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Wave cycles, i.e., pairs of local maxima and minima, play an important role in many engineering fields. Many cycle definitions are used for specific purposes, such as crest-trough cycles in wave studies in ocean engineering and rainflow cycles for fatigue life predicition in mechanical engineering. The simplest cycle, that of a pair of local maximum and the following local minimum is also of interest as a basis for the study of more complicated cycles. This paper presents and illustrates modern computational tools for the analysis of different cycle distributions for stationary Gaussian processes with general spectrum. It is shown that numerically exact but slow methods will produce distributions in almost complete agreement with simulated data, but also that approximate and quick methods work well in most cases.Of special interest is the dependence relation between the cycle average and the cycle range for the simple maximumYminimum cycle and its implication for the range distribution. It is observed that for a Gaussian process with rectangular box spectrum, these quantities are almost independent and that the range is not far from a Rayleigh distribution. It will also be shown that had there been a Gaussian process where exact independence hold then the range would have had an exact Rayleigh distribution. Unfortunately no such Gaussian process exists.
Wave cycles, i.e., pairs of local maxima and minima, play an important role in many engineering fields. Many cycle definitions are used for specific purposes, such as crest-trough cycles in wave studies in ocean engineering and rainflow cycles for fatigue life predicition in mechanical engineering. The simplest cycle, that of a pair of local maximum and the following local minimum is also of interest as a basis for the study of more complicated cycles. This paper presents and illustrates modern computational tools for the analysis of different cycle distributions for stationary Gaussian processes with general spectrum. It is shown that numerically exact but slow methods will produce distributions in almost complete agreement with simulated data, but also that approximate and quick methods work well in most cases.Of special interest is the dependence relation between the cycle average and the cycle range for the simple maximumYminimum cycle and its implication for the range distribution. It is observed that for a Gaussian process with rectangular box spectrum, these quantities are almost independent and that the range is not far from a Rayleigh distribution. It will also be shown that had there been a Gaussian process where exact independence hold then the range would have had an exact Rayleigh distribution. Unfortunately no such Gaussian process exists.
A Gaussian eld X dened on a square S of R 2 is considered. We assume that this eld is only observed at some points of a regular grid with spacing 1 n . We are interested in the normalized discretization error n 2 (M − M n ), with M the global maximum of X over S and M n the maximum of X over the observation grid. The density of the location of the maximum is given using Rice formulas and its regularity is studied. Joint densities with the value of the eld and the value of the second derivative are also given. Then, a kind of Slepian model is used to study the eld behavior around the unique point where the maximum is attained, called t * . We show that the normalized discretization error can be bounded by a quantity that converges in distribution to a uniform variable. The set where this uniform variable lies principally depends on the second derivative of the eld at t * . The bound is a function of this quantity which is approached by nite dierences in practice. The bound is applied both on simulated and real data. Real data are used in positioning by satellites systems quality assessment.
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