## 1. Introduction

*Q*in time

*t*during a storm. Following Borgman (1973), this probability can be expressed as

*Q*,

*D*is the storm duration, and

*z*in a sea state where

Borgman’s formulation (1) is the starting point of various statistical methods developed for predicting occurrences of extreme events in stormy seas (Krogstad 1985; Prevosto et al. 2000; Boccotti 2000; Isaacson and Mackenzie 1981; Guedes Soares 1988; Goda 1999; Arena and Pavone 2006, 2009; Fedele and Arena 2010). These assume that the effects of the sea state observed during time intervals of the short-term scales of *T _{s}* ~ 1–3 h can be accumulated to predict the wave conditions for the long-term scales of

*T*~ years. One of the drawbacks of such stochastic analyses is that, in short-crested seas, surface time series gathered at a fixed point tend to underestimate the true actual wave surface maximum that can occur over a given region of area

_{l}*E*

_{s}around

*Q*. A large crest observed in time at

*Q*represents a maximum observed at that point, but it may not even be a local maximum in the actual crest segment of a three-dimensional (3D) wave group. The actual crest representing the global maximum occurs at another point located without or within

*E*

_{s}. Certainly, the elevation of the actual crest is always larger than that measured at

*Q*. Thus, (1) underestimates the maximum wave surface height

*E*

_{s}, which is also not the highest crest height of the group, unless the area is large enough for all wave-group dynamics to develop fully. Indeed,

Note that the application of such advanced stochastic theories to realistic oceanic conditions has been limited because it requires the availability of wave surface data measurements collected both in space and time, in particular directional wave spectra (Baxevani and Richlik 2004). Only at large spatial scales, synthetic aperture radar (SAR) or interferometric SAR (INSAR) remote sensing provides sufficient resolution for measuring waves longer than 100 m (see, e.g., Marom et al. 1990; Marom et al. 1991; Dankert et al. 2003). However, it is insufficient to correctly estimate spectral properties at smaller scales. At such scales, up-to-date field measurements for estimating directional wave spectra are challenging or inaccurate even if a linear or two-dimensional (2D) wave probe-type arrays could be used, though expensive to install and maintain (Allender et al. 1989; O’Reilly et al. 1996). Recently, stereo video techniques have been proposed as an effective low-cost alternative for such precise measurements (Benetazzo 2006; Wanek and Wu 2006; Fedele et al. 2011a,b; Gallego et al. 2011; Bechle and Wu 2011; de Vries et al. 2011; Benetazzo et al. 2012). Indeed, a stereo camera view provides both spatial and temporal data whose statistical content are richer than that of a time series retrieved from wave gauges. For example, Gallego et al. (2011) have estimated directional spectra by a variational variant of the Wave Acquisition Stereo System (WASS) proposed by Benetazzo (2006). Further, WASS was used by Fedele et al. (2011a) to prove that in short-crested seas the maximum surface height over a given area is generally larger than that observed in time by point measurements (see also Forristall 2006). The fact that the spatial extremes are larger than those measured at a fixed point is not only because there are more waves in a spatial domain. The main reason is that fixed-point measurements cannot detect true extremes in short-crested seas. Theories due to Adler (1981) and Piterbarg (1995) follow from both reasons, especially from this essential difference between fixed-point versus true spatial picture. An extreme observed at a fixed probe in time in short-crested seas indicates that a wave crest section just propagated through the probe, and the probability that the actual extreme of that crest section coincides with the extreme observed in time is simply zero. It is only in long-crested seas that one can equate the extremes observed in time with the actual spatial extremes.

As pointed out by Baxevani and Richlik (2004), the occurrence of an extreme in a Gaussian field is analogous to that of a big wave that a surfer is in search of and always finds. Indeed, his likelihood to encounter a big wave increases if he moves around a large area instead of waiting to be hit by it. Indeed, if he spans a large area the chances to encounter the largest crest of a wave-group increase, in agreement with the findings of the recent European Union “MaxWave” project (Rosenthal and Lehner 2008).

In this work, the main focus is on characterizing the statistical properties of space–time extremes in short-crested sea states and their long-term predictions. The paper is structured as follows: First, the essential elements of the theory of Euler characteristics (EC; Adler 1981) are introduced. Then, their application is presented in the context of the Equivalent Power Storm (EPS) model of Fedele and Arena (2010). The statistical properties of space–time extremes are then derived. Further, the relative validity of the new model and its predictions are assessed by analyzing wave measurements and directional spectra retrieved from National Oceanic and Atmospheric Administration (NOAA) buoy 42003 (east Gulf of Mexico).

