Generalized crested products of Markov chains

We define a finite Markov chain, called generalized crested product, which naturally appears as a generalization of the first crested product of Markov chains. A complete spectral analysis is developed and the $k$-step transition probability is given. It is important to remark that this Markov chain describes a more general version of the classical Ehrenfest diffusion model. As a particular case, one gets a generalization of the classical Insect Markov chain defined on the ultrametric space. Finally, an interpretation in terms of representation group theory is given, by showing the correspondence between the spectral decomposition of the generalized crested product and the Gelfand pairs associated with the generalized wreath product of permutation groups.Comment: 18 page

Weighted spanning trees on some self-similar graphs

We compute the complexity of two infinite families of finite graphs: the Sierpi\'{n}ski graphs, which are finite approximations of the well-known Sierpi\'nsky gasket, and the Schreier graphs of the Hanoi Towers group $H^{(3)}$ acting on the rooted ternary tree. For both of them, we study the weighted generating functions of the spanning trees, associated with several natural labellings of the edge sets.Comment: 21 page

Focusing on soil-foundation heterogeneity through high-resolution electrical and seismic tomography

The reconstruction of the current status of a historic building is essential for seismic safety assessment and for designing the retrofitting interventions since different safety and confidence factors have to be assumed, depending on the level of information about the subsoil structure. In this work, we present an investigation of the shallow subsurface below and around a historic building affected by differential settlements in order to define its geometry and to characterise its stiffness at low strain. To this end, we employed high-resolution electrical resistivity and seismic (both P-wave and S-wave) tomographies. A three-dimensional electrical resistivity tomography survey was performed to obtain more information about the type and the maximum depth of the building foundation. Electrical resistivity and seismic tomographies were carried out alongside the building, aimed at imaging the top soils and the near-surface geometry. The corresponding inverted models pointed out a remarkable heterogeneity of the shallow subsoil below the building, which is partly founded on a weathered layer and partly on a more rigid lithotype. This heterogeneity is probably a concurrent cause of the building's instability under both static and seismic loading. Our results demonstrate that the man-made fillings and the top soils have to be thoroughly investigated to fully understand the soil-structure behaviour. In this light, the integration of non-invasive high-resolution geophysical techniques, especially tomographic methods, has been proved to properly address the problem of imaging the shallow subsoil

Focusing on Soil Foundation Heterogeneity through High-resolution Tomography

An historical building affected by differential settlements, which were triggered by an earthquake, is investigated by means of high-resolution tomography, both electrical and seismic. The objective is to image the geometric structure of the shallow soil below the building and to characterize its stiffness at low strain. A preliminary reconstruction of the geological units has been recovered through the combined use of electrical and seismic data, where the depth of the travertine bedrock varies significantly within the study site. The range of variation of the main geophysical parameters (resistivity, P- and S-wave velocities) inferred from these models has been set as reference point for tuning the results obtained from the geophysical survey performed near the building. The inverted tomographic models obtained from data acquired alongside the building exhibit heterogeneity of the shallow subsoil, which is partly founded on a weathered layer and partly on a more rigid lithotype, probably a fractured travertine or a gravel layer. Therefore the fill anthropic soils can play a relevant role for the structural stability in case of shallow foundations built on a heterogeneous subsoil

Tomographic inversion of time-domain resistivity and chargeability data for the investigation of landfills using a priori information

In this paper, we present a new code for the modelling and inversion of resistivity and chargeability data using a priori information to improve the accuracy of the reconstructed model for landfill. When a priori information is available in the study area, we can insert them by means of inequality constraints on the whole model or on a single layer or assigning weighting factors for enhancing anomalies elongated in the horizontal or vertical directions. However, when we have to face a multilayered scenario with numerous resistive to conductive transitions (the case of controlled landfills), the effective thickness of the layers can be biased. The presented code includes a model-tuning scheme, which is applied after the inversion of field data, where the inversion of the synthetic data is performed based on an initial guess, and the absolute difference between the field and synthetic inverted models is minimized. The reliability of the proposed approach has been supported in two real-world examples; we were able to identify an unauthorized landfill and to reconstruct the geometrical and physical layout of an old waste dump. The combined analysis of the resistivity and chargeability (normalised) models help us to remove ambiguity due to the presence of the waste mass. Nevertheless, the presence of certain layers can remain hidden without using a priori information, as demonstrated by a comparison of the constrained inversion with a standard inversion. The robustness of the above-cited method (using a priori information in combination with model tuning) has been validated with the cross-section from the construction plans, where the reconstructed model is in agreement with the original design

Time-lapse monitoring of an electrokinetic soil remediation process through frequency-domain electrical measurements

The electrokinetic (EK) method is an emerging technique for soil remediation, even though a monitoring system of the contaminant removal through geophysical methods has not been developed yet. In this paper, frequency-domain time-lapse measurements are used on heavy-metal contaminated sediments for monitoring an EK remediation process in a small-scale measuring cell. Our goal is to monitor the development of the electrokinetic process within the sediment and to evaluate the total time needed for the treatment. In fact, frequency-domain electrical monitoring provides complex resistivity spectra at different time steps that can be correlated to changes in the physical properties of the sediments. We perform laboratory spectral induced polarization (SIP) measurements on different samples before, during and after the EK treatment, using different electrolyte solutions (acids and tap water), commonly employed in EK remediation. Direct-current measurements (resistivity and chargeability) were also acquired on one sample for testing the reliability of the system by a comparison with a widespread commercial instrumentation for field measurements. Results indicate that resistivity is a diagnostic parameter as long as it is linked to changes in water saturation, pH and ionic concentration and not to the percentage of metal extraction. The resistivity exhibited well-defined signatures as a function of time that changes depending on the conditioning agent and the grain size distribution. These peculiarities were used to understand the physical processes occurring within the cell and consequently to assess the effectiveness of the electrokinetic treatment. Conversely, the polarization effect was negligible using acids as conditioning agents at the electrolyte chambers. Therefore, the SIP method is not effective under these conditions, being the polarization effect significant only when tap water was used at both ends of the measuring cell. In this case, we were able to correlate changes in water saturation with the time-shift observed on relaxation time distributions (RTDs) after inversion of SIP data and to observe, using normalized chargeability, that polarization is stronger at high pH values. On these basis, resistivity is suitable to monitor the development of the remediation, to optimise the energy levels required for treatment and to assess the end time of the EK process (time when metal mobilization ends). In fact, the end time of treatment can be associated with the time at which resistivity becomes stable. This time is highly dependent on the particular working conditions and sediment grain size as demonstrated by our experiments