8.2 The Fourier series  8. Spectral Analysis  8. Spectral Analysis

8.1 Introduction

Many data sources provide sets of measurements that show how a measured variable changes with time or with space. Examples are a series of measurements of wind along a flight segment flown by a research aircraft, a series of measurements of temperature or pressure at a ground site, or continuous measurements of cloud cover at locations below a satellite. In some cases, particularly those from turbulent fields, the measurements seem inherently stochastic: There is an apparent random component to the measurements, and the next measurement in the series cannot be predicted exactly from the preceding measurements.

The contributions to the variance having various frequencies or wavelengths can be determined using the methods of spectral analysis. Often, the resulting spectral density functions reveal important characteristics of the physical processes generating the fields. One can also use spectral analysis to characterize correlations between observations separated in space or in time, to predict the next point in the series, and to characterize the uncertainty in that prediction.

A special case is that where consecutive measurements are completely independent of each other, so that knowledge of the past history of the variable provides no information on the next measurement. This series is called white noise and often occurs in cases where the measured field is so uniform that the measured fluctuations are caused entirely by random errors in the measurements. An example is the repeated sampling of a uniform droplet concentration, where variations in the measurements are caused only by counting statistics and do not reflect real structure in the cloud.

The more interesting cases exhibit correlation among consecutive measurements. Figure 8.1 shows several time series, two simulated and two from hourly weather observations. All four series shown have about the same variance about the mean, but they differ in how consecutive measurements are correlated. For example, if a particular observation is smaller than the mean, there is an increased probability (in the series for pressure and temperature) that the next observation will also be smaller than the mean. Spectral analysis provides a tool for characterizing the differences among time series like these.

 

Figure 8.1: Examples of time series. P: hourly measurements of surface pressure at Denver Colorado during a 30-day period in wintertime; RAN P: simulated pressure values having the same mean and standard deviation but with variations from the mean generated by a random-number generator to represent white noise; G.RAN P: similar to RAN P but with fluctuations representing noise with a Gaussian probability distribution; T: temperature measurements corresponding to the same time period as the pressure measurements.

Measurements like those of temperature and pressure in Fig. 8.1 are influenced by atmospheric phenomena that have many different characteristic periods. For example, in the temperature measurements there is a strong diurnal cycle. In the pressure sequence, the diurnal cycle is hardly evident (although spectral analysis will reveal it clearly); instead the dominant scale is longer, and reflects the passage of weather disturbances with characteristic times of several days. The extension of these observations over several years reveals a strong annual cycle in temperature that is not present in pressure. Tidal motions contribute to the variance in the pressure measurements, although they are difficult to separate from diurnal effects. One purpose of spectral analysis is to characterize measurements such as these in ways that isolate the different contributions with different periods and so reveal the relative importance of different processes contributing to the variance in the observations.

 

Figure 8.2: Spectral density functions for the variance of the correspondingly labeled time series shown in Fig. 8.1, except that six years of hourly observations similar to the one-month period in Fig. 8.1 were used to determine these variance spectra. The spectral density functions $\Gamma(\nu)$are shown weighted by the frequency $\nu$.

Figure 8.2 shows spectral density functions for the variances in the time series of Fig. 8.1. Methods for calculating these density functions will be developed later in this chapter; for now, consider these as characterizing the distribution of variance in intervals of the logarithm of the frequency. The distributions for pressure and temperature are quite different from each other, and also from those for the random variables. The diurnal cycle, producing peaks with periods of 24/n hours where n=1, 2, 3, $\ldots$, is quite evident in the spectra for pressure and temperature, and the contributions having periods of a few days to a week are particularly strong in the pressure spectrum. The annual cycle is strong in the temperature but not in the pressure spectrum. Thus, the spectral density functions show the relative strengths of the diurnal and annual cycles and the influences of short-wave and long-wave traveling weather systems on the measured time series.

There are two alternative approaches to characterizing the variance in time series such as these:

• Describe the contributions to the variance from different fundamental components, e.g., by analyzing the strength of sine-wave contributions having different frequencies. This results in a variance or "power" spectrum like those shown in Fig. .
• Describe the variance in the product of measurements separated in space or time, as a function of the separation. This gives autocovariance or autocorrelation functions for the time series.

These two approaches are complementary in the sense that either description can be generated from knowledge of the other. Their formal equivalence can be proved for stationary processes, or processes characterized by probability distribution functions that do not change with time.

   
 8.2 The Fourier series  8. Spectral Analysis  8. Spectral Analysis