Approximate Radiative Transfer Research Articles

A microwave land emissivity model

Satellite observations using microwave radiometers operating near the window regions are strongly affected by surface emissivity. Presently, the measurements obtained over land are not directly utilized in numerical weather prediction models because of uncertainties in estimating the emissivity. This study develops a new model to quantify the land emissivity over various surface conditions. For surfaces such as snow, deserts, and vegetation, volumetric scattering was calculated using a two‐stream radiative transfer approximation. The reflection and transmission at the surface‐air interface and lower boundary were derived by modifying the Fresnel equations to account for cross‐polarization and surface roughness effects. Several techniques were utilized to compute the optical parameters for the medium, which is used in the radiative transfer solution. In the case of vegetation, geometrical optics is used because the leaf size is typically larger than the wavelength. For snow and deserts, a dense medium theory was adopted to take into account the coherent scattering of closely spaced particles. The emissivity spectra at frequencies between 4.9 and 94 GHz simulated and compared with the ground‐based radiometer measurements for bare soil, grass land, and snow conditions. It is shown that the features including the spectra, variability, and polarization agree well with the measurements. The simulated global distribution of land surface emissivity is also compared with the satellite retrievals from the Advanced Microwave Sounding Unit (AMSU). It is found that the largest discrepancies primarily occur over high latitudes where the snow properties are complex and least understood.

Three‐dimensional solar radiative transfer in small tropical cumulus fields derived from high‐resolution imagery

Since three‐dimensional (3‐D) radiative transfer in cloudy atmospheres is too expensive for large‐scale atmospheric models, approximate radiative transfer methods are used. The accuracy of these approximations for a large sample of realistic cloud fields has not been determined. This study examines 150 fields of small marine tropical cumulus to assess the magnitude of 3‐D effects on domain average solar fluxes and to evaluate the accuracy of these approximations in actual cumulus fields. The cloud fields are derived from Moderate‐Resolution Imaging Spectroradiometer Airborne Simulator (MAS) visible and thermal infrared imagery. Domain average broadband solar fluxes in these fields are simulated with a Monte Carlo radiative transfer model using four radiative transfer methods: full three‐dimensional, the independent pixel approximation (IPA), the tilted IPA, and the plane‐parallel approximation. The average 3‐D radiative effects of these cumulus cloud fields are small, with mean reflected flux errors up to 3 W/m2 for overhead Sun and less than 1 W/m2 for the daytime average. The mean errors for column‐absorbed fluxes are less than 0.3 W/m2 for all Sun angles. The small absolute flux errors are a result of the average field having small cloud fraction (10%), low cloud optical depth (4.4), and shallow clouds. Some individual fields have large 3‐D absolute reflected flux effects, and the normalized reflected flux errors are significant. The errors correlate well with the cloud fraction; so it may be possible to make corrections to the approximations.

Photo-oxidant chemistry in the polluted boundary layer under changing UV-B radiation

UV-B radiation is a driving factor for the chemistry of the polluted boundary layer. It is involved in the formation of radicals and consequently influences the formation and concentration of photo-oxidants. The 3-D mesoscale photochemical Metphomod model was employed to study the effect of changes in UV-B radiation on the concentration of photo-oxidants in the boundary layer over the Swiss Plateau. The model chemistry is based on the RACM mechanism and a two-stream approximation of radiative transfer. A summer (July) and a late winter (February) episode were simulated. All simulations were replicated with relatively large changes in the prescribed total ozone. The results for an increase in UV-B radiation show increases in PAN, HNO 3, and ozone at noon in NO x -rich areas and a decrease in NO x . In NO x -poor areas in summer the effect on ozone is weak and has a negative sign, the main effect being an increase in H 2O 2. The spatial variability of NO x concentrations in the Swiss Plateau in the summer case is such that the effect of increased UV-B radiation on ozone is spatially variable. The effect on the ozone production rate in summer is strongest positive at the surface in the NO x -rich regions in the morning and strongest negative at some altitude above ground in NO x -poor regions in the early afternoon. In the winter episode, NO x -rich conditions are found almost everywhere on the Swiss Plateau, the effect of increased UV-B radiation on the ozone production rate is positive all day long and is largest at 300 m above ground at noon. In this case, in contrast to the summer case, the increase in ozone is carried over to the next day. The model results for ozone are in good agreement with results from a case study and a time series analysis of surface ozone measurements. We estimate the effect of day-to-day changes in total ozone on surface ozone peaks to range from 4 to 6 ppb at most.

Systematic errors that are due to the monochromatic-equivalent radiative transfer approximation in thermal emission problems

An underlying assumption of data assimilation models is that the radiative transfer model used by them can simulate observed radiances with zero bias and small error. For practical reasons a fast parameterized radiative transfer model is used instead of a highly accurate line-by-line model. These fast models usually replace the spectral integration of the product of the transmittance and the Planck function with a monochromatic equivalent, namely, the product of a spectrally averaged transmittance and a spectrally averaged Planck function. The error of using this equivalent form is commonly assumed to be negligible. However, this error is not necessarily negligible and introduces a systematic height-dependent bias to the assimilation scheme. Although the bias could be corrected by a separate bias correction scheme, it is more effective to correct its source, the fast radiative transfer model. I examine the magnitude of error when the monochromatic-equivalent approach is used and demonstrate how a fast parameterized radiative model with Planck-weighted mean transmittances can effectively reduce if not eliminate these errors at source. I focus on channel 12 of the High-Resolution Infrared Radiation Sounder onboard the National Oceanic and Atmospheric Administration (NOAA)-14 satellite that, among all the channels of this instrument, displays the largest error.

