Analysis of radial basis function interpolation approach

The radial basis function （RBF） interpolation approach proposed by Freedman is used to solve inverse problems encountered in well-logging and other petrophysical issues. The approach is to predict petrophysical properties in the laboratory on the basis of physical rock datasets, which include the formation factor, viscosity, permeability, and molecular composition. However, this approach does not consider the effect of spatial distribution of the calibration data on the interpolation result. This study proposes a new RBF interpolation approach based on the Freedman＇s RBF interpolation approach, by which the unit basis functions are uniformly populated in the space domain. The inverse results of the two approaches are comparatively analyzed by using our datasets. We determine that although the interpolation effects of the two approaches are equivalent, the new approach is more flexible and beneficial for reducing the number of basis functions when the database is large, resulting in simplification of the interpolation function expression. However, the predicted results of the central data are not sufficiently satisfied when the data clusters are far apart.     

The choice of the spherical radial basis functions in local gravity field modeling

The choice of the optimal spherical radial basis function (SRBF) in local gravity field modelling from terrestrial gravity data is investigated. Various types of SRBFs are considered: the point-mass kernel, radial multipoles, Poisson wavelets, and the Poisson kernel. The analytical expressions for the Poisson kernel, the point-mass kernel and the radial multipoles are well known, while for the Poisson wavelet new closed analytical expressions are derived for arbitrary orders using recursions. The performance of each SRBF in local gravity field modelling is analyzed using real data. A penalized least-squares technique is applied to estimate the gravity field parameters. As follows from the analysis, almost the same accuracy of gravity field modelling can be achieved for different types of the SRBFs, provided that the depth of the SRBFs is chosen properly. Generalized cross validation is shown to be a suitable technique for the choice of the depth. As a good alternative to generalized cross validation, we propose the minimization of the RMS differences between predicted and observed values at a set of control points. The optimal regularization parameter is determined using variance component estimation techniques. The relation between the depth and the correlation length of the SRBFs is established. It is shown that the optimal depth depends on the type of the SRBF. However, the gravity field solution does not change significantly if the depth is changed by several km. The size of the data area (which is always larger than the target area) depends on the type of the SRBF. The point-mass kernel requires the largest data area.     

A new analytical approach to reconstruct the acceleration time history at the bedrock base from the free surface signal records

Acceleration time histories of earthquake events are typically measured in seismic stations that are placed close to the soil top surface. These acceleration records are often used as input data for seismic analysis. It may be used for base excitation in seismic analysis of above ground structures with shallow foundations.. However it may not be used for seismic analysis of underground structures, or even for above ground buildings with deep foundations and several underground stories. The required base excitation data of the latter should have been measured below the top surface, at a level that may be determined according to the specific analyzed building geometry or at the bedrock below. If the acceleration time history at the bedrock would have been known, the seismic wave propagation through the soil medium, from the bedrock towards the top surface, could have been carried out and the base excitation of the buried structure could be determined. Since there is no data on the acceleration time history at the bedrock, and the only given data is the acceleration records at the top surface, the goal of this paper is to provide an exact reverse analysis procedure to determine the unknown acceleration time history at the bedrock that would exactly produce the measured acceleration time history at the top surface. Once this goal is achieved, seismic analysis of buried structures may be carried out with the determined acceleration record at the bedrock as input. This paper presents an analytical exact solution of the inverse problem for determination of the acceleration, velocity and displacement time histories at the bedrock base of a layered geological medium that are compatible with the given acceleration record at the soil top surface. This new proposed method is based on analytical solutions of the initial-boundary value problems of the linear wave equation in the case of a layered medium. The relationship between waves in one layer and waves in another adjacent layer is derived considering the continuity of stresses and displacements at the common interface between the layers. The efficiency and accuracy of the proposed method is demonstrated through several examples involving the nonstationary response of the free surface. The case of the San Fernando Earthquake is studied. Excellent agreement is achieved between the recorded free surface time history and the reconstructed signal. This excellent agreement is obtained due to the exact analytical method used in deriving the inverse problem solution. This exact analytical method allows one to obtain an acceleration (velocity/displacement) distribution along all the layers at any time.     

Optical observations of Io's neutral clouds and plasma torus

A review of ground-based optical observations of Io's neutral clouds and plasma torus is provided. The physical processes determining the spatial distribution and intensity of torus emissions are described with reference to a model based on Voyager spacecraft data. The model is then compared to ground-based observations. Inconsistencies and variations in torus conditions over long timescales are emphasized. Periodicities in the torus evident in Voyager and ground-based are critically discussed.Processes determining the spatial characteristics of the neutral clouds are discussed. Observations of the slow sodium cloud are compared to model calculations. Special attention is paid to recent observations of high velocity neutrals and species in the upper atmosphere of Io itself. The article concludes with suggestions for future observations and research.     

