## Abstract

The current study advances research on the consequences of global climate change by utilizing the novel Gaidai multivariate risks evaluation methodology to conduct spatiotemporal analysis of areal windspeeds. Multidimensional structural and environmental dynamic systems that have been either physically observed or numerically simulated over a representative time-lapse are particularly suitable for the Gaidai risks evaluation methodology. Current research also presents a novel non-parametric deconvolution extrapolation method. As this study has shown, given in situ environmental input, it is possible to accurately predict environmental system hazard risks, based even on a limited underlying dataset. Furthermore, because of their complex nonlinear cross-correlations between various environmental system-critical dimensions or components and large dimensionality, environmental dynamic systems are difficult to handle using traditional methods for evaluating risks. In the North Pacific, close to the Hawaiian Islands, NOAA buoys gathered raw in situ wind speed data, which has been utilized in the current study. Areal ocean wind speeds constitute quite a complex environmental dynamic system that is challenging to analyze because of its nonlinear, multidimensional, cross-correlated nature. Global warming had impacts on ocean windspeeds in the recent decade. Developing novel state-of-the-art environmental system risk evaluation methods is a principal component of modern offshore structural analysis in light of adverse weather. The advocated novel risk/hazard assessment approach may be used for resilient island cities design, especially those that are near ocean shore and hence exposed to extreme weather.

## 1 Introduction

High dimensionality of environmental systems with a large number of random variables impacting environmental system dynamics presents considerable challenges for the system's prognostics [1–5]. The reliability of complex environmental systems can be easily and precisely determined by using either a substantial amount of measurement data or direct large-scale MCS (Monte Carlo Simulations) [6–13]. Many complex environmental systems may be beyond the reach of computational or experimental resources. Driven by the aforementioned contention, writers propose an innovative approach to risk/hazard assessment for environmental systems, to minimize expenses related to computation or measurement [14]. Ocean wind speeds are usually modeled as a stochastic/random piecewise ergodic process. Figure 1 presents NOAA (i.e., National Oceanic and Atmospheric Administration) Blended Sea Winds global ocean windspeed dataset [15].

For practical renewable energy industrial purposes, capturing full wind energy resource potential, especially over offshore areas, is a challenging task. The role of renewable energy technologies in the framework of an integrated strategy to mitigate climate change is a topic of much interest to researchers these days. Most studies utilize generic EVT (i.e., Extreme Value Theory)-based parametric approaches [16,17], while this study advocates a more robust non-parametric statistical approach. As ocean wind and wave processes are closely inter-correlated, investigating the extreme occurrences of ocean waves has been attempted with some success, or freak waves, though this is outside the scope of this study, with PDF (i.e., Probability Density Function) reflecting physical system dynamics. For illustration, in Refs. [18,19], ocean waves deviate from linear theory, as the author has demonstrated. Similarly, in Refs. [20–24], authors have shown that Rayleigh-type PDF reflects hurricane data extreme events. Furthermore, it appears that the effects of spectrum bandwidth on the PDF of extreme waves vary depending on whether they occur in shallow or deep waters [25,26]. Moreover, experiments have shown a significant increase in the frequency of extreme waves due to ocean processes like Wave-current or shoaling systems that push wave trains out of balance [27–31]. It has recently been discovered that no known theoretical PDFs, whether grounded in physical principles or not, can account for extreme wave statistics in a variety of scenarios, and none can account for them universally like Gumbel can [32–44]. Note that the methodology introduced by authors here is non-parametric and thus does not rely on Gumbel, Weibull, Pareto, or any other GEV (i.e., Generalized Extreme Value) type of distribution. The worst wind-related damages in wind farms are often caused by extreme wind speed events. To prevent damage to wind turbines and minimize cut-out occurrences in these facilities, a precise assessment of the frequency and severity of extreme windspeed events has been recently conducted using generative data augmentation techniques [45]. Samples from representative known distributions are bootstrapped, and the validity of forecasts for high wind speeds at large mean recurrence intervals can be evaluated [46], where sub-asymptotic Gumbel-type distribution was employed, taking into account imperfect convergence to the proper asymptote, is compared to the conventional asymptotic GEV and GP (i.e., Generalized Pareto) distribution.

The capacity of the Gaidai risks evaluation methodology to analyze multidimensional spatiotemporal environmental systems with an almost infinite number of critical dimensions and components is a key innovation. Figure 2 provides a schematic flowchart of the Gaidai risks evaluation methodology for practical use in engineering applications – particularly during the design phase.

