Mean Square Slope Research Articles

Revisiting the Cox and Munk wave-slope statistics using IASI observations of the sea surface

We use radiances collected from space by the Infrared Atmospheric Sounder Interferometer (IASI) when looking down at ocean surfaces during the day to remotely determine the probability distribution of wave slopes. This is achieved by using about 300 channels between 3.6 and 4.0μm and a physically-based approach which properly takes the contribution of the reflected solar radiation into account. Based on about 150 millions observations, the same number of wave-slope probabilities are retrieved for wind speeds (at 10 m) up to 15 m/s. We revisit and discuss the methodology proposed by Cox and Munk (CM) to derive their celebrated wave-slope probability distribution function (pdf) from photographs of the sun glitter. We propose an original and robust approach for accurate retrievals of the 7 parameters appearing in the Gram–Charlier representation of the pdf. Our results for the mean square slopes (MSSs) are fully compatible with those of CM, and with the more recent results by Bréon and Henriot, but our lower uncertainties enable to point out departures from the linear wind-speed dependencies and a slight overestimation of the upwind MSS described by the linear fit of CM at moderate wind speed. Our skewness and kurtosis coefficients show clear influences of the wind speed, with a steady decrease of the former and the alongwind kurtosis coefficient being maximal at moderate wind speeds, features that CM could not point out due to the limitations of their measurements. We revisit the renormalization procedure employed by CM to obtain the complete variances from truncated pdfs and show that it imposes stringent conditions on the kurtosis coefficients that allow to determine them accurately, with wind-dependent values specific to the local sea state. We also provide measurements of the shifted position of the most probable slope as well as a demonstration of a qualitative change of regime in the updown wind asymmetry of the wave-slope probability when the wind speed increases.

Effects of microplastics and surfactants on surface roughness of water waves

We study the flow physics underlying the recently developed remote sensing capability of detecting oceanic microplastics, which is based on the measurable surface roughness reduction induced by the presence of microplastics on the ocean surface. In particular, we are interested in whether this roughness reduction is caused by the microplastics as floating particles, or by surfactants which follow similar transport paths as microplastics. For this purpose, we experimentally test the effects of floating particles and surfactants on surface roughness, quantified by the mean square slope (MSS), with waves generated by a mechanical wave maker or by wind. For microplastics, we find that their effect on MSS critically depends on the surface area fraction of coverage. The damping by particles is observed only for fractions above O (5–10%), much higher than the realistic ocean condition. For surfactants, their damping effects on both mechanically generated waves and wind waves are quantified, which are shown to be much more significant than that by microplastics. Several new mechanisms/relations for roughness damping by surfactants are also identified. The implications of these experimental results to remote sensing are discussed.

Physics-Based Forward Modeling of Ocean Surface Swell Effects on SMAP L1-C NRCS Observations.

This paper examines the impact of ocean surface swell waves on near-coastal L-band high-resolution synthetic aperture radar (SAR) data collected using the National Aeronautics and Space Administration’s (NASA) Soil Moisture Active/Passive (SMAP) radar at 40° incidence angle. The two-scale model and a more efficient off-nadir approximation of the second-order small-slope-approximation are used for co- and cross-polarized backscatter normalized radar cross-section (NRCS) predictions of the ocean surface, respectively. Backscatter NRCS predictions are modeled using a combined wind and swell model where wind-driven surface roughness is characterized using the Durden–Vesecky directional spectrum, while swell effects are represented through their contribution to the long wave slope variance (mean-square slopes, or MSS). The swell-only MSS is numerically computed based on a model defined using the JONSWAP spectrum with parameters calculated using the National Data Buoy Center and Wave Watch III data. The backscatter NRCS model is further refined to include fetch-limited and low-wind corrections. The results show an improved agreement between modeled and observed HH-polarized backscatter NRCS when swell effects are included and indicate a relatively larger swell impact on L-band compared to higher radar frequencies. Preliminary investigations into the potential swell retrieval capabilities in the form of excess MSS are encouraging, however further refinements are required to make broadly applicable conclusions.

