Exceedance Probability Distribution Research Articles

Modeling Riverbed Elevation and Bedload Tracer Transport Resting Times Using Fractional Laplace Motion

AbstractRiverbed elevations play a crucial role in sediment transport and flow resistance, making it essential to understand and quantify their effects. This knowledge is vital for various fields, including river engineering and stream ecology. Previous observations have revealed that fluctuations in the bed surface can exhibit both multifractal and monofractal behaviors. Specifically, the probability distribution function (PDF) of elevation increments may transition from Laplace (two‐sided exponential) to Gaussian with increasing scales or consistently remain Gaussian, respectively. These differences at the finest timescale lead to distinct patterns of bedload particle exchange with the bed surface, thereby influencing particle resting times and streamwise transport. In this paper, we utilize the fractional Laplace motion (FLM) model to analyze riverbed elevation series, demonstrating its capability to capture both mono‐ and multi‐fractal behaviors. Our focus is on studying the resting time distribution of bedload particles during downstream transport, with the FLM model primarily parameterized based on the Laplace distribution of increments PDF at the finest timescale. Resting times are extracted from the bed elevation series by identifying pairs of adjacent deposition and entrainment events at the same elevation. We demonstrate that in cases of insufficient data series length, the FLM model robustly estimates the tail exponent of the resting time distribution. Notably, the tail of the exceedance probability distribution of resting times is much heavier for experimental measurements displaying Laplace increments PDF at the finest scale, compared to previous studies observing Gaussian PDF for bed elevation.

Bathymetric Modelling of High Mountain Tropical Lakes of Southern Ecuador

Very little is known on high mountain tropical lakes of South America. Thus, the main motivation of this research was obtaining base bathymetric data of 119 tropical lakes of the Cajas National Park (CNP), Ecuador, that could be used in future geomorphological studies. Eleven interpolation methods were applied with the intention of selecting the best one for processing the scattered observations that were collected with a low-cost fishing echo-sounder. A split-sample (SS) test was used and repeated several times considering different proportions of available observations, selected randomly, for training of the interpolation methods and accuracy evaluation of the respective products. This accuracy was assessed through the use of empirical exceedance probability distributions of the mean absolute error (MAE). A single best interpolation method could not be identified. Instead, the study suggested six better-performing methods, including the complex methods Kriging (ordinary), minimum curvature (spline), multiquadric, and TIN with linear interpolation but also the much simpler methods natural neighbour and nearest neighbour. A sensitivity analysis (SA), considering several data error magnitudes, confirmed this. This advocated that sophisticated interpolation methods do not always produce the best products as geomorphological characteristics of the study site(s) together with observation data characteristics are likely to play important roles in their performance. As such, this type of assessment should be carried out in any terrestrial mapping of bathymetry that is based on the interpolation of scattered observations. Upon the analysis of the relative hypsometric curves of the 119 study lakes, they were classified into three average form categories: convex, concave, and mixed. The separated accuracy analysis of these three groups of lakes did not help in identifying a single best method. Finally, the interpolated bathymetries of 114 of the study lakes were incorporated into the best DEM of the study site by equalising their elevation reference systems. It is believed that the resulting enhanced DEM could be a very useful tool for a more appropriate management of these very beautiful but fragile high mountain tropical lakes.

An effective two-stage optimization method on aerodynamic measures to mitigate wind loads considering uncertainties in wind direction and velocity

At present, deterministic optimization methods on aerodynamic measures are usually employed to mitigate wind loads at one certain wind direction, ignoring the effects of the uncertainties in wind direction and velocity on wind loads. The uncertain optimization method with the consideration of the joint probability distribution (JPD) of wind speed and direction is hardly to conduct because it is very time-consuming and costly. In order to overcome these shortcomings, this study proposes an effective two-stage uncertain optimization method considering uncertainties in wind direction and velocity to achieve a considerable reduction of wind loads. In the first stage, the exceedance probability distribution of wind loads is determined based on the JPD and the wind load coefficients on the building, and the wind loads in the most unfavorable range of wind directions are chosen as the optimization objective. In the second stage, the optimal values of the design variables are determined by an interval optimization algorithm using the surrogate model and non-dominated sorting genetic algorithm II in the most unfavorable range of wind directions, where the surrogate model is constructed by the radial basis function method to relate the wind load coefficients to the design variables of aerodynamic measures, and can significantly improve the efficiency of the optimization algorithm. To validate the effectiveness of the proposed method, a flat-roof building with four spoilers is investigated and the rotation angles of spoilers are set as the design variables. The results show that the proposed method reduces the uplift wind loads with a 50-years return period up to 15.48% compared to the deterministic optimization method for one certain wind direction.

