Radiative Transfer Method Research Articles

Impact of Jupiter’s Heating and Self-shadowing on the Jovian Circumplanetary Disk Structure

Abstract Deciphering the structure of the circumplanetary disk (CPD) that surrounded Jupiter at the end of its formation is key to understanding how the Galilean moons formed. Three-dimensional hydrodynamic simulations have shown that this disk was optically thick and significantly heated to very high temperatures owing to the intense radiation emitted by the hot, young planet. Analyzing the impact of Jupiter’s radiative heating and shadowing on the structure of the CPD can provide valuable insights into the conditions that shaped the formation of the Galilean moons. To assess the impact of Jupiter’s radiative heating and shadowing, we have developed a two-dimensional quasi-stationary CPD model and used a gray atmosphere radiative transfer method to determine the thermal structure of the disk. We find that the CPD self-shadowing has a significant effect, with a temperature drop of approximately 100 K in the shadowed zone compared to the surrounding areas. This shadowed zone, located around 10 Jupiter radii, can act as a cold trap for volatile species such as NH3, CO2, and H2S. The existence of these shadows in Jupiter’s CPD may have influenced the composition of the building blocks of the Galilean moons, potentially shaping their formation and characteristics. Our study suggests that the thermal structure of Jupiter’s CPD, particularly the presence of cold traps due to self-shadowing, may have played a crucial role in the formation and composition of the Galilean moons.

Subsweep: Extensions to the sweep method for radiative transfer

We introduce the radiative transfer postprocessing code Subsweep . The code is based on the method of transport sweeps, in which the exact solution to the scattering-less radiative transfer equation is computed in a single pass through the entire computational grid. The radiative transfer module is coupled to radiation chemistry, and chemical compositions as well as temperatures of the cells are evolved according to photon fluxes computed during radiative transfer. Subsweep extends the method of transport sweeps by incorporating sub-timesteps in a hierarchy of partial sweeps of the grid. This alleviates the need for a low, global timestep and as a result Subsweep is able to drastically reduce the amount of computation required for accurate integration of the coupled radiation chemistry equations. We succesfully apply the code to a number of physical tests such as the expansion of HII regions, the formation of shadows behind dense objects, and its behavior in the presence of periodic boundary conditions.

An implicit unified gas-kinetic particle method with large time steps for gray radiation transport

For a long time, efficient algorithms for high-dimensional equations, represented by photon radiation transport, have been one important topic in the development of computational methods for particle transport processes. In this paper, we present an implicit unified gas-kinetic particle (IUGKP) method for multiscale gray radiative transfer. Based on the integral solution of the radiative transfer equation, the photon transport processes are categorized into non-equilibrium transport processes with a large photon free path and equilibrium transport processes with a small photon free path. The long-path processes are solved by an implicit Monte Carlo (IMC) method, and the short-path processes are solved by an implicit diffusion system. The closure formulation of photon distribution is derived from the local integral solution of the radiative transfer equation to couple the IMC and diffusion system. The improvement of the proposed IUGKP method over UGKP method is that particles can be tracked continuously instead of just until the first collision, making simulation with large time steps possible. The IUGKP method has the properties of asymptotic-preserving (AP) and regime-adaptive (RA). The AP property states that the IUGKP method converges to the consistent numerical methods for the asymptotic limiting equations of RTE in the limiting regimes. The RA property states that the computational accuracy of the IUGKP method adapts to the regimes. In this paper, the mathematical proof of the AP and RA properties is presented, and the multiscale numerical tests are performed to demonstrate the accuracy and efficiency of the IUGKP method.

Parametric Analytical Modulation Transfer Function Model in Turbid Atmosphere with Application to Image Restoration

To address the issues of image blurring and color distortion in hazy conditions, an image restoration method based on a parametric analytical modulation transfer function model is proposed under turbid atmospheric conditions. A source database is established using a numerical radiative transfer method based on discrete ordinate. Through multivariate nonlinear fitting and linear interpolation, the quantitative relationships among critical spatial frequency, turbid atmospheric MTF, and key atmospheric optical parameters—such as optical thickness, single scattering albedo, and asymmetry factor—are examined. A fast and efficient parametric analytical MTF model for turbid atmospheres is developed and applied to restore images affected by fog. The results demonstrate that, within the applicable range of the model, the model’s maximum mean relative error and the root mean square error are 7.16% and 0.0454, respectively. The computational speed is nearly a thousand times faster than that of the numerical radiative transfer method, achieving high accuracy and ease of application. Images restored using this model exhibit enhanced clarity and quality, effectively compensating for the degradation in image quality caused by turbid atmospheres. This approach represents a novel solution to the challenges of image processing in complex atmospheric environments.

