Spherical Harmonic Discrete Ordinate Method Research Articles

Sensitivity studies of nighttime top-of-atmosphere radiances from artificial light sources using a 3-D radiative transfer model for nighttime aerosol retrievals

Abstract. By accounting for surface-based light source emissions and top-of-atmosphere (TOA) downward lunar fluxes, we adapted the spherical harmonics discrete ordinate method (SHDOM) 3-dimensional (3-D) radiative transfer model (RTM) to simulate nighttime 3-D TOA radiances as observed from the Visible Infrared Imaging Radiometer Suite (VIIRS) Day/Night Band (DNB) on board the Suomi-NPP satellite platform. Used previously for daytime 3-D applications, these new SHDOM enhancements allow for the study of the impacts of various observing conditions and aerosol properties on simulated VIIRS-DNB TOA radiances. Observations over Dakar, Senegal, selected for its bright city lights and a large range of aerosol optical depth (AOD), were investigated for potential applications and opportunities for using observed radiances containing VIIRS-DNB “bright pixels” from artificial light sources to conduct aerosol retrievals. We found that using the standard deviation (SD) of such bright pixels provided a more stable quantity for nighttime AOD retrievals than direct retrievals from TOA radiances. Further, both the mean TOA radiance and SD of TOA radiances over artificial sources are significantly impacted by satellite viewing angles. Light domes, the enhanced radiances adjacent to artificial light sources, are strong functions of aerosol properties and especially aerosol vertical distribution, which may be further utilized for retrieving aerosol layer height in future studies. Through inter-comparison with both day- and nighttime Aerosol Robotic Network (AERONET) data, the feasibility of retrieving nighttime AODs using 3-D RTM SHDOM over artificial light sources was demonstrated. Our study shows strong potential for using artificial light sources for nighttime AOD retrievals, while also highlighting larger uncertainties in quantifying surface light source emissions. This study underscores the need for surface light emission source characterizations as a key boundary condition, which is a complex task that requires enhanced input data and further research. We demonstrate how quality-controlled nighttime light data from the NASA's Black Marble product suite could serve as a primary input into estimations of surface light source emissions for nighttime aerosol retrievals.

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.

Cloud tomographic retrieval algorithms. I: Surrogate minimization method

A cloud tomographic retrieval algorithm relying on (i) the spherical harmonics discrete ordinate method for computing the radiative transfer and (ii) the surrogate minimization method for solving the inverse problem has been designed. The retrieval algorithm uses regularization, accelerated projected gradient methods, and two types of surrogate functions. The performances of the retrieval algorithm are analyzed on a few synthetic two- and three-dimensional problems.

A machine learning-based method to account for 3D Short-Wave radiative effects in 1D satellite observation operators

Data assimilation of Short-Wave (SW, ≤ 4μm) satellite radiance in cloudy regions could potentially improve cloud forecasts. Operational SW observation operators neglect cloud 3D radiative effects, which may cause nonnegligible errors in some circumstances, for example, with large solar zenith angles or for broken clouds. This study introduced a machine learning (ML)-based method to include some cloud 3D radiative effects in a 1D observation operator based on liquid water cloud simulations by the Weather Research and Forecasting model and radiative transfer simulations by the Spherical Harmonics Discrete Ordinate Method. The inputs of the ML correction method include the reflectance simulated by a 1D observation operator at the targeting column, cloud top height, cloud water content, and effective particle size at its adjacent columns, and other ancillary information (sun-viewing geometries, water vapor conditions, etc.). Preliminary results were presented for the channel 3 (0.75 ∼ 0.90µm) of the Advanced Geostationary Radiation Imager (AGRI) onboard FY-4A. Compared with 1D simulations, the ML correction method could reduce the biases from 6.9% ∼ 11.7% to 2.0% ∼ 4.7% and the root mean square errors from 22.3% ∼ 34.8% to 15.4% ∼ 27.4%. The elapsed CPU time for the ML correction method is approximately the same or two times that of the 1D observation operators, depending on whether atmospheric gases are included or not. In general, the ML correction method is computationally more efficient than traditional 3D radiative transfer solver, and could be used to correct current 1D observation operators’ simulations.