## 2. Euler characteristics and extremes

*z*, define the excursion set

*z*: namely,

*U*, minus the number of holes that pass through it, plus the number of hollows inside. For 2D random fields instead, the EC counts the number of connected components minus the number of holes of the respective excursion set. In one-dimension (1D), the EC simply counts the number of

*z*upcrossings, thus providing their generalization to higher dimensions (Adler 1981).

*z*depends on the domain size and it is well approximated by the expected EC of the excursion set

*z*increases, the holes and hollows in the excursion set

### a. Extremes of Gaussian fields

*XYD*(see Fig. 1). Drawing upon Adler and Taylor (2007), define

*x*and

*y*, respectively. These, as well as the parameter

*z*is given by

*z*can also be exceeded on the boundary surface

*z*can also be exceeded along the perimeter

*S*. In this case, the number of such occurrences follows the Rayleigh law of (7). The average number of 1D waves that exceed

*u*is given by

*V*of

*S*, or the perimeter

*P*, respectively. The three terms can be derived as follows: The probability that

*z*in

*V*is equal to the probability that all the 3D waves in

*V*have amplitudes less than or equal to

*z*. If one assume the stochastic independence among waves (which holds for large

*z*), then the first term in (11) can be expressed as

### b. Scale dimension of extremes

*V*away from the boundaries. For

*V*. The limiting case of

*D*,

*X*, or

*Y*is null: for example,

*D*= 0. In this case, the extreme can occur within an area

*E*=

_{s}*XY*and it is 2D. When the area’s boundaries are touched by the extreme, then

*E*collapses to a line (

_{s}*X*= 0 or

*Y*= 0). As an example, Fig. 2 shows the wave dimension

*H*

_{s}sequence recorded during the period 2007–09 by NOAA buoy 42003, moored off the east Gulf, for

*D*= 1 h and squared

*E*= 100

_{s}^{2}m

^{2}. Clearly, in milder or low sea states, extremes are quasi 3D because mean wavelengths (~30 m) and periods (~3 s) are much smaller than the lateral length

*L*and duration

*D*, respectively. As the intensity of the sea state increases, so do both the associated mean wavelengths (up to ~190 m) and periods (up to ~12 s) and the wave dimension reduces; at the highest sea states,

*X*–

*T*or

*Y*–

*T*of the volume

*V*.

Wave dimension *H*_{s} sequence recorded by NOAA buoy 42003 during 2007–09 (*D* = 1 h and *X* = *Y* = 100 m).

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0179.1

Wave dimension *H*_{s} sequence recorded by NOAA buoy 42003 during 2007–09 (*D* = 1 h and *X* = *Y* = 100 m).

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0179.1

Wave dimension *H*_{s} sequence recorded by NOAA buoy 42003 during 2007–09 (*D* = 1 h and *X* = *Y* = 100 m).

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0179.1

In the following sections, (15) is extended for a random wave field

### c. Space–time extremes during storms

*E*

_{s}=

*XY*during a time interval

*D*. Assuming that

*D*into

*t*axis and surrounding the interior

*t =*0 and

*D*, respectively, and the lateral surface

*z*via the quadratic equation

*C*

_{max}observed in time at point

*Q*. The expected maximum

*z*as

Note that the exceedance probability in (19) relies on the assumption of stochastic independence of large waves, which holds for weakly non-Gaussian fields dominated by second-order nonlinearities or short-crested seas considered in this work. Indeed, realizations of maxima typically occur at times and locations typically well separated to render them largely independent of one another in wind seas. Clearly, in long-crested sea states the areal effects are negligible and (19) reduces to the time Borgman formulation (1). However, in this case the wave surface is affected by nonlinear quasi-resonant interactions and fourth-order cumulants increase beyond the Gaussian threshold if the spectrum is narrow (see, e.g., Fedele et al. 2010). To account for such deviations, an obvious modification would be to simply replace in (1) the Rayleigh/Tayfun distribution with the Gram–Charlier (GC) type of models, such as those developed by Mori and Janssen (2006), Tayfun and Fedele (2007), or Fedele (2008). Indeed, GC models have been shown to describe the effects of quasi-resonant interactions on the wave statistics (see, e.g., Fedele et al. 2010). However, in such long-crested sea states individual waves are correlated (see, e.g., Janssen 2003) and (1), even with a GC model, loses its validity and yields conservative estimates as an upper bound. The space–time stochastic model proposed herein can be extended to smoothly bridge long- and short-crested conditions. This would require taking into account the correlation between neighboring waves, and it should depend upon the joint probability distribution of successive extremes (see, e.g., Fedele 2005). Such a model would be beneficial for estimating extreme waves in rapid development of long-crested sea states in time. Some work on marine accidents suggests that such conditions may occur (Tamura et al. 2009). The development of such a stochastic model is in progress and will be discussed elsewhere.

