A Comparative Study of the Statistical Distributions of Wave Heights

Distribution of wave heights and surface elevations of wind-driven waves are studied. Records of surface elevations obtained from both field observations and laboratory measurements are analyzed. Wave heights can be approximated by normal, two-parameter Weibull, and/or Rayleigh distribution. However, while the first two models may have almost equal probabilities to fit measured data quite satisfactorily, the Rayleigh distribution does not appear to be a good model for the majority of the cases studied. Surface elevations from field data are well described by the Gaussian model, but as with increasing wind speeds, water surface in a wind-wave flume deviates from normality, and the Edgeworth/s form of the type A Gram-Charlier series is then applied.

Derivation of a Formula for Mountain Height as a Function of Rank in Height

The relationship between mountain height and rank in height for a mountainous region is examined. A stochastic differential equation model is derived for the evolution of mountain elevations. The derivation is based on simple assumptions about tectonic and erosion processes in mountain elevation dynamics. At any given time, the model yields a CIR-type probability density for mountain heights. As data are often available for mountains of greatest elevation in a region, the tail of the CIR density is studied and compared with mountain height data for the highest mountains in the region. The tail density is proportional to the product of a power of height and an exponential function of height, i.e., hb-1exp(-ah) where h is mountain height and a and b are constants. The inverse distribution function of the tail probability density leads to a formula that relates rank in height to the corresponding mountain height. The formula provides, for example, a decreasing sequence of theoretical mountain heights for the region. The derived formula is tested against mountain height data sets for several mountainous regions in the British Isles, Continental Europe, Northern Africa, and North America. The derived formula provides an excellent fit to the mountain height data ranked by height.

Statistical Analyses of Wave Height Distribution for Multidirectional Irregular Waves over A Sloping Bottom

The main objective of this paper is to examine the influences of both the principal wave direction and the directional spreading parameter of the wave energy on the wave height evolution of multidirectional irregular waves over an impermeable sloping bottom and to propose an improved wave height distribution model based on an existing classical formula.The numerical model FUNWAVE 2.0,based on a fully nonlinear Boussinesq equation,is employed to simulate the propagation of multidirectional irregular waves over the sloping bottom.Comparisons of wave heights derived from wave trains with various principal wave directions and different directional spreading parameters are conducted.Results show that both the principal wave direction and the wave directional spread have significant influences on the wave height evolution on a varying coastal topography.The shoaling effect for the wave height is obviously weakened with the increase of the principal wave direction and with the decrease of the directional spreading parameter.With the simulated data,the classical Klopman wave height distribution model is improved by considering the influences of both factors.It is found that the improved model performs better in describing the wave height distribution for the multidirectional irregular waves in shallow water.

Dependence of Wave Height Distribution on Spectral Width and Wave Steepness

In this paper experimental wind wave data are analyzed. It is found that differences in spectral width will give rise to differences in wave height distribution. The effect of spectral width on the distribution is mainly in the high wave range. The effect of wave steepness is in low, medium and high wave ranges. In the high wave range the effect of spectral width is comparable to that of wave steepness. Differences in spectral width in the observations may give rise to discrepancies in the result when wave steepness is the only parameter in the distribution.

Wave Height Distribution for Spilling Waves in and outside the Surf Zone

The wave characteristics affecting coastal sediment transport include wave height, wave period and breaking wave direction. Wave height is a critical factor in determining the amount of sediment transport in the coastal area. The force of sediment transport is much more intense under breaking waves than under non-breaking waves. Breaking waves exhibit various patterns, principal- ly depending on the incident wave steepness and the beach slope. Based on the equations of con- servation of mass, momentum and energy, a theoretical model for wave deformation in and outside the surf zone was obtained, which is used to calculate the wave shoaling, wave set-up and set- down and wave height distributions in and outside the surf zone. The analysis and comparison were made about the breaking point location and the wave height decay caused by the wave breaking and the bottom friction. Flume experiments relating to the spilling wave height distribution across the surf zone were conducted to verify the theoretical model. Advanced wave maker, data sampling de- vices and data processing system were utilized in the flume experiments with a slope covered by sands of different diameters to facilitate the observation and research on the wave transformation and breaking. The agreement between the theoretical and experimental results is good.

