Maximum Expected Magnitude Assessment in a Geo Computer Environment: Case Study

Different approaches to seismic hazard assessment are compared. Each of them could be applied more or less successfully for territories which are stable in time and have a high level of seismic activity. A long-term seismic catalogue, not only including historical but also paleoearthquake data, is an essential requirement. But, in practice, such an ideal situation is very rare. Initial data is usually poor and short-term. Seismic hazard assessment could be more complicated for regions which are transient between relatively stable platforms and active mountain massifs. A new step in geoinformation technology for seismic hazard assessment based on a GEO computer environment is presented, its application is illustrated by the real case hazard evaluation for the territory of the Stavropol region, which is situated between the Russian platform and the Great Caucasus. The regional catalogue covers a time period of about 150 years. Though the region under consideration is not large, seismic activity is variable in space, from almost aseismic zones to rather active areas. GEO allows us to incorporate different techniques and all available information in the analysis, including those which are very difficult to formalize. The space distribution of the maximum expected earthquake magnitude is determined as a function of geological and geophysical data. An important feature of GEO is that it makes it possible to control the result of complicated algorithms through some relatively simple physical reasons.     

Storage-efficient method for generating contours focusing on roundness

A storage-efficient contour generation method, focusing on planar contours, is developed. Given cartographic elevations on a rectangular lattice, a continuous bivariate function, z = f(x, y), is determined by interpolating the elevation values. Then, we focus on a contour determined by z = constant. The contour curve is partitioned into multiple sections, each of which is exactly or approximately round. Three curvature types are introduced to evaluate the roundness of each section. The area and perimeter of the contour are computed by one-dimensional line integration using Green’s theorem. If the contour is open, it is divided into two curves starting from the same initial point, with the control points advancing in opposite directions. Two types of numerical experiments are performed to validate the effectiveness of the proposed method. One experiment uses an analytically defined elevation function and investigates the number of control points and computation time for a resulting computation error. The second experiment uses actual digital elevation model data of an isolated island in Japan and compares the proposed method with existing ones. Because the algorithm does not require lattice subdivision and the number of control points is drastically reduced, the proposed method is storage efficient.     

A reliable and efficient two-stream algorithm for spherical radiative transfer: Documentation of accuracy in realistic layered media

We present a fast and well documented two-stream algorithm for radiative transfer and particle transport in vertically inhomogeneous, layered media. The physical processes considered are internal production (emission), scattering, absorption, and Lambertian reflection at the lower boundary. The medium may be forced by internal sources as well as by parallel or uniform incidence at the top boundary. This two-stream algorithm is based on a general purpose multi-stream discrete ordinate algorithm released previously. It incorporates all the advanced features of this well-tested and unconditionally stable algorithm, and includes two new features: (i) corrections for spherical geometry, and (ii) an efficient treatment of internal sources that vary rapidly with depth. It may be used to compute fluxes, flux divergences and mean intensities (actinic fluxes) at any depth in the medium. We have used the numerical code to investigate the accuracy of the two-stream approximation in vertically inhomogeneous media. In particular, computations of photodissociation and warming/cooling rates and surface fluxes of ultraviolet and visible radiation for clear, cloudy and aerosol-loaded atmospheres are presented and compared with results from multi-stream computations. The O₃ +hv O(¹D) + O₂ and O₃ +hv O(³P) + O₂ photodissociation rates were considered for solar zenith angles between 0.0–70.0° and surface albedos in the range 0.0–1.0. For small and moderate values of the solar zenith angle and the surface albedo the error made by the two-stream approximation is generally smaller, <10%, than the combined uncertainty in cross sections and quantum yields. Surface ultraviolet and visible fluxes were calculated for the same range of solar zenith angles and surface albedos as the photodissociation rates. It was found that surface ultraviolet and visible fluxes may be calculated by the two-stream approximation with 10% error or less for solar zenith angles less than 60.0° and surface albedos less than 0.5. For large solar zenith angles and/or large surface albedos, conditions typical at high latitudes, the error made by the two-stream approximation may become appreciable, i.e. 20% or more for the photodissociation rates in the lower stratosphere and for ultraviolet and visible surface fluxes for large surface albedos. The two-stream approximation agrees well with multi-stream results for computation of warming/cooling rates except for layers containing cloud and aerosol particles where errors up to 10% may occur. The numerical code provides a fast, well-tested and robust two-stream radiative transfer program that can be used as a software tool by aeronomers, atmospheric physicists and chemists, climate modellers, meteorologists, photobiologists and others concerned with radiation or particle transport problems. Copies of the FORTRAN77 program are available to interested users.     

