(2005). Applied Multivariate Statistical Analysis. Technometrics: Vol. 47, No. 4, pp. 517-517.

Applied Multivariate Statistical Analysis

This book was published in 2004. The Interaction of Ocean Waves and Wind describes in detail the two-way interaction between wind and ocean waves and shows how ocean waves affect weather forecasting on timescales of 5 to 90 days. Winds generate ocean waves, but at the same time airflow is modified due to the loss of energy and momentum to the waves; thus, momentum loss from the atmosphere to the ocean depends on the state of the waves. This volume discusses ocean wave evolution according to the energy balance equation. An extensive overview of nonlinear transfer is given, and as a by-product the role of four-wave interactions in the generation of extreme events, such as freak waves, is discussed. Effects on ocean circulation are described. Coupled ocean-wave, atmosphere modelling gives improved weather and wave forecasts. This volume will interest ocean wave modellers, physicists and applied mathematicians, and engineers interested in shipping and coastal protection.

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The Interaction of Ocean Waves and Wind

Images of the ocean surface taken with active microwave sensors often contain much information on near-surface and subsurface processes. However, their interpretations may depend on a detailed understanding of the physics of electromagnetic scatter. In scattering theory, the surface hydrodynamics enters the equations via (1) the probability distribution function for either wave heights or slopes, and (2) the two-dimensional wave height/slope autocovariance or its Fourier transform, the wave vector spectrum. This paper advances an improved model for the ocean surface wave vector spectrum based on recent work by M. A. Donelan, by M. L. Banner, and by B. Jahne and their collaborators. The model addresses the range of surface wavelengths from fully developed wind waves to the gravity-capillary region. For gravity-capillary waves, the spectral equation satisfactorily represents the observational data of Jahne et al. taken in tanks at large fetches, in the range from approximately 50 to 1500 rad/m to within the accuracy of the data. From the spectrum, the two-dimensional autocovariance of the sea surface is computed and correlation lengths and curvatures obtained. When used with a modification of Holliday's formulation of microwave radar backscatter from a Gaussian sea, it quantitatively reproduces observational cross section data taken at vertical polarization from aircraft and spacecraft over the open ocean, with differences from the field data having a mean of −0.2 dB and a standard deviation of 1.7 dB, The range of parameters for which satisfactory fits are obtained includes: wind speeds from 1.5 to 24 m/s; frequencies from approximately 5.5 to 35 GHz; and incidence angles from 0° to greater than 60°. For horizontal polarization, the scattering calculations fail rather badly for larger incidence angles, as do all theories based on the Kirchhoff approximation. Additionally, in spite of the incorporation of an anisotropic angular distribution of wave energy, the observed azimuthal variation of radar scatter is not captured, indicating that the source of that variation lies elsewhere.

An improved model of the ocean surface wave vector spectrum and its effects on radar backscatter

We propose the use of support vector machines (SVMs) for automatic hyperspectral data classification and knowledge discovery. In the first stage of the study, we use SVMs for crop classification and analyze their performance in terms of efficiency and robustness, as compared to extensively used neural and fuzzy methods. Efficiency is assessed by evaluating accuracy and statistical differences in several scenes. Robustness is analyzed in terms of: (1) suitability to working conditions when a feature selection stage is not possible and (2) performance when different levels of Gaussian noise are introduced at their inputs. In the second stage of this work, we analyze the distribution of the support vectors (SVs) and perform sensitivity analysis on the best classifier in order to analyze the significance of the input spectral bands. For classification purposes, six hyperspectral images acquired with the 128-band HyMAP spectrometer during the DAISEX-1999 campaign are used. Six crop classes were labeled for each image. A reduced set of labeled samples is used to train the models, and the entire images are used to assess their performance. Several conclusions are drawn: (1) SVMs yield better outcomes than neural networks regarding accuracy, simplicity, and robustness; (2) training neural and neurofuzzy models is unfeasible when working with high-dimensional input spaces and great amounts of training data; (3) SVMs perform similarly for different training subsets with varying input dimension, which indicates that noisy bands are successfully detected; and (4) a valuable ranking of bands through sensitivity analysis is achieved.

