Analytical solution of radiative transfer in the coupled atmosphere-ocean system with a rough surface

Using the computationally efficient discrete-ordinate method, we present an analytical solution for radiative transfer in the coupled atmosphere-ocean system with a rough air-water interface. The theoretical formulations of the radiative transfer equation and solution are described. The effects of surface roughness on the radiation field in the atmosphere and ocean are studied and compared with satellite and surface measurements. The results show that ocean surface roughness has significant effects on the upwelling radiation in the atmosphere and the downwelling radiation in the ocean. As wind speed increases, the angular domain of sunglint broadens, the surface albedo decreases, and the transmission to the ocean increases. The downward radiance field in the upper ocean is highly anisotropic, but this anisotropy decreases rapidly as surface wind increases and as ocean depth increases. The effects of surface roughness on radiation also depend greatly on both wavelength and angle of incidence (i.e., solar elevation); these effects are significantly smaller throughout the spectrum at high Sun. The model-observation discrepancies may indicate that the Cox-Munk surface roughness model is not sufficient for high wind conditions.     

Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. II. The hybrid matrix operator--Monte Carlo method

A hybrid method is developed to solve the vector radiative transfer equation (VRTE) in a three-dimensional atmosphere-ocean system (AOS). The system is divided into three parts: the atmosphere, the dielectric interface, and the ocean. The Monte Carlo method is employed to calculate the impulse response functions (Green functions) for the atmosphere and ocean. The impulse response function of the dielectric interface is calculated by the Fresnel formulas. The matrix operator method is then used to couple these impulse response functions to obtain the vector radiation field for the AOS. The primary advantage of this hybrid method is that it solves the VRTE efficiently in an AOS with different dielectric interfaces while keeping the same atmospheric and oceanic conditions. For the first time, we present the downward radiance field in an ocean with a sinusoidal ocean wave.     

Assessment of the ultraviolet radiation field in ocean waters from space-based measurements and full radiative-transfer calculations

Quantitative assessment of the UV effects on aquatic ecosystems requires an estimate of the in-water radiation field. Actual ocean UV reflectances are needed for improving the total ozone retrievals from the total ozone mapping spectrometer (TOMS) and the ozone monitoring instrument (OMI) flown on NASA's Aura satellite. The estimate of underwater UV radiation can be done on the basis of measurements from the TOMS/OMI and full models of radiative transfer (RT) in the atmosphere-ocean system. The Hydrolight code, modified for extension to the UV, is used for the generation of look-up tables for in-water irradiances. A look-up table for surface radiances generated with a full RT code is input for the Hydrolight simulations. A model of seawater inherent optical properties (IOPs) is an extension of the Case 1 water model to the UV. A new element of the IOP model is parameterization of particulate matter absorption based on recent in situ data. A chlorophyll product from ocean color sensors is input for the IOP model. Verification of the in-water computational scheme shows that the calculated diffuse attenuation coefficient Kd is in good agreement with the measured Kd.     

Neutral points in an atmosphere-ocean system. 1: Upwelling light field

We have developed a Monte Carlo code that utilizes the complete Stokes vector to examine the structure of the degree of linear polarization in the complete observable solid angle at any level in an atmosphere-ocean system. By performing these calculations we are able to compute the positions of neutral points in the upwelling light above and beneath the ocean surface. The locations of these points in a single-scatter calculation and a Monte Carlo treatment are shown for various conditions. The presence of aerosols in the atmosphere and hydrosols in the ocean was found to have an effect on the location of these neutral points.     

How well can radiance reflected from the ocean-atmosphere system be predicted from measurements at the sea surface?

