Radiative Transfer Theory Research Articles

Asymptotic Behavior of the Solution for One Class of Nonlinear Integral Equations of Hammerstein Type on the Whole Axis

A class of nonlinear integral equations on the whole axis with a noncompact integral operator of Hammerstein type is investigated. This class of equations has applications in various fields of natural science. In particular, such equations are found in mathematical biology, in the kinetic theory of gases, in the theory of radiation transfer, etc. The existence of a nonnegative nontrivial and bounded solution is proved. The asymptotic behavior of the constructed solution on ±∞ is studied. In one important special case, the uniqueness of the constructed solution in a certain weighted space is established. At the end of the work, specific applied examples of the equations under study are given.

On a Nonlinear Problem for a System of Integro-Differential Equations of Radiative Transfer Theory

On a Nonlinear Problem for a System of Integro-Differential Equations of Radiative Transfer Theory

Monte Carlo simulations in anomalous radiative transfer: tutorial

Anomalous radiative transfer (ART) theory represents a generalization of classical radiative transfer theory. The present tutorial aims to show how Monte Carlo (MC) codes describing the transport of photons in anomalous media can be implemented. We show that the heart of the method involves suitably describing, in a "non-classical" manner, photon steps starting from fixed light sources or from boundaries separating regions of the medium with different optical properties. To give a better sense of the importance of these particular photon step lengths, we also show numerically that the described approach is essential in preserving the invariance property for light propagation. An interesting byproduct of the MC method for ART is that it allows us to simplify the structure of "classical" MC codes, utilized, for example, in biomedical optics.

An Improved Model for Light Transport in the Color Conversion Element of Light-Emitting Diodes

Light transport with fluorescence in a phosphor layer based on radiative transfer theory is an efficient tool for understanding the performance of a phosphor-converted light-emitting diode. In this work, the models based on radiative transfer theory including light conversion of phosphor particles are developed for calculating the light transport in planar remote phosphor layers. The models developed include an ordinary differential approximation and an improved model based on the differential approximation and an integral formulation of the transmission light flux for a phosphor layer with Fresnel boundaries. We calculate the light extraction efficiency (LEE) of a phosphor layer as a function of various parameters, such as the thickness of the layer and the concentration of phosphor. The present models are validated by comparing the results obtained by the present methods, the double spherical harmonics method of order one and a Monte Carlo method. Comparing the results obtained by those methods, one can see that the improvement based on the differential approximation and integral formulation of transmission light flux for calculating the LEE can yield better results for optically thin cases.

Polarization Reconstruction Algorithm of Target Based on the Analysis of Noise in Complex Underwater Environment

How to effectively eliminate interference such as scattering, absorption, and attenuation is a hot topic of underwater photoelectric detection at present. Around the hot issues, this paper carries out studying the method of polarization-imaging recovery in a dynamic complex underwater environment from the theory of underwater radiation transfer, and numerical simulation of imaging interference characteristics to the simulation of underwater environment experiment. First, by conducting the analysis and simulation of scattering characteristics of underwater suspension particles and bubble by using the theory of radiation transfer, and taking advantage of quantitative description on changing tendency of radiation intensity and polarization properties of light waves in turbid water under the condition of scattering interference. Second, by constructing an underwater target polarization reconstruction model on the basis of the Mueller matrix analysis, and taking target polarization characteristic into reconstruction model on the basis of classical Schechner’s model, automatically estimating polarization information of target by the method of covariance. Finally, by building a polarization imaging system in the simulated complex underwater environment that contains bubble and suspended particles, obtaining reconstructed results with different underwater environments and different materials of target. According to experiment results, and compared with other traditional methods, using the proposed method in this paper can get higher resolution and higher contrast of target in the reconstructed result.

