Approximating the Distribution of Pareto Sums

Heavy tailed random variables (rvs) have proven to be an essential element in modeling a wide variety of natural and human-induced processes, and the sums of heavy tailed rvs represent a particularly important construction in such models. Oriented toward both geophysical and statistical audiences, this paper discusses the appearance of the Pareto law in seismology and addresses the problem of the statistical approximation for the sums of independent rvs with common Pareto distribution F(x)=1 – x^– for 1/2 < < 2. Such variables have infinite second moment which prevents one from using the Central Limit Theorem to solve the problem. This paper presents five approximation techniques for the Pareto sums and discusses their respective accuracy. The main focus is on the median and the upper and lower quantiles of the sums distribution. Two of the proposed approximations are based on the Generalized Central Limit Theorem, which establishes the general limit for the sums of independent identically distributed rvs in terms of stable distributions; these approximations work well for large numbers of summands. Another approximation, which replaces the sum with its maximal summand, has less than 10% relative error for the upper quantiles when < 1. A more elaborate approach considers the two largest observations separately from the rest of the observations, and yields a relative error under 1% for the upper quantiles and less than 5% for the median. The last approximation is specially tailored for the lower quantiles, and involves reducing the non-Gaussian problem to its Gaussian equivalent; it too yields errors less than 1%. Approximation of the observed cumulative seismic moment in California illustrates developed methods.     

A new ellipsoidal gravimetric, satellite altimetry and astronomic boundary value problem, a case study: The geoid of Iran

A new gravimetric, satellite altimetry, astronomical ellipsoidal boundary value problem for geoid computations has been developed and successfully tested. This boundary value problem has been constructed for gravity observables of the type (i) gravity potential, (ii) gravity intensity (i.e. modulus of gravity acceleration), (iii) astronomical longitude, (iv) astronomical latitude and (v) satellite altimetry observations. The ellipsoidal coordinates of the observation points have been considered as known quantities in the set-up of the problem in the light of availability of GPS coordinates. The developed boundary value problem is ellipsoidal by nature and as such takes advantage of high precision GPS observations in the set-up. The algorithmic steps of the solution of the boundary value problem are as follows:

- Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and of the ellipsoidal centrifugal field for the removal of the effect of global gravity and the isostasy field from the gravity intensity and the astronomical observations at the surface of the Earth.

- Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the gravity intensity and the astronomical observations at the surface of the Earth the effect of the residual masses at the radius of up to 55 km from the computational point.

- Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and ellipsoidal centrifugal field for the removal from the geoidal undulations derived from satellite altimetry the effect of the global gravity and isostasy on the geoidal undulations.

- Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the geoidal undulations derived from satellite altimetry the effect of the water masses outside the reference ellipsoid within a radius of 55 km around the computational point.

- Least squares solution of the observation equations of the incremental quantities derived from aforementioned steps in order to obtain the incremental gravity potential at the surface of the reference ellipsoid.

- The removed effects at the application points are restored on the surface of reference ellipsoid.

- Application of the ellipsoidal Bruns’ formula for converting the potential values on the surface of the reference ellipsoid into the geoidal heights with respect to the reference ellipsoid.

- Computation of the geoid of Iran has successfully tested this new methodology.

High-resolution regional geoid computation without applying Stokes’s formula: a case study of the Iranian geoid

A new theory for high-resolution regional geoid computation without applying Stokess formula is presented. Operationally, it uses various types of gravity functionals, namely data of type gravity potential (gravimetric leveling), vertical derivatives of the gravity potential (modulus of gravity intensity from gravimetric surveys), horizontal derivatives of the gravity potential (vertical deflections from astrogeodetic observations) or higher-order derivatives such as gravity gradients. Its algorithmic version can be described as follows: (1) Remove the effect of a very high degree/order potential reference field at the point of measurement (POM), in particular GPS positioned, either on the Earths surface or in its external space. (2) Remove the centrifugal potential and its higher-order derivatives at the POM. (3) Remove the gravitational field of topographic masses (terrain effect) in a zone of influence of radius r. A proper choice of such a radius of influence is 2r=4×10⁴ km/n, where n is the highest degree of the harmonic expansion. (cf. Nyquist frequency). This third remove step aims at generating a harmonic gravitational field outside a reference ellipsoid, which is an equipotential surface of a reference potential field. (4) The residual gravitational functionals are downward continued to the reference ellipsoid by means of the inverse solution of the ellipsoidal Dirichlet boundary-value problem based upon the ellipsoidal Abel–Poisson kernel. As a discretized integral equation of the first kind, downward continuation is Phillips–Tikhonov regularized by an optimal choice of the regularization factor. (5) Restore the effect of a very high degree/order potential reference field at the corresponding point to the POM on the reference ellipsoid. (6) Restore the centrifugal potential and its higher-order derivatives at the ellipsoidal corresponding point to the POM. (7) Restore the gravitational field of topographic masses ( terrain effect) at the ellipsoidal corresponding point to the POM. (8) Convert the gravitational potential on the reference ellipsoid to geoidal undulations by means of the ellipsoidal Bruns formula. A large-scale application of the new concept of geoid computation is made for the Iran geoid. According to the numerical investigations based on the applied methodology, a new geoid solution for Iran with an accuracy of a few centimeters is achieved.Acknowledgments. The project of high-resolution geoid computation of Iran has been support by National Cartographic Center (NCC) of Iran. The University of Tehran, via grant number 621/3/602, supported the computation of a global geoid solution for Iran. Their support is gratefully acknowledged. A. Ardalan would like to thank Mr. Y. Hatam, and Mr. K. Ghazavi from NCC and Mr. M. Sharifi, Mr. A. Safari, and Mr. M. Motagh from the University of Tehran for their support in data gathering and computations. The authors would like to thank the comments and corrections made by the four reviewers and the editor of the paper, Professor Will Featherstone. Their comments helped us to correct the mistakes and improve the paper.     