There is interest in the prediction of the top-of-the-atmosphere (TOA) reflectance of the ocean-atmosphere system for in-orbit calibration of ocean color sensors. With the use of simulations, we examine the accuracy one could expect in estimating the reflectance ρ(T) of the ocean-atmosphere system based on a measurement suite carried out at the sea surface, i.e., a measurement of the normalized sky radiance ρ(B) and the aerosol optical thickness (τ(a)), under ideal conditions-a cloud-free, horizontally homogeneous atmosphere. Briefly, ρ(B) and τ(a) are inserted into a multiple-scattering inversion algorithm to retrieve the aerosol optical properties-the single-scattering albedo and the scattering phase function. These retrieved quantities are then inserted into the radiative transfer equation to predict ρ(T). Most of the simulations were carried out in the near infrared (865 nm), where a larger fraction of ρ(T) is contributed by aerosol scattering compared with molecular scattering, than in the visible, and where the water-leaving radiance can be neglected. The simulations suggest that ρ(T) can be predicted with an uncertainty typically Θ1% when the ρ(B) and τ(a) measurements are error free. We investigated the influence of the simplifying assumptions that were made in the inversion-prediction process, such as modeling the atmosphere as a plane-parallel medium, using a smooth sea surface in the inversion algorithm, using the scalar radiative transfer theory, and assuming that the aerosol was confined to a thin layer just above the sea surface. In most cases, these assumptions did not increase the error beyond ±1%. An exception was the use of the scalar radiative transfer theory, for which the error grew to as much as ~2.5%, suggesting that the use of ρ(B) inversion and ρ(T) prediction codes that include polarization would be more appropriate. However, their use would necessitate measurement of the polarization associated with ρ(B). We also investigated the uncertainty introduced by an unknown aerosol vertical structure and found it to be negligible if the aerosols were nonabsorbing or weakly absorbing. An extension of the analysis to the blue, which requires measurement of the water-leaving radiance, showed significantly better predictions of ρ(T) because the major portion of ρ(T) is the result of molecular scattering, which is known precisely. We also simulated the influence of calibration errors in both the Sun photometer and the ρ(B) radiometer. The results suggest that the relative error in the predicted ρ(T) is similar in magnitude to that in ρ(B) (actually it was somewhat less). However, the relative error in ρ(T) induced by error in τ(a) is usually much less than the relative error in τ(a). Currently, it appears that radiometers can be calibrated with an uncertainty of ~±2.5%, therefore it is reasonable to conclude that, at present, the most important error source in the prediction of ρ(T) from ρ(B) is likely to be error in the ρ(B) measurement.     

Reduction of deterministic coupled atmosphere-ocean models to stochastic ocean models: a numerical case study of the Lorenz-Maas system

Our starting point is the Lorenz-Maas coupled atmosphere-ocean model which was proposed by van der Schrier and van Veen and couples the ocean model by Maas with the Lorenz-84 model of the atmosphere. This 6-dimensional model (3-dimensional slow ocean and 3-dimensional fast atmosphere) is, to the knowledge of the authors, the simplest atmosphere-ocean model presently available which is derived from first principles in a controlled manner. This paper is an extensive numerical case study of the model, thereby implementing 'Hasselmann's program,' i.e. applying the various mathematical techniques of reducing the fully coupled deterministic model to a deterministic or stochastic model for the ocean alone, namely to:

the (deterministic) 'statistical model,' using the method of averaging,

the 'linear stochastic model,' based on the central limit theorem for the error in averaging,

the 'nonlinear stochastic model,' also known as 'Hasselmann's equation'. The long-term and bifurcation behaviour of these models are studied and compared. The general result is that in most situations the nonlinear stochastic model outperforms the other ones.     

Irradiance inversion theory to retrieve volume scattering function of seawater

An attempt to retrieve the volume scattering function (VSF) of source-free and no-inelastic-scattering ocean water is made from the upwelling irradiance Eu and downwelling irradiance Ed. It will be shown, from the radiative transfer equation, that the VSF of seawater can be calculated by the planar irradiances when the scattering phase function of the suspended particles in the backward direction and the molecular VSF are known. On the derivation of the hydrosol VSF, several optical properties such as the absorption coefficient a; the scattering coefficients of hydrosol, b, b(f), b(b) and those of the suspended particles, b(p), b(fp), b(bp); the beam attenuation coefficient c; the average cosines mu, mu(d), and mu(u); and the backscattering shape factor for the downwelling light stream, r(du), will also be obtained. On the derivation of those optical parameters, classical knowledge related to interrelationships between inherent optical properties and apparent optical properties and obtained with Monte Carlo numerical simulations is analytically verified. The present theory can be applied to surface waters and any wavelengths, except for waters and wavelengths with an extremely low b(b)/a ratio.     

