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Seismic properties play a fundamental role in the geological and petrophysical modeling of reservoirs due to their dependence on petrophysical properties. Most existing stochastic seismic inversion methods are based on Gaussian probability distribution functions and assume linear dependence. Examples include sequential Gaussian co-simulation (SGCS) and direct sequential simulation (DSS). In contrast, spatial stochastic co-simulation methods based on Bernstein copulas (BCCS) have recently been developed. These methods do not require a specific distribution type or linear dependence, thereby overcoming the limitations of traditional approaches. In this context, we propose a novel approach for the joint seismic inversion of elastic and petrophysical properties using parametric copulas within a Bayesian inference framework. A joint probability distribution is constructed using well-scale petrophysical and elastic property data, fitted to parametric copula functions and treated as prior information. The model parameters are then updated a posteriori using petrophysical properties scaled by a moving window averaging method and seismic properties upscaled using the Backus averaging method. The resulting posterior model is used within the inversion process to generate elastic property realizations at the seismic scale. The inverse problem is solved using a simulated annealing algorithm that minimizes a global objective function combining the root-mean-square (RMS) error between synthetic and observed seismic traces, and the semivariogram error between the simulated and target variogram models. For each elastic realization, a reflectivity series is computed and convolved with a seismic wavelet to generate a synthetic seismic trace. The best-fitting elastic realization is then used to simulate the corresponding petrophysical property using the same joint probability distribution. The proposed method was applied to a deepwater reservoir case study to estimate total porosity and acoustic impedance at the seismic scale. Results demonstrate that the use of parametric copulas reduces computational cost and execution time while enabling effective integration of nonlinear dependencies. The synthetic traces exhibit RMS errors below 8%, validating the accuracy and robustness of the copula-based inversion framework.

To investigate an appropriate wind load design for buildings considering dynamic air density changes, classical extreme value and copula theories were utilized. Using wind speed, air temperature, and air pressure data from 123 meteorological stations in Shandong Province from 2004 to 2017, a joint probability distribution model was established for extreme wind speed and air density. The basic wind pressure was calculated for various conditional return periods. The results indicated that the Gumbel and Gaussian mixture model distributions performed well in extreme wind speed and air density fitting, respectively. The joint extreme wind speed and air density distribution exhibited a distinct bimodal pattern. The higher the wind speed was, the greater the air density for the same return conditional period. For the 10-year return period, the air density surpassed the standard air density, exceeding 1.30 kg/m3. The basic wind pressures under the different conditional return periods were more than 10% greater than those calculated from standard codes. Applying the air density based on the conditional return period in engineering design could enhance structural safety regionally.

Determining the joint probability distribution of correlated non-normal geotechnical parameters based on incomplete statistical data is a challenging problem. This paper proposes a Gaussian copula-based method for modelling the joint probability distribution of bivariate uncertain data. First, the concepts of Pearson and Kendall correlation coefficients are presented, and the copula theory is briefly introduced. Thereafter, a Pearson method and a Kendall method are developed to determine the copula parameter underlying Gaussian copula. Second, these two methods are compared in computational efficiency, applicability, and capability of fitting data. Finally, four load-test datasets of load-displacement curves of piles are used to illustrate the proposed method. The results indicate that the proposed Gaussian copula-based method can not only characterize the correlation between geotechnical parameters, but also construct the joint probability distribution function of correlated non-normal geotechnical parameters in a more general way. It can serve as a general tool to construct the joint probability distribution of correlated geotechnical parameters based on incomplete data. The Gaussian copula using the Kendall method is superior to that using the Pearson method, which should be recommended for modelling and simulating the joint probability distribution of correlated geotechnical parameters. There exists a strong negative correlation between the two parameters underlying load-displacement curves. Neglecting such correlation will not capture the scatter in the measured load-displacement curves. These results substantially extend the application of the copula theory to multivariate simulation in geotechnical engineering.