## 2 Multivariate Gaidai Risk Evaluation Method for Series-Type Systems

*T*value, large with respect to autocorrelation and relaxation temporal scales of the dynamic environmental system. Let $X1,\u2026,XNX$ stand for the temporally consequent local maxima of the environmental system component $X(t)$, recorded at discrete, monotonically increasing time-moments $t1X<\u2026<tNXX$. In a similar way, the remaining system's primary 1D components $Y(t),Z(t),\u2026$ have local maxima $Y1,\u2026,YNY;$$Z1,\u2026,ZNZ$,… recorded in temporally increasing order. To keep things simple, we will assume that all local maxima in the environmental system are positive, and hence, all global maxima too. The objective now is to accurately estimate the probability/risk of environmental system failure/damage $PF$

*j*. The scaling parameter $0<\lambda \u22641$ may now be inserted in order to simultaneously decrease the limit values for all of the principal environmental dynamic system's components. Next, hazard/limit scaled limit vector $(\eta 1\lambda ,\u2026,\eta N\lambda )$ to be introduced, with $\eta j\lambda $ being equal to either $\eta X\lambda ,\eta Y\lambda ,\eta z\lambda ,\u2026,$ or their maxima, in case local maxima from different components occur at the same time $tj$, where $\eta X\lambda \u2261\lambda \u22c5\eta X,\eta Y\lambda \u2261\lambda \u22c5\eta Y,\eta z\lambda \u2261\lambda \u22c5\eta Z,\u2026$. Survival probability as $P(\lambda )$ of an environmental system becomes now smooth/differentiable $C1$ function of $\lambda $, with $P\u2261P(1)$, and failure/damage probability $PF\u22611\u2212P(1)$

*j*and solely reliant on the conditioning memory depth

*k*, because the MDOF system was assumed to be jointly piecewise ergodic and thus jointly quasi-stationary. Currently, the likelihood of non-exceedance can be roughly expressed as

Since design damage/failure/hazard probability/risk being of the small order of magnitude $o(1)$, holding $N\u226bk$, Eq. (5) follows from Eq. (1) by assuming $Prob(R1\u2264\eta 1\lambda )\u22481$. Convergence with regard to the conditioning parameter *k* is usual. Narrowband dynamic systems exhibit temporally successive/neighboring components as well as cascading/clustering hazards/failures/damages in significant environmental system's components. Due to the intrinsic interdependencies between nearby extreme events, local maxima appear as highly correlated groups or clusters of local maxima that are stored in the synthetic vector $R\u2192=(R1,R2,\u2026,RN)$ of the assembled system [47–55]. The system's vector $R\u2192$ is made up of *N*, the total number of local maxima of the system's critical component (Fig. 3) [56–60].

Using a running index of *j*, the unified system limit/hazard vector $(\eta 1,\u2026,\eta N)$ has each component $\eta j$ equal to either $\eta X,\eta Y,$$\eta Z,\u2026,$, depending on which principal component, $X(t),$$Y(t),Z(t),\u2026,$ corresponds to the current local maxima.

### 2.1 Self-Deconvolution Extrapolation Scheme.

straightforward extracting $pXA$ from the underlying dataset, namely from underlying time series $X(t)$

After the PDFs from each of the principal stochastic process components $X1(t)$, $X2(t)$, or $pX1,pX2$ are extracted independently, the convolution $pXB=conv(pX1,pX2)$ is applied.

*k*th element

Note that for self-deconvolution case $u=v$ and $m=n$. The vector $w$ is projected and stretched to a length twice that of the original PDF support domain. To sum up, the PDF support length $pX$ has increased twice, $(2n\u22121)\u22c5\Delta x\u22482n\u22c5\Delta x=2XL$, in comparison to the starting length $n\u22c5\Delta x=XL$. Since empirical PDFs are not smooth in the tail, a smoothing tail procedure is required. Næss-Gaidai (NG or modified four-parameter Weibull) approach has been also used here, since PDF tails resemble $exp{\u2212(ax+b)c+d}$ at $x\u2265x0$, where $a,b,c,d$ being four optimized parameters. The proposed approach may also make use of other extrapolation approaches; however, doing so would include some biases and assumptions. To get the PDF tail, deconvolution extrapolation does not have to adhere to any specific extrapolation-functional group or class. Given the CDF (i.e., Cumulative Distribution Function), it is more principal to assess the likelihood of exceedance or $CDF\xaf=1\u2212CDF$, than the marginal $PDF/CDF\xaf$.

