Pierson-Moskowitz Spectrum Research Articles

A computer simulation of relative backscattering coefficients of sea surfaces in satellite altimetry

AbstractThe purpose of this study is to establish a method for estimating the sea wave period by satellite radar altimetry. The backscatter coefficient, which is closely related to the period, is evaluated by computer simulation. Specifically, the following method is presented. The sea wave is modeled on the basis of the Pierson–Moskowitz spectrum. The average received power is determined by radar altimetry simulation of our proposal. It is shown that the backscatter coefficient of the sea surface can be calculated on a relative basis. Using the data for the relative received power calculated from various sea wave parameters, the relation between the significant wave height and the backscatter coefficient, and also the relation between the significant wavelength and the backscatter coefficient, are derived. Thus, the possibility of observing the sea wave period by using satellite radar altimetry is demonstrated. © 2007 Wiley Periodicals, Inc. Electron Comm Jpn Pt 2, 90(12): 121–128, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjb.20428

Doppler-shift approximations of encountered wave statistics

In this paper the joint density of waveheight and half-wavelength is considered for waves observed at a fixed time point and encountering waves that are overtaking a ship from behind. The densities for these two cases are related by a Doppler shift, expressed in terms of the relative velocity of the waves and the ship. Based on this observation, an approximation of the encountered density is proposed. This approximation is then investigated for a Gaussian sea having a Pierson–Moskowitz spectrum.

A Method for Requiring Block Size for Spectrum Measurement of Ocean Surface Waves

Spectrum measurement of field data of ocean surface waves is essential for the performance research, analysis and design of ship structure, and laboratory testing. There are two problems in measuring a power spectrum of wave data. One is how to estimate the upper bound of the mean-square error of a spectrum measurement of a block of wave data at a given wind speed (sea state). The other is how to quantitatively require the lower bound of the block size according to a given wind speed and a given degree of the mean-square error of spectrum measurement of a block of wave data. This paper presents four theorems and two corollaries with respect to the solutions to the above two problems. Theorems 1-4 describe the general solutions to the above problems based on two general spectral models of wave data. Corollaries 1 and 2 show the specific solutions to the problems based on the Pierson-Moskowitz spectrum. With Corollaries 1 and 2, the effect of large wind speed on the spectrum measurement of wave data is analyzed. The significance of the present results is examined and demonstrated by spectrum measurement of real field data of ocean waves

Doppler Spectra From a Two-Dimensional Ocean Surface at L-Band

An approximate time-harmonic three-dimensional electromagnetic boundary-integral method, the small-slope integral equation, is combined with a series expansion of the Creamer surface representation at second order with respect to the height, denoted by Creamer (2). The resulting model provides at low numerical cost simulations of the nonlinear ocean surface Doppler spectrum at L-band. As a result of approximations, the model is designed for a low-wind speed, typically up to 5 m/s. It is shown that applying directly a second-order model such as Creamer (2) to a semiempirical sea surface spectrum induces an unrealistic magnification of small-scale roughness that is involved in the scattering process at microwave frequencies. This paper thus proposes an undressed version of the Pierson-Moskowitz spectrum that corrects this artifact. Full-polarized Doppler simulations at L-band and 70deg incidence are presented. Effects of the surface nonlinearities are outlined, and the simulated Doppler spectra show correct variations with respect to wind speed and direction

Comprehensive study on the results of tension leg platform responses in random sea

Compliant offshore structures are used for oil exploitation in deep water. Tension leg platform (TLP) is a suitable type for very deep water. The nonlinear dynamic response of TLP under random sea wave load is necessary for determining the maximum deformations and stresses. Accurate and reliable responses are needed for optimum design and control of the structure. In this paper nonlinear dynamic analysis of TLP is carried out in both time and frequency domains. The time history of random wave is generated based on Pierson-Moskowitz spectrum and acts on the structure in arbitrary direction. The hydrodynamic forces are calculated using the modified Morison equation according to Airy's linear wave theory. The power spectral densities (PSDs) of displacements, velocities and accelerations are calculated from nonlinear responses. The focus of the paper is on the comprehen- sive interpretation of the responses of the structure related to wave excitation and structural characteristics. As an example a case study is investigated and numerical results are discussed.

