughout the time history. Similarly, although the shape of the error in the velocity and the displacement have not changed much after implementation of the

noise filter, they have significantly decreased in amplitude. In particular, the drift in the displacement has decreased by 40%. The improvement in the displacement accuracy is primarily due to the low-frequency correction of the modified Weiner filter. The changes made elsewhere in the frequency domain where the filter varies erratically have little effect. Hence, the modified Wiener filter is capable of reducing the noise level in the time histories without affecting the predominant harmonics of the signal. The long-period errors primarily due to the shift in the acceleration temporal mean are still present, but have decreased in amplitude.

The modified Wiener filter obtained for signal Q11 CNS is illustrated in Fig.

4.2.2.a. It is not a traditional type of filter, in the sense that it depends on the signal- to-noise ratio of the spectrum, and hence will be highly erratic in the frequency domain and different from one signal to the next. However, this figure illustrates the concept that the transfer function is close to unity in the region where the signal-to-noise ratio is large (i.e., below 25 Hz), and is close to zero in the region where the ratio is small (i.e., above 25 Hz). In particular, the filter decreases the high-frequency noise above 25 Hz by an average of 60%. Although the transfer function of the filter has an unusual form, the time-domain results (Figs. 3.4 and 4.2.1) prove that such a filter does reduce the error level.

Using the philosophy behind the modified Wiener noise filter, another class of noise filters can be defined. These are called exponential noise filters and are of the form:

'Vm E (l,N/2]. (4.2.12)

Just like the modified Wiener filter defined in Eq. 4.2.8, the exponential noise filters are signal-dependent, and decrease the noise as a function of the signal-to-noise ratio. However, they differ in that they approach zero faster as the ratio decreases, and approach unity faster as the ratio increases. The rate at which these filters approach zero or unity is controlled by the parameters a and (3.

For comparison, the exponential noise filters were also tested on synthetic signal Q11CNS for different values of a and (3. The values of the measure-of-error J for

the acceleration, velocity and displacement, and for different combinations of a and

f3

are listed in Table 4.2.1. It must first be noted that for most combinations of a and

f3

listed in this table, J has substantially decreased from the case where no noise filter is implemented; there is up to 60% improvement in the acceleration, 86% in the velocity and 93% in the displacement. In general, the exponential filters were more effective than the modified Wiener filter in decreasing the error levels in the time histories.

Table 4.2.1 indicates that there is a trade-off between the role of a and

f3

and their effect on the measures-of-error. For a constant value of a and increasing values of

f3,

the measure-of-error J in the displacement decreases and that in the acceleration increases. Conversely, for a constant value of

f3

and increasing values of a, the measures-of-error in the acceleration, velocity and displacement seem to decrease initially before increasing again. The relationship between the variations in a and {3, and in the J's, does not appear to be a simple one, but it can be noticed that as

f3

becomes large the value of the J's remains more or less constant regardless of the value assigned to a. More insight into the interaction between a and (3 can be gained by examining the transfer functions of the exponential noise filter obtained with Q11CNS for some of the cases listed in Table 4.2.1 and Fig. 4.2.2. The transfer function of the exponential noise filter for a= 1 and

f3

= 1 (Fig. 4.2.2.b) is very similar to that of the Wiener filter (Fig. 4.2.2.a), with the exception that the exponential filter decreases on the average more of the high- frequency noise, as corroborated by the slight drop in the J's. For a = 10 and

f3 =

1 (Fig. 4.2.2.e), most of the harmonics which have a high proportion of noise are removed, and in the process the amplitude of the harmonics where the signal- to-noise ratio is average are also significantly decreased. This alters the shape of the time histories and creates errors, as is reflected in the large values of J in Table 4.2.1.

Conversely, for a = 1 and

f3

= 10 (Fig. 4.2.2.d), the filter has a transfer function that is equal to unity almost everywhere except at the harmonics which have a very small signal-to- noise ratio, in which cases it is equal to zero.

In effect, increasing a significantly decreases the amplitudes of all of the har- monics which do not have a very high signal-to-noise ratio. Whereas, increasing (3

selectively removes all the harmonics which have an extremely small signal-to-noise ratio, and leaves all other harmonics unchanged although they may be contaminated by noise. A parametric study of the J's has shown that the combination a

=

2 and

f3

= 4 offers the best compromise between the two effects. The transfer function of this case is illustrated in Fig. 4.2.2.e for synthetic signal Q11 CNS. Compared to the transfer function of the Wiener filter (Fig. 4.2.2.a), the chosen exponential filter retains more of the spectrum below 25 Hz, in the range where the signal predom- inates, but removes on the average about the same amount of the high-frequency noise above 25 Hz. This is also reflected in the changes in J listed in Table 4.2.1.

The measure-of-error in the acceleration is approximately the same for both types of noise filters, yet the error in the displacement from the exponential filter dropped by an extra 80%. Hence, when the proper combination for a and

f3

is selected, the exponential noise filter significantly improves the correction effectiveness of the frequency-domain accelerogram processing method. In particular, for Q11CNS it decreases the noise-induced error in the acceleration by 25%, in the velocity by 85%

and in the displacement by 90%.

The time histories for Q11 CNS produced by implementation of the exponential filter with a

=

^{2 and}

f3 =

4 are shown in Fig. 4.2.3. Compared to the correspond- ing results obtained when no noise filter is implemented (Fig. 3.4), and when the modified Wiener filter is used {Fig. 4.2.1), the exponential noise filter considerably decreases the error in the processed time histories. However, according to the plots depicting the error between the noise-contaminated filtered signal and the exact signal, the exponential filter does not thoroughly remove the noise in the accelera- tion, which along with the shift in the temporal mean, still contributes to significant long-period errors in the displacement.

In the above, the noise filters are signal-dependent. Thus, the improvement in the correction effectiveness of the filter is in direct relation to the signal-to-noise ratio of the signal. For small events, in which the signal-to-noise ratio is relatively small, the noise filters have just shown to be useful in significantly decreasing the noise- induced errors. However, when large events are tested with the noise filters, the correction effectiveness of the processing procedure shows very little improvement.

This is expected for large events, since in this case the processing and digitization errors have a minimal effect on the time histories, and the transfer function of the noise filter is approximately unity. For instance, the measures-of-error J for the unfiltered large event simulation Q11CNL contaminated with noise are two orders of magnitude smaller than the values listed for the unfiltered signal Q11CNS in Table 4.2.1. Hence, there is very little room for improvement.

In the next section, a different and complementary approach to filtering is investigated. It will be shown to correct the long-period drifts in the time histories without affecting the harmonics that have a reasonable signal-to-noise ratio. If such an approach is adopted, then the noise filters presented in this section would only be useful in correcting the high-frequency errors which are prevalent mainly in the acceleration time histories. The high-frequency errors induced by digitization and processing are mostly of concern for small seismic events, and are not greatly reduced by the noise filters for large seismic events. Under such conditions, the noise filters described in this section are not part of the standard probabilistic frequency-domain processing method, but they could be implemented as an option if desired.

4.3 Spectral Substitution Method