8.8.4 Effect of a  8.8 Examples  8.8.2 Hourly measurements of

8.8.3 Pure random process

A pure random process without sequential correlations must have the autocorrelation function

$$\rho(\tau)=0, ~~~\tau\ne 0$$ (8.34)

  

$$\rho(\tau)=1, ~~~\tau=0.$$ (8.35)

 

Then the variance spectrum is

$$\Gamma(\nu) = \int V_{ff}(0)\rho(\tau)e^{i2\pi\nu\tau}d\tau.$$ (8.36)

 

The resulting variance spectrum will be zero for all frequencies unless $V_{ff}(0)=\infty$, so such a process must have infinite total variance. One way to represent such a process is to let the autocovariance function be a delta function (zero everywhere except for zero argument, infinite for zero argument, with integral equal to unity for any integration including zero argument):

$$V_{ff}=c\delta(\tau), \ \ c={\rm constant}$$ (8.37)

  

$$\Gamma(\nu) = c\int_{-\infty}^\infty\delta(\tau)e^{i2\pi\nu\tau}d\tau = c$$ (8.38)

  

$$V_{ff}(0)=\int_{-\infty}^\infty\Gamma(\nu)d\nu =\int_{-\infty}^\infty c~d\nu = \infty$$ (8.39)

  

Two examples of random-noise sequences, as they would be sampled in hourly observations, were shown in Fig. 8.1. Both give the same variance spectrum, as shown in Fig. 8.2. As expected, the spectrum is constant so the weighted spectrum increases linearly with frequency. The total variance of this sequence, for frequencies above 0.5 h^-1, is about the same as that in the sequence for hourly observations of pressure shown in Fig. 8.2, but the distribution in frequency is much different because here the highest frequencies make the dominant contribution. Flat regions in the variance spectrum (or linearly increasing regions in the spectrum weighted by frequency) can often be found in measured spectra, and are usually indications of noise.

   
 8.8.4 Effect of a  8.8 Examples  8.8.2 Hourly measurements of