1/f noise
1/f noise or pink noise is a mathematical equation that relates the power of the signal to the signal's frequency. A signal's power measurement will be 1/frequency, which is a power law relationship.
1/f Noise Media
A two-dimensional pink noise grayscale image, generated with a computer program. Some fields observed in nature are characterized by a similar power spectrum.
Relative intensity of pink noise (left) and white noise (right) on an FFT spectrogram with the vertical axis being linear frequency
\sigma^2(1) is approximately the area under the green curve. When \tau increases, S[g](\omega) shrinks on the x-axis, and the green curve shrinks on the x-axis but expands on the y-axis. When S[\dot x_f](\omega) \propto \omega^{-\alpha}, the combined effect of both is that \sigma^2(\tau) \propto \tau^{\alpha-1}.
Noise curves for a selection of gravitational-wave detectors as a function of frequency