Simulation of normal distributed smooth fields by Karhunen-Loéve expansion in combination with kriging

Simulation of multigaussian stochastic fields can be made after a Karhunen-Loéve expansion of a given covariance function. This method is also called simulation by Empirical Orthogonal Functions. The simulations are made by drawing stochastic coefficients from a random generator. These numbers are multiplied with eigenfunctions and eigenvalues derived from the predefined covariance model. The number of eigenfunctions necessary to reproduce the stochastic process within a predefined variance error, turns out to be a cardinal question. Some ordinary analytical covariance functions are used to evaluate how quickly the series of eigenfunctions can be truncated. This analysis demonstrates extremely quick convergence to 99.5% of total variance for the 2nd order exponential (‘gaussian’) covariance function, while the opposite is true for the 1st order exponential covariance function. Due to these convergence characteristics, the Karhunen-Loéve method is most suitable for simulating smooth fields with ‘gaussian’ shaped covariance functions. Practical applications of Karhunen-Loéve simulations can be improved by spatial interpolation of the eigenfunctions. In this paper, we suggest interpolation by kriging and limits for reproduction of the predefined covariance functions are evaluated.     

Simulation of normal distributed smooth fields by Karhunen-Loéve expansion in combination with kriging

Simulation of multigaussian stochastic fields can be made after a Karhunen-Loéve expansion of a given covariance function. This method is also called simulation by Empirical Orthogonal Functions. The simulations are made by drawing stochastic coefficients from a random generator. These numbers are multiplied with eigenfunctions and eigenvalues derived from the predefined covariance model. The number of eigenfunctions necessary to reproduce the stochastic process within a predefined variance error, turns out to be a cardinal question. Some ordinary analytical covariance functions are used to evaluate how quickly the series of eigenfunctions can be truncated. This analysis demonstrates extremely quick convergence to 99.5% of total variance for the 2nd order exponential (‘gaussian’) covariance function, while the opposite is true for the 1st order exponential covariance function. Due to these convergence characteristics, the Karhunen-Loéve method is most suitable for simulating smooth fields with ‘gaussian’ shaped covariance functions. Practical applications of Karhunen-Loéve simulations can be improved by spatial interpolation of the eigenfunctions. In this paper, we suggest interpolation by kriging and limits for reproduction of the predefined covariance functions are evaluated.     

Determination of S-wave velocity structure using microtremors and spac method applied in Thessaloniki (Greece)

Array measurements of microtremors at 16 sites in the city of Thessaloniki were performed to estimate the V_s velocity of soil formations for site effect analysis. The spatial autocorrelation method was used to determine phase velocity dispersion curves in the frequency range from 0.8–1.5 to 6–7 Hz. A Rayleigh wave inversion technique (stochastic method) was subsequently applied to determine the Vs profiles at all the examined sites. The determination of V_s profiles reached a depth of 320 m. Comparisons with V_s values from cross-hole tests at the same sites proved the reliability of the SPAC method. The accuracy of the V_s profiles, the ability to reach large penetration depths in densely populated urban areas and its low cost compared to conventional geophysical prospecting, make Mictrotremor Exploration Method very attractive and useful for microzonation and site effects studies. An example of its application for the site characterization in Thessaloniki is presented herein.     

Curve-Fitting Schemes for Fragmentation Data

Fragmentation measurements in the form of sieve passing and mass fraction data were used to test the capability of three different distributions to fit the observed data over a wide range in fragment size and mass. These distributions were based on Rosin-Rammler, lognormal and simple sigmoidal (S-shaped) functions, having 2 input parameters for the single-component versions and 5 input parameters for the two-component versions. Provided convergence was achieved in the non-linear curve-fitting technique, the two-component versions always provided superior fits to the observed data. However, these versions were very sensitive to variations in the values chosen for the input parameters. In this particular regard, the two-component sigmoidal function was the most robust. The present results also show that the two-component lognormal function provided the best fit to the fragmentation data in a general sense, and the two-component Rosin-Rammler function provided the worst fit. However, there was not a significant difference between any of the three methods.     

Martian Topography: Scaling, Craters, and High-Order Statistics

The high-order structure functions of Mars topography reveal three specific ranges of scales: (1) scaling range at small scales where the structure functions exhibit scaling behavior; (2) transition range where the structure functions continue to grow but do not reveal scaling; and (3) saturation range at large scales where the structure functions saturate. The scaling and saturation ranges are explored in detail in respect to scaling and intermittency. Analysis of the Mars Orbiter Laser Altimeter (MOLA) data and computer simulations suggest that there are two potential contributors to the small-scale scaling: (i) scale-invariant surface formation; and (ii) effects of discrete morphological forms such as craters. The crater effect also provides an explanation for the large-scale intermittency revealed using the normalized structure functions within the saturation range, which cannot be explained by the ‘scale-invariant’ concept. Overall, the obtained results suggest that the “crater” contribution to the structure function behavior often dominates over the effect of the scale-invariant surface formation.     