Radiative transfer in the atmosphere-ocean system: the finite-element method

The finite-element method has been applied to solving the radiative-transfer equation in a layered medium with a change in the refractive index, such as the atmosphere-ocean system. The physical processes that are included in the algorithm are multiple scattering, bottom-boundary bidirectional reflectivity, and refraction and reflection at the interface between the media with different refractive properties. The incident radiation is a parallel flux on the top boundary that is characteristic of illumination of the atmosphere by the Sun in the UV, visible, and near-IR regions of the electromagnetic spectrum. The necessary changes, compared with the case of a uniformly refracting layered medium, are described. An energy-conservation test has been performed on the model. The algorithm has also been validated through comparison with an equivalent backward Monte Carlo code and with data taken from the literature, and optimal agreement was shown. The results show that the model allows energy conservation independently of the adopted phase function, the number of grid points, and the relative refractive index. The radiative-transfer model can be applied to any other layered system with a change in the refractive index. The fortran code for this algorithm is documented and is available for applications.     

Ocean source and optical property estimation from explicit and implicit algorithms

We solve an inverse problem of ocean optics for estimating spatially dependent absorption and scattering coefficients and for determining sources such as fluorescence, bioluminescence, or Raman scattering. The solution requires in situ measurement of the downward and upward plane irradiances and scalar irradiances and a priori estimation of the angular shape of the volume scattering function. Both an explicit algorithm and an implicit one are developed from new two-stream radiative-transfer equations that utilize an asymptotic radiance approximation to close the set of equations. A comparison of numerical tests for the two algorithms is given.     

Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. I. Monte Carlo method

We have developed a powerful 3D Monte Carlo code, as part of the Radiance in a Dynamic Ocean (RaDyO) project, which can compute the complete effective Mueller matrix at any detector position in a completely inhomogeneous turbid medium, in particular, a coupled atmosphere-ocean system. The light source can be either passive or active. If the light source is a beam of light, the effective Mueller matrix can be viewed as the complete impulse response Green matrix for the turbid medium. The impulse response Green matrix gives us an insightful way to see how each region of a turbid medium affects every other region. The present code is validated with the multicomponent approach for a plane-parallel system and the spherical harmonic discrete ordinate method for the 3D scalar radiative transfer system. Furthermore, the impulse response relation for a box-type cloud model is studied. This 3D Monte Carlo code will be used to generate impulse response Green matrices for the atmosphere and ocean, which act as inputs to a hybrid matrix operator-Monte Carlo method. The hybrid matrix operator-Monte Carlo method will be presented in part II of this paper.     

Cross sections for surface plasmon polariton scattering from a nanoparticle in the dipole approximation

The differential and total cross sections for the scattering of surface electromagnetic waves in the optical frequency range (surface plasmon polaritons, SPPs) from a small particle into waves in the near-field zone (SPPs) and into waves propagating from the surface to the far-field zone have been calculated within the framework of the dipole approximation. The efficiencies of these scattering channels are compared as dependent on the main parameters of the system. An increase in the wavelength of the radiation exciting SPPs at a plane dielectric-metal interface (air-gold) may lead to a change in the most effective scattering channel.     

Relative importance of multiple scattering by air molecules and aerosols in forming the atmospheric path radiance in the visible and near-infrared parts of the spectrum

Single and multiple scattering by molecules or by atmospheric aerosols only (homogeneous scattering), and heterogeneous scattering by aerosols and molecules, are recorded in Monte Carlo simulations. It is shown that heterogeneous scattering (1) always contributes significantly to the path reflectance (rho(path)), (2) is realized at the expense of homogeneous scattering, (3) decreases when aerosols are absorbing, and (4) introduces deviations in the spectral dependencies of reflectances compared with the Rayleigh exponent and the aerosol angstrom exponent. The ratio of rho(path) to the Rayleigh reflectance for an aerosol-free atmosphere is linearly related to the aerosol optical thickness. This result provides a basis for a new scheme for atmospheric correction of remotely sensed ocean color observations.     

