Correlation Properties of Signals Backscattered From Fractal Profiles

A successful mathematical description of natural landscapes relies upon a class of random processes known as fractional Brownian motions (fBms), which may exhibit correlation with long-range dependence (LRD). In remote sensing applications, the sensor observes a certain real scene
B
and records data
I
for successive signal processing tasks. Assuming that
B
is modeled as an fBm, does the recorded signal
I
preserve the LRD character of
B
? More in general, can we relate the Hurst coefficient (an index of LRD) of the real scene to that of the recorded data? We address the problem in a simplified setup in which the data are related to (the slope of) the original scene through a zero-memory mapping. A mathematical framework is presented in which the above questions can be answered in the asymptotic regime of infinite data size. The effect of the finite sample size is also investigated. The mathematical model is also validated by real data, which are collected by a synthetic aperture radar that is mounted onboard of ERS-1/2 satellites.