The statistical properties of least-squares frequency analysis of unequally spaced data are examined. It is shown that, in the least-squares spectrum of gaussian noise, the reduction in the sum of squares at a particular frequency is aX22 variable. The reductions at different frequencies are not independent, as there is a correlation between the height of the spectrum at any two frequencies,f1 andf2, which is equal to the mean height of the spectrum due to a sinusoidal signal of frequencyf1, at the frequencyf2. These correlations reduce the distortion in the spectrum of a signal affected by noise. Some numerical illustrations of the properties of least-squares frequency spectra are also given.

Least - squares frequency analysis of unequally spaced data

Criticism and the Growth of Knowledge. Edited by I. Lakatos and A. Musgrave. Pp. viii, 282. £3·50, paperback £1. 1970. (Cambridge University Press.)

The Lomb-Scargle periodogram is a common tool in the frequency analysis of unequally spaced data equivalent to least-squares fitting of sine waves. We give an analytic solution for the generalisation to a full sine wave fit, including an offset and weights (χ 2 fitting). Compared to the Lomb-Scargle periodogram, the generalisation is superior as it provides more accurate frequencies, is less susceptible to aliasing, and gives a much better determination of the spectral intensity. Only a few modifications are required for the computation and the computational effort is similar. Our approach brings together several related methods that can be found in the literature, viz. the date-compensated discrete Fourier transform, the floating-mean periodogram, and the “spectral significance” estimator used in the SigSpec program, for which we point out some equivalences. Furthermore, we present an algorithm that implements this generalisation for the evaluation of the Keplerian periodogram that searches for the period of the best-fitting Keplerian orbit to radial velocity data. The systematic and non-random algorithm is capable of detecting eccentric orbits, which is demonstrated by two examples and can be a useful tool in searches for the orbital periods of exoplanets.

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The generalised Lomb-Scargle periodogram. A new formalism for the floating-mean and Keplerian periodograms

Interpretation theory in applied geophysics: Grant, F S ... Interpretation theory in applied geophysics Unknown Binding – January 1, 1965 5.0 out of 5 stars 1 rating. See all formats and editions Hide other formats and editions. The Amazon Book Review Book recommendations, author interviews, editors' picks, and more. Read it now. Enter your mobile number or email address below and we'll send you a ...

Interpretation Theory in Applied Geophysics Grant (McGraw-Hill, 140s.)

The 28 June Landers earthquake brought the San Andreas fault significantly closer to failure near San Bernardino, a site that has not sustained a large shock since 1812. Stress also increased on the San Jacinto fault near San Bernardino and on the San Andreas fault southeast of Palm Springs. Unless creep or moderate earthquakes relieve these stress changes, the next great earthquake on the southern San Andreas fault is likely to be advanced by one to two decades. In contrast, stress on the San Andreas north of Los Angeles dropped, potentially delaying the next great earthquake there by 2 to 10 years.

https://science.sciencemag.org/content/sci/258/5086/1328.full.pdf

Change in Failure Stress on the Southern San Andreas Fault System Caused by the 1992 Magnitude = 7.4 Landers Earthquake

Geodesy, the concepts

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.

Further Development and Properties of the Spectral Analysis by Least-Squares

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.

Approximate Spectral Analysis by Least-Squares Fit. Successive Spectral Analysis

The inherent precision of spirit leveling has preserved its utility as a geodetic measurement system for over a century. While various instrumental and procedural modifications designed to enhance this precision have been introduced over the years, the basic measurement system has remained virtually unchanged since the mid-nineteenth century. Possible systematic error has dictated the majority of the procedural and instrumental requirements associated with geodetic leveling; the physical source(s) of several of these errors remain poorly understood. Statistically independent random errors, which accumulate according to the square root of the survey distance, are generally controlled through redundancy and procedural randomization; they range from 0.5 mm L1/2 for the highest-order modern leveling to about 6 mm L1/2 for the lowest-order nineteenth-century geodetic surveys, where L is the survey distance in kilometers. Height differences are conceptually distinct from observed or measured elevation differences in the sense that the former are uniquely defined, whereas the latter are path dependent, a distinction that arises from the nonparallelism of the equipotential surfaces of the earth’s gravity field. The number of possible height systems is virtually limitless. They include the systems of geopotential numbers and dynamic heights; although neither of these systems is geometrically informative, each provides perfectly valid height characterizations that may be especially useful in the solution of certain physical problems. The most generally used system of heights is the orthometric height system; the resulting heights are true geometric heights above the geoid. Normal height systems are referred to the quasi-geoid rather than the geoid. Each of the various height systems meets the requirement of uniqueness, and none can be viewed as being conceptually superior. Conversion of the observed elevation differences obtained from leveling into uniquely defined height differences requires the application of a gravity-dependent correction. Because gravity coverage in North America was generally sparse until recently, an approximation for this correction, which provides for the effects of the poleward covergence of the equipotential surfaces, has usually been used on this continent. Heights have been traditionally referred to mean sea level as a datum, a usage that implies coincidence between mean sea level and the geoid (or quasi-geoid). Because the determination of mean sea level is dependent on the length of the observation period, because its definition makes no allowance for vertical crustal displacements or changes in eustatic sea level, and because its definition disregards the demonstrable existence of sea surface topography, local mean sea level generally departs from the geoid. This introduces errors in computed heights that probably equal or exceed those due to leveling. Repeated levelings continue to provide the best basis for determining terrestrial vertical displacements. These displacements are necessarily measured with respect to the time-dependent (instantaneous) geoid. Analysis of changes in the orthometric correction and changes in the geoid as a function of changes in gravity indicates that height determinations are almost insensitive to temporal variations in gravity. The simplest and probably the most accurate procedure for determining vertical displacements is based on direct comparisons between observed elevations derived from repeated levelings along the same line. The estimated errors associated with these displacements are a function of the precision of the utilized levelings and have been shown to be much smaller than those associated with heights. A smoothed representation of the vertical displacement field can be obtained by the fitting of a mathematical surface to the results of segmented relevelings. Surface-fitting techniques usually tend to subdue short-wavelength features, but they are especially useful in depicting artificially induced subsidence and broadly defined tectonic uplift or collapse.

Petr Vaníček

Papers

Geodesy, the concepts

Further Development and Properties of the Spectral Analysis by Least-Squares

Approximate Spectral Analysis by Least-Squares Fit. Successive Spectral Analysis

Geodetic leveling and its applications

Geodesy, the concepts