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.     

Retrieval of elements of the columnar aerosol scattering phase matrix from polarized sky radiance over the ocean: simulations

We have extended the Wang-Gordon [Appl. Opt. 32, 4598-4609 (1993)] and Gordon-Zhang [Appl. Opt.34, 5552-5555 (1995)] algorithms for retrieval of omega(0)P(?, where omega(0) is the aerosol single-scattering albedo and P(?) is the aerosol phase function for scattering through an angle ?, from measurement of the radiances exiting the top and the bottom of the atmosphere over the ocean, to include polarization. This permits derivation of the P(11)(?) and P(12)(?) elements of the Mueller scattering phase matrix P(?) from measurement of the linear polarization portion of the Stokes vectors associated with the radiance exiting the top and the bottom of the atmosphere. Simulations show that good retrievals are possible for aerosol optical thicknesses as large as 2; however, the atmosphere is required to be horizontally homogeneous. We study the influence of the elements of P(?) that cannot be determined in this manner. It is shown that including surface measurements of the linear polarization of the sky radiance improves the estimation of the radiance simultaneously exiting the top of the atmosphere (TOA) and also allows reasonably accurate estimates of the TOA polarization. This is important for in-orbit calibration of ocean-color sensors.     

Effects of ocean surface reflectance variation with solar elevation on normalized water-leaving radiance

Effects of the ocean surface reflection for solar irradiance on the normalized water-leaving radiance in the visible wavelengths are evaluated and discussed for various conditions of the atmosphere, solar-zenith angles, and wind speeds. The surface reflection effects on water-leaving radiance are simply due to the fact that the radiance that is backscattered out of the water is directly proportional to the downward solar irradiance just beneath the ocean surface. The larger the solar-zenith angle, the less the downward solar irradiance just beneath the ocean surface (i.e., more photons are reflected by the ocean surface), leading to a reduced value of the radiance that is backscattered out of the ocean. For cases of large solar-zenith angles, the effects of surface irradiance reflection need to be accounted for in both the satellite-derived and in situ measured water-leaving radiances.     

Moderate-resolution imaging spectroradiometer ocean color polarization correction

The polarization correction for the Moderate-Resolution Imaging Spectroradiometer (MODIS) instruments on the Terra and Aqua satellites is described. The focus is on the prelaunch polarization characterization and on the derivation of polarization correction coefficients for the processing of ocean color data. The effect of the polarization correction is demonstrated. The radiances at the top of the atmosphere need to be corrected by as much as 3.2% in the 412 nm band. The effect on the water-leaving radiances can exceed 50%. The polarization correction produces good agreement of the MODIS Aqua water-leaving radiance time series with data from another, independent satellite-based ocean color sensor, the Sea-Viewing Wide Field-of-View Sensor (SeaWiFS).     

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.     

Extension of Chandrasekhar's formula to a nonhomogeneous Lambertian surface and comparison with the 6S formulation

The classical Chandrasekhar's formula, which relates the surface albedo to the top of the atmosphere radiance, rigorously applies to a homogeneous Lambertian surface. For a nonhomogeneous Lambertian surface in a plane-parallel atmosphere, an extension of this formula was proposed in the 1980s and has been implemented recently in the 6S algorithm. To analyze this extension, the rigorous formula of the top of the atmosphere signal in a plane-parallel atmosphere bounded by a nonhomogeneous Lambertian surface is derived. Then the 6S algorithm extension is compared to the exact formula and approximations and their validity is examined. The derivation of the exact formula is based on the separation of the radiation fields into direct and diffuse components, on the introduction of the Green's function of the problem, and on integrations of boundary values of the radiation fields with Green's function.     

Normalized water-leaving radiance: revisiting the influence of surface roughness

Many spaceborne sensors have been deployed to image the ocean in the visible portion of the spectrum. Information regarding the concentration of water constituents is contained in the water-leaving radiance-the radiance that is backscattered out of the water and subsequently propagates to the top of the atmosphere. Recognizing that it depends on the viewing and Sun geometry, ways have been sought to normalize this radiance to a single Sun-viewing geometry--forming the normalized water-leaving radiance. This requires understanding both the bidirectional nature of the upwelling radiance just beneath the surface and the interaction of this radiance with the air-water interface. I believe that the latter has been incorrectly computed in the past when a water surface roughened by the wind is considered. The presented computation suggests that, for wind speeds as high as 20 m/s, the influence of surface roughness is small for a wide range of Sun-viewing geometries, i.e., the transmittance of the (whitecap-free) air-water interface is nearly identical (within 0.01) to that for a flat interface.     