Characterizing and Mitigating Digital Sampling Effects on the CYGNSS Level 1 Calibration

This article presents a detailed examination of fluctuating input noise power levels on the analog-to-digital convertor (ADC) sampling hardware of the NASA Cyclone Global Navigation Satellite System (CYGNSS) instruments and the associated impacts on the level 1 normalized bistatic radar cross section (NBRCS) estimation performance. The impact of external noise variations on both the CYGNSS science and navigation channels is quantified with respect to how the fluctuating received power impacts the low-level ADC sampling distribution and subsequent NBRCS estimation. This work demonstrates that there are clear quantifiable geospatially dependent noise variations linked to Global Navigation Satellite System (GNSS) space-based augmentation systems [notably Japanese Quasi-Zenith Satellite System (QZSS) and U.S. Wide Area Augmentation System (WAAS)] and that these additional noise sources significantly alter the digital sampling distribution of the CYGNSS instruments, actively degrading the level 1 NBRCS estimation if not corrected. A derivation of the theoretical correction for both the science and navigation channel ADC sampling variations is presented which is later tuned based on empirical performance metrics in an effort to minimize the induced calibration errors. The impacts of the enhancements outlined in this work on CYGNSS level 1 calibration are evaluated using one year of observations before and after the digital sampling corrections, using model European Centre for Medium-Range Weather Forecasts (ECMWF) wind and Wavewatch III mean square slope (MSS) surface validation datasets.

An Improved Two-Scale Model for Sea Surface Scattering in the Transition Range of Incidence Angles

An improved two-scale model (I-TSM) for the incidence angle transition range (5-11°) is proposed by analyzing the contributions of both quasi-specular and Bragg scatterings to the total sea surface scattering. With normalized radar cross section (NRCS) products of surface wave investigation and monitoring (SWIM) instrument, the mean square slope (MSS) and reflection coefficient of the geometrical optics (GO) model are modified to simulate the quasi-specular scattering in I-TSM, and a Bragg scattering contribution factor adaptive to the incidence angle and wind speed is fitted to provide the Bragg scattering component for I-TSM. At different incidence angles and wind speeds, I-TSM combines the quasi-specular and Bragg scatterings in variable proportions by the fitting factor. I-TSM avoids the critical angle selection and separation problems of the traditional TSM in the incidence angle transition range. Compared with the GO model and traditional TSM, I-TSM has significantly higher accuracies with root mean square error (RMSE) of 0.309dB, relative error (RE) of 2.019% and scattering index (SI) of 0.03 when SWIM NRCS products are the standard data in the validation. I-TSM will provide high-precision NRCS simulation data in the incidence angle transition range for scattering characteristics analysis and echo simulation of SWIM.

The Response of the Water Surface Layer to Internal Turbulence and Surface Forcing

We have carried out an experimental study of the turbulence kinetic energy dissipation rate (ϵ), temperature dissipation rate (χ), and turbulent heat flux (THF) within the water surface layer in the presence of non-breaking wave, surface convection, and horizontal heat and eddy fluxes that play a prominent role in the front. We noted that the non-breaking wave dominates ϵ values within the surface layer. While analyzing the vertical ϵ variability, the presence of a wave-affected layer from the water surface to a depth of z≈1.25λw is observed, where λw is the wavelength. ϵ associated with non-breaking waves ranged to 4.9×10−6–7×10−6 m2/s3 for the wavelength range of 0.038 m < λw < 0.098 m categorized as the gravity and gravity-capillary wave regimes. ϵ values increase for longer λw and non-breaking wave ϵ values represent their significant contribution to the ocean energy budget and dynamic of surface layer considering that the non-breaking wave covers the large fraction of ocean surface. We also found that the surface mean square slope (MSS) and wave generated ϵ have the same order of magnitude, i.e., MSS ∼ϵ. Besides, we have documented that the small-scale temperature fluctuation change (i.e., χ) is consistent with the large-scale temperature gradient change (i.e., d<T>/dz). The value of the THF is approximately constant within the surface layer. It represents that the measured THF near the water surface can be considered a surface water THF, challenging to measure directly.

