Theory of Intrinsic Random Functions

Chapter 3 discusses intrinsic random functions Y(x) of the space variable x, i.e. functions whose mean and variance of the increments Y(x + h) − Y(x) depend on h only. Half of this variance defines the variogram γ‎(h). The behaviour of the variogram near the origin, such as continuity, the nugget effect, etc., expresses the regularity of these functions. Regularisations of them, by grading and convolutions, produce new variograms, via the same rules as for transitive methods. When Z(v) and Z(v′) are the averages of these functions in v and v′, respectively, then the variance of Z(v) − Z(v′) by attributing to v′ the grade in v is called extension variance. Its formal expression is given and calculated for various patterns of sampling v, in dimensions 1, 2, or 3, via the de Wijsian scheme and the spherical scheme, and for various models of variograms, such as the semi-variogram.

Intrinsic random functions for mitigation of atmospheric effects in terrestrial radar interferometry

The benefits of terrestrial radar interferometry (TRI) for deformation monitoring are restricted by the influence of changing meteorological conditions contaminating the potentially highly precise measurements with spurious deformations. This is especially the case when the measurement setup includes long distances between instrument and objects of interest and the topography affecting atmospheric refraction is complex. These situations are typically encountered with geo-monitoring in mountainous regions, e.g. with glaciers, landslides or volcanoes.We propose and explain an approach for the mitigation of atmospheric influences based on the theory of intrinsic random functions of order k (IRF-k) generalizing existing approaches based on ordinary least squares estimation of trend functions. This class of random functions retains convenient computational properties allowing for rigorous statistical inference while still permitting to model stochastic spatial phenomena which are non-stationary in mean and variance. We explore the correspondence between the properties of the IRF-k and the properties of the measurement process. In an exemplary case study, we find that our method reduces the time needed to obtain reliable estimates of glacial movements from 12 h down to 0.5 h compared to simple temporal averaging procedures.