The detection of porosity changes within a soil matrix caused by internal erosion is beneficial for a better understanding of the mechanisms that induce and maintain the erosion process. In this paper, an electromagnetic approach using Spatial Time Domain Reflectometry (STDR) and a transmission line model is proposed for this purpose. An original experimental setup consisting of a coaxial cell which acts as an electromagnetic waveguide was developed. It is connected to a transmitter/receiver device both measuring the transmitted and corresponding reflected electromagnetic pulses at the cell entrance. A gradient optimization method based on a computational model for simulating the wave propagation in a transmission line is applied in order to reconstruct the spatial distribution of the soil dielectric permittivity along the cell based on the measured signals and an inversion algorithm. The spatial distribution of the soil porosity is deduced from the dielectric permittivity profile by physically based mixing rules. Experiments were carried out with glass bead mixtures of known dielectric permittivity profiles and subsequently known spatial porosity distributions to validate and to optimize both, the proposed computational model and the inversion algorithm. Erosion experiments were carried out and porosity profiles determined with satisfying spatial resolution were obtained. The RMSE between measured and physically determined porosities varied among less than 3% to 6%. The measurement rate is sufficient to be able to capture the transient process of erosion in the experiments presented here.
The problem of estimation of the unknown potential in a 1-dimensional wave equation via state observers is considered in this work. The potential is supposed to depend on the space variable only and be polynomial. The main observation information is the value of the solution of the wave equation in a subinterval of the domain, including also some of its higher-order spatial derivatives. The method we propose to estimate the potential includes turning it into a new state as in finite-dimensional parameter estimation approaches. However, in this infinite dimensions setting, this requires an indirect approach that is introduced, including an infinite-dimensional state transformation. Sufficient conditions allow the design of an internal semilinear observer for the resulting cascade system, corresponding to the observed subinterval, which estimates the potential in an exponentially fast manner.
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