Optimal archgrids: a variational setting

The paper deals with the variational setting of the optimal archgrid construction. The archgrids, discovered by William Prager and George Rozvany in 1970s, are viewed here as tension-free and bending-free, uniformly stressed grid-shells forming vaults unevenly supported along the closed contour of the basis domain. The optimal archgrids are characterized by the least volume. The optimization problem of volume minimization is reduced to the pair of two auxiliary mutually dual problems, having mathematical structure similar to that known from the theory of optimal layout: the integrand of the auxiliary minimization problem is of linear growth, while the auxiliary maximization problem involves test functions subjected to mean-square slope conditions. The noted features of the variational setting governs the main properties of the archgrid shapes: they are vaults over a subregion of the basis domain being the effective domain of the minimizer of the auxiliary problem. Thus, the method is capable of cutting out the material domain from the design domain; this process is built in within the theory. Moreover, the present paper puts forward new methods of numerical construction of optimal archgrids and discusses their applicability ranges.

Airborne Measurements of Surface Wind and Slope Statistics over the Ocean

AbstractAs reported in 1954, more than a half century ago, C. Cox and W. Munk developed an empirical model of the slope distribution of ocean surface waves that has been widely used ever since to model the optical properties of the sea surface and is of particular importance to the satellite remote sensing community. In that work, the reflectance of sunlight was photographed from a Boeing B-17G bomber and was then analyzed. In this paper, surface slope statistics are investigated from airborne scanning topographic lidar data collected during a series of field experiments off the coast of California and in the Gulf of Mexico, over a broad range of environmental conditions, with wind speeds ranging from approximately 2 to 13 m s−1. Unlike the reflectance-based approach of Cox and Munk, the slope distribution is computed by counting laser glints produced by specular reflections as the lidar is scanned over the surface of the ocean. We find good agreement with their measurements for the mean-square slope and with more recent (2006) results from Bréon and Henriot that were based on satellite remote sensing. Significant discrepancies for the higher-order statistics are found and discussed. We also demonstrate here that airborne scanning lidar technology offers a viable means of remotely estimating surface wind speed and momentum flux.

The Effect of Foam on Waves and the Aerodynamic Roughness of the Water Surface at High Winds

AbstractThis paper models the impact of the presence of foam on the short-wave component of surface waves and momentum exchange in the atmospheric boundary layer at high winds. First, physical experiments were carried out in a wind-wave flume in which foam can be artificially produced at the water surface. Tests were conducted under high-wind-speed conditions where equivalent 10-m wind speed ranged from 12 to 38 m s−1, with measurements made of the airflow parameters, the frequency–wavenumber spectra of the surface waves, the foam coverage of the water surface, and the distribution of the foam bubbles. Analysis of the resulting data indicates that the surface drag coefficient correlates with the fraction of foam coverage and the mean square slope (MSS) of the water surface, and that, at a certain wind speed, the MSS decreases with an increase in the fraction of foam coverage. Based on these results, we suggest a simple model for eddy viscosity in the turbulent boundary layer over a fractionally foam-covered wave surface. The measurements in a laboratory environment are shown to be in good agreement with the predictions of a quasi-linear model of the atmospheric boundary layer over a waved water surface that adopts this eddy viscosity. Adaptation of the proposed model to field conditions is discussed, and the synergetic effect of foam at the water surface and spray in the marine atmospheric boundary layer on ocean surface resistance at high winds is estimated so as to be able to explain the observed peaking dependence of the surface drag coefficient on the 10-m wind speed.