Cloud tomographic retrieval algorithms. II: Adjoint method

A cloud tomographic retrieval algorithm relying on (i) the spherical harmonics discrete ordinate method for radiative transfer calculation and (ii) the adjoint radiative transfer theory for computing the gradient of the objective function has been designed. In order to escape local minima and to increase the efficiency of the retrieval algorithm, the computation of the gradient of the objective function by the adjoint method has been combined with that of the gradient of a surrogate function. The retrieval algorithm uses regularization and accelerated projected gradient methods endowed with a step length procedure. The performances of the retrieval algorithm as compared to those of a retrieval algorithm based on the surrogate minimization method are analyzed on a few synthetic problems.

ON NONTRIVIAL SOLVABILITY OF ONE CLASS OF NONLINEAR INTEGRAL EQUATIONS WITH CONSERVATIVE KERNEL ON THE POSITIVE SEMI-AXIS

The work is devoted to a special class of nonlinear integral equations on the positive semi-axis with conservative kernel that corresponds to a nonlinear operator, for which the property of complete continuity in the space of bounded functions fails. In different special cases this class of equations has applications in particular branches of mathematical physics. In particular, this kind of equations can be met in the radiative transfer theory, kinetic theory of gases, kinetic theory of plasma and in the $p$-adic open-closed string theory. Using a combination of special iterations with the monotonic operator theory methods, that work in defined conical segments it is possible to prove a constructive existence theorem of nonnegative nontrivial bounded solution that has finite limit at infinity. The asymptotics of the constructed solution will also be studied. It is also given an example of nonlinear equation, for which the uniqueness of the solution in the space of bounded functions fails. At the end of the paper will consider some classes of equations both applied and pure theoretical character.

Retrieving the Infected Area of Pine Wilt Disease-Disturbed Pine Forests from Medium-Resolution Satellite Images Using the Stochastic Radiative Transfer Theory

Pine wilt disease (PWD) is a global destructive threat to forests which has been widely spread and has caused severe tree mortality all over the world. It is important to establish an effective method for forest managers to detect the infected area in a large region. Remote sensing is a feasible tool to detect PWD, but the traditional empirical methods lack the ability to explain the signals and can hardly be extended to large scales. The studies using physically-based models either ignore the within-canopy heterogeneity or rely too much on prior knowledge. In this study, we propose an approach to retrieve PWD infected areas from medium-resolution satellite images of two phases based on the simulations of an extended stochastic radiative transfer model for forests infected by pests (SRTP). A small amount of prior knowledge was used, and a change of background soil was considered in this approach. The performance was evaluated in different study sites. The inversion method performs best in the three-dimensional model LESS simulation sample plots (R2 = 0.88, RMSE = 0.059), and the inversion accuracy decreases in the real forest sample plots. For Jiangxi masson pine stand with large coverage and serious damage, R2 = 0.57, RMSE = 0.074; and for Shandong black pine stand with sparse and a small number of single plant damage, R2 = 0.48, RMSE = 0.063. This study indicates that the SRTP model is more feasible for pest damage inversion over different regions compared with empirical methods. The stochastic radiative transfer theory provides a potential approach for future monitoring of terrestrial vegetation parameters.

Analytical solution of the radiative transfer theory for the coherent backscattering from two-dimensional semi-infinite media.

An analytical solution for coherent backscattering (CBS) in two dimensions was derived by solving the radiative transfer equation. Particularly, the single scattered radiance from a semi-infinite medium containing perpendicularly illuminated cylinders was obtained. At the boundary, a refractive index mismatch was taken into account. Furthermore, the link between the radiance and the CBS was shown in the small angle approximation. An excellent agreement was found between Monte Carlo simulations and the analytical solution. Additionally, it was shown that the often applied solution in the spatial frequency domain for quantifying the CBS delivered significantly different results compared to the derived exact analytical solution.