Sliding-uplifting response of rigid blocks to base excitation

In the present paper a two-degree-of-freedom system is considered which allows the simulation of rigid blocks uplifting and sliding on frictional foundations; the monolateral constraint between block and base is schematized by means of a joint model, which allows the contact problem to be discretized. The joint model is governed by normal and shear constitutive laws, which have been derived by the phenomenological behaviour of stone blocks and rock joints, as given by rock mechanics. Furthermore, a numerical procedure has been developed in order to solve the non-linear equations governing the motion of the block-base system, and to analyse the dynamic response of this system under seismic excitation; particular attention has been paid to the influence of the vertical displacement on the slip response.     

On the geostatistical approach to the inverse problem

The geostatistical approach to the inverse problem is discussed with emphasis on the importance of structural analysis. Although the geostatistical approach is occasionally misconstrued as mere cokriging, in fact it consists of two steps: estimation of statistical parameters (“structural analysis”) followed by estimation of the distributed parameter conditional on the observations (“cokriging” or “weighted least squares”). It is argued that in inverse problems, which are algebraically undetermined, the challenge is not so much to reproduce the data as to select an algorithm with the prospect of giving good estimates where there are no observations. The essence of the geostatistical approach is that instead of adjusting a grid-dependent and potentially large number of block conductivities (or other distributed parameters), a small number of structural parameters are fitted to the data. Once this fitting is accomplished, the estimation of block conductivities ensues in a predetermined fashion without fitting of additional parameters. Also, the methodology is compared with a straightforward maximum a posteriori probability estimation method. It is shown that the fundamental differences between the two approaches are: (a) they use different principles to separate the estimation of covariance parameters from the estimation of the spatial variable; (b) the method for covariance parameter estimation in the geostatistical approach produces statistically unbiased estimates of the parameters that are not strongly dependent on the discretization, while the other method is biased and its bias becomes worse by refining the discretization into zones with different conductivity.     

Probability of radial anisotropy in the deep mantle

It is well established that the Earth's uppermost mantle is anisotropic, but observations of anisotropy in the deeper mantle have been more ambiguous. Radial anisotropy, the discrepancy between Love and Rayleigh waves, was included in the top 220 km of PREM, but there is no consensus whether anisotropy is present below that depth. Fundamental mode surface waves, for commonly used periods up to 200 s, are sensitive to structure in the first few hundred kilometers and therefore do not provide information on anisotropy below. Higher mode surface waves, however, have sensitivities that extend to and below the transition zone and should thus give insight about anisotropy at greater depths, but they are very difficult to measure. We previously developed a new technique to measure higher mode surface wave phase velocities with consistent uncertainties. These data are used here to construct probability density functions of a radially anisotropic Earth model down to approximately 1500 km. In the uppermost mantle, we obtain a high probability of faster horizontally polarized shear wave speed, likely to be related to plate motion. In the asthenosphere and transition zone, however, we find a high probability of faster vertically polarized shear wave speed. To a depth of 1500 km in the lower mantle, we see no significant shear wave anisotropy. This is consistent with results from laboratory measurements which show that lower mantle minerals are anisotropic but LPO is unlikely to develop in the pressure–temperature conditions present in the mid-mantle.     

Adaptation of prestack migration to multi-offset ground-penetrating radar (GPR) data

In adapting the prestack migration technique used in seismic imaging to the inversion of ground‐penetrating radar (GPR) from time‐ to depth‐sections, we show that the theoretical integral formulation of the inversion can be applied to electromagnetic problems, albeit with three assumptions. The first two assumptions concern the electromagnetic characteristics of the medium, primarily that the medium must be perfectly resistive and non‐dispersive, and the third concerns the antennae radiation pattern, which is taken to be 2D. The application of this adaptation of the inversion method is confirmed by migrating actual GPR measurements acquired on the test site of the Laboratoire Central des Ponts et Chaussées. The results show good agreement with the geometry of the structures in the medium and confirm that the possible departure from the assumption of a purely resistive medium has no visible effect on the information concerning the geometry of scattering and reflecting structures. The field experiments also show that prestack migration processing is sufficiently robust with regard to the assumption of a non‐dispersive medium. The assumption of a 2D antennae radiation pattern, however, produces artefacts that could be significant for laterally heterogeneous media. Nevertheless, where the medium is not highly laterally heterogeneous, the migration gives a clear image of the scattering potential due to the geometry of structural contrasts in the medium; the scatterers are well focused from diffraction hyperbolae and well localized. Spatial geometry has limited dimensional accuracy and positions are located with a maximum error equal to the minimum wavelength of the signal bandpass. Objects smaller than one wavelength can nevertheless be detected and well focused if their dielectric contrasts are sufficiently high, as in the case of iron or water in gneiss gravels. Furthermore, the suitability of multi‐offset protocols to estimate the electromagnetic propagating velocity and to decrease the non‐coherent noise level of measurements is confirmed. Our velocity estimation is based on the semblance calculation of multi‐offset migrated images, and we confirmed the relevance of this quantification method using numerical data. The signal‐to‐noise ratio is improved by summing multi‐offset results after the addition of random noise on measurements. Thus the adaptation of prestack migration to multi‐offset radar measurements significantly improves the resolution of the scattering potential of the medium. Limitations associated with the methods used here suggest that 3D algorithms should be applied to strongly laterally heterogeneous media and further studies concerning the waveform inversion are necessary to obtain information about the electric nature of the medium.     