Toward the Generation of a Wind Geophysical Model Function for Spaceborne GNSS-R

This paper presents a comprehensive procedure to improve the wind geophysical model function (GMF) for the Global Navigation Satellite System Reflectometry (GNSS-R) instrument onboard the TechDemoSat-1 satellite. The observable used to define the GMF is extracted from the measured delay-Doppler maps (DDMs) by correcting for the nongeophysical effects within the measurements. Besides the instrument and the geometric effects as provided in the bistatic radar equation, a calibration term that accounts for the uncalibrated receiver antenna gain and the unknown transmitter antenna gain is proposed to optimize the calculation of GNSS-R observables. Such calibration term is presented as a function of observing elevation and azimuth angles and is shown to remarkably reduce the measurement uncertainties. First, an empirical wind-only GMF is developed using the collocated Advanced Scatterometer (ASCAT) winds and European Centre for Medium-Range Weather Forecasts (ECMWF) model wind output. This empirical GMF agrees well with the model output. Then, the sensitivity of the observable to waves is analyzed using the collocated ECMWF wave parameters. The results show that it is difficult to include mean square slope (MSS) in the development of an empirical GMF, since the difference between ECMWF MSS and the MSS sensed by GNSS-R varies with incidence angle and wind speed. However, it is relevant to take significant wave height ( $H_{s}$ ) in account, particularly for low wind conditions. Consequently, a wind/ $H_{s}$ approach is proposed for improved wind retrievals.

Low-Frequency Mean Square Slopes and Dominant Wave Spectral Properties: Toward Tropical Cyclone Remote Sensing

Spectral properties near the dominant wave region influence significantly the surface roughness relevant to ocean remote sensing employing low-frequency microwave sensors. The critical parameters characterizing dominant waves are wind speed and dimensionless spectral peak frequency, which is the inverse wave age. The dimensionless spectral peak frequency can be expressed as an equivalent dimensionless fetch or duration. The connection between dominant waves and surface roughness is the spectral slope. This paper presents a surface wave spectral model designed for low-frequency microwave remote sensing, with special emphasis on tropical cyclone (TC) applications. The key elements of the spectral model are: 1) a general spectral function with coefficients accommodating a variable spectral slope and 2) a parametric function connecting the spectral slope and wind speed, which is established with the mean square slope (MSS) observations obtained inside hurricanes by the global positioning system reflectometry technique. In order to make use of the MSS observations inside hurricanes, parametric models of the spatial distributions of wind speed and dimensionless spectral peak frequency inside TCs are developed. The parametric models are based on the wind and wave similarity relationships derived from analyses of hurricane hunter measurements.

Study of Surface Wind and Mean Square Slope Correlation in Hurricane Ike With Multiple Sensors

This paper analyzes and compares multiple techniques for estimating the mean square slope (MSS) of surface waves during Hurricane Ike in the Gulf of Mexico and studies the correlation of the MSS estimates with wind speed measurements along the same tracks. Three separate instruments collected measurements in parallel, including a GPS reflectometry (GPS-R) receiver, a stepped frequency microwave radiometer (SFMR), and a wide swath radar altimeter (WSRA). These datasets were used to study the correlation between the flight level and near-surface wind and MSS during Hurricane Ike in 2008. The GPS-R, SFMR, and WSRA instruments recorded temporally and spatially coincident data during two passes over the hurricane eye. This paper estimates the ocean surface MSS using GPS-R for two eye transects using: a least squares model fitting technique, the reflected signal waveform width, and an integration of the reflected signals in an area around the peak. Subsequently, the correlations between the GPS-R and WSRA MSS estimates and the SFMR wind speed estimates are compared to reveal regions of high and low surface MSS to wind speed correlation. Finally, a relationship between wind and MSS was derived from the GPS-R and SFMR data and is compared to existing MSS/wind models, including the results obtained by Katzberg et al. for hurricane conditions.

Quantification of the Relationship Between Sea Surface Roughness and the Size of the Glistening Zone for GNSS-R

A formulation of the relationship between sea-surface roughness and extension of the glistening zone (GZ) of a Global Navigation Satellite System Reflectometry (GNSS-R) system is presented. First, an analytical expression of the link between GZ area, viewing geometry, and surface mean square slope (MSS) is derived. Then, a strategy for retrieval of surface roughness from the delay-Doppler map (DDM) is illustrated, including details of data preprocessing, quality control, and GZ area estimation from the DDM. Next, an example for application of the proposed approach to spaceborne GNSS-R remote sensing is provided, using DDMs from the TechDemoSat-1 mission. The algorithm is first calibrated using collocated in situ roughness estimates using data sets from the National Data Buoy Center, its retrieval performance is then assessed, and some of the limitations of the suggested technique are discussed. Overall, good correlation is found between buoy-derived MSS and estimates obtained using the proposed strategy ( $r=0.73$ ).

