# Spatio-temporal modelling of wind speed variations and extremes in the Caribbean and the Gulf of Mexico

## Abstract

The wind speed variability in the North Atlantic has been successfully modelled using a spatio-temporal transformed Gaussian field. However, this type of model does not correctly describe the extreme wind speeds attributed to tropical storms and hurricanes. In this study, the transformed Gaussian model is further developed to include the occurrence of severe storms. In this new model, random components are added to the transformed Gaussian field to model rare events with extreme wind speeds. The resulting random field is locally stationary and homogeneous. The localized dependence structure is described by time- and space-dependent parameters. The parameters have a natural physical interpretation. To exemplify its application, the model is fitted to the ECMWF ERA-Interim reanalysis data set. The model is applied to compute long-term wind speed distributions and return values, e.g., 100- or 1000-year extreme wind speeds, and to simulate random wind speed time series at a fixed location or spatio-temporal wind fields around that location.

## 1 Introduction

Due to the increased regulation pressure on maritime transport in terms of energy efficiency and emission control (DNV 2015), there is a growing interest in the study of the characteristics of wind/wave variation at sea. To develop solutions for utilizing renewable wind energy in ship propulsion (ABS 2013), it is greatly important to have access to reliable wind statistics along arbitrary ship routes. These statistics are used to estimate the average performance of a ship/offshore renewable energy unit and to assess the risk of extreme winds. Furthermore, extreme winds (and waves) in coastal areas can destroy infrastructures, cause human loss leading to great economical damages. Therefore, accurate modelling of wind speed variations is important for addressing related engineering problems.

Typically, the variation in wind speed at a certain location or in a certain region is described by its Cumulative Distribution Function (CDF). The long-term CDF of the wind speed, i.e., fraction of time when wind speed is below a threshold during one year, is often fitted with Weibull distributions; e.g., see DNV (2010) and Morgan et al. (2011). However, in the Caribbean sea, the Weibull model does not fit well with the observed data, particularly for regions where extreme wind speeds can arise during hurricanes or tropical storms (see, e.g., Mao and Rychlik (2016)). The limitation may compromise engineering safety when the Weibull model is used to determine the extreme design condition, i.e., the so-called 100/1000-year extreme wind speeds, for engineering structures in such regions.

However, the rarity of the high winds caused by, e.g., hurricanes and other storms, makes empirical estimation of extreme wind speeds difficult; instead, estimates of such extreme winds can be derived from stochastic models of wind speeds variability. In this study, a new model (named as the hybrid model) for wind speed variability is proposed. The proposed model is used to estimate the frequencies of extreme winds, i.e., winds exceeding a fixed high threshold, by means of the so-called Rice’s method, which was presented in details in, e.g., Azais and Wschebor (2009). Particular applications of this method were presented in, e.g., Baxevani and Rychlik (2006), Rychlik et al. (2011).

Methods based on the univariate extreme value theory (Coles 2001) are used to validate the proposed approach. For example, by fitting the generalized extreme value (GEV) distribution to yearly maximum wind speeds at a fixed location, the extreme wind speed estimated from the fitted GEV distribution is used as a reference value to validate the Rice’s method that uses the proposed spatio-temporal wind model. Alternatively, Peak over Threshold (POT) method (Davison and Smith 1990) could be employed. However, strong seasonal wind speed variation and correlation make the POT method very complex to use, and less suitable for our validation purpose. Methods of modelling extreme wind speeds due to hurricanes or tropical storms have been investigated by many researchers (see, e.g., Jagger and Elsner (2006) and Reich and Fuentes (2007) and references therein). An extensive publication list can be also found in Larsén et al. (2016). Recently, a large focus is also put on the modelling of multivariate extremes, spatial extremes (max-stable processes), and spatio-temporal extremes (see, e.g., Coles (1993)). In this study, an example of applying the proposed model to derive the distribution of maximum wind speeds in August in a region south of Haiti is presented in Section 7. However, more detailed studies of this problem are outside of the scope of this paper.

Spatio-temporal models are often based on well understood Gaussian models, but real data seldom follow Gaussian distributions perfectly. Hence, transformed Gaussian processes (fields) are often used for the modelling. Popular transformations proposed by Brown et al. (1984), Winterstein et al. (1994) etc. are frequently used in engineering literature. For example, the exponential transformation proposed by Brown et al. (1984) was successfully used in Mao and Rychlik (2016) to model wind speed distributions in the Northern Atlantic. However, for the Caribbean sea, the kurtosis of exponentially transformed wind speeds is often significantly exceeding the Gaussian threshold (three), and hence, this model underestimates the frequencies of high wind speeds. Consequently, the Hermite transformation proposed by Winterstein et al. (1994) was also tested. This transformation is a monotonic cubic polynomial, which is calibrated such that the first four moments (mean, variance, skewness, and kurtosis) of the transformed model are equivalent to the moments of the data. The following example shows that the Hermite transformed Gaussian model is overestimating frequencies of high wind speeds in the Caribbean sea. The reason is that the extreme storms are very rare.

### **Example**

*u*and empirical probabilities that wind speeds exceed a threshold

*u*have been estimated from the extracted data set. Those empirical characteristics are shown in Fig. 2, together with the expected characteristics, which are evaluated using the fitted transformed Gaussian models based on Brown’s exponential transformation and Winterstein’s Hermite polynomial transformation. The two models either underestimates or overestimates the empirical frequencies and wind speed distribution at the tail. Finally, the expected characteristics evaluated by the proposed hybrid model are also presented. They agree well with the empirical values.

In this paper, the exponentially transformed Gaussian model is used to model wind speeds at locations, where extraordinary meteorological events like extreme storms or lulls may rarely occur. Examples of such events can be seen in the time series shown in Fig. 1. The newly proposed model for *W*^{a} is a sum of a Gaussian field and several non-Gaussian random fields (with Laplace distributed amplitudes), which are independent and randomly spread in time and space. This type of model (named as the Hybrid model here) was used in Bogsjö et al. (2012) and Kvarnström et al. (2013) for signals with rarely occurring transients.