## 3. Prediction and properties of space–time extremes

*E*centered at point

_{s}*Q*during a storm. To do so, consider a time interval

*E*, and assume that the time series of significant wave heights

_{s}*Q*as well as the directional spectrum are given as measurements. Then, define a succession of storms where each storm, according to Boccotti (2000), is identified as a nonstationary sequence of sea states in which

*b*and peak amplitude

*a*at, say,

*h*varies in time

*t*according to a power law

*h*(

*t*) ~ |

*t*−

*t*

_{0}|

^{λ}, where

*a*and

*b*are realizations of two random variables: for example,

*A*and

*B*, respectively. Then, the storm-peak probability density function (pdf)

*u*,

*w*, and

*h*are regression parameters (see Fedele and Arena 2010). As a consequence, the analytical form of the storm-peak density

_{l}*z*, the probability that

*h*(

*t*) to that of the EPS storm (see Fedele and Arena 2010). As

*a*equal the actual maximum

*b*is chosen so that the expected maximum

*b*is given by imposing the exceedance probabilities of the actual and EPS storms to be equal at

*b*follows as

*b*depends upon the storm shape, but it slightly changes with the area

*E*as expected, because

_{s}*b*and the storm-peak density

*p*are unique temporal properties of the given location, as a result of the assumed spatial homogeneity. Thus, hereafter

_{A}*b*is estimated as

NOAA buoy 42003: (top) shape and exceedance probability of the maximum time crest height *C*_{max} of the observed actual storm and the associated EPS storm and (bottom) duration of EPS storms and conditional base regression *b _{m}* = 86.5 h,

*s*= −0.13 m

_{m}^{−1}, and

*a*

_{0}= 2.22 m).

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0179.1

NOAA buoy 42003: (top) shape and exceedance probability of the maximum time crest height *C*_{max} of the observed actual storm and the associated EPS storm and (bottom) duration of EPS storms and conditional base regression *b _{m}* = 86.5 h,

*s*= −0.13 m

_{m}^{−1}, and

*a*

_{0}= 2.22 m).

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0179.1

NOAA buoy 42003: (top) shape and exceedance probability of the maximum time crest height *C*_{max} of the observed actual storm and the associated EPS storm and (bottom) duration of EPS storms and conditional base regression *b _{m}* = 86.5 h,

*s*= −0.13 m

_{m}^{−1}, and

*a*

_{0}= 2.22 m).

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0179.1

*h*can be expressed as (Fedele and Arena 2010)

*z*can be derived a follows: Consider the number

*z*. Then,

*z*. Thus,

*z*at a given location in time. Drawing upon Fedele and Arena (2010) and from probabilistic principles, one can also estimate the most probable value of the peak significant wave height

*A*of the storm during which the maximum

*z*) over the area

*A*is in

*z*and area

*A*) is very close to

## 4. Long-term extremes in the east Gulf

*u*= 0.591,

*w*= 0.201 m, and

*h*= 0 m. Further, directional data available for the period 2000–09 are used to fit the wave parameters

_{l}Given *A* from (26) and predict the return period *R*. The associated return period *E*_{s} = *L*^{2}, with *L* = 10^{3} m, is computed from (39) and shown in Fig. 5 for EPS, GEV, and Gumbel. For comparisons, the associated time predictions of the return period *E*_{s} is larger than that expected at given point in time. Further, as the area increases the predictions tend to deviate from the time Borgman counterpart as shown in Fig. 6 (right), which reports the EPS predictions of *R* over increasing areas with *L* = 10^{2}, 10^{3}, and 10^{4} m, respectively. Over such large areas, the wave dimension *L* = 100 m). Thus, drawing upon Boccotti (2000), most likely *R* we need an estimate of the most probable value *A* of the storm during which such maximum *z*. This can be inferred using Eq. (40), which allows to predict the mean *A* given *R* for the predictions in Fig. 6 (top). For the largest area considered (*L* = 10^{4} m), this ratio increases to roughly 1.5–1.6, thus significantly exceeding the predictions at a given point in time (i.e., 0.9–1.1), in agreement with the stereo measurements of ocean waves (Fedele et al. 2011a). Given *R* (estimates from the

NOAA buoy 42003: predicted return period *μ*_{G} = −2.007 m and *σ*_{G} = 2.135 m; GEV parameters: *μ* = 2.656 m, *σ* = 0.422 m, and *k* = 0.353; Weibull parameters for EPS: *u* = 0.591, *w* = 0.201 m, and *h _{l}* = 0).

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0179.1

NOAA buoy 42003: predicted return period *μ*_{G} = −2.007 m and *σ*_{G} = 2.135 m; GEV parameters: *μ* = 2.656 m, *σ* = 0.422 m, and *k* = 0.353; Weibull parameters for EPS: *u* = 0.591, *w* = 0.201 m, and *h _{l}* = 0).