In-situ quantitative monitoring the organic contaminants uptake onto suspended microplastics in aquatic environments

Microplastics act as a source of organic contaminants in aquatic environments and thus affect their environmental fate and toxicity. Because of the weak and reversible interactions between microplastics and organic species, the organic coronas vary with their surrounding environments. Thus, in order to evaluate the possible environmental risks of microplastics, methods for evaluating the dynamic uptake of organic contaminants onto suspended microplastics in aquatic environments are greatly desired. In this work, a UV–vis spectroscopy-based approach was developed for in-situ monitoring organic contaminants uptake onto suspended microplastics after correcting the light scattering interference from microplastics suspensions and establishing the nonlinear relationship between concentration and light absorbance of organic species. The inverse adding-doubling method based on radiative transfer theory was adopted to correct the light scattering effect of suspensions. Then, the resulting mixed absorption spectra were decomposed to calculate the concentrations of the aqueous and adsorbed organic species simultaneously with a nonlinear calibration method. The uptake processes of bisphenol A and p-nitrophenol onto nylon 66 microparticles were monitored with this approach and confirmed by high-performance liquid chromatography analysis. The approach was validated by applying it to natural water samples, and the equilibrium adsorption capacity was found to be interfered mainly by the protein-like substances. This approach has high accuracy, good reproducibility, remarkable universality, and ease of handling, and also provides a potential tool for characterizing the corona formation process on suspended particles both in natural and artificial environments, such as eco-corona formation and engineering surface modification on nano/micro-particles.

Satellite retrieval of benthic reflectance by combining lidar and passive high-resolution imagery: Case-I water

Under the background of global change, increasing attention has been paid to the changes of benthic habitats in shallow ocean ecosystems (e.g., seagrass beds and coral reefs). Optical satellite remote sensing via both active and passive methods plays an important role in monitoring the health of benthic habitats by retrieving benthic reflectance spectra, but it remains difficult to accurately retrieve benthic reflectance spectra from only active or passive remote sensing because of the coupling between water column scattering and benthic reflectance. Here, we developed a semi-analytical model to retrieve benthic reflectance spectra in Case-I waters by combining active lidar and passive high-resolution imagery. Based on two-stream radiative transfer theory, the analytical relationship among the remote sensing reflectance (Rrs) and water column reflectance (Rw), benthic reflectance (Rb), diffuse attenuation coefficient (Kd), and water depth was established. The lidar data at a certain wavelength were applied to derive the water depth and the chlorophyll concentration (chl) along the lidar track. Then, the values of Kd at different wavelengths were estimated from the derived chl. In addition, we established a look-up-table (LUT) for the relationship between Rw and chl and water depth using Hydrolight simulation, and the Rw values at different wavelengths were then estimated by the lidar-derived chl and water depth. Finally, Rb(λ) values at different wavelengths along the lidar track were retrieved from the Rrs(λ) values observed by passive high-resolution imagery and the values of Rw(λ), Kd(λ), and water depth derived by lidar observation. The accuracy of the model was verified by using the Hydrolight simulated datasets, and the high correlation coefficient (R) revealed promising model performance for different benthic habitats, e.g., R > 0.9 for typical clean shallow water (chl = 0.5 mg/m3, H < 4 m) for the wavelength range of 400–640 nm. The model was further applied to real satellite data from ICESat-2 lidar and passive high-resolution satellite imagery (multispectral imagery of Sentinel-2 and hyperspectral imagery of Zhuhai−1) at two different benthic habitat sites (seagrass beds in Xincun Bay and coral reefs in the Huaguang Reef) in the South China Sea, and the results revealed that the model could reproduce the benthic spectra in both magnitude and shape. Overall, the proposed model can reliably yield benthic reflectance spectra along the lidar track without any requirement on prior knowledge, which should be beneficial for further benthic habitat health monitoring.