A Proof-of-Concept Algorithm for the Retrieval of Total Column Amount of Trace Gases in a Multi-Dimensional Atmosphere

An algorithm for the retrieval of total column amount of trace gases in a multi-dimensional atmosphere is designed. The algorithm uses (i) certain differential radiance models with internal and external closures as inversion models, (ii) the iteratively regularized Gauss–Newton method as a regularization tool, and (iii) the spherical harmonics discrete ordinate method (SHDOM) as linearized radiative transfer model. For efficiency reasons, SHDOM is equipped with a spectral acceleration approach that combines the correlated k-distribution method with the principal component analysis. The algorithm is used to retrieve the total column amount of nitrogen for two- and three-dimensional cloudy scenes. Although for three-dimensional geometries, the computational time is high, the main concepts of the algorithm are correct and the retrieval results are accurate.

A Spectral Acceleration Approach for the Spherical Harmonics Discrete Ordinate Method

A spectral acceleration approach for the spherical harmonics discrete ordinate method (SHDOM) is designed. This approach combines the correlated k-distribution method and some dimensionality reduction techniques applied on the optical parameters of an atmospheric system. The dimensionality reduction techniques used in this study are the linear embedding methods: principal component analysis, locality pursuit embedding, locality preserving projection, and locally embedded analysis. Through a numerical analysis, it is shown that relative to the correlated k-distribution method, PCA in conjunction with a second-order of scattering approximation yields an acceleration factor of 12. This implies that SHDOM equipped with this acceleration approach is efficient enough to perform spectral integration of radiance fields in inhomogeneous multi-dimensional media.

A Numerical Study of Effects of Radiation on Deep Convective Warm Based Cumulus Cloud Development with a 3-D Radiative Transfer Model

The effects of radiation heating and cooling on cumulus cloud development have been the focus of considerable attention for many years. However, it is still not clear how radiation impacts cloud droplet growth. Since cloud inhomogeneity has a great influence on radiation transmission, we coupled the 3D atmospheric radiative transfer model using the spherical harmonic discrete ordinate method with WRF-LES, which can improve the simulation accuracy of the inhomogeneous effect of clouds on radiation compared with that of the 1D radiation method. The shortwave and longwave radiation fluxes for upward and downward directions were simulated with different solar zenith angles. The comparison of 1D and 3D radiative solvers for deep convective cloud cases shows that the 3D radiative solver provides an accurate structure of solar and thermal radiation characteristics and the spatial distribution field. The solar radiation heating is likely to increase perpendicular to the solar incidence direction. For longwave radiation, the cooling effect on the cloud top and the heating effect on the cloud base are both more intense in the 3D radiation model. This study focuses on 3D cloud-radiative interactions in an inhomogeneous cloud field in a large eddy simulation, and the results suggest that compared with the widely used 1D radiative solver in WRF, the 3D radiation model can provide a precise description of the radiation field in an inhomogeneous atmosphere.

Spectral spherical harmonics discrete ordinate method

A new method for modeling the radiative transfer in inhomogeneous three-dimensional media illuminated by a Gaussian beam is described. This approach, called the Spectral Spherical Harmonics Discrete Ordinate Method (SSHDOM), uses the Fourier expansion method to transform the three-dimensional radiative transfer into an one-dimensional equation in the spectral domain, and the Spherical Harmonics Discrete Ordinate Method (SHDOM) for its solution. Specifically, (i) the source function is represented in the spectral domain through a spherical harmonic expansion, (ii) the spectral one-dimensional radiative transfer equation is integrated along discrete ordinates through a spatial grid, and (iii) the solution method is based on the Picard iteration. Both SSHDOM and SHDOM algorithms are implemented in a common computer code.

Effects of alumina particle clusters on UV radiation signature in the aircraft plume

An improved model of the aircraft plume is proposed to analyze the effects of alumina particle clusters on ultraviolet (UV) radiation signature. In the simulation, the scattering effect of particle clusters rather than individual particles on the radiation characteristics in the plume is considered. The modified diffusion-limited aggregation (DLA) method is used to generate the geometry of the cluster example and the scattering characteristics of the particle clusters are calculated by multi-sphere T-matrix method (MSTM) at first. Then the average optical properties of the particle clusters are calculated and applied to improved spherical harmonic discrete ordinate method (SHDOM), together with a new source term about the chemiluminescence emission in the plume, the improved radiation model is constructed carefully. The simulation results show that at a small observation angle, alumina particle clusters can significantly enhance the radiance of the plume comparing to the individual particle model. Meanwhile, the radiance distribution of a more realistic three-dimensional(3-D) plume is presented when both the radial and axial property variation of a whole plume structure are concerned in this model.