Hypersonic Boundary-Layer Transition on Cold-Wall Canonical Geometries with Quantified Distributed Roughness Elements

Given the challenges of obtaining a natural turbulent boundary layer on common test model geometries in a shock tunnel, this work aims to investigate the influence of roughness elements on the boundary layer with respect to transition to turbulence. The experiments were conducted in the High Enthalpy Shock Tunnel Göttingen at the German Aerospace Center on a 1100-mm-long, 7°-half-angle cone and a 602-mm-long flat plate at Mach 7.4. Roughness elements were applied on the nosetip of the cone and near the leading edge of the flat plate. The roughness elements were scanned with a laser profilometer, allowing their specification in terms of a roughness Reynolds number based on the 70th-percentile element height and an exceedance probability distribution. Transition was examined for the cone geometry using streamwise-aligned coaxial thermocouples on the 0° meridian. This assisted sizing the roughness elements required for transition to occur as far upstream as detectable. Breakdown of roughness-induced vortical structures generated by the roughness elements with a similar roughness Reynolds number was then examined using the flat plate geometry with temperature-sensitive paint applied downstream of the roughness elements. It was found that roughness-induced vortices required a finite distance (persistence length) to break down into turbulent structures. The persistence length was successfully reduced by interspersing roughness elements with smaller ones.

Experimental investigation of breaking regular and irregular waves slamming on an offshore monopile wind turbine

A series of experiments were carried out to investigate the breaking regular and irregular wave forces on a monopile foundation in a wave flume with a model scale factor of 1:40. Breaking regular wave slamming characteristic was firstly analyzed and comparisons were made between typical plunging and spilling breakers. For the irregular wave conditions, the wave slamming processes were recorded to illustrate the concentration and dissipation of wave energy. The effects of bottom slope, water depth, significant wave height and peak period on the wave slamming force and pressure distribution on the monopile were investigated. The results show that the distribution of maximum wave slamming pressure is affected by the water depth and bottom slope. Multiple harmonic components of the wave slamming force are observed and the predominant harmonic forces are first- and second- order components. The significant wave height and turbulence kinetic energy dissipation have significant effects on the wave slamming forces. A Gaussian distribution formula is proposed to fit the exceedance probability distribution of wave slamming force. Modified empirical formulas are proposed to calculate the wave slamming coefficient for the breaking regular and irregular waves on the monopile.

The Rayleigh-Haring-Tayfun distribution of wave heights in deep water

Nearly every known exceedance probability distribution applied to rogue waves has shown disagreement with its peers, suggesting that the underlying phenomenon may be too complex to grasp within a single distribution. In recent studies, this view has been supported by the realisation that there is no distribution suitable for wide ranges of sea states, even though distributions can perform well in a limited and small number of conditions. Towards the better quantitative assessment of rogue wave occurrence, the present work focuses on the apparent discrepancy among the majority of previous works. Furthermore, on the grounds of the uneven distribution of rogue waves found in North Sea observations, a new exceedance probability distribution for rogue waves is introduced. The proposed distribution is a geometrical composition of some of the most popular simple models for waves while introducing additional algebraic structures. The model also obeys empirical physical bounds obtained from the analysis of nearly 350,000 waves from storms recorded in North sea and supports the qualitative analysis of rogue wave occurrence based on the interaction among three sea state parameters.