Numerical method for wave height distribution within the artificial harbor with water depth of steep variation

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Long-Period Distribution of Wave Heights and Its Application

This paper reveals that the long-period statistic distribution of the characteristic heights of deep-water waves assumes the lognormal distribution. Thereafter, the largest wave-height which may occur in the service life of coastal structures is derived in this paper.

Effect of wave spectrum width on the probability density distribution of wind-wave heights

The probability distribution of wave heights under the assumption of narrowband linear wave theory follows the Rayleigh distribution and the statistical relationships between some characteristic wave heights, derived from this distribution, are widely used for the treatment of realistic wind waves. However, the bandwidth of wave frequency influences the probability distribution of wave heights. In this paper, a wave-spectrum-width parameter B was introduced into the JONSWAP spectrum. This facilitated the construction of a wind-wave spectrum and the reconstruction of wind-wave time series for various growth stages, based on which the probability density distributions of the wind-wave heights were studied statistically. The distribution curves deviated slightly from the theoretical Rayleigh distribution with increasing B. The probability that a wave height exceeded a certain value was clearly smaller than the theoretical value for B≥0.3, and the difference between them increased with the threshold value. The relation between the Hs/σ ratio and B was investigated statistically, which revealed that the Hs/σ ratio deviated from 4.005 and declined with B. When B reached 0.698 1, the Hs/σ ratio was 3.825, which is about 95.5% of its original value. This indicates an overestimation in the a potential method for improving the accuracy of the Hs extremely large waves under severe sea states. prediction of Hs from Hs=4.005σ, and provides remote sensing retrieval algorithm, critical for

On the long-term joint distribution of characteristic wave height and period and its application

By analysing the scatter diagrams of characteristic the wave height H and the period T on the basis of instrumental data from various ocean wave stations, we found that the conditional expectation and standard deviation of wave period for a given wave height can be better predicted by using the equations of normal linear regression rather than by those based on the log- normal law. The latter was implied in Ochi' s bivariate log-normal model(Ochi. 1978) for the long-term joint distribution of H and T. With the expectation and standard deviation predicted by the normal linear regression equations and applying proper types of distribution, we have obtained the conditional distribution of T for given H. Then combining this conditional P(T / H) with long-term marginal distribution of the wave height P(H) we establish a new parameterized model for the long-term joint distribution P(H,T). As an example of the application of the new model we give a method for estimating wave period associated with an extreme wave height.

Exact Statistical Distribution and Correlation of Human Height and Weight: Analysis and Experimental Confirmation

The statistical relationship between human height and weight is of especial importance to clinical medicine, epidemiology, and the biology of human development. Yet, after more than a century of anthropometric measurements and analyses, there has been no consensus on this relationship. The purpose of this article is to provide a definitive statistical distribution function from which all desired statistics (probabilities, moments, and correlation functions) can be determined. The statistical analysis reported in this article provides strong evidence that height and weight in a diverse population of healthy adults constitute correlated bivariate lognormal random variables. This conclusion is supported by a battery of independent tests comparing empirical values of 1) probability density patterns, 2) linear and higher order correlation coefficients, 3) statistical and hyperstatistics moments up to 6th order, and 4) distance correlation (dCor) values to corresponding theoretical quantities: 1) predicted by the lognormal distribution and 2) simulated by use of appropriate random number generators. Furthermore, calculation of the conditional expectation of weight, given height, yields a theoretical power law that specifies conditions under which body mass index (BMI) can be a valid proxy of obesity. The consistency of the empirical data from a large, diverse anthropometric survey partitioned by gender with the predictions of a correlated bivariate lognormal distribution was found to be so extensive and close as to suggest that this outcome is not coincidental or approximate, but may be a consequence of some underlying biophysical mechanism.