Linearizations of the Spherical Harmonic Discrete Ordinate Method (SHDOM)

Linearizations of the spherical harmonic discrete ordinate method (SHDOM) by means of a forward and a forward-adjoint approach are presented. Essentially, SHDOM is specialized for derivative calculations and radiative transfer problems involving the delta-M approximation, the TMS correction, and the adaptive grid splitting, while practical formulas for computing the derivatives in the spherical harmonics space are derived. The accuracies and efficiencies of the proposed methods are analyzed for several test problems.

Design and Verification of a New Monochromatic Thermal Emission Component for the I3RC Community Monte Carlo Model

Abstract The Intercomparison of 3D Radiation Codes (I3RC) community Monte Carlo model has been extended to include a source of photon emission from the surface and atmosphere, thereby making it capable of simulating scalar radiative transfer in a 3D scattering, absorbing, and emitting domain with both internal and external sources. The theoretical basis, computational implementation, verification of the internal emission, and computational performance of the resulting model, the “IMC+emission,” is presented. Thorough verification includes fundamental tests of reciprocity and energy conservation, comparison to analytical solutions, and comparison with another 3D model, the Spherical Harmonics Discrete Ordinate Method (SHDOM). All comparisons to fundamental tests and analytical solutions are accurate to within the precision of the simulations—typically better than 0.05%. Comparison cases to SHDOM were typically within a few percent, except for flux divergence near cloud edges, where the effects of grid definition between the two models manifest themselves. Finally, the model is applied to the established I3RC case 4 cumulus cloud field to provide a benchmark result, and computational performance and strong and weak scaling metrics are presented. The outcome is a thoroughly vetted, publicly available, open-source benchmark tool to study 3D radiative transfer from either internal or external sources of radiation at wavelengths for which scattering, emission, and absorption are important.

Cirrus Cloud Optical Thickness and Effective Diameter Retrieved by MODIS: Impacts of Single Habit Assumption, 3‐D Radiative Effects, and Cloud Inhomogeneity

AbstractFor inhomogeneous cirrus clouds, cloud optical thickness (COT) and effective diameter (De) provided by the Moderate Resolution Imaging Spectrometer (MODIS) Collection 6 cloud products are associated with errors due to the single habit assumption (SHA), independent pixel assumption (IPA), photon absorption effect (PAE), and plane‐parallel assumption (PPA). SHA means that every cirrus cloud is assumed to have the same shape habit of ice crystals. IPA errors are caused by three‐dimensional (3D) radiative effects. PPA and PAE errors are caused by cloud inhomogeneity. We proposed a method to single out these different errors. These errors were examined using the Spherical Harmonics Discrete Ordinate Method simulations done for the MODIS 0.86 μm and 2.13 μm bands. Four midlatitude and tropical cirrus cases were studied. For the COT retrieval, the impacts of SHA and IPA were especially large for optically thick cirrus cases. SHA errors in COT varied distinctly with scattering angles. For the De retrieval, SHA decreased De under most circumstances. PAE decreased De for optically thick cirrus cases. For the COT and De retrievals, the dominant error source was SHA for overhead sun whereas for oblique sun, it could be any of SHA, IPA, and PAE, varying with cirrus cases and sun‐satellite viewing geometries. On the domain average, the SHA errors in COT (De) were within −16.1%–42.6% (−38.7%–2.0%), whereas the 3‐D radiative effects‐ and cloud inhomogeneity‐induced errors in COT (De) were within −5.6%–19.6% (−2.9%–8.0%) and −2.6%–0% (−3.7%–9.8%), respectively.

Testing the Two-Layer Model for Correcting Near Cloud Reflectance Enhancement Using LES/SHDOM Simulated Radiances.

A transition zone exists between cloudy skies and clear sky, such that clouds scatter solar radiation into clear sky regions. From a satellite perspective, it appears that clouds enhance the radiation nearby. We seek a simple method to estimate this enhancement, since it is so computationally expensive to account for all 3-dimensional (3D) scattering processes. In previous studies, we developed a simple two-layer model (2LM) that estimated the radiation scattered via cloud-molecular interactions. Here we have developed a new model to accounts for cloud-surface interaction (CSI). We test the models by comparing to calculations provided by full 3D radiative transfer simulations of realistic cloud scenes. For these scenes, the MODIS-like radiance fields were computed from the Spherical Harmonic Discrete Ordinate Method (SHDOM), based on a large number of cumulus fields simulated by the UCLA Large Eddy Simulation (LES) model. We find that the original 2LM model that estimates cloud-air molecule interactions accounts for 64% of the total reflectance enhancement, and the new model (2LM+CSI) that also includes cloud-surface interactions accounts for nearly 80%. We discuss the possibility of accounting for cloud-aerosol radiative interactions in 3D cloud induced reflectance enhancement, which may explain the remaining 20% of enhancements. Because these are simple models, these corrections can be applied to global satellite observations (e.g. MODIS) and help to reduce biases in aerosol and other clear-sky retrievals.

