The Upcrossing Rate via the Characteristic Function

As was detailed in Chap. 4, a key function for a practical assessment of the extreme value distribution of stochastic response processes is the average rate of upcrossings of high levels by the response. An important class of such response processes can be expressed as a second-order stochastic Volterra series, that is, a stochastic Volterra series that has been truncated after the second-order term (Schetzen (1980) The Volterra and Wiener theories of nonlinear systems. Wiley, New York). A substantial amount of work has been done to derive methods for efficient analysis of this model, starting with the seminal paper by Kac and Siegert (J Appl Phys 18:383–397, 1947). Later, with the development of the offshore industry, this paper had an impact on investigations of the response statistics of large floating structures. Early contributions in this field of research were made, among many others, by Neal (Second order hydrodynamic forces due to stochastic excitation. In: Proceedings 10th ONR symposium, Cambridge, MA, 1974), Vinje (Int Shipbuilding Progress 30:58–68, 1983), Langley (Appl Ocean Res 6(4):182–186, 1984), Naess (J Ship Res 29(4):270–284, 1985; Probab Eng Mech 5(4):192–203, 1990), and Donley and Spanos (Dynamic analysis of non-linear structures by the method of statistical quadratization. Lecture notes in engineering, vol 57. Springer, Berlin, 1990).