Laboratory Study of the Nonlinear Transformation of Irregular Waves over A Mild Slope

This paper considers the nonlinear transformation of irregular waves propagating over a mild slope （1：40）. Two cases of irregular waves, which are mechanically generated based on JONSWAP spectra, are used for this purpose. The results indicate that the wave heights obey the Rayleigh distribution at the offshore location; however, in the shoaling region, the heights of the largest waves are underestimated by the theoretical distributions. In the surf zone, the wave heights can be approximated by the composite Weibull distribution. In addition, the nonlinear phase coupling within the irregular waves is investigated by the wavelet-based bicoherence. The bicoherence spectra reflect that the number of frequency modes participating in the phase coupling increases with the decreasing water depth, as does the degree of phase coupling. After the incipient breaking, even though the degree of phase coupling decreases, a great number of higher harmonic wave modes are also involved in nonlinear interactions. Moreover, the summed bicoherence indicates that the frequency mode related to the strongest local nonlinear interactions shifts to higher harmonics with the decreasing water depth.

Application of MEP Method to the Study of statistical Properties of Random Waves

The maximum entropy principle (MEP) method and the corresponding probability evaluation method are introduced, and the maximum entropy probability distribution expression is deduced in moment of the second order. Fully developed wave height distribution in deep water and wave height and period distribution for different depths in wind wave channel experiment are obtained from the MEP method, and the results are compared with the distribution and the experimental histogram. The wave height and period distribution for the Lianyungang port is also obtained by the MEP method, and the results are compared with the Weibull distribution and the field histogram.

Exact Statistical Distribution of the Body Mass Index (BMI): Analysis and Experimental Confirmation

Body Mass Index (BMI), defined as the ratio of individual mass (in kilograms) to the square of the associated height (in meters), is one of the most widely discussed and utilized risk factors in medicine and public health, given the increasing obesity worldwide and its relation to metabolic disease. Statistically, BMI is a composite random variable, since human weight (converted to mass) and height are themselves random variables. Much effort over the years has gone into attempts to model or approximate the BMI distribution function. This paper derives the mathematically exact BMI probability density function (PDF), as well as the exact bivariate PDF for human weight and height. Taken together, weight and height are shown to be correlated bivariate lognormal variables whose marginal distributions are each lognormal in form. The mean and variance of each marginal distribution, together with the linear correlation coefficient of the two distributions, provide 5 nonadjustable parameters for a given population that uniquely determine the corresponding BMI distribution, which is also shown to be lognormal in form. The theoretical analysis is tested experimentally by gender against a large anthropometric data base, and found to predict with near perfection the profile of the empirical BMI distribution and, to great accuracy, individual statistics including mean, variance, skewness, kurtosis, and correlation. Beyond solving a longstanding statistical problem, the significance of these findings is that, with knowledge of the exact BMI distribution functions for diverse populations, medical and public health professionals can then make better informed statistical inferences regarding BMI and public health policies to reduce obesity.

Influence of Wave Breaking on Wave Statistics for Finite-Depth Random Wave Trains

The influence of wave breaking on wave statistics for finite-depth random wave trains is investigated experimentally. This paper is to investigate the influence of wave breaking and water depth on the wave statistics for random waves on water of finite depth. Greater attention is paid to changes in wave statistics due to wave breaking in random wave trains. The results show skewness of surface elevations is independent of wave breaking and kurtosis is suppressed by wave breaking. Finally, the exceedance probabilities for wave heights are also investigated.

Computation of Long-Term Statistical Properties of Wave Agitation in Harbors

In this paper, the long-term statistical properties of wave height in an idealized square harbor with a partial opening are studied. The incident waves are propagated into the harbor numerically by the finite/infinite element method using three different wave models: (1) monochromatic wave train, (2) long-crested random wave train, and (3) short-crested random wave train. This study shows that for a given incident wave, the wave height in the harbor is affected by the wave model used. For long-term estimation of wave height exceedance probability, it is recommended that the waves be propagated into the harbor using the random wave model, and that wave heights be computed by use of the Rayleigh probability distribution.