Vibration analysis of functionally graded saturated porous beams using radiative energy transfer method

An energy flow model based on the radiative energy transfer method (RETM) is established for the high-frequency response prediction of finite functionally graded saturated porous beams. The Timoshenko beam theory and the Biot thoery are applied to derive the motion governing equations of the beam. The response of the beam loaded by a high-frequency point force is described by vibrational energy density and intensity. The energy response of the beam is contributed by the rays emitted from the actual source set at load point and then reflected through the fictitious sources at the boundaries. Fictitious sources are determined by the energy balance equations at the boundaries. Numerical examples show that the energy distributions of the saturated porous beams are well predicted by the proposed approach.

Integrating angular and domain decomposition with space-angle discontinuous Galerkin methods in 2D radiative transfer

A space-angle discontinuous Galerkin (saDG) method is used to solve the steady-state radiative transfer equation (RTE) for 2D problems involving absorption, emission, and scattering for a semitransparent medium. This approach discretizes both spatial and angular domains. Parallel computing is based on angular decomposition (AD), and domain decomposition (DD) techniques. The DD technique directly solves the entire domain using the MUMPS library, whereas the AD technique results in an iterative approach for scattering media. This study proposes a novel hybrid AD-DD method, combining the best aspects of both techniques. Numerical results investigate the scalability, performance, and efficiency of AD and DD techniques. It is shown that a hybrid AD-DD technique is superior to these individual techniques by taking advantage of their strengths. Numerical methods demonstrate the applicability of the method of the best combination of hybrid AD-DD to 2D scattering gray media with complex geometries or enclosures with circular and square obstacles.

Tetrahedral grids in Monte Carlo radiative transfer

Context. To understand the structures of complex astrophysical objects, 3D numerical simulations of radiative transfer processes are invaluable. For Monte Carlo radiative transfer, the most common radiative transfer method in 3D, the design of a spatial grid is important and non-trivial. Common choices include hierarchical octree and unstructured Voronoi grids, each of which has advantages and limitations. Tetrahedral grids, commonly used in ray-tracing computer graphics, can be an interesting alternative option. Aims. We aim to investigate the possibilities, advantages, and limitations of tetrahedral grids in the context of Monte Carlo radiative transfer. In particular, we want to compare the performance of tetrahedral grids to other commonly used grid structures. Methods. We implemented a tetrahedral grid structure, based on the open-source library TetGen, in the generic Monte Carlo radiative transfer code SKIRT. Tetrahedral grids can be imported from external applications or they can be constructed and adaptively refined within SKIRT. We implemented an efficient grid traversal method based on Plücker coordinates and Plücker products. Results. The correct implementation of the tetrahedral grid construction and the grid traversal algorithm in SKIRT were validated using 2D radiative transfer benchmark problems. Using a simple 3D model, we compared the performance of tetrahedral, octree, and Voronoi grids. With a constant cell count, the octree grid outperforms the tetrahedral and Voronoi grids in terms of traversal speed, whereas the tetrahedral grid is poorer than the other grids in terms of grid quality. All told, we find that the performance of tetrahedral grids is relatively poor compared to octree and Voronoi grids. Conclusions. Although the adaptively constructed tetrahedral grids might not be favourable in most media representative of astrophysical simulation models, they still form an interesting unstructured alternative to Voronoi grids for specific applications. In particular, they might prove useful for radiative transfer post-processing of hydrodynamical simulations run on tetrahedral or unstructured grids.

Polarization Characteristics of Massive HVI Debris Clouds Using an Improved Monte Carlo Ray Tracing Method for Remote Sensing Applications

