A Comment on the use of Bandpass Filtering to

Discover Atmospheric Oscillations

Andrew S. Milman

amilman@ieee.org

 

This article was published in the International Journal of Remote Sensing, 19, 2275-2282 (1998).

Introduction

About 20 years ago, Madden and Julian (1971, 1972, hereafter referred to as MJ) published an account of their discovery of oscillations in the tropical Pacific atmosphere with periods of about 45 days. Since then, numerous other authors have studied these and similar phenomena, reporting that they have found oscillations with periods ranging from 30 to 60 days. These collectively are called 30-60 day oscillations, or Madden-Julian oscillations, or intraseasonal oscillations. I will use the last term in this paper.

            The point of this comment is that, while there has been a lot of time and effort spent looking for intraseasonal oscillations, much of the work has employed a faulty mathematical technique, that of bandpass filtering the data, plotting the result, and trying to interpret it as representing fluctuations in some physical quantity. This practice, at least as it relates to intraseasonal oscillations, seems to have originated in the work of MJ and is now widespread in the meteorological literature. I show examples in Appendix A.

            The original reports by MJ concerned radiosonde measurements made at Canton Island (2º46' S, 171º43' W) in the tropical Pacific. They had a record of 3584 daily measurements-almost 10 years-and they analyzed the time series of the station pressure and of zonal winds at different altitudes. In the course of these analyses, they found peaks in the spectra and cross-spectra at frequencies centered about 0.0212 day-1, which corresponds to a period of about 47 days. They found that these peaks were statistically significant, as compared with a flat red-noise spectrum. In order to study them better, they filtered the time series with a bandpass filter centered on 0.0212 day-1, with half-power points at 0.011 day-1 and 0.029 day-1, corresponding to periods of 90 and 35 days, respectively. They then plotted these filtered time series and studied the relationships between the 150-mb zonal wind, 850-mb zonal wind, and surface pressure (MJ, 1971, Figure 5).

            MJ also calculated the complex coherency between time series from different stations, and used the high coherency between different variables at frequencies between 30 and 50 days as a justification for bandpass filtering at those frequencies. Given two time series, the coherency [from now on, I shall use just the word coherency unmodified to mean its magnitude, or what MJ (1971) called the "coherence-squared"] is a number between 0 and 1 that measures how similar the two time series are. If they are identical, the coherency is 1; if they are uncorrelated, it is 0. I will return to this later.

Bandpass filtering

The practice of bandpass filtering is often justified as follows. First, we take a time series v(t). Let V(f) be its Fourier transform, where f is frequency:

                                 (1)

The spectrum is estimated from the periodogram

                                 ( 2)

This always produces a noisy estimate of the true spectrum, so it is customary to smooth the periodogram over several adjacent frequencies (see Brockwell and Davis, 1991, §11.7). If there is a statistically significant peak somewhere in the spectrum, say at a frequency fc, some workers would argue that it is proper to bandpass-filter the time series v(t) to isolate the process that produced the peak in the spectrum. The bandpass-filtered signal is

                                   (3)

Here, H(f) is a bandpass filter, a function that is maximum at fc and goes to zero if f < f1 < fc or f > f2 > fc, where f2 - f1 is the bandwidth of the filter.

            The problem with this procedure is that, regardless of the nature of the process that generated v(t), we will get a time series that oscillates with frequency fc. It is easy to demonstrate this with a computer simulation, generating a white-noise time series with a random-number generator and then bandpass-filtering it. This always produces a sinusoid at the center frequency of the filter. Another way to look at this situation is to realize that, mathematically, any finite time series v(t) has a Fourier expansion in terms of sines and cosines. However, the presence of a Fourier component at some frequency does not necessarily imply that there is any physical process oscillating at that frequency.

            The amplitude of u(t) is determined by the requirement that its variance s2 be

                                     (4)

For a given spectrum, the amplitude of u(t) is determined entirely by the filter H(f) and the shape of the spectrum. Therefore no conclusions can be drawn from u(t) as to the relative importance of the bandpass-filtered time series.

                        The only situation where it makes sense to bandpass-filter a time series is when we know a priori that contains a signal that is confined to a certain band of frequencies, say between f1 and f2. In that case, we could bandpass-filter v(t), rejecting components with frequencies f < f1 or f > f2, thereby reducing the noise and possible interfering signals outside the band of interest. We cannot, however, use the bandpass-filtering method to show that such a signal exists; we have to determine its existence by other means.

Coherency

When there are two different time series vi(t), i = 1, 2, then coherency is often invoked as being independent evidence of the existence of waves and a justification for performing the bandpass filtering at the frequency where the coherency is largest. However, it is possible to have a high coherency without any oscillations being present; it is also possible to have a low coherency in the presence of a sine-wave component. The coherency tells us little about the presence or absence of propagating waves.