Radiative heat transfer between planar arrays of graphene plasmonic nanodisks

Graphene nanostructures, possessing extraordinary optical and thermal properties even beyond extended graphene, have recently entered the realm of thermal radiation. Here, using the many-body radiative transfer theory we study systematically radiative heat transfer between two finite-size planar arrays of doped graphene nanodisks, with attention to the many-body interaction (MBI) effect in such a plasmonic system. Focusing on heat exchanged between two finite-size square graphene disk arrays, we in particular identify two special regimes in which the heat transfer characteristics are akin to what are usually observed in coaxial disk dimers, and their extents depend on the array separation and the in-plane disk spacing. We then show that the MBI formed in this system can be responsible for remarkably inhibiting the heat flux, with respect to the case without disks in mutual interactions. Specifically, this inhibition arises when the purely strong intra-array MBI exists or when it coexists with the strong inter-array MBI, which is achievable via graphene with appropriately doping. Furthermore, the effect of graphene disk arrangements of arrays on the near- and far-field spectral flux is discussed. Last, a closed-form expression is offered for the energy density, including thermal emission contributions from the objects and the bath (environment). By tracing the energy density distribution above a single finite-size array of graphene nanodisks, we further analyze qualitatively how this array radiatively exchanges its energy with the adjacent one. Our results pave the way for plasmon-mediated thermal transport in networks of nanostructures.

Model equations of light scattering properties and a characteristic time of light propagation for polydisperse colloidal suspensions at different volume fractions.

We developed model equations of light scattering properties and a characteristic time of light propagation for polydisperse colloidal suspensions at different volume fractions. By the model equations, we examined numerical results using the first-order (dependent) scattering theory (FST) and radiative transfer theory in 600-980 nm wavelength. The model equations efficiently treat the interference of electric fields scattered from colloidal particles by a single effective coefficient, providing fast computation. Meanwhile, the FST provides accurate but complicated treatment. We found the interference effects on the scattering properties and characteristic time depend linearly on wavelength. Dimensionless analysis showed a simple mechanism of the interference effects, independently of wavelength and source-detector distance.

Modeling solar-induced fluorescence of forest with heterogeneous distribution of damaged foliage by extending the stochastic radiative transfer theory

Solar Induced chlorophyll Fluorescence (SIF) has been used as a novel proxy of photosynthetic activity, which carries information about plant physiological state from the remote sensing observations. It is of great interest and potential to use SIF to detect forest stresses, but the approach requires accurate modeling of the SIF emission within stressed forests. However, the existing radiative transfer approaches of SIF generally ignore the within-crown heterogeneity caused by pests. To account for within-canopy scattering with both high accuracy and efficiency, FluorESRT was proposed as a new model for simulating the SIF of forests with heterogeneous distribution of damaged foliage, which was the synergy of the SIF radiative transfer and the Stochastic Radiative Transfer (SRT) model for forest with vertical distribution of damage. The performance of FluorESRT for both homogeneous and heterogeneous cases were well validated not only by the one-dimensional (1D) model SCOPE but also by the three-dimensional (3D) model DART. As an analytically simple approach, FluorESRT can evaluate the sensitivity of canopy SIF and apparent reflectance to the level of pest damage. The impact of structural properties of damaged foliage on canopy fluorescence, as well as the response of hyperspectral signals on the pest damage was analyzed. According to our study, hyperspectral vegetation indices with the consideration of SIF showed higher sensitivity to pest damage in the early stage when the response of broad band vegetation indices was hardly detectable. The results indicated that the potential of FluorESRT to simulate SIF and hyperspectral apparent reflectance should play a critical role in early monitoring of pest damage.

