Data analysis and empirical wave distributions

From a sequence of wave data (sampling time and sampled value) WAT easily identifies turning points, i.e. local extremes; further a number of filtering and transformations can be made on the turning points. The turning points define the wave cycles and other wave characteristics and their empirical distributions.

Wave data with turning points,
i.e. local maxima and minima. Local maxima (= crests) and local minima (= troughs) are identified in a recorded wave data series. These form the turning points, which can be extracted and analysed further to give wave cycles, wave lengths, amplitudes, etc.

Rainflow cycles are identified;
small cycles can be removed. Rainflow cycles are identified. Small cycles can be removed by means of rainflow filtering. The figure shows rainflow cycles with amplitude exceeding one standard deviation of the wave data. The rainflow filter is a non-linear filter that retains the high peaks and low troughs at their original height. (This is in contrast to linear filters like the low-pass filter which smooths the waves process and brings the local extremes closer to the mean.)

A cycle plot shows pairs of maxima and minima. A cycle plot shows pairs of maxima and minima as points in the plane. Left plot shows the min/Max cycles and right plot shows trough/crest cycles.

The number N(u) of upcrossings of a level u is called the crossing spectrum and is here plotted as a function of u. The simulation routines can produce a wave with specified crossing spectrum and irregularity factor.

Significant values can be calculated. The significant alpha - value is defined as the average of the alpha - fractile highest waves. The figure shows significant values for trough-crest waves. alpha = 1/3 renders the common "one-third" significant value.

For fatigue analysis with the rain flow method a basic concept if the oscillation count which is a count of the number of times that the wave process crosses a band (u, v):
N(u,v) = number of cycles with min value < u and max value > v.
N(u,v) is an empirical distribution function. The figure shows the cycle plot of min and max-values, together with isolines in the empirical cycle count distribution.

WAT examples

	Local maxima (= crests) and local minima (= troughs) are identified in a recorded wave data series. These form the turning points, which can be extracted and analysed further to give wave cycles, wave lengths, amplitudes, etc.
	Rainflow cycles are identified. Small cycles can be removed by means of rainflow filtering. The figure shows rainflow cycles with amplitude exceeding one standard deviation of the wave data. The rainflow filter is a non-linear filter that retains the high peaks and low troughs at their original height. (This is in contrast to linear filters like the low-pass filter which smooths the waves process and brings the local extremes closer to the mean.)
	A cycle plot shows pairs of maxima and minima as points in the plane. Left plot shows the min/Max cycles and right plot shows trough/crest cycles.
	The number N(u) of upcrossings of a level u is called the crossing spectrum and is here plotted as a function of u. The simulation routines can produce a wave with specified crossing spectrum and irregularity factor.
	Significant values can be calculated. The significant alpha - value is defined as the average of the alpha - fractile highest waves. The figure shows significant values for trough-crest waves. alpha = 1/3 renders the common "one-third" significant value.
	For fatigue analysis with the rain flow method a basic concept if the oscillation count which is a count of the number of times that the wave process crosses a band (u, v): N(u,v) = number of cycles with min value < u and max value > v. N(u,v) is an empirical distribution function. The figure shows the cycle plot of min and max-values, together with isolines in the empirical cycle count distribution.