The paper is organized as follows. First, the transformed wind speeds *X* = *W*^{a} and methods of estimating the parameter *a* and the mean, variance and correlation structure of *X* are presented in Section 2. Then, Section 3 briefly introduces three types of random fields: Gaussian, Laplace, and the hybrid model. In particular, methods of estimating the parameters in the hybrid model are also discussed here and in Section 4. The technical details of the procedures are moved to appendices. Subsequently, Section 5 presents procedures for using the hybrid model in three practical applications, i.e., to derive the long-term wind speed distribution, to estimate the expected number of wind speeds that cross arbitrary values, and to simulate local wind speeds. In Section 6, the hybrid model is validated by three examples/locations, one in the Caribbean sea and two in the Gulf of Mexico, with respect to long-term wind distribution and 100/1000-year extreme wind prediction. Examples illustrating the hybrid model are given in Section 7. The estimation of spatio-temporal extreme wind speeds are also discussed, the Matlab code to simulate the spatio-temporal maximums using the hybrid model is given in the Appendix C.

## 2 Spatio-temporal modelling of the transformed wind speed *X* = *W* ^{a}

*W*(

**p**) =

*W*(

*t*

_{p},

*x*

_{p},

*y*

_{p}) denote the wind speed at a spatial-temporal location

**p**= (

*t*

_{p},

*x*

_{p},

*y*

_{p}). Obviously, the average characteristics of wind speed variation change with seasons

*t*

_{p}∈ [0,1] (unit: year), and geographical locations (

*x*

_{p},

*y*

_{p}). However, on shorter spatial and temporal scales, these characteristics can be considered to be constant. More precisely, within a small neighbourhood of

**p**with a spatial radius of several degrees and a time interval of a month, the variability of the wind speed is assumed to be homogeneous and stationary:

*X*

_{p}is a homogeneous field with a mean of

*m*

_{p}, a variance of \({\sigma }_{\textbf {p}}^{2}\) and a correlation function of

*ρ*

_{p}(

*t*,

*x*,

*y*), all of which depend on the location

**p**. In some cases, the index

**p**in

*X*

_{p}(

*t*,

*x*,

*y*),

*m*

_{p}, \({\sigma }^{2}_{\textbf {p}}\) and

*ρ*

_{p}(

*t*,

*x*,

*y*) will be neglected in order to simplify the notation.

The parameters *a*_{p}, *m*_{p} and \({\sigma }_{\textbf {p}}^{2}\) are estimated by the method of moments. First, the transformation exponent *a* is estimated to fulfill the criteria that the skewness of *W*^{a} is zero, and *m*,*σ*^{2} are equal to the mean and variance (monthly) of the transformed wind speeds. All parameters defining the model of Eq. 1 depend on the location **p** and are estimated “pointwise.” It should be noted that for the applications of predicting extreme wind speeds by the model, the parameters are smoothed over a spatial neighbourhood of about 3 degrees radius.

Furthermore, a non-homogeneous and non-stationary correlation function *ρ* of a spatio-temporal transformed wind speed \(W^{a_{\textbf {p}}}(\textbf {p})\) was presented in Rychlik and Mao (2014) and Rychlik (2015), and a global wind model for large regions was also established. In this study, only local non-Gaussian models of similar correlation functions are presented and validated. The case of a global model will be considered in the future work.

### 2.1 Correlation function *ρ*(*t*,*x*,*y*)

**p**, the correlation of wind speeds around the surrounding homogeneous region

**q**is written as

*λ*

_{ij}] is a positive-definite matrix. Equation 2 defines the spatio-temporal correlation of transformed wind speeds in the neighbourhood of the location

**p**.

### 2.2 The kernel *f* constructed from *ρ*(*t*,*x*,*y*)

*ρ*, a kernel

*f*can be constructed to describe the characteristics of wind speed time series and the shape of extreme wind speeds. It is defined such that

*f*∗

*f*(

**q**) =

*ρ*(

**q**), where ∗ defines a convolution operation. For a correlation function

*ρ*as in Eq. 2, the kernel

*f*is defined by

*X*

_{p}as in Eq. 1 by discretizing a spatio-temporal region

**S**as presented in the following Section 3. It can be shown that as the discretization steps tend to zero and

**S**grows without bounds, the approximation converges to a homogeneous standardized Gaussian field

*X*

_{G}of correlation

*ρ*defined in Eq. 2.

### 2.3 Estimation of Λ in the kernel

*X*is a locally stationary Gaussian field such as wind speeds in the North Atlantic, the matrix Λ could be estimated by the maximum likelihood method. However, for locations in the Caribbean Sea and the Gulf of Mexico, where tropical storms are rarely occurring, the Gaussian assumption to model

*X*fails. Consequently, a crude method, i.e., the method of moments, is used to estimate the matrix Λ. The property that Λ is proportional to the covariance matrix of the

*X*-gradient is employed here. More precisely, let

*X*(

*t*) =

*X*(

*t*,0,0),

*X*(

*x*) =

*X*(0,

*x*,0) and

*X*(

*y*) =

*X*(0,0,

*y*). The following derivatives can be defined as

*W*

^{a}at a geographical location (

*x*

_{p},

*y*

_{p}) and a time

*t*

_{p}. Then, let the position (

*x*

_{p},

*y*

_{p}) be fixed and only the time

*t*

_{p}vary, the above terms become four time series denoted by (

*X*,

*X*

_{1},

*X*

_{2},

*X*

_{3})(

*t*

_{p}).