TIME-DEPENDENT MULTI-GROUP MULTI-DIMENSIONAL RELATIVISTIC RADIATIVE TRANSFER CODE BASED ON SPHERICAL HARMONIC DISCRETE ORDINATE METHOD

We develop a time-dependent multi-group multidimensional relativistic radiative transfer code, which is required to numerically investigate radiation from relativistic fluids involved in, e.g., gamma-ray bursts and active galactic nuclei. The code is based on the spherical harmonic discrete ordinate method (SHDOM) that evaluates a source function including anisotropic scattering in spherical harmonics and implicitly solves the static radiative transfer equation with a ray tracing in discrete ordinates. We implement treatments of time dependence, multi-frequency bins, Lorentz transformation, and elastic Thomson and inelastic Compton scattering to the publicly available SHDOM code. Our code adopts a mixed frame approach; the source function is evaluated in the comoving frame whereas the radiative transfer equation is solved in the laboratory frame. This implementation is validated with various test problems and comparisons with results of a relativistic Monte Carlo code. These validations confirm that the code correctly calculates intensity and its evolution in the computational domain. The code enables us to obtain an Eddington tensor that relates first and third moments of intensity (energy density and radiation pressure) and is frequently used as a closure relation in radiation hydrodynamics calculations.

Discrete ordinate method with matrix exponential for stochastic radiative transfer in broken clouds

We present a coherent treatment of the stochastic radiative transfer in an atmosphere containing clouds. Our analysis includes the derivation of an nth-order stochastic model for the solar radiation problem and arbitrary cloud statistics, as well as its particularization to broken cloud fields in general, and in particular to homogeneous three-, two- and one-dimensional broken clouds. The stochastic radiative transfer equations, which are equivalent to the Levermore–Pomraning equations, are solved by using the discrete ordinate method with matrix exponential. Using the Spherical Harmonics Discrete Ordinate Method (SHDOM) as a benchmark model, we found that the accuracy of the stochastic model, dealing with a molecular atmosphere and its underlying surface, is at least twice that of the independent pixel approximation. The stochastic model can be used to analyze the effects of cloud inhomogeneities on trace gas retrievals in the solar regime.

A multi-dimensional vector spherical harmonics discrete ordinate method for atmospheric radiative transfer

The multi-dimensional scalar Spherical Harmonics Discrete Ordinate Method (SHDOM) is extended to the vector case. The vector model uses complex and real generalized spherical harmonics in the energetic representation of the Stokes vector, and retains some powerful features of the scalar model, as for example, the combination of the generalized spherical harmonic and the discrete ordinate representations of the radiance field, the use of a linear short characteristic method for computing the corner-point values of the Stokes vector, and the application of the adaptive grid technique. Results illustrating the accuracy of the vector model are shown for realistic simulated clouds.

An analysis of the short-characteristic method for the spherical harmonic discrete ordinate method (SHDOM)

An analysis of the short-characteristic method is described. The various approaches of the short-characteristic method are classified according to the degree of the interpolation scheme for the extinction/source function product. The accuracy of a difference scheme is quantified through two types of errors: the interpolation and the flux conservation errors. First- and high-order difference schemes equipped with interpolation and flux conservation error calculations are implemented in SHDOM. Results illustrating the accuracy of higher-order schemes are shown for realistic simulated clouds.