Tsunami trapping near circular islands

Trapping of long water waves that are induced by submarine earthquakes and that attack circular islands is studied by applying a theoretical model (Tinti andVannini, 1994) that is based on the linear shallow water approximation. The solution is computed as the superposition of the eigenmodes of the water basin. The tsunami trapping is seen in terms of the capability of the source to excite the trapped eigenmodes of the basin. The bottom depth dependence around the island is shown to be quite important in determining the trapping capability of the island: a depth profile that is downwardly concave as the distance from the island coasts increases is substantially more efficient in amplifying the incoming waves and in trapping their energy than a profile exhibiting an upward concavity.     

Higher-order approximation techniques for estimating stochastic parameter of a sediment transport model

Higher-order approximation techniques for estimating stochastic parameter of the non-homogeneous Poisson (NHP) model are presented. The NHP model is characterized by a two-parameter cumulative probability distribution function (CDF) of sediment displacement. Those two parameters are the temporal and spatial intensity functions, physically representing the inverse of the average rest period and step length of sediment particles, respectively. Difficulty of estimating the parameters has, however, restricted the applications of the NHP model. The approximation techniques are proposed to address such problem. The basic idea of the method is to approximate a model involving stochastic parameters by Taylor series expansion. The expansion preserves certain higher-order terms of interest. Using the experimental (laboratory or field) data, one can determine the model parameters through a system of equations that are simplified by the approximation technique. The parameters so determined are used to predict the cumulative distribution of sediment displacement. The second-order approximation leads to a significant reduction of the CDF error (of the order of 47%) compared to the first-order approximation. Error analysis is performed to evaluate the accuracy of the first- and second-order approximations with respect to the experimental data. The higher-order approximations provide better estimations of the sediment transport and deposition that are critical factors for such environment as spawning gravel-bed.     

波动方程有限差分法叠前深度偏移

从地震叠前反射椭圆方程出发,本文导出了基于波动理论的共偏移距地震剖面叠前偏移方程,然后对此方程进行参考速度场中的浮动坐标变换,获得了叠前深度偏移方程．为了解决叠前衍射方程中含有对深度的二阶导数引起波场延拓成像的不适定问题,文中采用低阶偏微分方程组近似描述全上行波的办法,得到了衍射方程的高阶近似方程,并给出了计算衍射方程和折射方程稳定的差分格式,最后用此方法编制的程序对某一碳酸岩地区的地震资料进行了试处理,效果良好．     

Representation of radiation damping by fractional time derivatives

When modelling unbounded domains, formulation of a matrix‐valued force–displacement relationship which can take radiation damping into account is of major importance. In this paper, a method to describe the dynamic stiffness by a system of fractional differential equations in the time‐domain is presented. Here, a doubly asymptotic rational approximation of the low‐frequency force–displacement relationship is used, whereas a direct interpretation of the asymptotic part as a fractional derivative is possible. The numerical solution of the corresponding system of fractional differential equations is demonstrated using the infinite beam on elastic foundation as an example. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical Investigations in the Analytical and Semi-Analytical Computation of Gravimetric Terrain Effects

A recently proposed method for the computation of the gravitational effect due to the topographic masses defined by a Digital Elevation Model (DEM) involves the representation of the surface relief by means of parts of bilinear surfaces. The so-called bilinear method delivers eventually the mathematical model for the gravitational attraction of a right rectangular prism, whose top is modeled by a bilinear surface. Scope of the paper is to assess the new method by conducting numerical tests using both real and synthetic data. The performance of the bilinear method is evaluated in terms of its computational efficiency as well as its precision by comparing it with other analytical methods available for the practical evaluation of gravitational terrain effects. The techniques considered for the assessment of the bilinear approximation are the vastly applied right rectangular prism method and the polyhedral modeling, a less popular but extremely flexible approach based on the closed expression for the gravity field of an arbitrarily shaped mass distribution defined by planar faces. The different geometric modeling of the topographic relief produces discrepancies to the gravitational attraction of up to several mGal. Thus the choice for the geometric representation of the terrain plays a fundamental role to the numerical computation of potential field quantities especially in the critical region surrounding the computation point.     

On the Validity of the Boussinesq Approximation for the Elder Problem

The Elder problem is an example of a density driven flow, motivated by experiments of a thermally driven convection in porous media. It is a mathematical benchmark problem used for code verification of density driven flow simulators and comparison of issues related to its numerical treatment. Its bifurcation diagram with respect to the Rayleigh number is investigated on a hierarchy of uniformly refined grids. Eleven stationary solutions are shown to exist for the Elder problem. Similar solutions can be found using the Boussinesq approximation. Despite this similarity the corresponding bifurcation diagrams are shown to be topologically not equivalent. This gives rise to serious doubts on the validity of Boussinesq approximation for this model problem. Grid convergence is investigated for the numerically obtained solutions.     

小波分析在大地测量中的应用及其进展

回顾了近10年来大地测量中小波分析应用研究的主要进展，并对今后重力场中的小波应用研究提出了若干建议。