*X*

_{1},

*X*

_{2},

*X*

_{3})(

*t*

_{p}) at the position

**p**be denoted by Σ = [

*λ*

_{ij}],

*i*,

*j*= 1,2,3. Using the Theorem 7.6 as in Lindgren (2013), it can be shown that

*σ*

^{2}=

*V*(

*X*). It is assumed that at any geographical location (

*x*

_{p},

*y*

_{p}), the above four wind time series can be considered to be stationary for a period of approximately one month. Consequently, for

**p**= (

*x*

_{p},

*y*

_{p}),

*t*

_{p}=

*j*/12 + 1/24 (

*t*with unit year), mean and variance

*m*

_{p}, \({\sigma }_{\textbf {p}}^{2}\) and Σ

_{p}of the transformed wind speeds

*W*

^{a}(

**p**) can be estimated by standard statistical methods.

## 3 Stochastic models for the transformed wind speed field *X*

In the neighbourhood of **p**, the transformed wind speed field is modeled by a homogeneous random field *X*(*t*,*x*,*y*) as in Eq. 1 with mean *m*_{p}, variance \({\sigma }_{\textbf {p}}^{2}\) and the correlation function *ρ*_{p} defined in Eq. 2. For locations in the Northern part of the North Atlantic, the variable *X*(*t*,*x*,*y*) can be modelled by a stationary Gaussian field, which is uniquely defined by a mean *m*, a variance *σ*^{2} and the correlation function *ρ* given in Eq. 2. For locations in the Caribbean Sea and the Gulf of Mexico with rarely occuring tropical storms, non Gaussian models requiring additional parameters should be used to model the transformed wind speeds, but keep using the same correlation *ρ*.

*X*

_{M}(of mean zero, variance one and the same correlation

*ρ*) will be investigated to describe the random wind fields

*X*, which is modelled by scaling one of these standardized fields by

*σ*

_{p}and adding

*m*

_{p}:

*X*

_{M}represents the Gaussian field

*X*

_{G}, the Laplace field

*X*

_{LMA}and the hybrid field

*X*

_{H}, respectively. Furthermore, these fields

*X*

_{M}are symmetrical with skewness zero. Hence, the same estimates of the parameters

*a*,

*m*,

*σ*, and

*ρ*can be used in Eq. 1 for any model of

*X*

_{M}.

### 3.1 Gaussian moving average model *X* _{G}

**p**= (

*t*

_{p},

*x*

_{p},

*y*

_{p}), the transformed Gaussian model is given by

*X*(

*t*,

*x*,

*y*) =

*m*

_{p}+

*σ*

_{p}

*X*

_{G}(

*t*,

*x*,

*y*), where the Gaussian field

*X*

_{G}is approximated by a moving average of Gaussian noise as follows:

*Z*

_{ijk}are independent zero mean, variance one, Gaussian variables and

*f*is a deterministic kernel function defined by Eq. 3. For the approximation, a stationary and homogeneous wind region

**S**⊂

*R*

^{3}has to be chosen first. Choice of region

**S**is discussed in Section 4.2. The region

**S**is dicretized using a grid, i.e.

**S**∋ (

*t*

_{k},

*x*

_{i},

*y*

_{j}). The discretization steps of

**S**are

*dt*,

*dx*,

*dy*. (The detailed definition of

**S**will be given later.)

*dt*,

*dx*,

*dy*are chosen such that

It should be noted that the kernel used here is very simple and cannot describe real wind field structures and dynamics in great detail. More complex models could be constructed to describe the wind speed variability on different time scales; see e.g., Rychlik and Mao (2014), Rychlik (2015), where the wind speeds in the North Atlantic were modelled using four kernels of various scales but of a similar type to that in Eq. 3: a diurnal pattern due to different temperatures at day and night, a pattern considering the frequency of depressions and anti-cyclones, an annual pattern and a pattern of fast variability (noise).

### 3.2 Laplace moving average model (LMA) *X* _{LMA}

*Z*

_{ijk}to have variable variance. More precisely,

*Z*

_{ijk}is multiplied by the square root of independent gamma distributed random factor

*K*

_{ijk}as follows:

*K*

_{ijk}are defined to fulfill two conditions: the mean value of

*K*

_{ijk}has to be one and the kurtosis of

*X*

_{LMA}should be equal to the kurtosis of the observed wind speeds. Obviously, many other possible models are available to construct stationary non-Gaussian fields using the moving average method, e.g., to replace the Gaussian field in Eq. 6 by other infinitely divisible noises. The infinite divisibility assumption is convenient to allow for a variable discretization step while still keeping the same noise type (see, e.g., Feller (1966) for basic properties of infinite divisible distributions).

Moving averages of Laplace noise \(\sqrt {K_{ijk}}Z_{ijk}\) have many attractive mathematical properties (see, e.g., Cambanis et al. (1995) and Podgórski and Wallin (2016)). For example, conditionally on values of factors \(\sqrt {K_{ijk}}\), the LMA field becomes a zero mean nonstationary Gaussian field. In particular, the very large factors *K*_{ijk} will result in unusual high variability of the wind speed field. Intuitively, it would be convenient to reorder the factors *K*_{ijk} in such a way that larger factors can be considered first. This is achieved through an alternative definition of LMA using series expansions of infinitely divisible distributions (see, e.g., in Bondesson (1982)). In this study, this definition is used to model *X*_{LMA}.

*f*defined in Eq. 3 will be used again, along with a large region

**S**∋ (0,0,0) to define

*X*

_{LMA}. The size of

**S**depends on

*f*through the matrix Λ (the choice of

**S**will be discussed in Section 4). Here, a symmetrical Laplace field will be used, and hence, only one additional parameter,

*𝜗*> 0, is needed to define its distribution. Now, the Laplace moving average field at

**q**= (

*t*,

*x*,

*y*) is approximated by the following sum of random functions modelling weather events with extreme values equal to \(Z_{i}\sqrt {R_{i}}\) and located at

**U**

_{i}∈

**S**:

*Z*

_{i},

*R*

_{i}and

**U**

_{i}are independent. The

*Z*

_{i}have a standard normal distribution, whereas the

**U**

_{i}are uniformly spread in

**S**. The distribution of

*R*

_{i}depends on the parameter

*𝜗*. They are independent but not equally distributed. Given the parameter

*𝜗*and the region

**S**,

*R*

_{i}can be simulated as follows:

*ζ*

_{i}are i.i.d. standard exponential random variables independent of

*γ*

_{i}, which are locations of the

*i*-th point in a Poisson process, i.e., \({\gamma }_{i}={\sum }_{j = 1}^{i} G_{j}\), where

*G*

_{j}are independent standard exponential distributed random variables. Note that \(Y_{i}=Z_{i}\sqrt {\zeta _{i}}\) forms a sequence of independent standard Laplace distributed random variables (Kotz et al. 2001), i.e., of probability density

*f*(

*y*) = 0.25exp(−|

*y*|/2). This motivates the name Laplace moving average. The random functions

*X*

_{i}have the following amplitudes,

*V*

_{1},…,

*V*

_{N}are independent uniformly on [0,1] distributed variables and also independent of

*Y*

_{1},…,

*Y*

_{N}.