Monte Carlo and discrete-ordinate simulations of spectral radiances in a coupled air-tissue system

We perform a detailed comparison study of Monte Carlo (MC) simulations and discrete-ordinate radiative-transfer (DISORT) calculations of spectral radiances in a 1D coupled air-tissue (CAT) system consisting of horizontal plane-parallel layers. The MC and DISORT models have the same physical basis, including coupling between the air and the tissue, and we use the same air and tissue input parameters for both codes. We find excellent agreement between radiances obtained with the two codes, both above and in the tissue. Our tests cover typical optical properties of skin tissue at the 280, 540, and 650 nm wavelengths. The normalized volume scattering function for internal structures in the skin is represented by the one-parameter Henyey-Greenstein function for large particles and the Rayleigh scattering function for small particles. The CAT-DISORT code is found to be approximately 1000 times faster than the CAT-MC code. We also show that the spectral radiance field is strongly dependent on the inherent optical properties of the skin tissue.     

Effects of ocean waves on airborne lidar imaging

The effects of ocean waves on lidar imaging of submerged objects are investigated. Two significant consequences of wave focusing or defocusing are quantified: (a) intensification of near-surface backscatter in which the mean return is increased relative to that for a flat interface, and (b) spatial-temporal modulations of the backscattered return. For the former, mean returns can be as much as 50% larger than flat surface returns at shallow depth. For the latter, the strong modulations induced by wave motion present a dominant clutter field that significantly affects the imaging of shallow objects. Both effects are compensated at greater depths by beam spreading caused by multiple scattering, which diminishes the intensity of the wave focusing.     

Simulation of three-dimensional elastoacoustic wave propagation based on a Discontinuous Galerkin spectral element method

In this article, we present a numerical discretization of the coupled elastoacoustic wave propagation problem based on a discontinuous Galerkin spectral element approach in a three-dimensional setting. The unknowns of the coupled problem are the displacement field and the velocity potential, in the elastic and the acoustic domains, respectively, thereby resulting in a symmetric formulation. After stating the main theoretical results, we assess the performance of the method by convergence tests carried out on both matching and nonmatching grids, and we simulate realistic scenarios where elastoacoustic coupling occurs. In particular, we consider the case of Scholte waves, the scattering of elastic waves by an underground acoustic cavity, and a problem of marine seismic exploration. Numerical simulations are carried out by means of the code SPEED , available at http://speed.mox.polimi.it .     

Degree of radiance and polarization of the upwelling radiation from an atmosphere-ocean system

The radiance and degree of linear polarization of the upward radiation emerging from the top of the terrestrial atmosphere bounded by a ruffled ocean surface are computed in the wavelength region ranging from 0.40 to 0.80 microm with the aid of the adding method. The ruffled ocean surface is treated as an interacting interface, where the radiation transmitted diffusely from below the ocean surface into the atmosphere is also taken into account. Computational results show that the simultaneous measurement of radiance and polarization degree from space makes it possible to derive atmospheric and oceanic parameters.     

Linearization of Markov chain formalism for vector radiative transfer in a plane-parallel atmosphere/surface system

The Markov chain formalism for polarized radiative transfer through a vertically inhomogeneous atmosphere is linearized comprehensively with respect to the aerosol and polarizing surface properties. For verification, numerical results are compared to those obtained by the finite difference method. We demonstrate the use of the linearized code as part of a retrieval of aerosol and surface properties for an atmosphere overlying a black and Fresnel-reflecting ocean surface.     

Neutral points in an atmosphere-ocean system. 2: Downwelling light field

We use a Monte Carlo code that calculates the complete Stokes vector to predict the degree of polarization in the complete observable solid angle at any level in an atmosphere-ocean system. Using the Stokes vector components, we can find the positions of neutral points in a simulated plane-parallel atmosphere-ocean system for various conditions. We examine the locations and behavior of these neutral points for an observer placed directly above and beneath the air-water boundary and show how their positions are influenced by different atmospheric and oceanic conditions.