Feasibility of Oil Slick Detection Using BeiDou-R Coastal Simulation

Oil spills, which can cause severe immediate and long-term harm to marine ecological environments for decades after the initial accident, require rapid and accurate monitoring. Currently, optical and radar satellite images are used to monitor oil spills; however, remote sensing generally needs a long revisit period. Global Navigation Satellite System reflected signals (GNSS-R) can provide all-weather and all-day ocean monitoring and is therefore more suitable for oil spill monitoring. To assess the feasibility of the BeiDou Navigation Satellite System reflected signals (BeiDou-R) in detecting oil slicks, a BeiDou-R coastal simulated experiment is performed in this study on the oil slick distribution of an oil pipeline explosion accident. We set up an observation point and selected observation satellites, and a delay-Doppler map (DDM) of an oil-slicked sea surface under coastal scenarios was created by combining the mean-square slope (MSS) model for oil-slicked/clean surfaces and the Zavorotny–Voronovich (Z–V) scattering model. DDM simulation of the coastal scenarios effectively represents the scattering coefficient distribution of the presence of an oil slick. Theoretical analysis revealed that oil slicks can be detected within a radius of less than 5 km around the specular reflection point (SP) for BeiDou-R coastal simulation.

Effect of High-Frequency Sea Waves on Wave Period Retrieval from Radar Altimeter and Buoy Data

Wave periods estimated from satellite altimetry data behave differently from those calculated from buoy data, especially in low-wind conditions. In this paper, the geometric mean wave period T a is calculated from buoy data, rather than the commonly used zero-crossing wave period T z . The geometric mean wave period uses the fourth moment of the wave frequency spectrum and is related to the mean-square slope of the sea surface measured using altimeters. The values of T a obtained from buoys and altimeters agree well (root mean square difference: 0.2 s) only when the contribution of high-frequency sea waves is estimated by a wavenumber spectral model to complement the buoy data, because a buoy cannot obtain data from waves having wavelengths that are shorter than the characteristic dimension of the buoy.

Polarized reflectance and transmittance distribution functions of the ocean surface.

Two aspects of ocean modelling are treated: representation of ocean waves considering all size-classes of waves and tracing of light-interactions at the wavy sea surface. Nonlinear wave profiles are realized accounting for a wide range of climatologically relevant sea states and wind speeds. Polarized ray tracing is used to investigate air-incident and whitecap-free reflectance and transmittance distributions with high angular resolution subject to sea-characterizing parameters, such as significant wave height, peak wave period, wind speed, and surface roughness. Wave-shadowing effects of incident and multiple reflected rays are fully considered. Their influence mostly starts with incidence angles greater than 60°, i.e., when the sun is near the horizon, and is especially pronounced for steep sea states. The net effect of multiple reflections is a redistribution of reflectance and transmittance fractions in their respective hemispheres and a slight increase of the net transmission of light into the sea. Revised reflectance and transmittance distribution functions, RDF and TDF, are provided depending on surface roughness in terms of the mean-square slope; reference is made to other sea state parameters. In comparison with the slope statistics approach, uncertainties related to sun near the horizon are reduced and on average this study yields somewhat higher reflectance values with some variability related to the sea state. By means of provided data, irradiance and radiance reflectances can be computed using desired sky radiance distributions, e.g., clear sky, overcast or partly cloudy sky, as well as wind or sea state information including wave propagation direction.