It can be shown that mean of *X*_{LMA} is zero, variance \(\mathbb V(X_{LMA})\approx 1\), skewness zero and the correlation is approximately equal to *ρ* as in Eq. 2. Note that the sum in Eq. refLMA approaches the LMA field as the region **S** grows to *R*^{3} having variance one and correlation as in Eq. 2.

### 3.3 The Hybrid model *X* _{H}

*X*

_{i}(

**q)**have mean zero and decreasing variances

*X*

_{i}in Eq. 8 appears in order of decreasing variance, only the first

*N*terms will be used to model “weather anomalies,” e.g., large storms or lengthy wind lulls. Furthermore, since the variability of transformed wind speeds under the “normal” conditions is very well described using Gaussian fields,

*X*

_{G}, as defined in Eq. 6, is adopted to approximate \({\sum }_{i=N + 1}^{\infty } X_{i}(\textbf {q})\) in Eq. 8. This leads to the following definition of the hybrid model:

*X*

_{G}is the Gaussian field in Eq. 6 and

*X*

_{i}are defined in Eq. 8. The parameters

*N*and

*𝜗*are related to frequency of storms and kurtosis of the transformed wind speeds.

Both the Gaussian field *X*_{G} and the weather anomalies *X*_{i} are defined using the same kernel *f* presented in Eq. 3. The value of *p*, 0 ≤ *p* ≤ 1, is chosen such that variance of *X*_{H} within the homogeneity region (**S**) is one. This causes the correlation function *ρ* of *X*_{H} to coincide with that given in Eq. 2. Obviously, if *N* = 0, then *X*_{H} = *X*_{G}, whereas *X*_{H} tends to *X*_{LMA} as *N* tends to infinity. By introducing the two extra parameters *p* and *N*, the hybrid model will significantly better model the actual wind speeds. It is demonstrated by the following example.

### **Example**

*u*. The expected upcrossing frequencies were evaluated for transformed Gaussian (

*N*= 0, hybrid

*N*= 6 and LMA (

*N*= 1000)). It is shown that the transformed Gaussian model underestimates, while LMA overestimates the frequencies of high wind speeds events. The frequencies computed using transformed hybrid model agree very well with the observed ones. Actually, the value

*N*= 6 is estimated by minimizing a distance between the observed and computed upcrossing frequencies (see Appendix B for details).

Finally, the Gaussian, LMA and the hybrid models predict the 1000-year extreme wind speeds to be 16.5, 33, and 46 m/s, respectively. Since during years 2000–2015, a maximum wind of 27 m/s was recorded, it seems that the 1000-year wind predicted by the Gaussian and LMA models are very likely too low, and the hybrid model gives better fit than the Gaussian and LMA models.

### 3.4 Estimation of parameters in the hybrid model

*W*

^{a}have skewness zero and kurtosis about three. Estimation of parameters

*a*,

*m*

_{p}, \({\sigma }^{2}_{\textbf {p}}\) in Eq. 1 and correlation

*ρ*has been discussed in Section 2. Figures 4 and 5 present the estimation of these parameters for the Caribbean Sea.

However, in the Caribbean and the Gulf of Mexico, the kurtosis of transformed wind speeds may significantly exceed three in hurricane season. Hence, the hybrid model is used here. In this model, the parameter *N*, i.e., the number of storms and lulls in the stationary region **S**, is one of the most important parameter. If *N* = 0, the hybrid model becomes the transformed Gaussian model, while for very large *N* it is equivalent to the transformed LMA model. In addition to *N*, the other two more parameters are *p* and *𝜗*. Given the values of *N*, they are estimated by solving the condition that the kurtosis of *X*_{H} is equal to the kurtosis of the observed transformed wind speeds, while *p* is used to ensure that variance of *X*_{H} is one.

The difficult part of constructing the hybrid model is to find an appropriate value of *N*. It is estimated by setting that frequencies of crossings of high wind levels given by the hybrid model have to well agree with the observed crossing frequencies in the wind data. This is an important requirement for engineering safety assessment (Rychlik et al. 1997). Algorithms to estimate *N*, *p* and *𝜗* are given in the Appendix B. Finally, it should be noted that the hybrid models are fitted point-wise and monthly, i.e., the parameters vary in terms of geographical locations and months. However, in applications, the parameters are smoothed over suitable neighbourhood of a location of interest.

### **Remark**

The hybrid model and the proposed algorithm to estimate parameters are working very well at various locations, where wind speeds have positive skewness. At such locations, the transformation parameter, i.e., the exponent *a* < 1. However, there are locations with negative skewness of *W* and hence *a* > 1, as shown in the bottom plots of Fig. 4. At those locations, the hybrid model gives lighter tails of wind speed distributions than what the standard extreme value analysis would suggest. Consequently, when skewness is negative, it is proposed to change methods to estimate *N* and *𝜗* by requiring that only crossings of high wind levels are well approximated; see the Appendix B for more details.

## 4 Stationary and homogeneous wind region **S**

The region **S** depends on the shape of the kernel *f* defined in Eq. 3. It should contain (0,0,0) and should be at least large enough that \({\int }_{\textbf {S}} f(\textbf {q})^{2}\,d\textbf {q}\approx 1\). In this study, **S** is defined using the average geometry (size) of a spatio-temporal windy weather region. The geometry parameters are defined as follows.

