![]() Note that | A ws( f )| 2 is not the cospectrum (whose integral over all frequencies is the covariance between w and s), rather it is the sum of the squares of the co- and quadrature spectra (refer to Stull 1988 for details). (1), | A ws( f )| 2 is the square of the absolute value of the cross spectrum, and | S ww( f )| and | S ss( f )| are the autospectral densities for the variables w and s, respectively. See Finnigan (2004) for a discussion of the errors induced as a consequence of multiple coordinate rotations in a Cartesian coordinate system. In addition, wind speeds and vertical velocities can be tilt corrected without introducing errors. Wind speed was chosen for this application because it is free of wind direction trend and meander effects from large-scale motions that contaminate turbulent momentum flux computations. However, coherence can be computed using any paired data that share a common sampling rate and record length. The records used for the present illustration are tilt-corrected vertical velocities ( w) and horizontal wind speeds ( s) obtained from concurrent three-dimensional (3D) sonic anemometer–thermometer (sonic) measurements. Time series data records of any two continuous variables suitable for computing a covariance, if of sufficient length for computing a stable fast Fourier transform (fft), can be transformed into the frequency domain for computation of a dimensionless squared spectral coherence. (2004) also provide excellent background material. Kaimal and Finnigan (1994), Stull (1988), and Lee et al. Refer to Koopmans (1974), Bendant and Piersol (1986), or other time series analysis texts for definitions and terminology associated with spectrum analysis. Note that a certain amount of familiarity with spectral analysis is assumed here. Alternatively, transforming from the time to the frequency domain and computing the squared spectral coherence (CH) provides frequency-stratified results that can be tested for statistical significance using the F distribution. ![]() Significance tests that require equivalent sample size estimates based on the products of autocorrelation functions, such as those suggested by Sciremammano (1979), are based on assumptions that are untenable when working with high-frequency turbulence data. Consequently, standard correlation coefficient significance tests that rely on an independent sampling assumption cannot be used directly on time-domain turbulence data. Unlike random numbers, data records collected sequentially over constant, discrete time intervals in the ABL are not independent of their neighbors but are autocorrelated over time periods related to each variable’s integral scale ( Lumley and Panofsky 1964). In particular, time series summary statistics do not include sufficient information to determine whether the magnitude of a flux estimate exceeds results that could be expected to arise from uncorrelated (random) turbulent motions. However, examining time-mean summary statistics offers only limited insight into boundary layer processes. Similar covariance measurements are used to estimate the fluxes of carbon dioxide or other atmospheric constituents (e.g., Aubinet et al. ![]() These covariances frequently serve as heat or momentum flux estimates describing the state of the atmospheric boundary layer (ABL). Time series data acquired at fixed time intervals by a triaxis sonic anemometer–thermometer (sonic) are frequently used to compute time-mean statistics, to include temperature and velocity component covariances. A record length and transform size that yield equivalent degrees of freedom in the range between 10 and 100 produces the most consistent and reliable significance test results. The sampling period should be sufficiently long to span major flux-generating scales of motion up to the cross-spectral gap but not so long that extraneous larger scale motions are included. Successful application of this coherence significance test requires selection of a properly sized data record and a Fourier transform with appropriate windowing. The test is illustrated using sonic anemometer–thermometer data acquired in an urban setting and over a salt playa. A statistically significant coherence peak also indicates the likely presence of a significant turbulent flux. This significance test determines the probability with which coherence peaks are likely to arise out of random turbulence. A simple coherence significance test designed to determine whether coherence peaks exceed critical values expected from chance random number correlations is described. The squared spectral coherence, a frequency-domain analog of the squared correlation coefficient, identifies the frequencies at which two variables most strongly covary.
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