A new approach to the problem of the lunar libration

Modern techniques permit a new approach to the solution of the problem of the lunar physical libration. The equations which express the phenomenon can be reconstructed and solved with the use of computers, while higher accuracy in its detection has been achieved with the development of a new observational method.     

Application of the BME approach to soil texture mapping

In order to derive accurate space/time maps of soil properties, soil scientists need tools that combine the usually scarce hard data sets with the more easily accessible soft data sets. In the field of modern geostatistics, the Bayesian maximum entropy (BME) approach provides new and powerful means for incorporating various forms of physical knowledge (including hard and soft data, soil classification charts, land cover data from satellite pictures, and digital elevation models) into the space/time mapping process. BME produces the complete probability distribution at each estimation point, thus allowing the calculation of elaborate statistics (even when the distribution is not Gaussian). It also offers a more rigorous and systematic method than kriging for integrating uncertain information into space/time mapping. In this work, BME is used to estimate the three textural fractions involved in a texture map. The first case study focuses on the estimation of the clay fraction, whereas the second one considers the three textural fractions (sand, silt and clay) simultaneously. The BME maps obtained are informative (important soil characteristics are identified, natural variations are well reproduced, etc.). Furthermore, in both case studies, the estimates obtained by BME were more accurate than the simple kriging (SK) estimates, thus offering a better picture of soil reality. In the multivariate case, classification error rate analysis in terms of BME performs considerably better than in terms of kriging. Analysis in terms of BME can offer valuable information to be used in sampling design, in optimizing the hard to soft data ratio, etc.     

Geometric approach to statistical analysis on the simplex

 The geometric interpretation of the expected value and the variance in real Euclidean space is used as a starting point to introduce metric counterparts on an arbitrary finite dimensional Hilbert space. This approach allows us to define general reasonable properties for estimators of parameters, like metric unbiasedness and minimum metric variance, resulting in a useful tool to better understand the logratio approach to the statistical analysis of compositional data, who's natural sample space is the simplex.     

周期径向荷载作用下圆弧格构拱动力稳定性分析

基于结构弹性稳定理论,根据能量法推导出以位移为基本未知量的圆弧格构拱总势能,从Hamilton原理出发,建立了考虑剪切变形的圆弧格构拱的动力稳定微分方程.利用Galerkin方法将其转化为二阶常微分Mathieu型参数共振方程,求得周期解所包围的动力不稳定区域,探讨了圆弧格构拱发生参数共振的动力稳定性问题,分析了缀条面...     

Hydromechanics for the formation and development of radial sandbanks (I)

Based upon the long-term observation of field data, a two-dimensional numerical model is applied to simulating the tidal flow covering from the neap tide to spring tide in the radial sandbank area in the southern Yellow Sea. From the development of tidal current ridges under the hydrodynamic action, multi-purpose analysis and study are carried out, which include the propagation process of tidal wave, the distributions of tidal wave energy rate and tidal range, the tidal ellipses and traces. It is shown that the tidal current is the major dynamic factor for the formation and development of the radial sandbanks, and the differences of tidal wave energy rate and current strength determine the distinct plane shapes of ridges and troughs in this region.     

Adaptation of the ionospheric radio channel parameters to the current conditions using observations of the signals of reference radio stations

A method of adaptation of the ionospheric channel parameters to the current conditions is proposed. The observational data on the temporal behavior of signals of radio stations with known location and set of working frequencies (the so-called reference stations) are used as information describing the current state of the channel. A facility that makes possible to determine by simple means the moments of radio rises and declines on the reference paths has been created. A technique that relates for these instants working frequencies of the stations with maximum usable frequencies of the paths has been developed. By means of an approximate solution of the inverse problem using the method of characteristics for the calculation of radiowave propagation and the monthly average semiempirical model of the ionosphere, the height profile of electron density in a certain spatial region is refined. The efficiency of the proposed method of adaptation of the ionospheric channel parameters using the data of vertical sounding is assessed and the expediency of practical application of such adaptation is shown.     

Primitive functions used to compute the potential and its derivatives of homogeneously magnetized two-dimensional bodies

Summary A method of computing the potential, and its first derivatives, of a homogeneously magnetized body is presented, in which the body is approximated by a polygon with sides parallel to the coordinate axes. The potential, or derivatives of the potential, of such a body can be computed as the sum of values of primitive functions at the vertices of polygon.