Monte Carlo Simulations of Radiative Transfer in Cloudy Atmosphere over Sea Surfaces

A rigorous and modular Monte Carlo radiative transfer model MCRT has been developed to compute radiance for realistic cloudy atmosphere over sea surfaces in the visible and near-IR spectra. The simulation methods of the sample's trajectories are consistently described in the whole simulation spatial domain, which consists of a number of voxel cells for atmosphere and square cells for the sea surface. An inverse transformation method is applied in an atmospheric scattering event simulation and a rigorous accept/reject method is formulated for simulating a reflection event at Cox-Munk (CM) anisotropically and Nakajima-Tanaka (NT) isotropically roughened sea surface. Both methods have been implemented within the model and were tested as two individual modules achieving high accuracy. The model as a whole system was also validated by other codes. The mean difference is about 0.084% and 0.528% in comparison to the Spherical Harmonics code (SHARM) and Monte Carlo Atmospheric Radiative Transfer Simulator (MCARaTS) respectively for 1D atmosphere and the NT model. The comparisons of the Spherical Harmonic Discrete Ordinate Method (SHDOM) show that agreements are obtained in the sun glint regions for 1D atmosphere and the CM model and the mean differences are below 0.478% for a 3D cloud field with lambertian. In general, the major advantage of MCRT is that it could simulate more accurate reflection direction at sea surfaces in a shorter time and hence more accurate radiance in complex cloudy atmosphere over sea surfaces with high numerical efficiency especially on a small PC.

Computational Cost and Accuracy in Calculating Three-Dimensional Radiative Transfer: Results for New Implementations of Monte Carlo and SHDOM

Abstract This paper examines the tradeoffs between computational cost and accuracy for two new state-of-the-art codes for computing three-dimensional radiative transfer: a community Monte Carlo model and a parallel implementation of the Spherical Harmonics Discrete Ordinate Method (SHDOM). Both codes are described and algorithmic choices are elaborated. Two prototype problems are considered: a domain filled with stratocumulus clouds and another containing scattered shallow cumulus, absorbing aerosols, and molecular scatterers. Calculations are performed for a range of resolutions and the relationships between accuracy and computational cost, measured by memory use and time to solution, are compared. Monte Carlo accuracy depends primarily on the number of trajectories used in the integration. Monte Carlo estimates of intensity are computationally expensive and may be subject to large sampling noise from highly peaked phase functions. This noise can be decreased using a range of variance reduction techniques, but these techniques can compromise the excellent agreement between the true error and estimates obtained from unbiased calculations. SHDOM accuracy is controlled by both spatial and angular resolution; different output fields are sensitive to different aspects of this resolution, so the optimum accuracy parameters depend on which quantities are desired as well as on the characteristics of the problem being solved. The accuracy of SHDOM must be assessed through convergence tests and all results from unconverged solutions may be biased. SHDOM is more efficient (i.e., has lower error for a given computational cost) than Monte Carlo when computing pixel-by-pixel upwelling fluxes in the cumulus scene, whereas Monte Carlo is more efficient in computing flux divergence and downwelling flux in the stratocumulus scene, especially at higher accuracies. The two models are comparable for downwelling flux and flux divergence in cumulus and upwelling flux in stratocumulus. SHDOM is substantially more efficient when computing pixel-by-pixel intensity in multiple directions; the models are comparable when computing domain-average intensities. In some cases memory use, rather than computation time, may limit the resolution of SHDOM calculations.

Multidimensional Longwave Forcing of Boundary Layer Cloud Systems

Abstract The importance of multidimensional (MD) longwave radiative effects on cloud dynamics is evaluated in an eddy-resolving model (ERM)—the two-dimensional analog to large-eddy simulation (LES)—framework employing multidimensional radiative transfer [Spherical Harmonics Discrete Ordinate Method (SHDOM)]. Simulations are performed for a case of unbroken, marine boundary layer stratocumulus and a broken field of trade cumulus. “Snapshot” calculations of MD and independent pixel approximation (IPA; 1D) radiative transfer applied to simulated cloud fields show that the total radiative forcing changes only slightly, although the MD effects significantly modify the spatial structure of the radiative forcing. Simulations of each cloud type employing MD and IPA radiative transfer, however, differ little. For the solid cloud case, relative to using IPA, the MD simulation exhibits a slight reduction in entrainment rate and boundary layer total kinetic energy (TKE) relative to the IPA simulation. This reduction is consistent with both the slight decrease in net radiative forcing and a negative correlation between local vertical velocity and radiative forcing, which implies a damping of boundary layer eddies. Snapshot calculations of the broken cloud case suggest a slight increase in radiative cooling, although few systematic differences are noted in the interactive simulations. This result is attributed to the fact that radiative cooling is a relatively minor contribution to the total energetics. For the cloud systems in this study, the use of IPA longwave radiative transfer is sufficiently accurate to capture the dynamical behavior of boundary layer clouds. Further investigations are required to generalize this conclusion for other cloud types and longer time integrations.