Slope Probability Density Function Research Articles

Design of one-dimensional band-limited uniform diffusers of light

We present two methods for generating numerically a one-dimensional random surface, defined by the equation x3=ζ(x1), that has a prescribed probability density function of slopes. This problem arises in the design of a diffuser that scatters light uniformly within a range of scattering angles and produces no scattering outside this range. Numerical calculations of the scattering of light from random surfaces generated by these approaches show that the scattered intensity, indeed, has a rectangular distribution as a function of the scattering angle.

Scanning-laser glint measurements of sea-surface slope statistics

A scanning-laser glint meter designed for field measurements of sea-surface slope statistics is described. A narrow laser beam is scanned in a line, and specular reflections (glints) are counted in bins according to their slope angle. From normalized glint histograms, moments to the fourth order are calculated, and slope probability density functions are approximated with a Gram-Charlier expansion. Field measurements with this instrument show good agreement with previous results when the stability (essentially air-sea temperature difference) is near neutral (zero). Under conditions of negative stability (warm ocean), both the mean-square slope and the probability density function kurtosis increase.

The Probability Density Function of Ocean Surface Slopes and Its Effects on Radar Backscatter

Abstract Based on Longuet-Higgins’s theory of the probability distribution of wave amplitude and wave period and on some observations, a new probability density function (PDF) of ocean surface slopes is derived. It is where ζx and ζy are the slope components in upwind and crosswind directions, respectively; σ2u and σ2c are the corresponding mean-square slopes. The peakedness of slopes is generated by nonlinear wave–wave interactions in the range of gravity waves. The skewness of slopes is generated by nonlinear coupling between the short waves and the underlying long waves. The peakedness coefficient n of the detectable surface slopes is determined by both the spectral width of the gravity waves, and the ratio between the gravity wave mean-square slope and the detectable short wave mean-square slope. When n equals 10, the proposed PDF fits the Gram Charlier distribution, given by Cox and Munk, very well in the range of small slopes. When n → ∞, it is very close to the Gaussian distribution. Radar backscat...

Tilt modulation of high resolution radar backscatter cross sections: unified full wave approach

The unified full wave approach is used to determine the tilt modulation of the like- and cross-polarized (high-resolution) radar backscatter cross sections for the rough sea surface. Real or synthetic aperture radars (SARs) with small effective footprints (resolution cells) are considered. Since the unified full-wave approach accounts for Bragg scattering as well as specular point scattering in a self-consistent manner, it is not necessary to adopt a two-scale model for the rough sea surface. The sea surface slope probability density function is assumed to be Gaussian. The backscattering cross sections are evaluated for all angles of incidence (normal to grazing). For tilts in the plane of incidence, the modulation of all the cross sections is largest at angles of incidence of 10 degrees . The cross-section modulation due to tilts perpendicular to the plane of incidence critically depends on the incident and scattered polarizations. The effective filtering of the large-scale spectral components of the rough sea surface by the high-resolution radar is accounted for, and the dependence of the cross-section tilt modulation on the size of the effective footprint is determined.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Nonstationary surface roughness measurement by light scattering

A noncontacting method of measuring the mean-square slope of realistic, non-stationary surface roughness is suggested. The method is based on the relationship between the mean scattered power and the two-dimensional slope probability density function, which is valid in the high-frequency limit.

Correction to "Relationship between slope probability density function and the physical optics integral in rough surface scattering"

Correction to "Relationship between slope probability density function and the physical optics integral in rough surface scattering"

Relationship between slope probability density function and the physical optics integral in rough surface scattering

The integral for the average scattered power from a rough surface obtained from physical optics is shown to be proportional to the joint probability density function for the surface slopes in the high-frequency limit.

Statistical Properties of Ramplike Random Processes

The work of Lampard and Redman [5] on the statistical properties of the integral of a binary random process is generalized and extended to include more general ramplike random processes z(t) . These ramplike processes are defined to be ramps of random slope that switch slope at random times. Both the intervals between switching times and the slopes of the ramp segments are assumed to be independent random variables. The general properties of the one-dimensional and multidimensional probability density functions of z(t) are discussed. A Laplace transform technique is employed to obtain the Laplace transform of the characteristic function of these probability density functions in terms of the Laplace transform of the characteristic function of the ramp slopes multipled by the probability functions of the switching times. For Poisson switching times, these expressions are relatively compact. However, the inversion problem is formidable, and closed form results are presently available only for a binary slope probability density function (a previously known result). Due to the properties of characteristic functions, the moments of z(t) are obtained by a single Laplace inversion of the derivative of the Laplace transform expressions. These moments may be used to characterize the probability density functions of z(t) with a series expansion involving the Gaussian density function and its derivatives. Examples are given for the cases of Gaussian and rectangular slope probability density functions with Poisson switching times.

Symposium on Radar and Radiometric Observations of Venus during the 1962 Conjunction: Radar scattering from Venus and the Moon

view Abstract Citations (93) References (6) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Symposium on Radar and Radiometric Observations of Venus during the 1962 Conjunction: Radar scattering from Venus and the Moon Muhleman, D. O. Abstract A statistical theory for the radar backscatter function for a planetary surface will be derived. The physical basis of the theory involves combining two random variables which represent height variations and horizontal scattering lengths to form the probability distribution function for surface slopes. Two classes of distribution functions representing the two random variables are studied: those related to the normal distribution, and those related to the exponential distribution. These two classes result in slightly different distribution functions for slopes. Each of the slope functions contains a single parameter which is shown to be the ex- pectation value of the slopes, i.e., the mean effective slope. The probability density function of slopes is to be directly related to the radar backscatter func- tion under the assumption of geometrical optics. These functions are applied to radar observations of the moon at wavelengths of 3.6, 10, 12.5, and 68 cm and it is shown that the exponential statistics yield slightl superior results. The theory is applied to radar observations of Venus made at wavelengths of 12.5 and 68 cm in an attempt to deduce the mean slope of Venus and the Venusian rotational rate. It will be shown that Venus appears very smooth at radar wavelengths; i.e., the mean slope parameters are small relative to those for the moon. Values of the intrinsic angular velocity are computed from the 12.5-cm radar spectral observations of Venus and are found to be between the limits of 4.5X 10- and 2.OX 10-2 rad/sec. The spread in this result arises from the difficulties in determining the wavelength dependence for the slope parameters determined from the radar data. Publication: The Astronomical Journal Pub Date: February 1964 DOI: 10.1086/109225 Bibcode: 1964AJ.....69...34M full text sources ADS |