            MJ (1972) calculated the coherency between the surface pressures of different islands in the Pacific, using Canton island as a reference, and averaging over frequencies of 0.02 to 0.03 day-1. They concluded from the calculated coherency and phase between Canton Island and other stations that there are eastward-propagating disturbances in the tropical Pacific Ocean.

                        The coherency is defined as follows. Assume that v1(t) and v2(t) refer to time series of the same quantity at different locations. The cross-periodogram of v1 and v2 is defined to be

                                    ( 5)

where Vi(f) is the Fourier transform of vi(t) and Vi* denotes the complex conjugate of Vi. The cross-periodogram is just as irregular as the periodogram of a single time series is; it is usually averaged over adjacent frequencies also. The complex coherency is estimated from

                                (6)

|c(f)|2 is what MJ (1971) called the "coherence-squared." The complex phase of c may also be important, as it would correspond to the relative phases of the oscillation at locations 1 and 2.

            Note that the magnitude of the estimated coherency is identically equal to 1 unless it is averaged over adjacent frequencies (see Brockwell and Davis, 1991, §11.6). We need to see how this smoothing affects our estimate of the coherency between two time series.

            If v1(t) and v2(t) are incoherent, the phase of c(f) will be a random function of the frequency, so averaging over a sufficiently wide range of frequencies will produce zero coherency. On the other hand, if they are partially coherent over some range of frequencies between f1 and f2, then the estimated coherency in that band

                                    (7)

will be different from zero. If f2 - f1 is small, |^c| will be close to 1, regardless of the true relationship between v1(t) and v2(t). But if f2 - f1 large enough, |^c| will be close to 0, except in the unlikely situation that v1(t) and v2(t) are coherent and have the same relative phase over a large bandwidth.

            Now suppose that there is a spectral peak at a frequency fc, where f1 < fc < f2. If there are waves propagating with frequencies near fc, and not with other frequencies, one might expect a narrow peak in |^c| near fc. I shall give a simple example to show that this peak may not, in fact, appear. Let v1 and v2 each be the sum of a cosine term and white noise:

                         (8)

where si is a white random process, i = 1, 2, and s1 is uncorrelated with s2. Figure 1 shows one realization of such a pair of signals. Figure 1-a shows the spectrum of one of the time series. The amplitude of the cosine term was chosen so that it contains about one-half of the total power. Panels b to e of Figure 1 show the coherency averaged over different ranges of frequency. None of these shows any particularly convincing evidence for a spectral component with a wavenumber of 200. This example shows that we cannot form any judgment about the possible presence of waves with frequencies near fc based solely on an estimate of the coherency at the frequency fc, regardless of the bandwidth used to smooth |^c|. This will always happen unless the two signals are coherent over a relatively wide bandwidth. However, when there is a wave propagating through the atmosphere, we would expect that the wavelength should depend on frequency in some way (as is the case with sound waves, for example), in which case the phase of c should change as the frequency changes, and the coherency will again be small.

            So there may be a propagating wave even where the coherency is small. On the other hand, there may be a high coherency and no propagating wave. I illustrate this possibility schematically in Figure 2. Here, the observed coherency between (say) Yap and Guam might be due to the annual motions of the sun north and south of the equator, rather than to waves propagating from one island to the other. Measuring the coherency between different islands cannot distinguish between these two possibilities.

            There is also the problem that random events will contribute to the bandpass-filtered time series. For example,   MJ measured the coherency in the surface pressure at different locations; they discovered a high coherency when they averaged over frequencies between 0.028 and 0.020 day­­1, corresponding to periods of 36 to 50 days. They used the high coherency as justification for bandpass-filtering the different time series using theses same frequency limits and then interpreting the results as being due to physical oscillations of some sort.

            However, there are many factors that control the variability of surface pressure. Even if there is an oscillation with a period near 45 days, there are other factors that affect the surface pressure and that are uncorrelated with the 45-day oscillation. Let me call this second component noise-like, in the sense that, for the purposes of the present discussion, it behaves like random noise. I can represent this in a manner similar to equation , where the cosine term corresponds to the 45-day oscillation, and the si, to the noise-like component. Let P0 represent the spectral power, integrated between 0.020 and 0.028 day­­1, due to the oscillation, and PN be the power in the noise-like component, integrated over the same band of frequencies. It is not hard to show that, under these circumstances, the coherency is P0 / (P0+PN).