Atmospheric correction of ocean color sensors: analysis of the effects of residual instrument polarization sensitivity

We provide an analysis of the influence of instrument polarization sensitivity on the radiance measured by spaceborne ocean color sensors. Simulated examples demonstrate the influence of polarization sensitivity on the retrieval of the water-leaving reflectance rho(w). A simple method for partially correcting for polarization sensitivity-replacing the linear polarization properties of the top-of-atmosphere reflectance with those from a Rayleigh-scattering atmosphere-is provided and its efficacy is evaluated. It is shown that this scheme improves rho(w) retrievals as long as the polarization sensitivity of the instrument does not vary strongly from band to band. Of course, a complete polarization-sensitivity characterization of the ocean color sensor is required to implement the correction.     

Polarimetric measurements of long-wave infrared spectral radiance from water

Polarimetric measurements of the thermal infrared spectral radiance from water are reported and are compared with calculations from a recently published model over the spectral range of 600-1600 cm(-1) (6.25-16.67-mum wavelength). In this spectral range, warm water viewed under a dry, clear atmosphere appears vertically polarized by 6-12%. The measured spectral degree of polarization agrees with calculations within the measurement uncertainty (~0.5% polarization in spectral regions with high atmospheric transmittance and 1.5% polarization in spectral regions with low atmospheric transmittance). Uncertainty also arises from temporal changes in water and air temperatures between measurements at orthogonal polarization states, indicating the desirability of simultaneous measurements for both polarization states.     

Infrared radiance and solar glint at the ocean-sky horizon

An analytic model is developed for the mean and clutter infrared radiance emitted from the ocean surface near the horizon and in the presence of solar glint. The model is based on the identification of a characteristic facet dimension over which the ocean surface is essentially flat. Fluctuations in the facet orientation generated by the water wave motion are modeled by a parameterized wave height power spectral density that provides the two orthogonal wave slope variances. The mean and root-meansquare facet radiances are calculated with Gaussian probability-density functions for the wave slopes. One can determine the number of facets within the field of view of a single detector by estimating the exposed ocean area and dividing by the facet area. This estimation takes into account shadowing effects of the swell wave, the swell wavelength, and the transverse detector field of view. The number of exposed facets together with the central-limit theorem permits computation of the radiance clutter as a function of look-down angle below the horizon. Vertical radiance profiles, parameterized by the azimuthal offset from the solar position, are calculated over a sensor look-down angle range of ±50 mrad about the horizon. The results of this analysis are compared with infrared radiance measurements of the ocean surface near the horizon and in the presence of solar glint. Agreement between the measured and calculated values of the mean and clutter radiances is good.     

Degree of linear polarization in spectral radiances from water-viewing infrared radiometers

Infrared radiances from water become partially polarized at oblique viewing angles through both emission and reflection. I describe computer simulations that show how the state of polarization for water varies with environmental conditions over a wavelength range of 3-15 mum with 0.05-mum resolution. Polarization at wavelengths longer than approximately 4 mum generally is negative (p, or vertical) and increases with incidence angle up to approximately 75 degrees , beyond which the horizontally polarized reflected atmospheric radiance begins to dominate the surface emission. The highest p polarization (~4-10%) is found in the atmospheric window regions of approximately 4-5 and 8-14 mum. In the 3-5-mum spectral band, especially between 3 and 4 mum, reflected atmospheric radiance usually is greater than surface emission, resulting in a net s polarization (horizontal). The results of these simulations agree well with broadband measurements of the degree of polarization for a water surface viewed at nadir angles of 0-75 degrees .     

Four-stream solution for atmospheric radiative transfer over a non-Lambertian surface

An analytical model characterizing the atmospheric radiance field over a non-Lambertian surface divides the radiation field into three components: unscattered radiance, single-scattering radiance, and multiple-scattering radiance. The first two components are calculated exactly. A δ-four-stream approximation is extended to calculate the azimuth-independent multiple-scattering radiance over a non-Lambertian surface, which is modeled by a statistical bidirectional reflectance distribution function (BRDF). Accuracy is assessed with respect to the exact results computed from a Gauss-Seidel iterative algorithm. Experiments comparing the results obtained with Lambertian and non-Lambertian surfaces show that incorporating the BRDF into the four-stream approximation significantly improves the accuracy in calculating radiance as well as radiative flux.     

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.     

Extension of Chandrasekhar's formula to a homogeneous non-Lambertian surface and comparison with the 6S formulation

The classical Chandrasekhar's formula relating the surface reflectance to the top of the atmosphere radiance rigorously applies to a Lambertian surface. For a homogeneous non-Lambertian surface in a plane-parallel atmosphere, an extension of this formula was proposed in the 1980s and has been recently implemented in the second simulation of the satellite signal in the solar spectrum (6S) algorithm. To analyze this extension, the rigorous formula of the top of the atmosphere signal is derived in a plane-parallel atmosphere bounded by a homogeneous non-Lambertian surface. Then the 6S algorithm extension is compared with the exact formula and approximations and their validity are pointed out. The methods used for the derivation of the exact formula are classical. They are based on the separation of direct and diffuse components of the radiation fields, on the introduction of the Green's function of the problem, and on integrations of boundary values of the radiation fields with the Green's function.     