Passive and Active Multiple Scattering of Forests Using Radiative Transfer Theory With an Iterative Approach and Cyclical Corrections

In this article, a unified framework of vegetation scattering using radiative transfer (RT) theory for passive and active remote sensing of vegetated land surfaces, especially those associated with moderate-to-large vegetation water contents (VWCs), e.g., forest field, is presented. The framework allows for modeling passive and active microwave signatures of the vegetated field with the same physical parameters describing the vegetation structure. RT equations are solved by a numerical iterative approach for both passive and active configurations. This approach allows including higher order scattering, which represents multiple scattering. In fields such as forests with large VWCs, associated with large scattering albedo and optical thickness, multiple scattering effects are critical. In the active iterative approach, cyclical terms are identified and backscattering enhancement is included by doubling contributions from cyclical terms. The method is applied to aspen trees in forest fields to compute the brightness temperatures and backscattering coefficients for passive and active remote sensing configurations, respectively. In the passive configuration, for forest field with VWC of 15 kg/ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{m}^{2}$ </tex-math></inline-formula> , the deviation between the zeroth-order brightness temperature, i.e., the tau–omega model results, and multiple scattering results around 40° observation angle, can be as large as 50 K for vertical polarization and 35 K for horizontal polarization. In the active configuration, the deviation between first-order results, which is identical to the distorted Born approximation, and the multiple scattering results around 40° incidence angle, is about 1.6 dB for VV and 0.7 dB for HH polarization. Multiple scattering is shown to be crucial for accurate forward modeling, especially over forested areas. The proposed approach is thus suitable for vegetation scattering with large VWCs. Furthermore, the proposed model is validated with the passive and active L-band sensor (PALS) acquired in SMAPVEX12 measurements in 2012, which demonstrates the applicability of this model.

On summable solutions of a class of nonlinear integral equations on the whole line

In this paper, we study a class of nonlinear integral equations with a noncompact monotone Hammerstein-Nemytskii operator on the whole real line. This class of equations is widely used in various fields of natural science. In particular, such equations arise in mathematical biology and in the theory of radiative transfer. A constructive existence theorem for a nonnegative nontrivial summable and bounded solution is proved. We also study the asymptotic behavior of the solution at $\pm\infty$. At the end of the paper, specific examples of the indicated equations are given, that satisfy all the conditions of the proved existence theorem. In an important particular case, it is possible to prove a uniqueness theorem in a certain class of essentially bounded functions.

A Physics-Assisted Convolutional Neural Network for Bathymetric Mapping Using ICESat-2 and Sentinel-2 Data

Machine learning methods for water depth estimation using remote sensing require accurate prior depth measurements. The successful operation of the ICESat-2 mission provides depth in shallow water directly; however, its spatial coverage is limited. Machine learning has been used to link optical remote sensing images and ICESat-2 data for bathymetric mapping. Compared with other machine learning models, convolutional neural network (CNN) models utilize the local spatial correlation between adjacent pixels and can thus reduce the effect of environmental noise. However, existing CNN and other machine learning models rely on data mining to build a general relationship between water depth and spectral information, and they ignore the known physical law. In this paper, we propose a physics-assisted convolutional neural network (PACNN) model. This model incorporates knowledge from radiative transfer theory into a normal CNN model by building a series of spectral feature input layers. In the PACNN model, the spectral information, which is directly related to water depth, is emphasized. Multitemporal ICESat-2 data and Sentinel-2 images were used to validate the model. In experiments with data from three study areas, the PACNN model outperformed the existing CNN model, achieving an accuracy of over 98%. The proposed method can effectively solve the underestimation in deeper water (20–30 m) and reduce the variance of estimates. The superiority of the PACNN model demonstrates how a machine learning model can be assisted by physics theory.

A Four-Component Parameterized Directional Thermal Radiance Model for Row Canopies

