Replace Spectrum Tail with Diagnostic Tail

If for any reason the values of spectrum tail (a high frequency section of the wave spectrum) are not reliable, then spectrum tail may be replaced with tail of empirical spectrum (diagnostic tail). It is not recommended to use a diagnostic tail for measured data, unless either data are recorded with a low sampling frequency which leads to no data in higher frequency range, or higher frequency data are contaminated with noise. In these cases, a higher section of the spectrum can be replaced with tail of empirical spectrum as (Siadatmousavi et al. 2012):

\(S_{yy}(f)=S_{yy}(f_{tail}) \times (\frac{f}{f_{tail}})^{(-n)}\) for \(f>f_{tail}\)

where, \(S_{yy}(f)\) is a water surface elevation power spectral density, f is a frequency, \(f_{tail}\) is a frequency that a tail applied after that, and n is a power coefficient. The f_tail typically set at 2.5f_m, where \(f_m=1/T_{m01}\) is a mean frequency (Ardhuin et al. 2010). A value of n depends on deployment conditions, however, typically it is -5 for deep and -3 for shallow water (e.g. Kaihatu et al. 2007, Siadatmousavi et al. 2012).

For more details on this topic refer to Karimpour and Chen (2017) and Karimpour (2018).


  • Ardhuin, F., Rogers, E., Babanin, A. V., Filipot, J. F., Magne, R., Roland, A., … & Collard, F. (2010). Semiempirical dissipation source functions for ocean waves. Part I: Definition, calibration, and validation. Journal of Physical Oceanography, 40(9), 1917-1941.
  • Kaihatu, J. M., Veeramony, J., Edwards, K. L. & Kirby, J. T. 2007 Asymptotic behaviour of frequency and wave number spectra of nearshore shoaling and breaking waves. J. Geophys. Res. 112, C06016
  • Karimpour, A., & Chen, Q. (2017). Wind Wave Analysis in Depth Limited Water Using OCEANLYZ, a MATLAB toolbox. Computers & Geosciences, 106,181-189.
  • Karimpour A., (2018), Ocean Wave Data Analysis: Introduction to Time Series Analysis, Signal Processing, and Wave Prediction, KDP.
  • Siadatmousavi, S. M., Jose, F., & Stone, G. W. (2011). On the importance of high frequency tail in third generation wave models. Coastal Engineering.