The Wave Heights Distribution of Random Wave Based on Ocean Basin

The wave height parameter in ocean waves is one of the important information for a marine structure design. The present paper investigates the results of wave heights distribution from laboratory-generated for single sea state. Data of the random wave time series collected at the ocean basin are analyzed using the wave spectrum and compared with the theoretical spectrum in this study. The random wave data is varied with four sea states consisting of sea states 3, 4, 5 and 6 obtained from laboratory measurements. The parameter conditions of generated sea waves are represented by a value of significant wave height and wave peak period in the range of sea states. The individual wave heights data in each sea state are presented in the form of exceedance probability distribution and the predictions using a linear model. This study aims to estimate the wave heights distribution using the Rayleigh and Weibull distribution model. Furthermore, the accuracy of the wave heights distribution data's prediction results in each sea state has been compared and examined for both models. The applied linear models indicate similar and reasonable estimations on the observed data trends.

Probabilistic modeling of fatigue crack growth and experimental verification

Stochastic fatigue crack growth (FCG) modeling is vital for fatigue reliability and durability analyses of engineering components. To characterize the statistical properties of FCG process, the Yang–Manning model was introduced to calculate the crack exceedance probability and fatigue life distribution to reach any crack size; however, it is only applicable for failure probability analyses of pre-cracking specimens. According to this, a new probabilistic fatigue life model considering the distribution of initial crack sizes was proposed. Experimental data of Al 2024-T3 alloy and 30NiCrMoV12 steel were utilized for model validation and comparison. Results show that the proposed probabilistic model works well for FCG analysis with independent cracks. In addition, the Yang–Manning model was extended to consider the crack coalescence issue under multiple cracks condition, which shows a sound conformity to experimental data.

Analysis of Discharge Variability in the Naryn River Basin, Kyrgyzstan

Changing climate and land-use practices influencing the natural stream flow processes in the Naryn river basin of Kyrgyzstan. Variations in stream flow regime over 33-years (1980 to 2012) were investigated using daily discharge data of three hydro-stations (Naryn, Ych-Terek and Uzunakmat), located in the Naryn River Basin. Mean monthly discharge (MMD), mean annual discharge (MAD), standard deviation (SD) and coefficient of variation (CV) were calculated to know the spatio-temporal variability. Similarly, Pearson’s correlation coefficient (r) was used to know the relationship between discharge and rainfall. Advanced time-series graph, exceedance probability and frequency distribution were computed using Hydrognomon (V.4.0.3) software to observe the variability and trends in discharge. The results from statistical calculations and software-based computations highlight the monthly, annual, and long term spatio-temporal discharge variability, extreme events, distribution and changes in stream flow records. This study preciously creates the frequency and trends of seasonal discharge, annual discharge cycle, and range of highest and lowest discharge flows. The weak and negative relationship (-0.2121, -0.4238) between rainfall and discharge propose for more investigation of climatic parameters and the topography of Tian Shan Mountain perhaps influencing discharge variability due to melting of glacier at high altitude. The flow regime of the Naryn river basin over the past 33-years perhaps changed due to climatic fluctuations, with the seasonal snowmelt timing (Post-Spring, Summer, Pri-Autumn), precipitations period (March-October), and large-scale land-use alterations.

A parameterization of DNV-GL storm profile for the calculation of design wave of marine structures

This paper proposes a parameterization of the Det Norske Veritas and Germanischer Lloyd (DNV GL) storm profile for the development of analytical solution for the calculation of the associated return period. This solution allows the possibility of calculating the design wave for marine structures by means of a very simple approach. The basic concept is to substitute the sequence of actual storms at a given site with a sequence of trapezoidal storms (TS) defined on the basis of the DNV GL storm temporal evolution and derive analytically the return period referring to the TSs sequence. The DNV GL storm is represented by a trapezium which may be totally defined by means of three parameters: the trapezium height related to the storm intensity, and its minor and major bases, which represent two durations Dp, D* pertaining to the maximum significant wave height Hsmax and to 50% of Hsmax, respectively. It is assumed that the parameters Dp, D* are uncorrelated to the storm intensity and are the same for all the storms. With this assumption, the return period is obtained in a closed form as a function of the ratio n between Dp and D*, D* itself, and of the exceedance probability distribution of significant wave height. The model is validated by comparison with Equivalent Triangular Storm (ETS) model by processing buoys data at four locations representative of different wave climates: Haltenbanken in North Sea, Alghero in Mediterranean Sea, stations 41001 and 46006 in Atlantic and Pacific Oceans, respectively. Results reveal that although its significant simplicity, the TS model provides reliable estimate of return values which are in agreement with those achieved by ETS model. Further, the employment of TS model with the values of parameters n and D* corresponding to the DNV GL profile is suitable for any location.