As a signature phenomenon of massive hypervelocity impacts (HVIs) in space, debris clouds provide critical optical information for satellite remote sensing and the assessment of large-scale impacts. However, studies of the optical scattering properties of debris clouds remain limited, and existing vector radiative transfer (VRT) methods struggle to accurately simulate the optical characteristics of these complex scatterers. To address this gap, this paper presents an improved Monte Carlo VRT program (PGS–MC) for multicomponent polydisperse scatterers to precisely evaluate the radiation and polarization characteristics of complex scatterers. Based on the Monte Carlo ray tracing (MCRT) method, our program introduces a particle grouping strategy (PGS) to further emphasize the importance of accounting for optical property discrepancies between different materials and particle sizes, thus significantly improving the fidelity of VRT simulations. Moreover, our program, developed using the compute unified device architecture (CUDA), can be run parallelly on graphics processing units (GPUs), which effectively reduces the computational time. The validation results indicated that the developed PGS–MC program can accurately and efficiently simulate the polarization of complex 3D scatterers. A further investigation showed that the polarization characteristics of debris clouds are highly sensitive to parameters such as the angle between the incident and detection directions, number density, particle size distribution, debris material, and wavelength. In addition, the polarization imaging of debris clouds offers distinct advantages over intensity imaging. This study offers guidance for analyzing the VRT properties of massive HVI debris clouds. Additionally, it provides a practical tool and concrete ideas for modeling the polarization characteristics of various complex scatterers, such as aircraft contrails and clouds, etc.

A hybrid radiative energy transfer and image source method for high-frequency vibrational coupled plates

A hybrid approach composed of the radiative energy transfer method (RETM) and image source method (ISM) is proposed in this study for estimating the vibrational energy of coupled plates loaded by high-frequency point excitation. The vibrational energy at a receiver is represented by energy density and intensity, which are superposed by incoherent rays emitted by the load point in the analyzed domain and reflected/transmitted by the domain boundaries. The energy rays reflected/transmitted at the boundaries are described by two modes: diffuse reflection/transmission, which is a basic assumption in RETM, and the other obeys Snell’s law of reflection/refraction, which is deduced by Fermat’s principle and often applied in ISM. The local energy response distribution in the loading subsystem is described by ISM. The energies in other subsystems are described by RETM. The energy transfer relationship between subsystems is expressed by the energy transfer coefficient. At the boundary of coupled plates, a Fredholm equation of the second type is established through the balance among the outgoing energy of the diffuse reflection fictitious sources and the incident energy of actual sources, other diffuse reflection sources, and specular reflection image sources, which can be used to determine the intensity of the diffuse reflection fictitious sources. The average energy transfer coefficient in the transmission direction half-space is used to express the energy transfer relationship associated with diffuse reflection sources, while the energy transfer relationship associated with specular reflection sources is expressed directly by the energy transfer coefficient. Numerical tests show that the energy distributions and energy flow fields of typical coupled plates are well predicted by the proposed hybrid RETM-ISM approach. Compared with RETM, hybrid RETM-ISM is more consistent with the FEM solution.

The Influence of Spatial Heterogeneity of Urban Green Space on Surface Temperature

Urban green space (UGS) has been recognized as a key factor in enhancing the urban ecosystem balance, particularly in arid areas. It is often considered an effective means to mitigate the urban heat island (UHI) effect. In this study, the reference comparison method was utilized to optimize the process of nighttime lighting data; the random forest classification method was employed to extract UGS data; and the radiative transfer method was applied in land surface temperature (LST) inversion. Additionally, moving window analysis was conducted to assess the robustness of the results. The objective of this research was to analyze the spatial distribution characteristics of UGS and LST and to explore their bivariate local spatial autocorrelations by calculating four landscape metrics, including the aggregation index (AI), edge density (ED), patch density (PD), and area-weighted mean shape index (Shape_am). It was found that the distribution of UGS in the study area was uneven, with higher temperatures in the eastern and western regions and lower temperatures in the central and southern regions. The results also revealed that ED, PD, and Shape_am were negatively correlated with LST, with correlation coefficients being −0.469, −0.388, and −0.411, respectively, indicating that UGS in these regions were more effective in terms of cooling effect. Conversely, AI was found to be positively correlated with LST (Moran’ I index of 0.449), indicating that surface temperatures were relatively higher in regions of high aggregation. In essence, the fragmented, complex, and evenly distributed green patches in the study area provided a better cooling effect. These findings should persuade decision makers and municipal planners to allocate more UGS in cities for UHI alleviation to improve quality of life and enhance recreational opportunities.