            So if c = 0.33, which is statistically significant, we might think that bandpass filtering the time series is justified. But in this case, PN = 2P0: the filtered time series is affected twice as much by the noise-like component as it is by the oscillation. What physical significance could such a product possibly have?

Conclusions

In summary, many researchers, using the methods of MJ, seem unwittingly to be using methods that create sinusoids in their data. The basic method seems to be to take a time series, bandpass-filter it, plot the result, and remark upon the oscillations that are now evident in the data. None of the authors I am aware of even mentions the possibility that the oscillations they measure were created by this bandpass-filtering process. Some of these authors-starting with MJ-have also calculated coherency and used a high coherency to justify this method. However, two signals in different locations can be coherent without having anything propagate from one location to the other, while an actual wave may not affect the coherency significantly. As I have shown, coherency is difficult to interpret, since it will depend on the bandwidth over which it is integrated. Furthermore, random events that are uncorrelated with the possible oscillations in question will often contribute more to the bandpass-filtered data than the oscillations themselves do.

            Therefore, bandpass filtering does not, as many believe, necessarily isolate any particular physical phenomenon, nor does a high coherency prove the existence of propagating waves.     There is, however, an analytical method for measuring the power in propagating waves that does not use coherency-or bandpass filtering. This will be the subject of a forthcoming paper.

References

Brockwell, P. J., and R. A. Davis (1991): Time Series: Theory and Methods, Second Edition, Springer-Verlag, New York.

Julian, P. R. (1971): 'Some aspects of variance spectra of synoptic scale tropospheric wind components in midlatitudes and in the tropics,' Monthly Weather Review, 99, 954-965.

Madden, R. A., and P. R. Julian (1971): 'Detection of a 40-day oscillation in the zonal wind in the Tropical Pacific,' J. of the Atmospheric Sciences, 28, 702-708.

Madden, R. A., and P. R. Julian (1972): 'Description of global-scale circulation cells in the tropics with a 40-50 day period,' J. of the Atmospheric Sciences, 29, 1109-1123.

Appendix A. Papers on Bandpass-filtered Time Series

This method of finding oscillations in the atmosphere by bandpass-filtering various time series is fairly common. I have discovered the following examples in a cursory review of recent literature. I include the following list of articles to show that bandpass filtering is a widespread practice.

Anderson, J. R. and R. D. Rosen (1983), 'The latitude-height structure of 40-50 day variations in atmospheric angular momentum,' J. of the Atmospheric Sciences, 40, 1584-1591.

Anderson, J. R. (1986): 'A diagnostic study of the tropical 40-50 day oscillation,' Proceedings of the Eleventh Annual Climate Diagnostics Workshop, October 14-17, 1986, US Dept of Commerce, pp 267-270.

Dunkerton, T. J. (1993): 'Observation of 3-6-day meridional wind oscillations over the Tropical Pacific, 1973-1992: Vertical structure and interannual variability,' J. of the Atmospheric Sciences, 50, 3292-3307.

Gao, X. H., and J. L. Stanford (1990): 'Low-frequency oscillations in total ozone measurements,' J. Geophysical Research, 95(D9), 13,797-13,806.

Gutzler, D. S. (1991): 'Interannual fluctuations of intraseasonal variance of near-equatorial zonal winds,' J. Geophysical Research, 96, Supplement, 3173-3185.

Hong, Y., and H.-S. Lim, 'Evidence for low-frequency waves of tropical rainfall inferred from microwave brightness temperature,' Monthly Weather Review, 122, 1364-1370 (1994).

Lau, K.-M., and P. H. Chan (1985): 'Aspects of the 40-50 day oscillation during the northern winter as inferred from outgoing longwave radiation,' Monthly Weather Review, 113, 1889-1909 .

McAfee, J. R., K. S. Gage, B. B. Balsley, W. L. Ecklund, D. A. Carter, and G. C. Reid (1989): 'Intraseasonal oscillations in the upper troposphere observed with the Christmas Island wind profiler,' Proceedings of the Fourteenth Annual Climate Diagnostics Workshop, October 16-20, 1989, US Dept of Commerce, pp 132-137.

Marcus, S. L., M. Ghil, and J. O. Dickey, 'The extratropical 40-day oscillation in the UCLA general circulation model. Part I: Atmospheric angular momentum,' J. Atmospheric Sciences, 51, 1431-1446 (1994).

Randel, W. H. (1993): 'Global Normal-mode Rossby waves observed in stratospheric ozone data,' J. of the Atmospheric Sciences, 50, 406-420.

Reed, R. J., D. C. Norquist, and E. E. Recker (1977): 'The structure and properties of African wave disturbances as observed during Phase III of GATE,' Monthly Weather Review, 105, 317-333.

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