### 4.1 Size of windy weather region

In the Caribbean and the Gulf of Mexico, wind speeds are modelled by the transformed hybrid model. The model consists of a Gaussian part describing the “every day” variability of wind and a number of random functions *X*_{i} modelling storms and lulls occurring in the region **S**. At a location **p**, let a windy weather be defined as *X*(*t*,*x*,*y*) > *m*. Furthermore, let *τ* be the average duration of uninterrupted windy weather at location **p** = (*t*_{p}, *x*_{p}, *y*_{p}), where *t*_{p} is discretized by month for the estimation.

*L*

_{x}and

*L*

_{y}, respectively. More precisely,

*τ*,

*L*

_{x}and

*L*

_{y}are the average length of excursions above the median wind speed (i.e.,

*W*≥

*m*

^{1/a}) in the processes

*X*(

*t*) =

*X*(

*t*,0,0),

*X*(

*x*) =

*X*(0,

*x*,0) and

*X*(

*y*) =

*X*(0,0,

*y*), respectively. If

*X*(

*t*),

*X*(

*x*) and

*X*(

*y*) were stationary Gaussian processes, then by Rice’s formula Eq. 22 and the Theorem 7.6 given in Lindgren (2013), the following relations can be obtained:

*V*(

*X*) denotes the variance of the random variable

*X*. When wind speeds should be described by the hybrid model, i.e.,

*N*> 0, the above parameters

*τ*,

*L*

_{x}and

*L*

_{y}should be seen as spectral parameters. The parameters can be fair approximations of the observed average sizes of windy weather regions since most of the time the hybrid model is generated using a Gaussian field

*X*

_{G}. The parameters

*τ*,

*L*

_{x},

*L*

_{y}depend on

*a*, which varies with season/month (

*t*

_{p}) and geographical position (

*x*

_{p},

*y*

_{p}) of the location

**p**; see Fig. 6 for an illustration.

### 4.2 Choice of **S**

**p**, the region

**S**| is approximately 8 ⋅10

^{4}[hour⋅

*d*

*e*

*g*

^{2}]. If the spatio-temporally homogeneous properties of

*W*in a neighbourhood field of

**p**are needed,

**S**must be enlarged to contain the neighbourhood. For example, to simulate time series of wind speeds during one month, i.e., 720 h, the size of

**S**has to be enlarged to

*N*must also be increased by a factor of \(|\tilde {\textbf {S}}|/|\textbf {S}|= 1+\frac {360}{1.75\tau }\). The new value of

*N*is the number of wind anomalies placed at random in \(\tilde {\textbf {S}}\) in order to describe the wind fields in the larger region. The intensity of extraordinary wind events

*N*/|

**S**| is presented in Fig. 7. However, the value of

*𝜗*presented in Fig. 8 and |

*S*| used in Eq. 9 to define amplitudes \(Z_{i}\sqrt {R_{i}}\) of

*X*

_{i}do not change.

The wind field in the neighbourhood of **p** for the period of time within a month can be simulated by the hybrid model in Eq. 11, where the Gaussian part can be simulated on any grid using some standard methods, e.g., Cholesky decomposition of the covariance matrix. Here, the size of the field matrix is the limiting factor. Since the estimated values of *N* in the Caribbean and the Gulf of Mexico are small, for example for a region as in Eq. 14 and a time period of 1 month it would not exceed 100, simulations of \({\sum }_{i = 1}^{N} X_{i}(\textbf {q})\) is numerically very simple task for any grid.

Note that only local models are considered in this study. If more than one region are considered, the simulations of wind speed fields in these regions are independent, even if these regions are overlapped.

## 5 Validation statistics and practical applications of the hybrid model

In the following, three statistics of wind speeds are presented: (1) the long-term wind speed distribution at a fixed location, (2) the crossing rates for the prediction of 100/1000-year extreme wind speeds using the Rice’s method, and (3) the simulation of local fields and time series of wind speeds at specific locations. These statistics are used to validate the proposed hybrid model, as well as to demonstrate the practical applications of the hybrid model.

### 5.1 Long-term wind speed distributions

*W*depends on the season, to avoid ambiguity when discussing the distribution of

*W*, the time span and region from which the wind observations were gathered need to be clearly specified. The long-term CDF at

**p**can be retrieved by averaging

*W*(

**p**) distributions. For example, the yearly distribution of

*W*at (

*x*

_{p},

*y*

_{p}) is given by

**A**is given by

**A**| is the area of the region

**A**. If

*W*is described by the transformed Gaussian model then

*x*) is the cumulative distribution of a standard Gaussian variable. However, if the transformed

*W*

^{a}is described by the hybrid model, then the computations of \(\mathbb {P}(W(\textbf {p})\le w)\) is slightly more complicated. The detailed calculation procedure is given in the Appendix A.

### 5.2 Estimation of 100/1000-year extreme wind speeds

The long-term distribution of *W* is commonly used to describe the variability of wind speeds. An important applications of the distribution is to estimate expected values of various functions of wind speed, e.g., the average available wind energy for harvesting at a particular wind farm. Another category of wind characteristics consists of the extreme wind speed statistics, and such characteristics are relevant to, e.g., maritime-safety-related activities. For such purposes, the statistical wind characteristics are most often described in terms of the distribution of the maximum wind speeds over a given period of time *T*, e.g., one year.

*M*

_{T}denote the maximum wind speed during a period

*T*, where

*T*is usually one year, 1 month or one hurricane season. The probability distribution of

*M*

_{T}describes the long-term variability of

*M*

_{T}. Somewhat simpler statistics are the so-called 100/1000-year extreme (return) wind speeds denoted by

*w*

_{100}and

*w*

_{1000}, respectively. These extreme values are quantiles of the

*M*

_{T}distribution; e.g., the 100-year extreme wind is defined by

*T*= 1 year, the wind speed level of

*w*

_{100}will be exceeded in average once per 100 years.