Testing Approximate Infrared Scattering Radiative-transfer Methods for Hot Jupiter Atmospheres

The calculation of internal atmospheric (longwave) fluxes is a key component of any model of exoplanet atmospheres that requires radiative-transfer (RT) calculations. For atmospheres containing a strong scattering component such as cloud particles, most 1D multiple-scattering RT methods typically involve numerically expensive matrix inversions. This computational bottleneck is exacerbated when multitudes of RT calculations are required, such as in general circulation models (GCMs) and retrieval methods. In an effort to increase the speed of RT calculations without sacrificing too much accuracy, we investigate the applicability of approximate longwave scattering methods developed for the Earth science community to hot Jupiter atmospheres. We test the absorption approximation and variational iteration method (VIM) applied to typical cloudy hot Jupiter scenarios, using 64-stream DISORT calculations as reference solutions. We find the four-stream VIM variant is a highly promising method to explore for use in hot Jupiter GCM and retrieval modeling, and it shows excellent speed characteristics, with typical errors ∼1% for outgoing fluxes and within ∼50%, but with larger errors in the test case of a deep cloud layer, for vertical heating rates. Other methods explored in this study were found to typically produce similar error characteristics in vertical heating rates.

Numerical investigations and design optimization of cryogenic plate-fin heat exchanger in Electric Arc Furnace using radiative heat transfer model

The most crucial element affecting cryogenic systems’ energy efficiency is heat exchangers. In this article, a model for cryogenic plate-fin heat exchanger (PFHE) under thermal-hydraulic performance with various blends is proposed. To reduce five objective functions total heat transfer rate, total weight, total mass flow rate, number of entropy generating units, and whole annual cost the study carried out the PFHE’s optimal design. This study innovatively focuses on optimizing PFHE design for cryogenic conditions using Opposition Learning with the Beetle Swarm algorithm. The proposed geometry addresses non-uniform temperature distribution, emphasizing thermal-hydraulic properties in various blends. Additionally, a second-order semi-discretization radiative heat transfer approach enhances boiler simulation accuracy. A multi-objective optimization of Opposition Learning with Beetle Swarm (MO-OBBS) optimization approach was suggested in the article for the design of PFHE. Despite the significant potential for renewable energy the low-temperature waste heat treatment process is rarely examined from a sustainability perspective. Consequently, the study proposed a second-order semi-discretization radiative heat transfer method to simulate a furnace using a water heat recovery process. The cooling panel’s numerical analysis was conducted using computed temperature fields. The performance of the various algorithms is compared using the Freidman rank-sum test to determine the statistical significance of the findings obtained. In logarithmic coordinates, the proposed method’s faster convergence is compared to GQB-PSO, SLTLBO, Jaya, CKKO, and OLE-SCA algorithms. The proposed method exhibits a maximum average increase of 115.39% for PFHE design. When the numerical simulation results are compared to measurements of output water temperature, the accuracy is highly improved.

Improving Monte Carlo radiative transfer in the regime of high optical depths: The minimum scattering order

Radiative transfer simulations are a powerful tool that enables the calculation of synthetic images of a wide range of astrophysical objects. These simulations are often based on the Monte Carlo method, as it provides the needed versatility that allows the consideration of the diverse and often complex conditions found in those objects. However, this method faces fundamental problems in the regime of high optical depths which may result in overly noisy images and severely underestimated flux values. In this study, we propose an advanced Monte Carlo radiative transfer method, namely, an enforced minimum scattering order that is aimed at providing a minimum quality of determined flux estimates. For that purpose, we extended our investigations of the scattering order problem and derived an analytic expression for the minimum number of interactions that depends on the albedo and optical depth of the system, which needs to be considered during a simulation to achieve a certain coverage of the scattering order distribution. The method is based on the utilization of this estimated minimum scattering order and enforces the consideration of a sufficient number of interactions during a Monte Carlo radiative transfer simulation. Moreover, we identified two notably distinct cases that shape the kind of complexity that arises in such simulations: the albedo-dominated case and the optical depth-dominated case. Based on that, we analyzed the implications related to the best usage of a stretching method as a means to alleviate the scattering order problem. We find that its most suitable application requires taking into account the value of the albedo as well as the optical depth. Furthermore, we argue that the derived minimum scattering order can be used to assess the performance of a stretching method with regard to the scattering orders its usage promotes. Finally, we stress the need for developing advanced pathfinding techniques to fully solve the problem of Monte Carlo radiative transfer simulations in the regime of high optical depths.

