8.15 The "maximum-entropy" method  8. Spectral Analysis  8.13 The Fast Fourier

8.14 The cospectrum

Equation (8.4) for the variance spectrum can be written

$$\hat\Gamma(\nu) = {{1}\over{T}} \hat f^*(\nu)\hat f(\nu)$$ (8.68)

 

where $\hat f^*(\nu)$ is the complex conjugate of $\hat f(\nu)$. An analogous characterization of the covariance between two variables X and Y is:

$$\hat\Psi(\nu) = {{\hat X^*(\nu)\hat Y(\nu)}\over{T}}$$ (8.69)

 

where $\hat X(\nu)$ and $\hat Y(\nu)$ are the Fourier transforms of the functions X(t) and Y(t). Unlike the variance spectrum, $\hat\Psi(\nu)$ has both real and imaginary parts, which are called respectively the cospectrum $C(\nu)$ and the quadrature spectrum $Q(\nu)$:

$$\Psi(v) = C(\nu) + i Q(\nu) \ .$$ (8.70)

 

The cospectrum often finds application in atmospheric research because it characterizes the contributions to a product from different frequencies. For example, the cospectrum of vertical wind and water vapor density shows the contributions to the vertical flux of water vapor from different wavelength components in a turbulent field.

Two other functions that can be determined from $C(\nu)$ and $Q(\nu)$are the squared coherence spectrum $r^2(\nu)$and the phase spectrum $\phi(\nu)$:

$$r^2(\nu) = {{\overline{C(\nu)^2} + \overline{Q{\nu}^2}}\over{\overline{\Gamma_X(\nu)}\overline{\Gamma_Y(\nu)}}}$$ (8.71)

 

and

$$tan\bigl(\phi(\nu)\bigr) = {{Q(\nu)}\over{C(\nu)}} \ ,$$ (8.72)

 

where the overbars denote averages over some frequency interval and $\Gamma_X$ and $\Gamma_Y$ are the variance spectra for X and Y, respectively. Because the equation for the coherence is identically unity for any individual frequency, averaging of nearby estimates is necessary if these results are to be meaningful. After such averaging, the coherence can be used to test for consistency in the phase relationship between the two variables. If the phases change randomly for components in the selected interval, the coherence will tend to zero as the number of estimates increases. If there is coherence between the two variables, the phase function characterizes the phase between the two components.

   
 8.15 The "maximum-entropy" method  8. Spectral Analysis  8.13 The Fast Fourier