## 1. Introduction

[2] The verification of coastal and harbor structures may require the use of Level III verification methods. These methods are usually complex and require the use of numerical simulation techniques (e.g., Monte Carlo techniques) [*Losada*, 2002].

[3] In coastal engineering, the main variables to be simulated are sea-state variables such as significant wave height, wind, and sea level, which characterize the sea state in a time domain in which processes are assumed to be stationary. For this purpose, generally speaking, the duration should not exceed O(1hr). This research focuses on the evolutionary behavior of the sea-state variables, i.e., on long-term analysis.

[4] From a physical point of view, the temporal evolution of sea-state variables is conditioned by phenomena operating on different time scales.

[5] Processes with a time scale of O(day)-O(weeks), such as synoptic phenomena and the cycles of spring and neap tides, produce dependence among the variables that originate and autocorrelation in each variable. The clearest example related to sea states is the passage of a storm. The storm will generate wind speeds and wave heights that are larger than average, and therefore, it is expected that these variables will be correlated during a storm. At the same time, the evolution of these variables (and others) over time is determined by the intensity and path of the storm, so there are physical reasons to expect that these variables will present significant autocorrelation within the time scale of the storm.

[6] O(year) scale processes, such as seasons, produce variations in the intensity and frequency of the O(day)-O(week) scale phenomena and thus cause temporal variations in sea-state variables. In the same way, O(>year) scale processes, such as interannual variability, influence the characteristics of each year (e.g., they create drier or wetter years and years with more or less wave action) and also produce temporal variations in sea-state variables.

[7] Regarding the statistical tools used in the long-term analysis of sea-state variables, it is important to note that such studies can be univariate or multivariate, may or may not include auto-correlation, and can be stationary or non-stationary. Table 1 summarizes the characteristics of a study: whether the variables are dependent on other variables (i.e., whether they are correlated with other variables), whether the variables are self-dependent (i.e., exhibit autocorrelation or time dependence), or whether they are dependent on time (i.e., whether their distribution is non-stationary). The long-term (climate) behavior of sea-state variables includes such characteristics and, consequently, should be studied using non-stationary multivariate models that represent the time dependence (or auto-correlation) of the variables.

Connection With | NO | YES |
---|---|---|

Other variables | Univariate | Multivariate |

Same variable Time | Without auto-correlation Stationary | With auto-correlation Non-stationary |

[8] In Figure 1, various physical phenomena evolving in different time scales are associated with statistical models that have been used in this study to appropriately model the sea-state variables for these time scales.

[9] The maximum time scale that the simulation must take into account to be applied to engineering is the period used to verify the system. This period is generally the useful life of the system, which is 10–50 years, although it can be a shorter duration when the aim is to verify construction processes or evaluate other short-term phenomena.

[10] With regard to the simulation of times series for significant wave heights (*H*_{s} or *H*_{m0}), there are currently two lines of research: one that focuses on simulating storms and another that simulates complete series of values.

[11] The method most widely used to simulate storms involves developing joint or conditioned distributions for the random variables of storm occurrence, intensity, and duration. Based on these distributions, new time series are simulated assuming a standard shape for the storm.

[12] In general, storm occurrence is modeled using a Poisson distribution and storm intensity using a generalized Pareto distribution (GPD). It is common to condition the duration of a storm to its intensity. Some examples of this type of approximation are presented by *DeMichele et al.* [2007], *Payo et al.* [2008], and *Callaghan et al.* [2008]. Although stationary functions are generally used for this purpose, non-stationary functions can also be employed, such as those proposed by *Luceño et al.* [2006], *Méndez et al.* [2006, 2008], and *Izaguirre et al.* [2010]. A less frequent alternative in storm simulation is to assume that it is a Markov process and to use a multivariate distribution of extremes to model the time dependence of the variable while the storm lasts [*Coles*, 2001, chap. 8]. This technique is used by *Smith et al.* [1997], *Fawcett and Walshaw* [2006], and *Ribatet et al.* [2009].

[13] *Monbet et al.* [2007] review simulation methods for complete time series applied to wind and waves. The methods currently used can be classified as parametric and non-parametric.

[14] The Translated Gaussian Process (TGP) method [*Walton and Borgman*, 1990; *Borgman and Scheffner*, 1991; *Scheffner and Borgman*, 1992] is the most widely used non-parametric method. This method uses the spectrum of the normalized variable. According to *Monbet et al.* [2007], non-parametric methods such as those based on resampling (called resampling methods) are less frequently used and are not discussed in this article.

[15] The most frequently used parametric methods are based on autoregressive models. Studies employing such methods include *Guedes Soares and Ferreira* [1996], *Guedes Soares et al.* [1996], *Scotto and Guedes Soares* [2000], *Stefanakos* [1999], *Stefanakos and Athanassoulis* [2001], and *Cai et al.* [2007] for univariate series; for multivariate series, relevant studies include *Guedes Soares and Cunha* [2000], *Stefanakos and Athanassoulis* [2003], *Stefanakos and Belibassakis* [2005], and *Cai et al.* [2008]. As in the TGP, before autoregressive models can be used, the series must be normalized. For this purpose, non-stationary models of the mean and the standard deviation, like those proposed by *Athanassoulis and Stefanakos* [1995], *Stefanakos* [1999], and *Stefanakos et al.* [2006], are used.

[16] The current methods present the following limitations:

[17] (a) Methods of normalizing variables are either stationary [e.g., *Cai et al.*, 2007, 2008] or non-stationary. However, they focus on the center of the data distribution, generally using the non-stationary mean and standard deviation for normalization [e.g., *Guedes Soares et al.*, 1996; *Athanassoulis and Stefanakos*, 1995].

[18] (b) Parametric time dependence models are linear [e.g., *Guedes Soares et al.*, 1996], piecewise linear [e.g., *Scotto and Guedes Soares*, 2000], or non-linear but are limited to the extremes [e.g., *Smith et al.*, 1997].

[19] (c) Generally speaking, the simulation is only evaluated using the mean, the standard deviation and the autocorrelation.

[20] This article proposes a simulation method for non-stationary univariate series with time dependence. This method involves the use of a non-stationary parametric mixture distribution to model the univariate distribution of the variable and of copulas to model their time dependence.

[21] The rest of this paper is structured in three sections and seven annexes. In section 2, the proposed model is presented together with the procedure for simulating new time series. In section 3, the model parameters are fitted to a data series of significant wave heights, new series are simulated and the results obtained are discussed. Finally, in section 4, the conclusions are summarized. The derivation of the equations associated with the presented model is illustrated in the appendices at the end of the paper, along with a list of the abbreviations used throughout the paper (Appendix G).