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0179.1

NOAA buoy 42003: predicted return period *μ*_{G} = −2.007 m and *σ*_{G} = 2.135 m; GEV parameters: *μ* = 2.656 m, *σ* = 0.422 m, and *k* = 0.353; Weibull parameters for EPS: *u* = 0.591, *w* = 0.201 m, and *h _{l}* = 0).

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0179.1

NOAA buoy 42003: predicted return periods *L* = 10^{3} m) estimated with G, GEV, and EPS models (regression parameters as in Fig. 4).

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0179.1

NOAA buoy 42003: predicted return periods *L* = 10^{3} m) estimated with G, GEV, and EPS models (regression parameters as in Fig. 4).

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0179.1

NOAA buoy 42003: predicted return periods *L* = 10^{3} m) estimated with G, GEV, and EPS models (regression parameters as in Fig. 4).

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0179.1

NOAA buoy 42003: (top) predicted return period *L* = 0 (time), 10^{2}, 10^{3}, and 10^{4} m estimated with the EPS model (regression parameters as in Fig. 4); (middle) significant wave height

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0179.1

NOAA buoy 42003: (top) predicted return period *L* = 0 (time), 10^{2}, 10^{3}, and 10^{4} m estimated with the EPS model (regression parameters as in Fig. 4); (middle) significant wave height

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0179.1

NOAA buoy 42003: (top) predicted return period *L* = 0 (time), 10^{2}, 10^{3}, and 10^{4} m estimated with the EPS model (regression parameters as in Fig. 4); (middle) significant wave height

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0179.1

Finally, to confirm the above long-term predictions the *H*_{s} sequence of hourly sea states recorded by NOAA buoy 42003 during the period 2007–09 has been analyzed. In particular, Fig. 7 (top) reports the short-term (*D =* 1 h) expected maximum surface height *X* = *Y* = 10^{3} m) for each hourly sea state. The associated

NOAA buoy 42003 (east Gulf): (top) short-term expected maximum surface height *L* = 10^{3} m) for each hourly sea state (period 2007–09) in terms of the ratio *β* is ~3 for all the analyzed sea states.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0179.1

NOAA buoy 42003 (east Gulf): (top) short-term expected maximum surface height *L* = 10^{3} m) for each hourly sea state (period 2007–09) in terms of the ratio *β* is ~3 for all the analyzed sea states.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0179.1

NOAA buoy 42003 (east Gulf): (top) short-term expected maximum surface height *L* = 10^{3} m) for each hourly sea state (period 2007–09) in terms of the ratio *β* is ~3 for all the analyzed sea states.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0179.1

## 5. Conclusions

The stochastic model developed herein extends the Borgman time-domain model (1) to space–time extremes and demonstrates the increased likelihood of large waves over a given area in short-crested seas (see also Baxevani and Richlik 2004). The proposed model was applied to several storms recorded by the NOAA buoy 42003. The results reveal that given a return period, the associated threshold *z* exceeded by the maximum surface height *L* = 10^{4} m),

The present EPS model provides another “hand on the elephant” for the subject of extreme waves (see, e.g., Boccotti 1981, 2000; Fedele 2008; Fedele and Tayfun 2009; Gemmrich and Garrett 2008) by demonstrating that the occurrence of large waves over an area can be explained in terms of extremes in space–time. In particular, the proposed model is of relevance as a practical tool for identifying safer shipping routes and for improving the design and safety of offshore facilities.

The correlation or stochastic dependence of wave extremes is not an issue for the statistics of maxima because realizations of maxima typically occur at times and locations typically well separated to render them largely independent of one another in wind seas. However, under conditions conducive to the rapid development of long-crested sea states such as those studied numerically by Waseda et al. (2011), stochastic dependence can be an important factor in analysis. In this regard, the space–time stochastic model proposed here can be extended to smoothly bridge long- and short-crested conditions by taking into account the correlation between neighboring waves (see, e.g., Fedele 2005).

## APPENDIX A

### Wave Parameters

*W*.

## APPENDIX B

### Scale Dimension of Extremes

*N*waves whose parent distribution follows an exceedance distribution of the form

*P*and

_{S}*P*in (7) and (8) for 2D and 3D waves, respectively. Thus,

_{V}*β*is related to the mean wavelengths and periods as well as the volume’s geometry by equating the expected maximum

*N*“beta waves” to the true maximum

*N*beta waves is given by

*N*are chosen as

*N*is the average number of waves of dimension

## APPENDIX C

### Derivation of

*z*). Then, the probability of exceedance can be rewritten as

*z*in

*z*via the quadratic equation

## APPENDIX D

### Function G(λ, a)

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