Elastic lateral dynamic impedance functions for a rigid cylindrical shell type foundation

Elastic lateral dynamic impedance functions are defined as the ratio of the lateral dynamic force/moment to the corresponding lateral displacement/rotation at the top ending of a foundation at very small strains. Elastic lateral dynamic impedance functions have a defining influence on the natural frequencies of offshore wind turbines supported on cylindrical shell type foundations, such as suction caissons, bucket foundations, and monopiles. This paper considers the coupled horizontal and rocking vibration of a cylindrical shell type foundation embedded in a fully saturated poroelastic seabed in contact with a seawater half‐space. The formulation of the coupled seawater–shell–seabed vibration problem is simplified by treating the shell as a rigid one. The rigid shell vibration problem is approached by the integral equation method using ring‐load Green's functions for a layered seawater‐seabed half‐space. By considering the boundary conditions at the shell–soil interface, the shell vibration problem is reduced to Fredholm integral equations. Through an analysis of the corresponding Cauchy singular equations, the intrinsic singular characteristics of the problem are rendered explicit. With the singularities incorporated into the solution representation, an effective numerical method involving Gauss–Chebyshev method is developed for the governing Fredholm equations. Selected numerical results for the dynamic contact load distributions, displacements of the shell, and lateral dynamic impedance functions are examined for different shell length–radius ratio, poroelastic materials, and frequencies of excitation. Copyright © 2016 John Wiley & Sons, Ltd.     

Thermodynamic properties of tooeleite,Fe63+(As3+O3)4(SO4)(OH)4·4H2O

Tooeleite, nominally Fe₆³⁺(As³⁺O₃)₄(SO₄)(OH)₄·4H₂O, is a relatively uncommon mineral of some acid-mine drainage systems. Yet, if it does occur, it does so in large quantities, indicating that some specific conditions favor the formation of this mineral in the system Fe-As-S-O-H. In this contribution, we report the thermodynamic properties of synthetic tooeleite. The sample was characterized by powder X-ray diffraction, scanning electron microscopy, extended X-ray absorption fine-structure spectroscopy, and Mössbauer spectroscopy. These methods confirmed that the sample is pure, devoid of amorphous impurities of iron oxides, and that the oxidation state of arsenic is 3+. Using acid-solution calorimetry, the enthalpy of formation of this mineral from the elements at the standard conditions was determined as −6196.6 ± 8.6 kJ mol⁻¹. The entropy of tooeleite, calculated from low-temperature heat capacity data measured by relaxation calorimetry, is 899.0 ± 10.8 J mol⁻¹ K⁻¹. The calculated standard Gibbs free energy of formation is −5396.3 ± 9.3 kJ mol⁻¹. The log K_sp value, calculated for the reaction Fe₆(AsO₃)₄(SO₄)(OH)₄·4H₂O + 16H⁺ = 6Fe³⁺ + 4H₃AsO₃ + SO₄²⁻ + 8H₂O, is −17.25 ± 1.80. Tooeleite has stability field only at very high activities of aqueous sulfate and arsenate. As such, it does not appear to be a good candidate for arsenic immobilization at polluted sites. An inspection of speciation diagrams shows that the predominance field of Fe³⁺ and As³⁺ overlap only at strongly basic conditions. The formation of tooeleite, therefore, requires strictly selective oxidation of Fe²⁺ to Fe³⁺ and, at the same time, firm conservation of the trivalent oxidation state of arsenic. Such conditions can be realized only by biological systems (microorganisms) which can selectively oxidize one redox-active element but leave the other ones untouched. Hence, tooeleite is the first example of an “obligatory” biomineral under the conditions prevailing at or near the Earth's surface because its formation under these conditions necessitates the action of microorganisms.     

Fourier-based multi-scale representation and progressive transmission of cartographic curves on the internet

ABSTRACT

Automatic generation of multi-scale representations from the same spatial data source has been the research focus in map generalization for a long time. Based on the Fourier technique, this paper proposes a continuous, multi-scale representation model for progressive transformation of cartographic curves on the Internet. In our method, all the curves, whether closed or open, are depicted as periodical functions which are further expressed as Fourier series. The convergence degrees of the Fourier series are explored for different kinds of curves, and truncating frequencies are derived based on the similarity between the original and reconstructed curves. Using information theory and the Radical Law in cartography, the relationship between map scales and Fourier frequencies is established. Based on the proposed multi-scale model, we also introduce the principles and implementation of a progressive transmission method. Our method is evaluated using the contours from a topographic map. The results show that our model is a valid approach to multi-scale representation of cartographic curves.