Petr Vaníček

Further development and properties of the spectral analysis by least-squares

The concept of spectral analysis using least-squares is further developed to remove any undesired influence on the spectrum. The influence of such a ‘systematic noise’ can be eliminated without the necessity of knowing the magnitudes of the noise constituents. The technique can be used for irregularly spaced as well as equidistantly spaced data. The response to random noise is found to be constant in the frequency domain and its expected level is derived. Presence of random noise in the analyzed time series is shown to transform the spectrum merely linearly. Examples of applications of the technique are presented.

Approximate spectral analysis by least-squares fit

An approximate method of spectral analysis called ‘successive spectral analysis’ based upon the mean-quadratic approximation of an empirical function by generalised trigonometric polynomial with both unknown frequencies and coefficients is developed. A few quotations describing some properties of the method as well as one of the possible methods for numerical solution are given.

Reformulation of Stokes's theory for higher than second‐degree reference field and modification of integration kernels

An argument is put forward in favor of using a model gravity field of a degree and order higher than 2 as a reference in gravity field studies. Stokess approach to the evaluation of the geoid from gravity anomalies is then generalized to be applicable to a higher than second-order reference spheroid. The effects of truncating Stokess integration and of modifying the integration kernels are investigated in the context of the generalized approach. Several different modification schemes, starting with a Molodenskij-like modification and ending with the least squares modification, are studied. Particular attention is devoted to looking at both global and local biases and mean square errors of the individual schemes.

Higher-degree reference field in the generalized Stokes-Helmert scheme for geoid computation

In this paper we formulate two corrections that have to be applied to the higher-degree reference spheroid if one wants to use it in conjunction with the Stokes-Helmert scheme for geoid determination. We show that in a precise geoid determination one has to apply the correction for the residual topographical potential and the correction for the earth ellipticity. Both these corrections may reach several decimetres; we show how their magnitudes vary within Canada and we give their global ranges.

Global displacements caused by point dislocations in a realistic Earth model

We define dislocation Love numbers [h nm ij , l nm ij , k nm ij , l nm t , ij ] and Greens functions to describe the elastic deformation of the Earth caused by a point dislocation and study the coseismic displacements caused in a radially heterogeneous spherical Earth model. We derive spherical harmonic expressions for the shear and tensile dislocations, which can be expressed by four independent solutions : a vertical strike slip, a vertical dip slip, a tensile opening in a horizontal plane, and a tensile opening in a vertical plane. We carry out calculations with a radially heterogeneous Earth model (1066A). The results indicate that the dominating deformations appear in the near field and attenuate rapidly as the epicentral distance increases. The shallower the point source, the larger the displacements. Both the Earths curvature and vertical layering have considerable effects on the deformation fields. Especially the vertical layering can cause a 10% difference at the epicentral distance of 0.1°. As an illustration, we calculate the theoretical displacements caused by the 1964 Alaska earthquake (m w = 9.2) and compare the results with the observed vertical displacements at 10 stations. The results of the near field show that the vertical displacement can reach some meters. The far-field displacements are also significant. For example, the horizontal displacements (u ψ ) can be as large as 1 cm at the epicentral distance of 30°, 0.5 cm at about 40°, magnitudes detectable by modern instrument, such as satellite laser ranging (SLR), very long baseline interferometry (VLBI) or Global Positioning System (GPS). Globally, the displacement (u r ) caused by the earthquake is larger than 0.25 mm.

Transformation of coordinates between two horizontal geodetic datums

The following topics are discussed in this paper: the geocentric coordinate system and its different realizations used in geodetic practice; the definition of a horizontal geodetic datum (reference ellipsoid) and its positioning and orientation with respect to the geocentric coordinate system; positions on a horizontal datum and errors inherent in the process of positioning; and distortions of geodetic networks referred to a horizontal datum. The problem of determining transformation parameters between a horizontal datum and the geocentric coordinate system from known positions is then analysed. It is often found necessary to transform positions from one horizontal datum to another. These transformations are normally accomplished through the geocentric coordinate system and they include the transformation parameters of the two datums as well as the representation of the respective network distortions. Problems encountered in putting these transformations together are pointed out.

To the problem of noise reduction in sea-level records used in vertical crustal movement detection

Abstract The aim of this investigation is to develop a simple technique that would allow us to use the sea-level records for detecting contemporary vertical crustal movements of duration from several months to several years. The choice of auxiliary data needed for any such analysis is restricted to the regularly available meteorological data to make this approach possible in routine search for precursory movements in earthquake-prone areas. A linear mathematical model is designed to evaluate the effect of atmospheric temperature and pressure variations, river discharge, long periodic tides and Chandlerian motion. Spectra of the residual sea-level variations are also shown. It is concluded that local episodic crustal movements of a magnitude larger than some 10 cm may be detectable by this approach. If finer resolution is needed then it it necessary to also account for steric level, wind, and sea-current variations, for which data are largely non-existent.

The Role of Coordinate Systems, Coordinates and Heights in Horizontal Datum Transformations

This paper reviews the fundamental definitions of geodetic and geocentric coordinate systems, whilst clarifying the distinction between coordinates and coordinate systems. It is then argued that the transformation of coordinates from a local geodetic datum to a geocentric datum should first employ a change of the coordinate system using a six- or four-parameter transformation, followed by further modeling of the distortion in the coordinates. It is also argued that the horizontal coordinate transformation should not include height information, since this forms an entirely different coordinate in another coordinate system.

A comparison of present sea level linear trends from tide gauge data and radiocarbon curves in Eastern Canada

Abstract A comparison is made of the results for determinations of vertical crustal movements by means of two independent sources of data widely used in geodesy and geology: tide gauge sea level linear trends and gradients of radiocarbon curves. As a reference, the map of vertical crustal movements of Canada is added which was computed using not only tide gauge data but also geodetic levelling data. These results show similar trends computed by both techniques but an average value of 0.1 m/century greater for the sea level trends. The map of vertical crustal movements is in good agreement with the trends provided by the gradients of radiocarbon curves.

An Alternative Algorithm to FFT for the Numerical Evaluation of Stokes's Integral

Stokess kernel used for the evaluation of a gravimetric geoid is a function of the spherical distance between the point of interest and the dummy point in the integration. Its values thus are obtained from the positions of pairs of points on the geoid. For the integration over the near integration zone (near to the point of interest), it is advantageous to pre-form an array of kernel values where each entry corresponds to the appropriate locations of the two points, or equivalently, to the latitude and the longitude-difference between the point of interest and a dummy point. Thus, for points of interest on the same latitude, the array of the Stokes kernel values remains the same and may only be evaluated once. Also, only one half of the array need be evaluated because of its longitudinal symmetry: the near zone can be folded along the meridian of the point of interest.Numerical tests show that computation speed improves significantly after this algorithm is implemented. For an area of 5 by 10 arc-degrees with the grid of 5 by 5 arc-minutes, the computation time reduces from half an hour to about 1 minute. To compute the geoid for the whole of Canada (20 by 60 arc-degrees, with the grid of 5 by 5 arc-minutes), it takes only about 17 minutes on a 400MHz PC computer.Compared with the Fast Fourier Transform algorithm, this algorithm is easier to implement including the far zone contribution evaluation that can be done precisely, using the (global) spectral description of the gravity field.