*W*(

*t*), let

*N*

_{T}(

*u*) be the number of level crossings in a time interval

*T*, i.e., the number of times

*t*at which

*W*(

*t*) =

*u*for

*t*∈

*T*= [

*t*

_{1},

*t*

_{2}]. The Rice’s method uses the following bound:

*u*reaches very high values, e.g., extreme values. The means to evaluate \(\mathbb E[N_{T}(u)]\) are given in the Appendix A.

The transformed Gaussian and the hybrid model can be used to evaluate the frequencies/distributions of extreme wind speeds by the generalized Rice’s formula (Azais and Wschebor 2009) (see, e.g., Brodtkorb et al. (2000)). The estimation of the maximum distribution as in Eq. 16 is the typical/simplest application of this methodology to study the frequencies of extreme events. Furthermore, the so-called Slepian models (Lindgren and Rychlik 1991; Podgórski et al. 2015) allow for simulating time series of wind or wind surfaces in the vicinity of an extreme event. With the simulated wind information, physical interaction between wind and an engineering structure can be modelled to estimate the frequencies of potentially harmful events, which can trigger undesired/dangerous responses.

### 5.3 Simulation of local fields and time series of wind speeds

First, the generation of time series at three fixed locations will be demonstrate to validate the hybrid model in the following Section 6. Then, the simulation of temporal wind speed fields of 4-h interval in a neighbourhood of a fix location will be demonstrated by the model in Section 7. Such simulations could be used to study the responses of engineering structures to wind loads. For example, this model can be used to estimate the long-term (e.g., 30 years) distribution of wind speeds along arbitrary ship routes. Then, Rice’s method can be employed to get the maximum wind speed during those 30-year sailing routes. The obtained wind information is essential for estimating the pay-back times of wind-assisted propulsion devices (Nelissen et al. 2016) to be installed onboard ships and for assessing the safety of the ships after installation. Based on such simulated wind information, a ship could also choose to sail in a more economic way so as to encounter more beneficial wind conditions along its routes. Furthermore, wind speed field simulations provide important information regarding the choice/design of wind farms in specific regions. The wind correlations provided by this model could also be used for conditional prediction of wind speeds when some spatial or temporal wind information is known for given areas of interest.

## 6 Validation of the hybrid model for wind in the caribbean and the gulf of Mexico

Here, the hybrid models are estimated using 16 years ERA-Interim wind data (Dee et al. 2011) from ECMWF for the years 2000–2015. The parameters of the models are estimated for each month, and they also vary in space. The spatio-temporal variability of the transformation exponent *a*, the mean *m*, the standard deviations *σ* and the duration of windy weather *τ* are shown in Figs. 4, 5, and 6. An additional important parameter *N*, which describes the frequency of storms, is shown in Fig. 7. The wind speeds at the three locations represented in Fig. 1 (upper plot) will be used to illustrate and validate the accuracy of the hybrid model.

In the following, the accuracy of the fitted hybrid models will be investigated by comparing the yearly exceedance probabilities \(\mathbb P(W>u)\), and the expected number of upcrossings during one year \(\mathbb E[N^{+}(u)]=\mathbb E[N(u)]/2\). For model validation, the theoretical long-term distribution \(\mathbb P(W>u)\) is computed using Eq. 30, and the upcrossing rate \(\mathbb E[N^{+}(u)]\) is estimated using Eqs. 27–29 based on the fitted hybrid models; see the Appendix A for more details. These estimates are then compared with the empirical values based on 16 years of hindcast wind data.

### 6.1 Validation at the location west of Haiti coast (72 W, 17.25 N)

At this location, the transformation exponents *a* during the hurricane and winter seasons are very different, as shown in Fig. 4. In winter, *a* is far above 1.2, whereas in summer, it is far below 0.7. In addition, a highest wind speed of 47 m/s (due to the hurricane Dean) has been observed at this location; see the lower plot in Fig. 1. Data of this type present a challenge for automatic estimation procedures.

### 6.2 Validation at the location south of New Orleans (90 W,28.50 N)

At this location, the transformation exponents *a* are also distinctly different between the hurricane and winter seasons. However, in the winter, *a* is approximately equal to 1, i.e., no transformation is needed. It is well-known that the coasts of Louisiana and Mississippi often experience extreme winds when hurricanes are passing this region. For example, in 2005, the passage of Hurricane Katrina through this region produced maximum wind speeds above 50 m/s. However, no such extreme wind speeds are recorded in the 2000–2015 ERA-Interim data; a highest wind speed of only 25 m/s was recorded on 2008-09-12. (One possible reason for this is that wind speeds are given every 6 h in the dataset.) Thus, an interesting question is whether the proposed hybrid model for estimating the 100/1000-year extreme wind speeds will result in a higher wind speed.

### 6.3 Validation at the location west of the Florida coast (84W, 28.5N)

Here, wind speeds vary in a similar way as in the North Atlantic; i.e., there is no significant difference between the values of *a* in the hurricane and winter seasons. However, since the kurtosis of the transformed wind speeds *X* = *W*^{a} for the months of June and July is greater than four, the hybrid model should nevertheless be used for this location. (In the North Atlantic, the kurtosis of *X* is approximately 3.)

## 7 Extreme prediction and simulation of wind speeds by the hybrid model

In this section, three examples illustrating the utilization of the hybrid model will be presented. The first example is the extreme wind prediction in the regions of the Caribbean sea and the Gulf of Mexico, and the second example is the simulation of wind speeds as local fields and time series. In the third example, simulating wind speed fields will be used to estimate spatio-temporal wind extremes of the fields. The examples have quite practical applications within the maritime community.