Hybrid retrieval of grass biophysical variables based-on radiative transfer, active learning and regression methods using Sentinel-2 data in Marakele National Park

Biophysical variables such as leaf area index (LAI) and leaf chlorophyll content (LCC) are cited as essential biodiversity variables. A comprehensive comparison and integration of retrieval methods is needed for the estimation of biophysical variables such as LAI and LCC over a multispecies grass canopy. This study tested an assortment of five potentially robust, nonparametric regression methods (NPRMs) for inversion of radiative transfer model (RTM) to retrieve grass LAI and LCC in the Marakele National Park (MNP) of South Africa. The NPRMs used were, namely (i) Partial least squares regression (PLSR), (ii) Principle components regression (PCR), (iii) Kernel ridge regression (KRR), (iv) Random forest regression (RFR), and (v) K-nearest neighbours regression (KNNR). Furthermore, the study attempted to constrain the inversion process by using active learning (AL) techniques which ensured the selection of informative samples from a large pool of RTM simulations. Results show the most accurate grass LAI and LCC retrievals had lower relative root mean squared errors (RRMSEs) of 39.87% and 16.58% respectively. These findings have significant implications for the development of transferable rangeland monitoring systems in protected mountainous regions.

An Algorithm Developed for Smallsats Accurately Retrieves Landsat Surface Reflectance Using Scene Statistics

Closed-form Method for Atmospheric Correction (CMAC) is software that overcomes radiative transfer method problems for smallsat surface reflectance retrieval: unknown sensor radiance responses because onboard monitors are omitted to conserve size/weight, and ancillary data availability that delays processing by days. CMAC requires neither and retrieves surface reflectance in near real time, first mapping the atmospheric effect across the image as an index (Atm-I) from scene statistics, then reversing these effects with a closed-form linear model that has precedence in the literature. Five consistent-reflectance area-of-interest targets on thirty-one low-to-moderate Atm-I images were processed by CMAC and LaSRC. CMAC retrievals accurately matched LaSRC with nearly identical error profiles. CMAC and LaSRC output for paired images of low and high Atm-I were then compared for three additional consistent-reflectance area-of-interest targets. Three indices were calculated from the extracted reflectance: NDVI calculated with red (standard) and substitutions with blue and green. A null hypothesis for competent retrieval would show no difference. The pooled error for the three indices (n = 9) was 0–3% for CMAC, 6–20% for LaSRC, and 13–38% for uncorrected top-of-atmosphere results, thus demonstrating both the value of atmospheric correction and, especially, the stability of CMAC for machine analysis and AI application under increasing Atm-I from climate change-driven wildfires.

Deriving fine-scale patterns of sea surface temperature in coral reef habitats using the Landsat 8 thermal infrared sensor

The available sea surface temperature (SST) products are too coarse to assess the fine-scale (<1 km) SST variations related to coral bleaching. In this study, we proposed an optimal SST inversion model using Landsat 8 thermal infrared sensor (TIRS) images to derive fine-scale SST patterns in the coral reef habitats of the Xisha Islands, South China Sea. Our study included two parts: 1) six SST inversion models were developed using the radiative transfer method and the split window (SW) algorithm in the hot season and cool season, from which the optimal SST inversion model was determined; and 2) the optimal model was applied to 47 Landsat 8 TIRS images to derive the SST spatial and temporal pattern among the geomorphic zones of six reefs in hot and cool season conditions. Compared with the measured sea temperature data and the verified MODIS SST products, the SST6 model using the SW algorithm was optimal, with an RMSE of approximately 0.64°C in the hot season. The average SST results usually had a pattern of reef flat > lagoon > reef slope/offshore sea. The reef flat was usually approximately 0.05°C–0.2°C hotter than the lagoon in the hot season. The SST in the lagoon also increased from south to north and the shallow lagoon was usually warmer than the deep lagoon in the hot season. Our results suggested that scleractinian corals in the reef flat and the lagoon were more susceptible to bleaching-level thermal stress than other geomorphic zones. During the cool season, the SST fluctuated markedly among coral reefs and geomorphic zones.

Simulating ionization feedback from young massive stars: impact of numerical resolution

ABSTRACT Modelling galaxy formation in hydrodynamic simulations has increasingly adopted various radiative transfer methods to account for photoionization feedback from young massive stars. However, the evolution of H ii regions around stars begins in dense star-forming clouds and spans large dynamical ranges in both space and time, posing severe challenges for numerical simulations in terms of both spatial and temporal resolution that depends strongly on gas density (∝n−1). In this work, we perform a series of idealized H ii region simulations using the moving-mesh radiation-hydrodynamic code arepo-rt to study the effects of numerical resolution. The simulated results match the analytical solutions and the ionization feedback converges only if the Strömgren sphere is resolved by at least 10–100 resolution elements and the size of each time integration step is smaller than 0.1 times the recombination time-scale. Insufficient spatial resolution leads to reduced ionization fraction but enhanced ionized gas mass and momentum feedback from the H ii regions, as well as degrading the multiphase interstellar medium into a diffuse, partially ionized, warm (∼8000 K) gas. On the other hand, insufficient temporal resolution strongly suppresses the effects of ionizing feedback. This is because longer time-steps are not able to resolve the rapid variation of the thermochemistry properties of the gas cells around massive stars, especially when the photon injection and thermochemistry are performed with different cadences. Finally, we provide novel numerical implementations to overcome the above issues when strict resolution requirements are not achievable in practice.