Information-theoretic portfolio decision model for optimal flood management

The increasing impact of flooding urges more effective flood management strategies to guarantee sustainable ecosystem development. Recent catastrophes underline the importance of avoiding local flood management, but characterizing large scale basin wide approaches for systemic flood risk management. Here we introduce an information-theoretic Portfolio Decision Model (iPDM) for the optimization of a systemic ecosystem value at the basin scale by evaluating all potential flood risk mitigation plans. iPDM calculates the ecosystem value predicted by all feasible combinations of flood control structures (FCS) considering environmental, social and economical asset criteria. A multi-criteria decision analytical model evaluates the benefits of all FCS portfolios at the basin scale weighted by stakeholder preferences for assets’ criteria as ecosystem services. The risk model is based on a maximum entropy model (MaxEnt) that predicts the flood susceptibility, the risk of floods based on the exceedance probability distribution, and its most important drivers. Information theoretic global sensitivity and uncertainty analysis is used to select the simplest and most accurate model based on a flood return period. A stochastic optimization algorithm optimizes the ecosystem value constrained to the budget available and provides Pareto frontiers of optimal FCS plans for any budget level. Pareto optimal solutions maximize FCS diversity and minimize the criticality of floods manifested by the scaling exponent of the Pareto distribution of flood size that links management and hydrogeomorphological patterns. The proposed model is tested on the 17,000 km2 Tiber river basin in Italy. iPDM allows stakeholders to identify optimal FCS plans in river basins for a comprehensive evaluation of flood effects under future ecosystem trajectories.

Experimental study on sloshing reduction effects of baffles linked to a spring system

The liquid sloshing phenomenon which damages the structures of various systems and affects their global motion of the system is one of the most important problems in naval architecture and ocean engineering. Thus far, several efforts have been attempted to reduce the fluid impact load induced by sloshing, such as the installation of additional equipment like baffles to avoid the resonance period with the tank or to restrain violent sloshing flow. In this paper, a concept of “moving baffles” is introduced, which is containing a spring system as one of the attempts for reducing sloshing impact. To observe the sloshing reduction effect of the moving baffles, a series of sloshing experiments considering the sway motion of a rectangular tank was conducted, and the experimental results for the case without any additional equipment were compared to those for the case with the baffles linked to two types of spring systems with different stiffness. The image, pressure, and force data were obtained through the sloshing experiments, and the effectiveness of the spring baffles was examined by comparing the statistical analysis results of the extremal probability and exceedance probability distribution.

Health protection planning for extreme weather events and natural disasters.

There is a need to develop cost-effective methods to support public health policy makers plan ahead and make robust decisions on protective measures to safeguard against severe impacts of extreme weather events and natural disasters in the future, given competing demands on the social and healthcare resources, large uncertainty associated with extreme events and their impacts, and the opportunity costs associated with making ineffective decisions. The authors combine a physics-based method known as nonextensive statistical mechanics for modeling the probability distribution of systems or processes exhibiting extreme behavior, with a decision-analytical method known as partitioned multiobjective risk method to determine the optimal decision option when planning for potential extreme events. The method is illustrated using a simple hypothetical example. It is shown that partitioning the exceedance probability distribution of health impact into three ranges (low severity/high exceedance probability, moderate severity/medium exceedance probability, and high severity/low exceedance probability) leads to the correct estimation of the conditional expected impact in each range. Multiobjective optimization is used to determine the optimal decision option based on the perspective of the policy maker. This method constitutes a robust generic framework for the quantification of impacts and supporting decision-making under scenarios of extreme and catastrophic health risks.