### 7.1 Estimation of 100/1000-year extreme winds speed

Various methods are available for statistical extreme value prediction. For example, in Walshaw (2000), the parameters in a GEV distribution were assumed to be random, and a Bayesian approach was used to estimate the extreme wind speeds. By contrast, in Payer and Kuchenhoff (2004), not only the yearly maximum but also the *r* highest values during each year were used to fit the GEV distribution. This approach is particularly useful if there are only a few years of data available. When only very limited data are available, an alternative approach is to compute the probability \(\mathbb P(M_{T}>u)\) from a parametric wind speed variability model, e.g., a transformed Gaussian or the hybrid model; see Rychlik et al. (2011). Obviously, extrapolation beyond the range of the available data can lead to severely biased estimates if the assumed model does not hold in the region of extrapolation. This is why a careful validation of the accuracy of proposed models is always needed.

Estimation of 100/1000-year extreme wind speeds (denoted by *W*_{100} and *W*_{1000}) at three locations

Estimation method | (72.75W, 17.25N) | (72.75W, 16.5N) | (90W, 28.5N) | (84W, 28.5N) | ||||
---|---|---|---|---|---|---|---|---|

| | | | | | | | |

Rice method based on the hybrid model | 39 m/s | 69 m/s | 24 m/s | 46 m/s | 38 m/s | 76 m/s | 26 m/s | 44 m/s |

Fitted GEV distribution | 55 m/s | 200 m/s | 31 m/s | 84 m/s | 33 m/s | 51 m/s | 29 m/s | 58 m/s |

Rice method for transformed Gaussian | 18 m/s | 20 m/s | 15 m/s | 16 m/s | 23 m/s | 27 m/s | 24 m/s | 28 m/s |

### 7.2 Simulation of time series of wind speeds using the hybrid model

Here and in the following subsections, the wind speeds in the neighbourhood of the location (72.75W, 16.5N) will be simulated using hybrid model. The fitted parameters for the August are *a* = 0.6, *m* = 6.8, *σ* = 1.5, and *p* = 0.92, which means that 84% of variance is modelled by Gaussian component. Furthermore *N* = 6, *𝜗* = 0.03. The parameters defining the spatio-temporal dependence are *τ* = 1.08 day, *L*_{x} = 10.1,*L*_{y} = 6.2 degree, defined in Eq. 12, and drift speed *v*_{x} = − 0.11,*v*_{y} = 0.02 [deg/hour], defined in Section 7.3.1, while correlation between the spatial derivatives of transformed wind speed 0.04 is negligible.

The parameters *τ*, *L*_{x}, *L*_{y} and velocities *v*_{x}, *v*_{y} describe average geometry of extreme wind speed regions and their dynamics. The estimated values are in terms of the covariance matrix of the gradients. The values given above are quite typical for these parameters at this region. The parameter *N* describes the average number of weather anomalies, which are about 1.6 per month in a region of 25 deg^{2}. For example, Fig. 7 presents the variability of this parameter for various months. It should be noted that the intensity of extraordinary wind events is higher than intensity of cyclones. The parameters *σ*, *𝜗* and *a* influence the tails of wind speed distribution, and hence the frequencies of extreme wind speeds. The low value of the parameter *a* makes tails heavier. For example, if a hybrid model predicts an extreme value of *x*_{max}, it is corresponding to the extreme wind speed of (*m* + *σ**x*_{max})^{1/a}. (Note that using models with very low *a* is questionable for extreme wind prediction.) Furthermore, *𝜗* is the scale parameter to determine the highest value of *X*_{i}(**q**) components. It can be deduced, by combining Eqs. 3, 9, and 36. The highest value of *X*_{i}(**q**) is about \(c\sqrt {\vartheta } Y_{i}\), where *c* is a constant and *Y*_{i} is Laplace distributed variables.

**S**has been increased using Eq. 14. The parameter

*N*= 6 is also increased by a factor of 1 + 360/(1.75 × 30) giving the new value of

*N*≈ 50. Figure 14 presents 16 (out of 500) simulated time series. The simulations are compared with wind speeds extracted from the ECMWF hindcast data. It demonstrates that the hybrid model can generate extreme windy episodes as observed in the data. In order to compare the time series for low and moderately high speeds, these plots have been zoomed and presented in the right plots of Fig. 14. It is shown that the simulated time series of wind speeds are smoother than the observed values. This is caused by simplicity of the chosen kernel. In Rychlik (2015) and Rychlik and Mao (2014), a more complex kernel was presented and could be used even here.

Finally, the problem of uncertainties in predicting extreme wind speeds can be illustrated by the simulation study. The hybrid model is used to simulate wind speed time series in August for 500 years. The wind speed that is exceeded once in 500 years is taken as an empirical estimate of the 500-year extreme wind speed. This procedure has been repeated ten times. The estimated 500-year extreme wind speeds vary between 38 and 102 m/s. Note that 500-year extreme wind speed evaluated by the hybrid model at the location is 51 m/s.

### 7.3 Spatio-temporal simulation of local wind speeds using the hybrid model

In this section, a local spatio-temporal hybrid model will be used to simulate wind speed fields in a small region of about 6 × 6 degrees square and a time period of 8 h. Obviously, observed wind speeds at a fixed position does not reflect dynamics of wind moving systems. The joint spatial and temporal data are needed to investigate movements of wind systems. Here, the gradient is used to define local spatio-temporal dependence. However, such a local information is not sufficient to investigate the movement of cyclones. For this type of dynamics, a global spatio-temporal model is needed. Development of such a model is planned but not discussed here.

*X*

_{G}(

**q**) in Eq. 11 are simulated, and then transformed into wind speed fields, which are presented in the left plots (with 4-h interval) of Fig. 16. The other approach is to employ the hybrid model as in Eq. 11, i.e., multiplying the above simulated

*X*

_{G}(

**q**) by a factor

*p*, adding the \({\sum }_{i = 1}^{N} X_{i}(\textbf {q})\) and transforming them into wind speed fields as follows:

*X*

_{i}with the kernel

*f*defined in Eq. 3. And then the amplitudes and the tops of these storms are generated and placed at random in this region. The amplitudes of the storms change in time and the tops of the storms are moving with constant velocity. The simulation results are presented in the right plots of Fig. 16.