Transmission system operator (TSOs) need to project the system state at the seasonal scale to evaluate the risk of supply-demand imbalance for the season to come. Seasonal planning of the electricity system is currently mainly adressed using climatological approach to handle variability of consumption and production. Our study addresses the need for quantitative measures of the risk of supply-demand imbalance, exploring the use of sub-seasonal to seasonal forecasts which have hitherto not been exploited for this purpose. In this study, the risk of supply-demand imbalance is defined using exclusively the wind energy production and the consumption peak at 7 pm. To forecast the risks of supply-demand imbalance at monthly to seasonal time horizons, a statistical model is developed to reconstruct the joint probability of consumption and production. It is based on a the conditional probability of production and consumption with respect to indexes obtained from a linear regression of principal components of large-scale atmospheric predictors. By integrating the joint probability of consumption and production over different areas, we define two kind of risk measures: one quantifies the probablity of deviating from the climatological means, while the other, which is the value at risk at 95% confidence level (VaR95) of the difference between consumption and production, quantifies extreme risks of imbalance. In the first case, the reconstructed risk accurately reproduces the actual risk with over 0.80 correlation in time, and a hit rate around 70–80%. In the second case, we find a mean absolute error (MAE) between the reconstructed and real extreme risk of 2.5 to 2.8 GW, a coefficient of variation of the root mean square error (CV-RMSE) of 3.8% to 4.2% of the mean actual VaR95 and a correlation of 0.69 and 0.66 for winter and fall, respectively. By applying our model to ensemble forecasts performed with a numerical weather prediction model, we show that forecasted risk measures up to 1 month horizon can outperform the climatology often used as the reference forecast (time correlation with actual risk ranging between 0.54 and 0.82). At seasonal time horizon (3 months), our forecasts seem to tend to the climatology.

The method of the joint probability distribution function was applied in order to estimate the normal structure factor amplitudes of the anomalous scatterer substructure in a FEL experiment. The two-wavelength case was examined. In this, the prior knowledge of the moduli | F 1 + | , | F 1 − | , | F 2 + | , | F 2 − | was used to predict the value of | F 0 a | , which is the structure factor amplitude arising from the normal scattering of the heavy atom anomalous scatterers. The mathematical treatment provides a solid theoretical basis for the RIP (Radiation-damage Induced Phasing) method, which was originally proposed in order to take the radiation damage induced by synchrotron radiation sources into account. This was further adapted to exploit FEL data, where the crystal damage is usually more massive.

In the present study, a total of 360 FE analyses were carried out on tubular X-joints strengthened with collar plates under brace tension under laboratory testing conditions (20 °C) and various fire conditions. The generated FE models were validated based on 31 tests. The FE analyses produced a comprehensive dataset that encapsulated resistance metrics, with detailed simulations of welds, contacts, and the incorporation of non-linear geometrical and material attributes. Twelve theoretical probability density functions (PDFs) were matched to the constructed histograms, with the maximum likelihood (ML) technique utilized to assess the parameters of these fitted PDFs. The theoretical PDFs, rigorously evaluated against the Anderson–Darling, Kolmogorov–Smirnov, and Chi-squared tests, identified the Generalized Petrov distribution as the optimal model for capturing the resistance behaviors of X-joints under tensile load and varying fire conditions. The findings have led to the proposition of five detailed theoretical PDFs and cumulative distribution functions (CDFs), introducing a novel perspective for assessing and reinforcing the structural resilience of strengthened CHS X-joints in engineering practices.

The determination of the so-called design block is one of the central elements of the Austrian guideline for rockfall protection ONR 24810. It is specified as a certain percentile (P95–P98, depending on the event frequency) of a recorded block size distribution. Block size distributions may be determined from the detachment area (in situ block size distribution) and/or from the deposition area (rockfall block size distribution). Deposition areas, if present, are generally accessible and measurable without technical aids. However, most measuring methods are subjective, uncertain, not verifiable, or inaccurate. Also, rockfall blocks are often fragmented due to the preceding fall process. The in situ block size distribution is (also) required for meaningful rockfall modelling. The statistical method seems to be the most efficient and cost-effective method to determine in situ block size distributions with many blocks within the whole range of block sizes. In the current literature, joint properties are often described by the lognormal and exponential distribution functions. Today, we can model synthetic rock masses on the basis of discrete fracture networks. They statistically describe the geometric properties of the joint sets. This way, we can carry out exact rock mass block surveys and determine in situ block size distributions. We wanted to know whether the in situ block size distributions derived from the synthetic rock mass models can be described by probability distribution functions, and if so, how well. We fitted various distribution functions to three determined in situ block size distributions of different lithologies. We compared their correlations using the Kolmogorov–Smirnov test and the mean-squared error method. We show that the generalized exponential distribution function best describes the in situ block size distributions across various lithologies compared to 78 other distribution functions. This could lead to more certain, accurate, verifiable, holistic, and objective results. Further investigations are required.