System Identification of a Coupled Two DOF Moored Floating Body in Random Ocean Waves

Dynamics of a large moored floating body in ocean waves involves frequency dependent added mass and radiation damping as well as the linear and nonlinear mooring line characteristics. Usually, the added mass and radiation damping matrices can be estimated either by potential theory-based calculations or by experiments. The nonlinear mooring line properties are usually quantified by experimental methods. In this paper, we attempt to use a nonlinear system identification approach, specifically the reverse multiple input-single output (R-MISO) method, to coupled surge-pitch response (two-degrees-of-freedom) of a large floating system in random ocean waves with linear and cubic nonlinear mooring line stiffnesses. The system mass matrix has both frequency independent and frequency dependent components whereas its damping matrix has only frequency dependent components. The excitation force and moment due to linear monochromatic waves which act on the system are assumed to be known that can either be calculated or obtained from experiments. For numerical illustration, a floating half-spheroid is adopted. The motion as well as the loading are simulated assuming Pierson-Moskowitz (PM) spectrum and these results have been analyzed by the R-MISO method yielding frequency dependent coupled added mass and radiation damping coefficients, as well as linear and nonlinear stiffness coefficients of mooring lines satisfactorily.

Non-linear analysis of a dynamically positioned platform in stochastic seaway

Safety is the topmost priority considered by the designers of floating systems or any other structural systems. Reliability, economy and environmental pollutions and the liability of the structure in the case of accidents are also considered by the designers. The paper concentrates on the non-linear behavior of a moored floating platform in stochastic seaway generated using the Pierson–Moskowitz spectrum. Second-order wave forces (slow drift force) acting on the structure is considered as they contribute to a major percentage of the excursion of a large platform. Wave drift damping and skin friction damping have also been considered. It has been shown that the principal frequency of the second-order motion of the platform due to drift forces closely matches the natural frequency of the system in surge motion. This has been subsequently used in tuning a PID (proportionate integral and differential)-based control system for the surge mode, where reduction in the order of 90% has been observed.

Frequency of maxima of non-narrow banded stochastic processes

In many engineering problems one deals with quantities that must be considered to be of a stochastic or random nature. This is true for the natural environment such as wind, waves, and earthquakes, which are the driving mechanism behind the loadings on a wide variety of land-based and offshore structures. From a design perspective it is important to determine the expected highest value of a stochastic process, and structural fatigue life. The relevant procedures are reasonably well established for processes that have narrow band spectra, but it is much less clear how to deal with non-narrow band cases. In this paper it is shown that the extremes of a Gaussian, non-narrow band process are asymptotically equal to the extremes calculated according to the narrow band formula. Also demonstrated is that fatigue estimates may, with good accuracy, be based on the narrow band formula unless the bandwidth becomes extremely large. These statements are illustrated by examples of a process with (1) a low pass box spectrum, and (2) a Pierson–Moskowitz wave amplitude spectrum. It is also shown that the bandwidth parameter may in some cases be counter-intuitive, as the Pierson–Moskowitz spectrum has a larger bandwidth parameter than the box spectrum, even when both cover the same frequency range.

Ocean wave parameters estimation using backpropagation neural networks

In the present study, various ocean wave parameters are estimated from theoretical Pierson–Moskowitz spectra as well as measured ocean wave spectra using backpropagation neural networks (BNN). Ocean wave parameters estimation by BNN shows that the correlations are very close to one. This substantiates the use of neural networks (NN). For Indian coast, Scott spectra are used as it reasonably represents the measured spectra. The correlations of NN and Scott spectra are also compared. Once the network is trained, the ocean wave parameters can be estimated for unknown measured spectra, whereas significant wave height and spectral peak period are required to first generate the Scott spectra and then estimate other ocean wave parameters.