Detailed measurements of interfacial dynamics in air-water pipe flow

Stratified air-water flow in a horizontal pipe is investigated experimentally using particle image velocimetry and conductance probes. This flow regime is characterized by a complex interplay between a turbulent airflow and propagating waves at the interface. The waves are generated by interfacial shear and pressure forces exerted by the faster flowing airflow. The goal of this study is to characterize the waves by means of statistical and spectral methods, and to explore the influence of different wave regimes on the airflow.Two cases in which the air bulk velocity increases from 2.4 m/s (case A) to 3.5 m/s (case B), while the liquid velocity remains constant at 0.26 m/s, are assessed in detail. Case A belongs to a region of flow conditions in which wave amplitudes grow as a consequence of increasing gas flow rates, i.e., wave growth regime. Meanwhile, case B is in a regime of saturated wave amplitudes. In the first case, the interface was populated by small amplitude 2D waves of relatively small steepness (ak ≈ 0.07). These waves obey Gaussian statistics and are thus considered to be linear. In the second case, the waves are larger, steeper (ak ≈ 0.13) and considerably more irregular. They display non-linear behaviour (steep crests and long troughs) and their exceedance probability distribution deviates significantly from Gaussian statistics. Bicoherence maps show evidence of both overtone and sub-harmonic interactions.Airflow velocity fields acquired by PIV were subjected to a conditional phase-averaging method based on a steepness criterion. The phase-averaged vorticity field shows evidence of shear-layer separation above the steeper waves of case B. Hence, in addition to non-linear mode interactions and micro-breaking, shear-layer separation may contribute to the transition from the growth regime to the saturation regime.

Functional Topology of Evolving Urban Drainage Networks

AbstractWe investigated the scaling and topology of engineered urban drainage networks (UDNs) in two cities, and further examined UDN evolution over decades. UDN scaling was analyzed using two power law scaling characteristics widely employed for river networks: (1) Hack's law of length (L)‐area (A) [ ] and (2) exceedance probability distribution of upstream contributing area (δ) [ ]. For the smallest UDNs (<2 km2), length‐area scales linearly (h ∼ 1), but power law scaling (h ∼ 0.6) emerges as the UDNs grow. While plots for river networks are abruptly truncated, those for UDNs display exponential tempering [ ]. The tempering parameter c decreases as the UDNs grow, implying that the distribution evolves in time to resemble those for river networks. However, the power law exponent ɛ for large UDNs tends to be greater than the range reported for river networks. Differences in generative processes and engineering design constraints contribute to observed differences in the evolution of UDNs and river networks, including subnet heterogeneity and nonrandom branching.

Comparative study of wet channel network extracted from LiDAR data under different climate conditions

Abstract Temporal streams are vitally important for hydrology and riverine ecosystems. The identification of wet channel networks and spatial and temporal dynamics is essential for effective management, conservation, and restoration of water resources. This study investigated the temporal dynamics of stream networks in five watersheds under different climate conditions and levels of human interferences, using a systematic method recently developed for extracting wet channel networks based on light detection and ranging elevation and intensity data. In this paper, thresholds of canopy height for masking densely vegetated areas and the ‘time of forward diffusion’ parameter for filtering digital elevation model are found to be greatly influential and differing among sites. The inflection point of the exceedance probability distribution of elevation differences in each watershed is suggested to be used as the canopy height threshold. A lower value for the ‘time of forward diffusion’ is suggested for watersheds with artificial channels. The properties of decomposed and composite probability distribution functions of intensity and the extracted intensity thresholds are found to vary significantly among regions. Finally, the wet channel density and its variation with climate for five watersheds are found to be reasonable and reliable according to results reported previously in other regions.