Path-radiance correction by polarization observation of Sun glint glitter for remote measurements of tropospheric greenhouse gases

High-accuracy remote measurement of greenhouse gases is hampered by contamination of the field of view by the path radiance of solar radiation scattered from clouds and aerosols. A method is proposed for eliminating the effect of path radiance by differentiating two components of polarized light. The polarization of path radiance is measured directly at the wave-number region of strong water-vapor absorption. Using this measurement, we eliminate the components of path radiance involved in other bands, which are used for greenhouse gas measurements, by differentiating two components of the polarized light. It is shown that the effect of path radiance on retrieving the column amount of gases potentially can be reduced to below 0.1%.     

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.     

Columnar aerosol properties over oceans by combining surface and aircraft measurements: simulations

We report an algorithm that can be used to invert the radiance exiting the top and bottom of the atmosphere to yield the columnar optical properties of atmospheric aerosol under clear sky conditions over the oceans. The method is an augmentation of a similar algorithm presented by Wang and Gordon [Appl. Opt. 32, 4598 (1993)] that used only sky radiance, and therefore was incapable of retrieving the aerosol phase function at the large scattering angles that are of critical importance in remote sensing of oceanic and atmospheric properties with satellites. Well-known aerosol models were combined with radiative transfer theory to simulate pseudodata for testing of the algorithm. The tests suggest that it should be possible to retrieve the aerosol phase function and the aerosol single-scattering albedo accurately over the visible spectrum at aerosol optical thicknesses as large as 2.0. The algorithm is capable of retrievals with such large optical thicknesses because all significant orders of multiple scattering are included. We believe that combining an algorithm of this type with surface-based and high-altitude aircraft-based radiance measurements could be useful for studying aerosol columnar optical properties over oceans and large lakes. The use of the retrieval method is possible over the ocean because, unlike the land surface, the albedo of the ocean is low and spatially uniform.     

Polarized radiance distribution measurement of skylight. II. Experiment and data

Measurements of the skylight polarized radiance distribution were performed at different measurement sites, atmospheric conditions, and three wavelengths with our newly developed Polarization Radiance Distribution Camera System (RADS-IIP), an analyzer-type Stokes polarimeter. Three Stokes parameters of skylight (I, Q, U), the degree of polarization, and the plane of polarization are presented in image format. The Arago point and neutral lines have been observed with RADS-IIP. Qualitatively, the dependence of the intensity and polarization data on wavelength, solar zenith angle, and surface albedo is in agreement with the results from computations based on a plane-parallel Rayleigh atmospheric model.     

Experimental validation of the MODTRAN 5.3 sea surface radiance model using MIRAMER campaign measurements

Radiometric images taken in mid-wave and long-wave infrared bands are used as a basis for validating a sea surface bidirectional reflectance distribution function (BRDF) being implemented into MODTRAN 5 (Berk et al. [Proc. SPIE5806, 662 (2005)]). The images were obtained during the MIRAMER campaign that took place in May 2008 in the Mediterranean Sea near Toulon, France. When atmosphere radiances are matched at the horizon to remove possible calibration offsets, the implementation of the BRDF in MODTRAN produces good sea surface radiance agreement, usually within 2% and at worst 4% from off-glint azimuthally averaged measurements. Simulations also compare quite favorably to glint measurements. The observed sea radiance deviations between model and measurements are not systematic, and are well within expected experimental uncertainties. This is largely attributed to proper radiative coupling between the surface and the atmosphere implemented using the DISORT multiple scattering algorithm.     

Reduction of skylight reflection effects in the above-water measurement of diffuse marine reflectance: comment

Use of a vertical polarizer has been suggested to reduce the effects of surface reflection in the above-water measurements of marine reflectance. We suggest using a similar technique for airborne or spaceborne sensors when atmospheric scattering adds its own polarization signature to the upwelling radiance. Our own theoretical sensitivity study supports the recommendation of Fougnie et al. [Appl. Opt. 38, 3844 (1999)] (40-50 degrees vertical angle and azimuth angle near 135 degrees , polarizer parallel to the viewing plane) for above-water measurements. However, the optimal viewing directions (and the optimal orientation of the polarizer) change with altitude above the sea surface, solar angle, and atmospheric vertical optical structure. A polarization efficiency function is introduced, which shows the maximal possible polarization discrimination of the background radiation for an arbitrary altitude above the sea surface, viewing direction, and solar angle. Our comment is meant to encourage broader application of airborne and spaceborne polarization sensors in remote sensing of water and sea surface properties.