## 3 Results

In order to illustrate the efficacy of the Gaidai risks evaluation approach in the North Pacific, this section makes use of the 10-min average wind speed measurements in the m/s dataset from the NOAA buoy, close to the Hawaii islands. Wind speed dynamics is widely acknowledged to be a multidimensional, highly nonlinear, cross-correlated dynamic system that can be challenging to analyze. Environmental system risk/hazard assessment techniques are also principal for marine structures that operate in a specific offshore region of interest and are occasionally exposed to severe offshore weather.

Considering three NOAA offshore wind measurement sites, (1) Station 51101 (LLNR 28006.3)—NW of Hawaii; (2) Station 51002—SW of Hawaii; Station (3) 51000 – northern (N) Hawaii, had been chosen in the current study, its respective measured 10-min average windspeed values in m/s have been chosen as three environmental system's critical components (or dimensions) $X,Y,Z$ thus constituting an example of 3D (i.e., three dimensional) environmental system. Critical limits either $\eta X$, $\eta Y$, or $\eta Z$ for the 1D system's critical/key components have been set equal to the whole analyzed dataset's maximum windspeeds, during a decade observation period between years 2009–2018, monitored synchronously for all three chosen windspeed measurement buoy's locations. Figure 4 provides North Pacific NOAA buoy locations [34], and the Area of interest indicated by a blue circle.

Figure 5 shows the NOAA data buoy, comprising sensors that track and gather information on oceanic and atmospheric conditions. The information is then converted into an electrical signal, which is either recorded in the onboard data unit or transmitted to land. The parameters/sensors feature of the 3-meter NOAA buoy are as follows:

Site elevation: MSL (i.e., Mean Sea Level);

Height of air temperature: 3.4 meters above MSL;

The height of the anemometer is 3.8 meters above MSL;

Barometer elevation: 2.4 meters above MSL;

Sea temperature: 2 meters below MSL;

Ocean depth: 2 km.

To integrate the three measured time series, $X,Y,Z$ the scaling that follows was carried out in compliance with Eq. (2). This led to the key components of all three environmental systems being non-dimensional and having equal failure, danger, and damage limitations of 1. After that, all of the local maxima from the three measured time series were merged into one synthetic time series by sorting all of the sets $max{Xj,Yj,Zj}$ according to the chronologically increasing occurrences of occurrence of these system-critical component's local maxima. This was done by retaining the local maxima in the temporally rising order: $R\u2192=(max{X1,Y1,Z1},\u2026,max{XN,YN,ZN})$. A non-dimensional example of an assembled/synthetic vector $R\u2192$ is shown in Fig. 6. This dataset is an assembly of the raw daily greatest highest daily hindcast wind speed dataset's local maxima.

Using the PDF tail, the target failure/damage probability/risk level was extrapolated to a 100-year return period. It should be highlighted that a synthetic vector $R\u2192$ has no specific physical importance on its own because it is composed of completely different primary components of the environmental system. The “shorter” data record was created by taking the 10th data point from the «full/longer» wind speed dataset. The “shorter” dataset's $fX1$ PDF tail is shown in Fig. 7(a). It was created by deconvolution and then linearly extrapolated in the PDF's terminal tail section to encompass the $X1$ range that corresponds to the “full/longer” dataset. The final unscaled results by the Gaidai hazards assessment approach are shown in Fig. 7(b); this comprises the deconvolution-based extrapolation of the «shorter» decimal log scale $fX$ tail and the PDF tail and NG extrapolation of the «full/longer» data.

It can be seen that the NG prediction and the Gaidai risk assessment approach agree fairly well. Figure 7(b) illustrates how well the Gaidai risks assessment approach works, despite being based on a “shorter” dataset and producing a distribution that is fairly similar to that of datasets based on a “full/longer” dataset. Extrapolation results, based on the «shorter» dataset, have been found to differ less than 10% from those based on the «full/longer» dataset, for 100-year return period of interest [12]. Note that a 10% accuracy deviation is well accepted for practical engineering reliability/design purposes. Figure 7(b) presents extrapolations towards a 100-year return period, which corresponds to $\lambda =1$, with $\lambda =0.05$ being the extrapolation cut-on value. $p(\lambda )$ is directly related to the desired failure/hazard probability/risk $1\u2212P$ from Eq. (1) after using Eq. (5). Therefore, $1\u2212P\u22481\u2212Pk(1)$ may be evaluated by Eq. (5) environmental system failure/hazard/damage probability/risk. It should be emphasized that the parameter *N* in Eq. (5) denotes the total number of local maxima in the unified synthetic vector $R\u2192$. Convergence of the conditioning memory depth parameter $k=3$ is convergent. Note that the above-described novel methodology possesses clear advantages of being able to utilize available MC simulated/measured multidimensional datasets efficiently, due to its capacity to handle the multidimensionality of environmental systems and its ability to carry out precise extrapolation from a relatively small underlying dataset. Non-stationary systems with a distinct underlying trend can benefit from the application of the Gaidai hazards assessment approach. The measurements can be brought to a quasi-stationary state by subtracting the underlying pattern, if one is aware of it. If not, appropriate trend analysis would be required in addition to the Gaidai technique for evaluating risks [62–77]. The second-order difference plot is derived from the Poincare plot (SODP). Within the underlying dataset, SODP tracks statistical trends of sequential differences.