Hurricane risk assessment for offshore wind plants

This study describes a framework for estimation of structural damage to offshore wind plants during hurricanes. Related risk assessment is fundamentally dependent on the estimation of hurricane-generated wind speed exceedance probabilities at selected hub heights of wind turbines in the plant and on estimation of associated wind turbine loads. As part of a framework for risk assessment introduced here, synthetic storm tracks are first simulated over the ocean using available historical tropical storm data; then, a hurricane intensity evolution model based on thermodynamic and atmospheric environmental variables is developed for each simulated track as it approaches the chosen wind plant site. Based on this intensity model, a turbulent wind field can be simulated at any location of interest along the hurricane track. The simulated turbulent wind field can then be used to estimate wind speed exceedance probability distributions and, when combined with partially correlated waves, it can also be used in analyzing the response of individual turbines in a wind plant. A framework for the overall risk assessment is presented; individual components that are part of such a framework are described briefly in illustrative applications. Finally, a brief discussion is presented that addresses issues related to the concept of “robustness” checks that are often considered in the context of safe performance-based design. This concept employed in the oil and gas industries provides for a secondary margin for “beyond design-level” external conditions out to more extreme conditions as might accompany hurricanes.

Shallow water effects on high order statistics and probability distributions of wave run-ups along FPSO broadside

Ship hydrodynamics in shallow water becomes especially complicated since the nonlinearities in both the incident waves and the wave–hull interactions will be affected by the water depth. For a ship-shaped Floating Production, Storage and Offloading unit (FPSO) operating in shallow water, the broadside often suffers from the wave run-up and green water incidents in non-collinear harsh ocean environments. By applying the methods of ordinary moments and L-moments and the empirical Weibull distribution on the data measured in a series of model experiments, the high order statistics and the exceedance probability distribution of the run-ups along the FPSO broadside are evaluated and the effects of the shallow water depth and the incident environments are analyzed in this paper. It is seen that both the incident waves and the wave run-ups are non-Gaussian in shallow water and that the wave run-up characteristics are significantly influenced by the water depth and the incident environments, while the contribution due to the vessel vertical motions is negligible for the FPSO used in this study. The exceedance probabilities of the wave run-ups show that the broadside will be more likely to suffer from serious wave run-up and green water incidents in shallower water, in a higher incident wave and a non-collinear environment, especially so at locations around the FPSO midship within a range of 3/8Lpp ∼ 5/8Lpp. The dependency of the shape and scale parameters of the wave run-up probability distributions on the locations and the environment is quantified by model tests. The present study leads to the conclusion that the wave run-up characteristics and the shallow water effects should be considered carefully in determining the wave loads and the freeboard of a large FPSO in non-collinear environment conditions.

Geomorphologic Characteristics of Indiana Basins

In this study, ten Indiana basins were selected to investigate their geomorphologic characteristics. The drainage area of these basins covers a wide range. Stream networks of these basins are classified according to both Strahler’s and Shreve’s ordering systems. Based on these ordering systems, more than twenty primary and secondary geomorphologic parameters were determined for each study basin. Horton’s laws of stream numbers, stream length and stream area were checked and found valid for these Indiana basins. Values of cumulative drainage area were used to investigate the exceedance probability distribution of mean annual discharge and energy fluxes in the study watersheds. The results of the study show that discharge and energy fluxes were found to follow power law distributions. Exponent of the resulting power laws does not show the universality suggested by previous investigations.

Sedimentary bed evolution as a mean‐reverting random walk: Implications for tracer statistics

AbstractSediment tracers are increasingly employed to estimate bed load transport and landscape evolution rates. Tracer trajectories are dominated by periods of immobility (“waiting times”) as they are buried and reexcavated in the stochastically evolving river bed. Here we model bed evolution as a random walk with mean‐reverting tendency (Ornstein‐Uhlenbeck process) originating from the restoring effect of erosion and deposition. The Ornstein‐Uhlenbeck model contains two parameters, a and b, related to the particle feed rate and range of bed elevation fluctuations, respectively. Observations of bed evolution in flume experiments agree with model predictions; in particular, the model reproduces the asymptotic t−1 tail in the tracer waiting time exceedance probability distribution. This waiting time distribution is similar to that inferred for tracers in natural gravel streams and avalanching rice piles, indicating applicability of the Ornstein‐Uhlenbeck mean‐reverting model to many disordered transport systems with tracer burial and excavation.