Hydrodynamic interactions and relative motions of two floating platforms with mooring lines in side-by-side offloading operation

The hydrodynamic interaction and mechanical coupling effects of two floating platforms connected by elastic lines are investigated by using a time-domain multi-hull/mooring/riser coupled dynamics analysis program. Particular attention is paid to the contribution of off-diagonal hydrodynamic interaction terms on the relative motions during side-by-side offloading operation. In this regard, the exact method (CMM: combined matrix method) including all the vessel and line dynamics, and the 12×12 hydrodynamic coefficients in a combined matrix is developed. The performance of two typical approximation methods (NHI/No Hydrodynamic Interaction: iteration method between two vessels without considering hydrodynamic interaction effects; SMM/Separated Matrix Method: iteration method between two vessels with partially considering hydrodynamic interaction effects, i.e. ignoring off-diagonal cross-coupling terms in the 12×12 hydrodynamic coefficient matrix) is also tested for the same side-by-side offloading operation in two different environmental conditions. The numerical examples show that there exists significant discrepancy at sway and roll modes between the exact and the approximation methods, which means that the cross-coupling (off-diagonal block) terms of the full hydrodynamic coefficient matrix play an important role in the case of side-by-side offloading operation. Therefore, such approximation methods should be used with care. The fender reaction forces, which exhibit large force with contact but no force without contact, are also numerically modeled in the present time-domain simulation study.     

Asymptotic Analysis Methods for Structural Reliability

- Applying Laplace asymptotic approximation of integral, asymptotic analysis methods of structural reliability are proposed in generalized and orthogonal random space. Analytic results show that the proposed methods are simple in calculation and accurate enough for problems of continuous random variables.     

The effects of a porous-elastic seabed on interfacial wave propagation

A theoretical model for the decay of progressive interfacial gravity waves propagating above a porous bed is developed assuming potential flow in a two-layer system with a free surface and a sharp interface. A new wave dispersion relation for two-layer flow above a quasi-static porous seabed is derived and investigated. The solutions for the nonlinear wave profile are derived using a perturbation method and the effects of geometric and flow parameters including bed characteristics, depth ratios and the densities of the two fluids are studied and discussed. Comparisons with existing analytical solutions for viscous interfacial wave attenuation over a rigid bed demonstrate the relative importance of the porous bed as a mechanism for wave decay. It is shown that the influence of a porous seabed on wave propagation is significant when the depth of the lower layer, normalised by the wavenumber, is less than π.     

A smoothing algorithm to enhance finite-element tsunami modelling: An application to the 5 February 1783 Calabrian case,Italy

Numerical simulations are essential tools for studying tsunami generation and evolution and finite-element (FE) methods are widely used, especially because of their capability in modeling water waves in basins with complex bathymetry and irregular coastlines. This paper presents the numerical simulation of an historical Italian tsunami that affected the Tyrrhenian coasts of Calabria and Sicily on 5 February 1783 following a strong destructive earthquake that was the first of a terrible sequence of seismic shocks terrifying the Calabrian population for more than two months. The numerical model is an FE model based on the nonlinear nondispersive shallow-water approximation of the Navier-Stokes equations. Since FE discretization schemes may lead to solutions undesirably affected by noise over coarse grids, in this study numerical noise is controlled by suitably smoothing the FE solution at regular time steps t _s. The performance of our smoothing algorithm is tested for significant linear cases for which an analytical solution is available.     

Impulse approximation improved

Improved formulas of impulse approximation method for stellar perturbations are derived. The method proposed involves a deflection of the stellar path. It is also applicable to an arbitrary time interval. A comparison of the classical vs improved method is presented both in qualitative discussion and numerical results for Oort cloud cometary orbits.     

Probability density functions for non-linear random waves and responses

Non-linear waves have been a research topic for a long time, but have not yet entered common ocean engineering practice. In this article the problem of calculating the probability density function (pdf) for non-linear waves is investigated. The ‘saddle point approximation’ is applied to a second-order wave model to estimate the pdf. The performance of the approximation is compared with the performances of the more commonly used methods for different types of input spectra. In addition, a non-parametric transformation method, with extrapolation of the level crossing intensity, is proposed to estimate the pdf. One simulation and three different data sets are used to investigate the performance of the model.     

Wave kinematics of partial reflection from a vertical wall

A third-order perturbation approximation for the partial reflection from a vertical wall is presented in this paper. The wave parameters are expressed in terms of the amplitude of incident waves. The reflection coefficient is defined as the ratio of the height of reflected waves to incident waves. The numerical results demonstrate the significant influences of reflected coefficient on the wave profile and wave frequency bifurcation. For example, the critical angle of wave frequency bifurcation with partial reflection is about 7.5 degrees, not 21 degrees as reported previously for fully reflection.