If the components of gradients *X*_{x}, *X*_{y} are uncorrelated (as in the current example), the velocity of *X*_{i} movement is equal to median propagation velocity of the wind speed field given in Eq. 18. It will be briefly reviewed in Section 7.3.1. Obviously, the storms move in variable velocities and usually faster than (*v*_{x}, *v*_{y}) (see, e.g., Dorst (2014)). Development of more realistic models for the storm dynamics is planned in future.

#### 7.3.1 Velocities of a wind speed field

In the classical paper by Longuet-Higgins (1957), velocities were introduced to study the movements of random surfaces. Alternative definitions can be found in, e.g., Baxevani et al. (2003). Here, the so-called velocity in a fixed direction will be defined below and used throughout this paper.

**p**, i.e.,

*W*(

*t*,

*x*,

*y*) =

*W*(

*t*

_{p}+

*t*,

*x*

_{p}+

*x*,

*y*

_{p}+

*y*). The velocities

*V*

_{x}and

*V*

_{y}in the

*x*and

*y*directions, respectively, are defined by

*W*

_{t},

*W*

_{x}, and

*W*

_{y}are the partial derivatives of

*W*. Some simple calculus is needed to show that

*a*. The median velocities

*V*

_{x}and

*V*

_{y}will be denoted by

*v*

_{x}and

*v*

_{y}, respectively. Now, if

*X*is a homogeneous Gaussian field, then the medians of the velocities

*V*

_{x}and

*V*

_{y}are given by

*λ*

_{ij}is the

*i*,

*j*element of the matrix Λ of Eq. 4, which defines the correlation structure of the wind field

*X*.

The drift speeds *v*_{x} and *v*_{y} are approximately half of the average speed of cyclone movements at this region; see, Dorst (2014) for the movement speeds statistics of cyclones. When cyclones are not present in the region, most likely *v*_{x} and *v*_{y} could describe the movement of windy weather. In the Appendix C, some MATLAB code is given and can be used (after small adaptations) to simulate the wind speed fields.

#### 7.3.2 Estimation of spatio-temporal extremes by means of simulated wind speeds fields

In this section, an example is presented to demonstrate the application of wind speed simulations by the hybrid model for the spatio-temporal extreme predictions. A spatial region of longitude between 68.5W and 76.5W, latitude between 14.5N and 20.5N, is chosen for wind simulations in August. The wind speed variability in this region is assumed to be homogeneous and stationary (The assumption might be not accurate because of islands in this region.) The hybrid model is fitted at the location (72.25W, 16.5N) in August. Wind speeds have been simulated based on the hybrid model over the region on a mesh resolution of 0.5 degree and 2 hours. Then, maximum wind speeds are picked up from the simulated spatio-temporal wind speed fields. In total, 500 spatio-temporal maximum wind speeds are simulated and the GEV distribution is also fitted to the data.

The figure shows that the spatio-temporal maximums are considerably higher than the one observed at a fixed location. It should be noted that the hybrid model at the location (72.75W, 16.5N) is used instead of the location (72.75W, 17.25N), where the maximum wind speed in the whole data set was observed. For practical applications, smoothed over region model for simulations of spatio-temporal wind maximums should be used.

Some MATLAB scripts to simulate spatio-temporal extreme wind speeds is given in the Appendix C. In the program, the zero mean variance one Gaussian field *X*_{G}(**q**) is replaced by a standard Gaussian variable (*X*_{G}(0,0,0)). This approximation is used because simulations of Gaussian field on a large number of positions are very time consuming. Since *X*_{G} and *X*_{i} are independent and the size of extreme maximums is defined by the sum \({\sum }_{i = 1}^{N} X_{i}(\textbf {q})\) (easy and fast to simulate), this leads to fast and accurate simulations of spatio-temporal maximums.

## 8 Conclusions

Due to the strong tropical storms and hurricanes that can occur in the Caribbean sea and the Gulf of Mexico, the spatio-temporal wind model used in the North Atlantic, which is based on a transformed Gaussian field, cannot properly describe the wind variability in this region. Therefore, a hybrid spatio-temporal model, combining a transformed Gaussian field with Laplace moving averages, is proposed to describe the wind speed variability in this region. The hybrid model encompasses the Gaussian and Laplace models as limiting cases.

The capability of the hybrid model has been demonstrated through the estimation of the yearly long-term distribution and the distributions of yearly maximum wind speeds at three locations in the Caribbean sea and the Gulf of Mexico. The wind speed distributions computed using the hybrid models agree well with the empirical distributions estimated from the ECMWF hindcast wind database. The application of Rice’s method to the fitted hybrid models yields 100/1000-year extreme wind speed predictions, which agree well with those derived using generalized extreme value (GEV) distributions fitted to 16 consecutive yearly maximum wind speeds.

The proposed model can be used to evaluate the available wind energy at a fixed location. It provides a means of predicting frequencies of extreme wind speeds, which are needed in safety analysis for maritime operations and for planning of coastline protection, at locations where there are no long time series of measured wind speeds available. The model can be also used to estimate the long-term distribution of wind speeds and upcrossing frequencies of extreme wind speeds along arbitrary ship routes. Those are essential for a ship’s safety and route planning considering relevant responses due to wind loads. It can be used to estimate the spatio-temporal extreme/maximum wind speeds as well.

In addition, further research is needed for the future development of the hybrid model, which could allow for variable (more realistic) cyclones movements in the chosen regions. Furthermore, a global non-homogeneous and non-stationary hybrid model could be developed, in which the parameter *N* will be random (Poisson distributed).

## Notes

### Acknowledgments

Support from the Chalmers Area of Advance of ICT, Energy and Transport is gratefully acknowledged. The authors also give many thanks to an anonymous referee for insightful comments and to ECMWF for providing access to the data used for the wind modelling in this paper.

### Funding information

This research was supported by the Swedish Research Council through Grant 340-2012-6004 and by the Knut and Alice Wallenberg stiftelse.

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