Directional brightness temperature (DBT) acquired by remote sensing instruments plays a significant role in characterizing the directional anisotropy of land surface, especially for row canopies. The difference between shaded vegetation and sunlit vegetation is ignored in the existing models. In this article, a four-component parameterized directional thermal radiance model (FCPMod) has been proposed to describe the DBT of the row canopy by considering the four components including the sunlit/shaded soil and sunlit/shaded leaf, the improved multiple scattering within the canopy, and the sensor&#x2019;s field of view (FOV). First, the sensor&#x2019;s FOV is divided into many tiny rectangles along the row direction and the probabilities of four components in each tiny rectangle are estimated based on the radiative transfer (RT) theory and the bidirectional gap probability. Second, the DBTs are weighted by the four components&#x2019; probabilities and brightness temperatures of tiny rectangles. Third, a modified multiple scattering model is proposed to improve the modeling accuracy by considering the contribution of the multiple scattering radiance between soil and canopy. The sensitivity analysis results show that the proposed method performed well compared to the FRA97 model proposed by Fran&#x00E7;ois <i>et al.</i> (1997) over continuous canopy and the RT model (FovMod) proposed by Ren <i>et al.</i> (2013) over row canopy. Finally, the field validations on a maize row canopy show that the proposed FCPMod performed better than about 0.4 K compared with the FovMod.

Vulnerability of Passive Microwave Snowfall Retrievals to Physical Properties of Snowpack: A Perspective From Dense Media Radiative Transfer Theory

The uncertainty of passive microwave retrievals of snowfall is notoriously high where the high-frequency surface emissivity is significantly reduced and varies markedly in response to changes of snowpack physical properties. Using the dense media radiative transfer theory, this article studies the potential effects of terrestrial snow-cover depth, density, and grain size on high-frequency channels 89 and 166 GHz of the radiometer onboard the Global Precipitation Measurement (GPM) core satellite, which are commonly used to capture the snowfall scattering signals. Integrating the inference across all feasible grain sizes, ranges of snowpack density and depth are identified over which the snowfall scattering signatures can be time varying and potentially obscured. Using 10 years of reanalysis data, the seasonal vulnerability of snowfall retrievals to changes of snowpack emissivity in the Northern Hemisphere is mapped in a probabilistic sense and connections are made with uncertainties of the GPM passive microwave snowfall retrievals. It is found that among different snow classes, relatively light Arctic tundra snow in fall, with a density below 260 kg m^-3, and shallow prairie snow during the winter, with a depth of less than 40 cm, can reduce the surface emissivity and obscure the snowfall passive microwave signatures. It is demonstrated that, during the winter, the highly vulnerable areas are over Kazakhstan, and Mongolia with taiga and prairie snow. In the fall, these areas are largely over tundra and taiga snow in north of Russia and the Arctic Archipelagos as well as prairies in Canada and the Great Plains in the United States.

Physics‐Based Narrowband Optical Parameters for Snow Albedo Simulation in Climate Models

AbstractAccurate snow albedo simulation is a prerequisite for climate models to produce reliable climate prediction. Climate models would benefit from schemes of snowpack radiative transfer that are responsive to changing atmospheric conditions. However, the uncertainties in the narrowband snow optical parameters used by these schemes have not been evaluated. Conventional methods typically compute these narrowband parameters as irradiance‐weighted averages of the spectral snow optical parameters, with the single scattering albedo being additionally weighted by the optically thick snowpack albedo. We first evaluate the effectiveness of the conventional methods as adopted by the widely used Community Land Model (CLM). Snow albedo calculations using the CLM narrowband optical parameters are relatively accurate for very thin snow (e.g., a bias of 0.01 for a 2‐cm snowpack). The error, however, becomes larger as snowpack thickens (with biases of up to 0.05 for semi‐infinite snowpack), because the snow radiative transfer is highly nonlinear and is most significant at wavelengths &lt;1 μm. In this study, we propose a novel method to retrieve broadband optical parameters according to snow radiative transfer theory, reducing the albedo biases to &lt;0.003 for 2 cm snowpacks and &lt;0.005 for thick snowpacks. We find little impact in changing incident spectra on narrowband snow albedo. These newly derived narrowband optical parameters improve snow albedo accuracy by a factor of 10, allowing to trace the impacts of aerosol pollution in snow. The parameters are independent of which two‐stream approximation is used, and are thus applicable to sea ice, glaciers, and seasonal snow cover.