The Construction of a Three-Dimensional Antenna Gain Matrix and Its Impact on Retrieving Sea Surface Mean Square Slope Based on Aircraft Wave Spectrometer Data

AbstractIn this study, a rotating frequency-modulated continuous wave (FMCW) radar is installed on an aircraft to retrieve the sea wave spectra. Because the aircraft attitude angles produce the incorrect antenna gain used in the radar equation, the incorrect normalized radar cross section (NRCS) of the sea surface will be acquired. To eliminate the effect of the angles, a three-dimensional matrix of the radar antenna gain is constructed by means of coordinate transformation and interpolation, based on a large set of configurations of the aircraft attitude angles (roll, pitch, etc.). With the application of the matrix, the NRCS of the sea surface is corrected and the calculating time is reduced. Then the sea surface mean square slope (MSS) is obtained from the echoes of the airborne wave spectrometer. Considering a weak periodicity of MSS due to low sea state, four images are presented to show the variation of the MSS after aircraft attitude angle correction. The results indicate that the accurate incidence angle of the antenna beam center is critical for retrieving the sea surface MSS, and that the magnitude of the MSS from three cycles of radar echoes can be changed by as much as 40% within 5° of the attitude angles. Furthermore, the MSS becomes more periodic and regular after correction.

Revisiting the Short-Wave Spectrum of the Sea Surface in the Light of the Weighted Curvature Approximation

Existing models for the short-wave spectrum of the sea surface are not consistent with microwave satellite data when multi-bands and multi-incidence data sets are considered. We devise a simple parametric model for the short-wave omnidirectional spectrum of the sea surface on the basis of a three-band (C, Ku, and Ka) and multi-incidence (low, moderate, and large) data set and an improved analytical scattering model, namely the non-Gaussian Weighted Curvature Approximation. This spectrum is also constrained by several optical measurements which provide a priori conditions on the total and filtered mean-square slopes. It is compared with classical models such as Elfouhaily and Kudryavtsev unified curvature spectra. Significant differences are observed at wave numbers corresponding to the range of decimeter scales. The new spectrum is by construction fully consistent with the omnidirectional normalized radar cross section of the multi-band data set.

Dependence of mean square slope on wave state and its application in altimeter wind speed retrieval

Mean square slope (MSS) of sea surface is a parameter describing the sea surface roughness and plays a key role in understanding the sea surface dynamics. Although MSS is influenced by many factors such as wind speed and sea surface waves, it has been traditionally parameterized by wind speed only. In this study, the surface wave impact on MSS is investigated using a collocated data set of altimeter and buoy measurements. It is found that the MSS detected by Ku-band altimeter is closely related to the wind wave components and increases with the degree of wind wave development; however, it is almost independent of the swell. The wave effect on MSS is mainly ascribed to the contribution of longer waves rather than shorter waves. An analytical spectral MSS model is proposed, and it is shown that the MSS calculated from the model agrees well with the altimeter-measured MSS when the pseudo-wave age is adjusted to the wave age corresponding to wind waves. This model is applied to derive wind speeds from altimeter data including the normalized radar cross section and the significant wave height of sea surface waves. With the collocated data set, it is shown that this new algorithm performs better than previous empirical algorithms.

On some consequences of the canonical transformation in the Hamiltonian theory of water waves

We discuss some consequences of the canonical transformation in the Hamiltonian theory of water waves (Zakharov, J. Appl. Mech. Tech. Phys., vol. 9, 1968, pp. 190–194). Using Krasitskii's canonical transformation we derive general expressions for the second-order wavenumber and frequency spectrum and the skewness and the kurtosis of the sea surface. For deep-water waves, the second-order wavenumber spectrum and the skewness play an important role in understanding the so-called sea-state bias as seen by a radar altimeter. According to the present approach but in contrast with results obtained by Barrick & Weber (J. Phys. Oceanogr., vol. 7, 1977, pp. 11–21), in deep water second-order effects on the wavenumber spectrum are relatively small. However, in shallow water in which waves are more nonlinear, the second-order effects are relatively large and help to explain the formation of the observed second harmonics and infra-gravity waves in the coastal zone. The second-order effects on the directional-frequency spectrum are as a rule more important; in particular it is shown how the Stokes-frequency correction affects the shape of the frequency spectrum, and it is also discussed why in the context of the second-order theory the mean-square slope cannot be estimated from time series. The kurtosis of the wave field is a relevant parameter in the detection of extreme sea states. Here, it is argued that in contrast perhaps to one's intuition, the kurtosis decreases while the waves approach the coast. This is related to the generation of the wave-induced current and the associated change in mean sea level.