Capturing photoionization shadows in streamer simulations using the discrete ordinates method

Numerical simulations of streamer propagation involving photoionization are presented, utilizing an ANSYS Fluent implementation that employs unstructured meshes and automatic mesh refinement. Two approximate methods for radiative transfer, used to handle computation of the photoionization source terms, are compared: the Eddington approximation and the discrete ordinates (DOs) method. The former is commonly employed in streamer simulations, while the latter is well-established in other branches of computational physics, such as radiative heat transfer. A 2D test case with two distinct regions, where streamer propagation can be triggered thanks to the protruded electrodes, is introduced. The two regions are partially separated by an opaque solid insulator barrier to study the effects of photoionization shadows on streamer inception and propagation. The primary positive streamer is initiated by placing a neutral plasma patch close to one of the electrode protrusions, while the secondary positive streamer, in the other region of the computational domain, is initiated by photoionization originating from the primary streamer zone. The Eddington approximation results in an excessively high photoionization source in the secondary streamer inception zone, as it fails to capture the shadowing effects of the opaque dielectric barrier. Consequently, this leads to a fast secondary streamer inception process, followed by rapid streamer propagation. On the other hand, the DOs method accurately captures the shadow, leading to a delayed secondary streamer inception. It is also shown that both methods exhibit very similar results when the dielectric barrier is transparent and the shadow is absent. This work demonstrates that using the DOs method for streamer simulations offers considerable advantages over the Eddington approximation, especially in cases involving more complex geometries where shadows need to be captured for accurate streamer inception and dynamics.

Analysis Of An Inductive Coupling Wireless Power Transfer System With a Finite Element Method For Charging Application Of Electric Vehicles

With the development of technology, Wireless Power Transfer (WPT) systems have also started to be used in charging applications of electric vehicles. The basic methods of WPT systems are based on power transmission systems with laser, microwave, magnetic induction and magnetic resonance. Factors such as the limited use of fossil fuels and environmental pollution have led researchers to see electric vehicles as a solution and to charge these vehicles with wireless systems. Most commercial applications of wireless power transmission are currently limited to close contact transmission distances. There are challenges in increasing coverage, routing transmissions and safely exploiting sufficiently strong electric fields. To provide power efficiently, highly directional transmitters must be applied. Otherwise, with an omnidirectional transmission, only a small part of the transmitted power will reach the receiver. In order to distribute power, especially over long distances, it is often necessary to resort to radiation transfer methods that tightly combine electric and magnetic fields. In this study, important studies on WPT systems were investigated and examined. Then, a WPT transformer is modeled with Finite Element Method based ANSYS-Maxwell-3D. The results obtained using the simulation method are presented in comparison with the research findings. Determining the efficiency of wireless power transfer used in electric vehicle charging applications is the expected result of the study.

Lattice Boltzmann method for radiative transfer in two-layered slab with graded-index and Fresnel reflecting surfaces

Analysis of transport of radiative transfer in an inhomogeneous two-layered slab with graded refractive indices and bounded by Fresnel's reflecting walls is carried out using the lattice Boltzmann method. The radiative intensity in each layer is found as a superposition of two radiative waves, which undergo multiple reflections and refractions at the boundaries and interface. The accuracy and efficiency of the methodology is first tested for steady-state problems with constant or variable refractive indices and for time-resolved reflectance and transmittance in a single layer with graded-index. The effect of the governing parameters is examined and the results show that these parameters significantly influence radiative transfer. Especially, the curvature of the incoming beam due to refractive index variation changes when considering the increasing or decreasing variation of the refractive index. The smaller refractive index gap at the interface induces smaller reflection and radiative intensity discontinuity at the interface, while increasing the scattering albedo has no significant effect on the transmittance signal but corresponds to increasing the reflectance signal which takes more time to show the steady state profile. Moreover, the developed LBM is an efficient, powerful, and accurate solver for radiative transport in inhomogeneous layered media with graded-index.