Effect of wave directional spread on the radiation stress: comparing theory and observations

Nearshore circulation and shoreline setup are forced through the wave radiation stress, which is a function of the frequency-directional wave spectrum. The radiation stress components often are approximated by assuming a spectrum that is narrow-band in frequency and direction. These narrow-band approximations overestimate the true radiation stress components in 8-m water depth on the Outer Banks, NC where locally generated (broad in frequency and direction) sea conditions are prevalent. Using the empirical Pierson-Moskowitz spectrum, frequency-spread is shown not to be the cause of this overestimation. Analytic expressions derived by Battjes [J. Mar. Res. 30 (1972) 56], for the ratio of true to narrow-band approximated radiation stress, which depend only on the directional spread, are compared to field data. The observed ratio of true to narrow-band approximated radiation stress is a strong function of the observed bulk (energy-weighted) directional spread, and closely follows the analytic expressions. Thus the narrow-band radiation stresses approximation can be augmented with a directional-spread dependence. These results are expected to apply to other geographic areas where waves are often locally generated.

Improved joint probability distribution for ocean wave heights and periods

Expressions have been derived, from the standpoint of Gaussian random theory, for the joint and marginal distributions of wave envelope amplitude and local wave period when the sampling frequency is equal to the local wave frequency. The new marginal p.d.f. for wave envelope amplitudes shows substantial non-zero probability density at very small amplitudes. The new joint and marginal distributions are found to compare favourably with data obtained from both high frequency measurements of real ocean waves in extreme storms and from measurements of numerical simulations of moderately broadbanded processes with Pierson-Moskowitz spectra. The new p.d.f. of wave envelope amplitudes is found to provide a better approximation to the p.d.f.s of the simulated zero down-crossing wave amplitude than either the traditional Rayleigh p.d.f., applicable for infinitesimal bandwidth; or the narrow bandwidth approximation given by M. S. Longuet-Higgins (Proc. R. Soc. Lond. A, 389: 241-258, 1983). The reason for this improvement is that our method takes into account that small waves are likely to have shorter periods than large waves, rather than assuming a constant wave period. There are, however, limitations to the approach adopted which assumes that the individual wave amplitudes can be obtained from the amplitude of the wave envelope. These limitations become more severe as the bandwidth increases. The results obtained apply not only to sea waves, but to any Gaussian linear random process.

Local and far-field surface elevation around a vertical cylinder in unidirectional steep wave groups

The problem of diffraction of a unidirectional incident wave group by a bottom-seated cylinder is considered. We assume the amplitude of the incoming wave to be small in comparison with other linear scales of the problem, and develop the corresponding second-order perturbation theory. We use the Fourier transform to treat time variation and separate spatial variables when solving the non-homogeneous second-order problem. The resulting set of non-homogeneous Bessel equations is solved numerically. Solutions for various types of incoming wave spectrum are obtained including the Gaussian spectrum and the Pierson–Moskowitz spectrum. To validate the method, problems with gradually decreasing bandwidth of Gaussian spectrum are solved and it is shown that the corresponding solution approaches that for the monochromatic case. The Pierson–Moskowitz spectrum with a set of realistic physical parameters is used as an example of extreme wave interaction with an offshore structure. The corresponding first- and second-order solutions are obtained and the effect of non-linearity on the solution is discussed with the emphasis on the growth of maximum free-surface elevation on the cylinder’s surface and generation of high frequency free radiated waves.

Study of the Electromagnetic Scattering from the Very Rough Fractal Sea Surface**The project supported by National Natural Science Foundation of China under Grant No. 60101001 and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China

In this paper, based on the fundamental formulae of the first-order and second-order Kirchhoff approximation and with consideration of the shadowing effect, the backscattering enhancement of the one-dimensional very rough fractal sea surface with Pierson-Moskowitz spectrum is studied under the second-order Kirchhoff approximation at microwave frequency. The numerical results are compared with those of the first-order Kirchhoff approximation and integral equation method. The dependencies of the bistatic scattering cross section and the backscattering enhancement on the incident angle, fractal dimension, and windspeed over the sea surface are analyzed in detail.

