On observations of correlation functions and power spectra in rain: obfuscation by advection and sampling

Rain is often characterized using statistical approaches. Among the most common are temporal correlations and power (variance) spectra from time series measurements at a single location. Likewise, temporal observations over a network are used to deduce a radial distribution function and spatial power spectra. In such studies the potential effects of advection on the results are ignored. Moreover, observations involve filtering of the data. In time, this may involve sampling over a sufficiently long period so as to increase statistical confidence in the measurement. The same is also true for spatial observations over a network which must contain a sufficient number of instruments for a reliable characterization of the spatial variability. This also usually includes some form of averaging over time as well. Temporal averaging amounts to a low pass filter that attenuates contributions from higher frequencies. In contrast, the finite dimension of a network acts as a high-pass filter that tends to suppress the lower wavenumbers much larger than the dimension of the network. In this work the effects of both the advection of the rain and the observational filtering are considered for the simplest case of wide-sense statistically stationary and homogeneous rain along one-dimension for rain exponentially correlated in both space and time. It is found that advection and filtering can significantly shift the portrayal of the rain from the true structures. Consequently, rainfall characterizations from observations should not be over-generalized to other situations.