Robust support vector method for hyperspectral data classification and knowledge discovery

Backscattering experiments at microwave frequencies were conducted off the west coast of Scotland in the summer of 1991. Using a dual-polarization, eight-frequency, X band, coherent scatterometer mounted on the bow of a boat, we measured time-resolved backscattering from ocean waves at a range of grazing angles from 10° to 70°. From the grazing-angle-dependent signals and their Doppler spectra, we are able to differentiate Bragg scattering from non-Bragg scattering and resolve “peak separation” between the vertical and horizontal polarizations. We observe instances of “super” events, i.e., instances when the horizontal polarization return power equals or exceeds the vertical polarization power at particular frequencies. We find that “super” events occur not only at low grazing angles but at any grazing angle for upwind viewing directions and obtain statistics for such occurrences as a function of grazing angle. We study the coherence properties of scatterers and find strong evidence that at low grazing angles, lifetime-dominated, non-Bragg scattering contributes noticeably to returns of both polarizations, but is dominant in providing returns for the horizontal polarization. We examine “spiking” events and find that they can be related to, but need not be limited to, breaking wave events. By comparing the data of upwind runs with cross-wind and circle runs, we obtain wind direction dependence of Doppler spectra, which further assists in the identification of scattering mechanisms.

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X band microwave backscattering from ocean waves

Brewster angle damping and local multipath effects are considered as sources of polarization differences in low grazing angle sea scatter characteristics. The authors show that at least five observed polarization differences can be explained by local multipath interference effects that occur due to the illumination of discrete nonlinear ocean surface features, such as bores and small scale breaking waves. The illumination gain factor (IGF) is defined at a point in space, as the total power at that point relative to the power in the incident plane wave. The IGF resulting from local multipath from the sea surface forward of a discrete scatterer produces strong interference patterns that can vary both with grazing angle and scatterer height. As a result, IGF values up to a factor of 16 (12 dB) can occur for horizontal polarization (HH) when the interference is constructive; a corresponding strong cancellation occurs for destructive interference. These extreme variations can cause strong HH NRCS amplitude modulations due either to a change of local wave slope or a change of scatterer shape with time. However, Brewster angle damping of the forward scatter path for grazing angles below 20/spl deg/ occurs for vertical (VV) polarization, and reduces the VV IGF in magnitude and dynamic range, eliminating such strong modulations. This effect scales with radar wavelength, and higher wave features are required to produce equivalent effects for radar frequencies far below 10 GHz. As an illustration, six radar bands are compared: L (1.4 GHz), S (3.5 GHz), C (5 GHz), X (10 GHz), K/sub u/ (15 GHz), and K/sub a/ (35 GHz), for a sea water dielectric. X-band results indicate that 12-dB IGFs can occur for water surface features just a centimeter above the mean surface. As an application of these results, the influence of these HH and VV IGF patterns is modeled for discrete scatterers distributed uniformly along an ocean gravity wave. The dynamic range of the HH IGF for a distribution of bore scatterers up to 5 cm high is found to he significantly larger than for VV at all locations on the long wave. Moreover, the IGF HHVV polarization ratio forward of the crest, where the largest number of small scale breaking wave scatterers occurs, is larger than at all other regions of the long wave, of the order of 20 dB. These results suggest that HH polarization may be sensitive to small scale breaking features on the ocean surface at low grazing angles, and thus may be a sensitive measure of air-sea fluxes.

A model for Brewster angle damping and multipath effects on the microwave radar sea echo at low grazing angles

Reports on experiments featuring horizontally (HH) and vertically (VV) polarized X-band marine radar sea scatter imagery in the coastal environments of Bermuda and La Jolla, under light to moderate winds and the absence of long gravity waves. These conditions allowed the study of fundamental scattering mechanisms from small scale roughness and short waves of a few meters wavelength or less, shorter than the radar pulse. While a large fraction of radar data collected revealed the presence of slick bands, the authors analyze the radar echo of ambient background outside of the slicks. Sea scatter data were digitally recorded for 360/spl deg/ azimuthal coverage for grazing angles between 1-3.5/spl deg/, and were converted to normalized radar cross section (NRCS or /spl sigma//spl deg/) images. The HH and VV polarizations show quite different spatial texture, with HH exhibiting a discrete character and VV being more nearly spatially homogeneous. Grazing angle dependencies are different for HH and VV: upwind-downwind differences of mean NRCS show ratios of just 4-6 dB for VV, but are equal or greater than the 16-dB noise-imposed limit for HH for the low wind conditions. HH NRCS grazing angle characteristics are shown to correlate with biological activity indicators of the coastal waters, with a fourth power dependence in relatively unproductive waters, to a nearly quadratic in productive waters. Arguments are presented that suggest different scattering mechanisms for the two polarizations: evenly distributed Bragg scatter patches for VV and scatter from small asymmetric bore features for HH. A multipath illumination model for small bore features is outlined, and scale sizes for the small scale breakers inferred from the experimental results presented are between 2 and 4 cm in height, with crest widths between 24 and 48 cm.