Numerical simulation of radar surveillance for the ship target and oceanic clutters in two‐dimensional model

A fast iterative approach of generalized forward‐backward method with the spectrum acceleration algorithm (GFBM/SAA) is developed for solving electromagnetic field integral equation and is applied to numerical simulation of radar surveillance of the ship target and oceanic clutters in two‐dimensional (2‐D) model. Randomly rough surface is realized by the Monte Carlo method using the 1‐D Pierson‐Moskowitz spectrum. Numerical results of bistatic and backscattering from the ship target and oceanic clutters demonstrate the functional dependencies upon the situation of radar surveillance such as the incidence and observation angles, radar altitude, ship RCS, and other oceanic conditions.

Numerical simulation of radar surveillance for the ship target in oceanic clutters

A fast iterative approach of generalized forward-backward method with the spectrum acceleration algorithm (GFBM/SAA) is developed for solving electromagnetic field integral equation and is applied to numerical simulation of radar surveillance of the ship target in oceanic clutters. Randomly rough surface is realized by the Monte Carlo method using the Pierson-Moskowitz spectrum. Numerical results of bistatic and back-scattering from the ship target and oceanic clutters demonstrate the functional dependencies upon the situation of radar surveillance such as the incidence and observation angles, radar altitude, ship RCS and other oceanic conditions.

Stochastic model for joint wave and wind loads on offshore structures

The stochastic wave load environment of offshore structures is of such a complicated nature that any engineering analysis requires extensive simplifications. This concerns both the transformation of the wave field velocities and accelerations to forces on the structure and the probabilistic description of the wave field itself. In this paper the last issue is in focus. The modeling follows the traditional structure of subdividing the time development of the wind driven wave process into sea states within each of which the wave process is modeled as a stationary process. The wave process of each sea state is modeled as an affinity in height and time of a Gaussian process defined by a normalized dimensionless spectrum of Pierson-Moskowitz type. The affinity factors are the so-called significant wave height H s and the characteristic zero upcrossing time T z . Based on measured data of ( H s , T z ) from the North Sea a well fitting joint distribution of ( H s , T z ) is obtained as a so-called Nataf model. Since the wave field is wind driven, there is a correlation between the time averaged wind velocity pressure Q and the characteristic wave height in the stationary situation. Using the Poisson process model to concentrate on those load events that are of importance for the evaluation of the safety of the structure, that is, events with Q larger than some threshold q 0, available information about the wind velocity pressure distribution in high wind situations can be used to formulate a Nataf model for the joint conditional distribution of ( H s , T z , Q) given that Q> q 0. The distribution of the largest wave height during a sea state is of interest for designing the free space between the sea level and the top side. An approximation to this distribution is well known for a Gaussian process and by integration over all sea states given Q> q 0, the distribution is obtained that is relevant for the free space design. However, for the forces on the members of the structure also the wave period is essential. Within the linear wave theory (Airy waves) the drag term in the Morison force formula increases by the square of the ratio between the wave height and the wave length, and the mass force term increases proportional to the ratio of the wave height and the square of the period. For a strongly narrow band Gaussian process Longuet-Higgins has derived a joint distribution of the height and the period. However, simulations show that the Pierson–Moskowitz spectrum (or any other standard spectrum for wind driven sea waves of similar bandwidth such as the JONSWAP spectrum) does not provide a sufficiently narrow banded process for the distribution of Longuet-Higgins to make a good fit. Surprisingly it turns out that the random time L between two consecutive 0-upcrossings and the random wave height H observed between the two 0-upcrossings behave such that L and the ratio H/ L are practically uncorrelated and both normally distributed except for clipping the negative tails. This result is of global nature and is therefore very difficult if not impossible to obtain by analytical mathematical reasoning. Finally, by combining all the derived distributional models into a Rosenblatt transformation, a first order reliability analysis of a tubular offshore platform can be made with respect to static pushover. Correction for non-linear wave theory can be taken into account crudely by using the deterministic 5th order Stokes wave in the limit state formulation. A dynamic analysis will be more complicated, of course, but the provided distributional information and the demonstrated modeling principles are judged as generally applicable.