Figure 8 presents a second-order SODP plot, which may be used for identification of the underlying dataset's patterns, and comparison with other pertinent datasets, such as entropy AI (i.e., Artificial Intelligence) recognition techniques [78]. Extreme windspeed at an offshore location was predicted using the MCS and MCP (i.e., Measure–Correlate–Predict) approach. Extreme windspeeds computed from the combined PDF acquired by the MCP and MCS approaches agree well with the actual in situ data in a nation with a mixed environment like Japan, when compared to the combined PDF derived by the MCP method alone [79]. As a result, it was shown that in mixed climates, the 50-year recurrent windspeed error of the MCS technique is substantially smaller than that of the MCP approach. Figures 6 and 7 represent the application of the Gaidai risks evaluation method to the raw underlying dataset, i.e., to the unfiltered measured dataset. Note that various extrapolation methods can be plugged into the Gaidai risks evaluation method, hence not limited only to the four-parameter Weibull-type or Gumbel-type methods [80–85]. Recently novel deconvolution extrapolation method has been developed by authors and has been described in the previous section. The deconvolution method's major advantage is the fact that it is not parametric, hence extrapolation is less prone to optimization errors. Note that the advocated multimodal Gaidai risks evaluation method is mathematically exact; hence, inaccuracies may arise either from the underlying dataset itself or from the chosen extrapolation method [86–91].

## 4 Discussion

Presented case study, primarily aimed to further benchmark state-of-the-art Gaidai multivariate reliability methodology, along with novel deconvolution extrapolation scheme. Existing reliability techniques, when applied to raw measured time series, are not always applicable to environmental systems with spatiotemporal complexity and nonlinear cross-correlations between numerous principal components. The primary benefit of the suggested methodology is its capacity to evaluate high-dimensional nonlinear environmental dynamic system's reliability. Presented study analyzed 10-min average windspeeds, measured by NOAA buoys around the Hawaiian Islands in the North Pacific, during the decade 2009–2018. The probability of extreme events with a 100-year return period had been forecasted. A detailed description of the theoretical foundation of the suggested Gaidai hazards assessment technique has been provided. Because environmental systems are highly dimensional and complex, there is a constant need to create precise, inventive, and dependable solutions that make efficient use of the limited number of underlying datasets that are now available [92–99].

Gaidai risks evaluation methodology, presented in this work, has already been used in a range of practical applications, however only for the system's one-dimensional components, yielding on average accurate predictions. The main goal of this study was to develop and enhance a multidimensional, flexible, and dependable methodology for evaluating risks. The advocated methodology may be applied to a variety of environmental nonlinear dynamic system reliability evaluations. Predicted values have been cross-validated versus the alternative extrapolation method, and reasonable agreement has been reported. A primary constraint of the recommended methodology was the joint quasi-stationarity of the system. If there is an underlying trend in the environmental system, it must be identified first.

## Author Contribution Statement

All authors contributed equally.

## Conflict of Interest

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent is not applicable. This article does not include any research in which animal participants were involved.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Abbreviations

- CI =
confidence interval

- CDF =
cumulative distribution function

- EVT =
extreme value theory

- GEV =
generalized extreme value (theory)

- MDOF =
multi-degree-of-freedom (system)

- MUR =
mean up-crossing rate

- NAG =
Numerical Algorithm Group

- NG =
Næss-Gaidai extrapolation scheme (modified four-parameter Weibull)

- NOAA =
National Oceanic and Atmospheric Administration (USA)

- MC =
Monte Carlo (method)

- MCS =
Monte Carlo Simulations

- PDF =
probability density function

- POT =
peaks over the threshold