Studies of dual polarized low grazing angle radar sea scatter in nearshore regions

Spatial measurements of capillary-gravity waves in the ocean were obtained using a scanning slope sensor mounted on a free-drifting buoy intended to minimize the flow disturbance. The data provide direct calculation of the wavenumber spectra of surface curvature in the capillary-gravity wave range. The results indicate that 1) a pronounced peak at the wavenumber k = 9 rad cm−1 is evident in the curvature spectra for wind speeds below 6 m s−1; 2) the slopes of the curvature spectra are 1 and −1 on the two sides of the spectral peak, 3) the spectral density and mean-square roughness properties increase linearly with wind speed; and 4) these observations suggest a spectral function of the form χ(k) = Au*c−2cmkmk−4, which is proportional to u*k−3 in the short gravity wave region and u*k−5 in the capillary wave region, where u* is the wind friction velocity, cm the minimum phase velocity of surface waves, and km the corresponding wavenumber. Capillary-gravity wave wavenumber spectra obtained from the ...

A Study of the Wavenumber Spectra of Short Water Waves in the Ocean

We report on measurements of low grazing angle radar sea spikes in a laboratory wave tank. Using an X band radar with 10-cm range resolution and multiple-range bin capability, the sea spikes were observed to have a horizontal/vertical polarization ratio much larger than unity. This is similar to observations in earlier field experiments, but is in contrast to a ratio of unity recently observed for moderate grazing angles (Jessup et al., 1990). Our experiment was designed to validate the low grazing field results and to establish design criteria for spatial-temporal measurement instrumentation; single-point measurements currently used as surface truth cannot provide water surface profiles needed for modeling of radar scatter from deterministic surface features. Surface scattering features were generated using a paddle-generated wave, combined with wind waves for a series of wind speeds. The surface feature responsible for the sea spike echo was visually identified as a short wave crest, occurring on the windward side of the paddle wave, driven to extreme steepness by the passage of the faster paddle wave crest. Long decorrelation times observed in the field are also seen in these studies, of the order of 500 ms for horizontal polarization and less than 5 ms for vertical. We suggest that rather than Bragg scatter, the occurrence of similar ubiquitous features on the ocean are responsible for horizontally polarized low grazing angle sea scatter at the X band for low to moderate winds, and perhaps at other frequencies as well.

Laboratory studies of radar sea spikes at low grazing angles

An ultrawide-band radar system centered at 10 GHz has been developed for sea-scatter research and was recently deployed on a research pier in the North Atlantic. The radar is based on a time-domain reflectometer module for a sampling oscilloscope. Using transient excitation of a traveling wave-tube amplifier, the system generates 200-ps wide pulses with a 10-GHz center frequency and a peak power of approximately 1 KW. The resulting range resolution is approximately 3 cm. The system was used to investigate low-grazing angle, multipath effects in the ocean environment using a trihedral corner reflector, mounted 45 cm above the water surface. The ultrahigh-range resolution of the system allows spatial separation of the direct and indirect echoes from the trihedral. In addition, a comparison of the indirect vertically- (VV) and horizontally- (HH) polarized echoes illustrates the effects of Brewster angle damping. The implications of these effects for sea-clutter statistics are briefly discussed.

D.B. Trizna

Papers

A model for Brewster angle damping and multipath effects on the microwave radar sea echo at low grazing angles

Studies of dual polarized low grazing angle radar sea scatter in nearshore regions

A Study of the Wavenumber Spectra of Short Water Waves in the Ocean

Laboratory studies of radar sea spikes at low grazing angles

Ultrawide-band radar observations of multipath propagation over the sea surface