Acoustic propagation in a shallow water waveguide with a rough surface: Comparison of results from PE and coupled unperturbed normal mode methods

The PE method is an accepted standard for accurate acoustic propagation modeling. We have developed an alternative method (Henyey et al., this session) based on flat-surface normal modes, with coupling due to surface waves. Results for the complex field obtained with the new mode coupling method are compared with the solutions from a rough surface PE code [A. D. Rosenberg, J. Acoust. Soc. Am. 105, 144–153 (1999)]. A variety of rough surfaces are considered in the comparison. Mode coupling is most easily tested using a single mode starting field with a sinusoidal surface whose wavelength is chosen to produce a resonance with another mode. As an example of a randomly rough surface, a modified Pierson–Moskowitz spectrum of surface waves is considered. Comparisons with the PE method include closeness of the results, accuracy of energy conservation, computational speed, and ease of use. [Work supported by ONR.]

A Sea State Parameterization with Nonarbitrary Wave Age Applicable to Low and Moderate Wind Speeds

A simple coupled flux and sea state model is developed. It is applicable to low, moderate, and high wind speeds, with a nonarbitrary wave age. It is fully consistent with an atmospheric flux parameterization. The flux model includes the influence of capillary waves on surface stress, which dominate surface stress on a wave-perturbed surface for U10 < 7 m s−1. The coupled model is verified for conditions of local equilibrium and it is used to examine the influences of surface tension and capillary waves on the equilibrium sea state at low and moderate wind speeds (U10 < 7 m s−1). Capillary waves can directly influence characteristics of the airflow (roughness length and friction velocity), and surface tension can directly influence wave characteristics (period, phase speed, and wave age). The influences of capillary waves and surface tension on wave characteristics are found to be noticeable for U10 < 6 m s−1, and are likely to be significant in most applications when U10 < 5 m s−1. The conditions under which relations valid for higher wind speeds break down are discussed. The new sea state parameterization is an improvement over previous relations (which apply well for U10 > 7 m s−1) because it is also applicable for low and moderate winds. The mean wind speed over most of the world's oceans is less than 7 m s−1, so there is considerable need for such a parameterization. The nonarbitrary wave age is particularly important because of the influence of wave age on the shape and size of waves, as well as fluxes of momentum, heat, and moisture. Additional results include a criterion for the capillary cutoff, significant wave height, and dominant wave period (as functions of wind speed and wave age). Consistency with a flux parameterization is crucial because of the influence of atmospheric stability on surface stress, which influences wave characteristics. The modeled significant wave height for local equilibrium is shown to be consistent with several parameterizations. The modeled dominant period is a good match to that of the Pierson–Moskowitz spectrum when U10 > 13 m s−1; however, for lower wind speeds, the modeled period differs significantly from that of the Pierson–Moskowitz spectrum. The new model is validated with field observations. The differences are largely due to the capillary cutoff, which was not considered by Pierson and Moskowitz.

Verification of Hasselmann's energy transfer among surface gravity waves by direct numerical simulations of primitive equations

The temporal evolution of nonlinear wave fields of surface gravity waves is studied by large-scale direct numerical simulations of primitive equations in order to verify Hasselmann's theory for nonlinear energy transfer among component gravity waves. In the simulations, all the nonlinear interactions, including both resonant and non-resonant ones, are taken into account up to the four-wave processes. The initial wave field is constructed by combining more than two million component free waves in such a way that it has the JONSWAP or the Pierson–Moskowitz spectrum. The nonlinear energy transfer is evaluated from the rate of change of the spectrum, and is compared with Hasselmann's theory. It is shown that, in spite of apparently insufficient duration of the simulations such as just a few tens of characteristic periods, the energy transfer obtained by the present method shows satisfactory agreement with Hasselmann's theory, at least in their qualitative features.

Radar image studies of an ocean‐like surface

AbstractHigh‐resolution radar images of a 1‐D ocean‐like surface are formed through back‐projection tomography. The surface profile is described by a Pierson–Moskowitz spectrum, and satisfies an impedance boundary condition. Backscatter results from 10 to 14 GHz excited by a tapered beam at incidence angles from 60 to 80° are considered. Radar images from approximate and numerical scattering models demonstrate dominant Bragg scattering components, small multiple interaction effects associated with the long‐wave portion of the surface spectrum, and significant polarization dependencies. © 2001 John Wiley &amp; Sons, Inc. Microwave Opt Technol Lett 30: 381–384, 2001.