5. Variance Spectra

Variance spectra are also plots and perhaps belong in the preceding chapter, but they are discussed here in greater detail than the preceding plots and so seemed to fit better in a separate chapter.

5.1 Background regarding spectral variance

5.1.1 Terminology

The plots discussed here as “variance spectra” are often referred to as “power spectra.” That term is not used here because the spectra representing variance in the data.frame measurements from NCAR/EOL/RAF netCDF files are not power. Even in the case of wind (with variance dimensions $\mathrm{m}^{2}\mathrm{s}^{-1}/Hz$), the variance spectrum is better described as a kinetic-energy spectrum. For this reason, the plots discussed in this chapter will be called “variance spectra” and the plotted quantity will be called the spectral variance.

5.1.2 Transformations among spectra; “proper” spectra

Consider the cumulative distribution function for variance $C(\nu)$, the fraction of the variance that is contributed by frequencies smaller than $\nu$. The differential distribution function with respect to frequency is then

$$\bgroup\color{fgcolor} P(\nu)=\frac{dC(\nu)}{d\nu}\,\,\,\,. \egroup$$

Consider how this distribution function would change if defined in terms of a new variable $x(v)$ starting with the specific variable $x=\ln\nu$. The differential distribution function would then be $T(x)$ specified by

$$\bgroup\color{fgcolor} T(x)=\frac{dC(x)}{dx}=\frac{dC(\nu)}{d\nu}\frac{d\nu}{dx}=\nu P(\nu) \egroup$$

$\nu P(\nu)$ thus gives the spectral density in terms of $\ln\nu$ and so in terms of $\ln(10)\log_{10}\nu\approx2.30\log_{10}\nu$. In the following, two options for variance spectra are emphasized: $P(\nu)$ vs $\nu$ on a linear plot and $\nu P(\nu)$ vs $\log_{10}\nu$ with a logarithmic abscissa and either a linear or logarithmic ordinate scale. These are regarded here as “proper” displays because the area under segments of the plotted curves represent contributions to the variance so it is possible to estimate the contributions to variance from various intervals in frequency by using the areas on the plot. This direct representation is compromised in the case where the variable $\nu P(\nu)$ is plotted on a logarithmic scale because then it is necessary to consider the logarithmic ordinate when evaluating areas. This minor inconvenience nevertheless is less significant than the problems that arise from using a linear ordinate scale, in which case the ordinate range obscures relationships and the common “ $-5/3$” slope seen in logarithmic plots becomes a parabolic line that is difficult to interpret. For that reason, plots of spectral variance here will emphasis plots of $\nu P(\nu)$ vs $\log_{10}(\nu)$ on log-log scales. It is suggested that plots of $P(\nu)$ vs $\nu$ on log-log scales should be avoided because the connection between area on the plot and variance is lost, making the plot harder to interpret. In addition, “ $-5/3$” spectra are steep, the range of ordinate values is higher, and the plots are therefore more difficult to interpret than those plotting $\nu P(\nu)$ vs $\nu$ on a log-log scale.

5.1.3 Methods used to estimate spectral variance

The function “Ranadu::VSpec()” includes three methods that can be selected to estimate the spectral variance:

1. The function “spectrum()” from the “stats” R package, which estimates a periodogram using Fourier-series estimation and then optionally smooths the spectrum using modified Daniell smoothers. The amount of smoothing can be controlled with the “spans” parameter in the function call.
2. The “Welch” method as implemented by the R package “bspec”. In this method, the time series is divided into subsets and the spectra resulting from the subsets are averaged to reduce the variance in the result. The degree of smoothing is controlled by the “segLength” parameter that specifies the length of the segments to use.
3. The “maximum entropy” method in which the spectrum is represented by “poles” and evaluated from the contributions of those poles across the frequency spectrum. A higher number of poles and a smaller value of the “resolution” will lead to more structure in the result.

In addition, it is possible to smooth the resulting spectrum further by specifying a value for the parameter “smoothBins”, in which case the frequency range will be partitioned into the specified number of logarithmically spaced bins and the values of the spectral density will be averaged in each bin.

5.2 Generating plots of variance spectra

See “?Ranadu::VSpec” for details regarding using this function. The essential inputs are a data.frame that includes at least the variables “Time”, “TASX”, and the variable for which the variance spectrum is desired. TASX is needed to interpret the scale both in terms of frequency and wavelength. An example is shown in Fig. 5.1, using the default specifications.

getNetCDF('/Data/SOCRATES/SOCRATESrf08h.nc', Start=45600, End=50100) %>%
  VSpec('WIC') + theme_WAC()

Figure 5.1: Variance spectrum for measurements of vertical wind during SOCRATES flight 8, 4:56:00 - 5:01:00.

The following demonstrates how to combine plotted spectra. The three lines on this plot were generated using the three methods of spectral estimation available in VSpec(), all for the WIC variable:

D <- getNetCDF('/Data/SOCRATES/SOCRATESrf08h.nc', Start=45600, End=50100) %>%
     Rmutate(WIC2=WIC, WIC3=WIC) ## duplicate the variable
g <- VSpec(D, 'WIC', VLabel='spectrum')
g <- VSpec(D, 'WIC2', method='Welch', VLabel='Welch',
           segLength=128, smoothBins=50, add=g)
VSpec(D, 'WIC3', method='MEM', VLabel='MEM', add=g) +
  theme_WAC()

Figure 5.2: Variance spectra for the same data shown in the preceding plot but generated by the three methods indicated in the legend.

Another option that may be of use, although the result is not a “proper” spectrum in the sense used above, is to plot with weighting by $\nu^{5/3}$ and additional change of variables so that the resulting ordinate matches the eddy dissipation rate in a case where the measurements are indeed from an inertial subrange. Figure 5.3 illustrates this plot. The variable plotted is $(2\pi/V)(CP(\nu)\nu^{5/3})^{3/2}$, with $V$ the airspeed and $C=1.5$ for lateral spectra; this quantity should equal the eddy dissipation rate in an inertial subrange.

VSpec(D, 'WIC', EDR=TRUE) + theme_WAC(1)  ## theme_WAC(1) => smaller title

Figure 5.3: An eddy-dissipation-rate plot for the same data shown in the preceding plot. but generated by the three methods indicated in the legend.

## to suppress the title, add "+ ggtitle(”)