Drought is one type of natural disaster that impacts huge areas over an extended period. Multi-variable indices have been established to evaluate numerous factors at once and obtain more information regarding drought. In this study, a combined Joint Multivariable Index (CJMI) was defined for Iran based on three indices: SPI, PDSI, and SRI, covering the period from 1990 to 2021, aiming to consider all aspects of drought. By examining various marginal distribution functions (such as Log-normal, Logistic, Exponential, Weibull, and Gamma), the most suitable distribution function was determined based on two criteria: AIC and BIC. The best-fit distribution function for most stations was found to be log-normal and exponential, while the gamma distribution function was optimal only for the Mashhad station with AIC = 503.24 and BIC = 513 Subsequently, return periods (AND) and joint risk of the trivariate were evaluated. Return periods and joint risk analysis of the three variables were compared based on three copula functions: Frank, Gumbel, and Clayton, for periods of 50 and 100 years. The obtained values for risk analysis were categorized into three levels, with the highest risk values observed in the northwestern region and stations such as Ardabil, Urmia, Tabriz, and Tehran exceeding 0.70. The overall results of this research indicate that the CJMI index obtained contains more uncertainties than individual indices and better analyzes the primary data approach.

This study introduces methodologies for constructing joint probability distribution functions utilizing the Copula function and neural networks, and evaluates their efficacy in marine and civil engineering projects. Through an analytical comparison of both models using a numerical example, it is revealed that the neural network model exhibits superior adaptability to large sample sizes. This adaptability is attributed to the neural network's ability to learn complex relationships within the data, which is especially beneficial when dealing with large datasets. The neural network model also demonstrates higher accuracy in constructing joint probability distribution functions compared to the Copula function model. In marine and civil engineering, the adaptability and accuracy of neural networks are of paramount importance due to the variable and complex nature of weather patterns. A practical engineering application is presented, wherein a joint probabilistic distribution neural network model of wind velocity and rain intensity is established for the Lanzhou–Xinjiang high-speed railroad in China. This model illustrates the promising application of neural networks in engineering projects where weather factors play a critical role. Subsequent to the construction of the joint probability distribution functions, a feature importance analysis is incorporated to quantify the contribution of different weather parameters such as wind velocity and rain intensity to the joint distribution function. This analysis provides an objective assessment of the relative importance of various weather factors and offers data-driven insights that are essential for engineering applications where weather conditions are a significant consideration. The study concludes by highlighting the potential benefits of neural network models in marine and civil engineering, suggesting areas for future exploration.

Accurate and reliable assessment of wind energy potential has important implication to the wind energy industry. Most previous studies on wind energy assessment focused solely on wind speed, whereas the dependence of wind energy on wind direction was much less considered and documented. In this paper, a copula‐based method is proposed to better characterize the direction‐related wind energy potential at six typical sites in Hong Kong. The joint probability density function (JPDF) of wind speed and wind direction is constructed by a series of copula models. It shows that Frank copula has the best performance to fit the JPDF at hilltop and offshore sites while Gumbel copula outperforms other models at urban sites. The derived JPDFs are applied to estimate the direction‐related wind power density at the considered sites. The obtained maximum direction‐related wind energy density varies from 41.3 W/m2 at an urban site to 507.9 W/m2 at a hilltop site. These outcomes are expected to facilitate accurate micro‐site selection of wind turbines, thereby improving the economic benefits of wind farms in Hong Kong. Meanwhile, the developed copula‐based method provides useful references for further investigations regarding direction‐related wind energy assessments at various terrain regions. Notably, the proposed copula‐based method can also be applied to characterize the direction‐related wind energy potential somewhere other than Hong Kong.