Progress in bidirectional reflectance modeling and applications for surface particulate media: Snow and soils

Surface particulate media such as soil and snow have significant impacts on global systems because of their contributions to the radiation balance of the Earth system, their roles in the hydrologic cycle, and their extensive spatial distributions. At regional and global scales, characteristics of soils and snow cover can best be mapped and analyzed using satellite‐based sensors that acquire both spectral and angular reflectance information. Advances in surface reflectance models over the past decade now provide improved interpretation of remote sensing imagery over snow. However, advances have been slower in remote sensing of soil properties and a number of key issues remain. This article begins with an overview of the similarities and differences between particulate media modeling approaches for soil and snow. We review common modeling methods: radiative transfer approximations, numerical solutions, and geometrical optics. Model inversion and validation experiments are summarized and research priorities ...

Interactions of Radiation, Microphysics, and Turbulence in the Evolution of Cirrus Clouds

A two-dimensional cirrus cloud model has been developed to investigate the interaction and feedback of radiation, ice microphysics, and turbulence-scale turbulence, and their influence on the evolution of cirrus clouds. The model is designed for the study of cloud-scale processes with a 100-m grid spacing. The authors have incorporated a numerical scheme for the prediction of ice crystal size distributions based on calculations of nucleation, diffusional growth, advection, gravitational sedimentation, and turbulent mixing. The radiative effect on the diffusional growth of an individual ice crystal is also taken into account in the model. The model includes an advanced interactive radiative transfer scheme that employs the d-four-stream approximation for radiative transfer, the correlated k-distribution method for nongray gaseous absorption, and the scattering and absorption properties of hexagonal ice crystals. This radiation scheme is driven by ice water content and mean effective ice crystal size that represents the ice crystal size distribution. To study the effects of entrainment and mixing on the cloud, a second-order turbulence closure has been developed and incorporated into the model. Simulation results show that initial cloud formation occurs through ice nucleation associated with dynamic and thermodynamic forcings. Radiation becomes important for cloud evolution once a sufficient amount of ice water is generated. Radiative processes enhance both the growth of ice crystals at the cloud top by radiative cooling and the sublimation of ice crystals in the lower region by radiative heating. The simulated ice crystal size distributions depend strongly on the radiation fields. In addition, the radiation effect on individual ice crystals through diffusional growth is shown to be significant. Turbulence begins to play a substantial role in cloud evolution during the maintenance and dissipation period of the cirrus cloud life cycle. The inclusion of turbulence tends to generate more intermediate-to-large ice crystals, especially in the middle and lower parts of the cloud. Incorporation of the second-order closure scheme enhances instability below the initial cloud layer and brings more moisture to the region above the cloud, relative to the use of the traditional eddy mixing theory.

Variability of Mars' North Polar Water Ice Cap: I. Analysis of Mariner 9 and Viking Orbiter Imaging Data

Previous studies interpreted differences in ice coverage between Mariner 9 and Viking Orbiter observations of Mars' north residual polar cap as evidence of interannual variability of ice deposition on the cap. However, these investigators did not consider the possibility that there could be significant changes in the ice coverage within the northern residual cap over the course of the summer season. Our more comprehensive analysis of Mariner 9 and Viking Orbiter imaging data shows that the appearance of the residual cap does not show large-scale variance on an interannual basis. Rather we find evidence that regions that were dark at the beginning of summer look bright by the end of summer and that this seasonal variation of the cap repeats from year to year. Our results suggest that this brightening was due to the deposition of newly formed water ice on the surface. We find that newly formed ice deposits in the summer season have the same red-to-violet band image ratios as permanently bright deposits within the residual cap. We believe the newly formed ice accumulates in a continuous layer. To constrain the minimum amount of deposited ice, we used observed albedo data in conjunction with calculations using Mie theory for single scattering and a delta-Eddington approximation of radiative transfer for multiple scattering. The brightening could have been produced by a minimum of (1) a ∼35-μm-thick layer of 50-μm-sized ice particles with 10% dust or (2) a ∼14-μm-thick layer of 10-μm-sized ice particles with 50% dust.

Effects of Cloud Heterogeneities on Shortwave Radiation: Comparison of Cloud-Top Variability and Internal Heterogeneity

This paper examines the processes through which cloud heterogeneities influence solar reflection. This question is important since present methods give numerical results only for the overall radiative effect of cloud heterogeneities but cannot determine the degree to which various mechanisms are responsible for it. This study establishes a theoretical framework that defines these mechanisms and also provides a procedure to calculate their magnitude. In deriving the framework, the authors introduce a one-dimensional radiative transfer approximation, called the tilted independent pixel approximation (TIPA). TIPA uses the horizontal distribution of slant optical thicknesses along the direct solar beam to describe the radiative influence of cloud heterogeneities when horizontal transport between neighbors is not considered. The effects for horizontal transport are then attributed to two basic mechanisms: trapping and escape of radiation, when it moves to thicker and thinner cloud elements, respectively. Using the proposed framework, the study examines the shortwave radiative effects of cloud-top height and cloud volume extinction coefficient variations. It is shown and explained that identical variations in cloud optical thickness can cause much stronger heterogeneity effects if they are due to variations in geometrical cloud thickness rather than in volume extinction coefficient. The differences in albedo can exceed 0.05, and the relative differences in reflectance toward the zenith can be greater than 25% for overhead sun and 50% for oblique sun. The paper also explains a previously observed phenomenon: it shows that the trapping of upwelling radiation causes the zenith reflectance of heterogeneous clouds to increase with decreasing solar elevation.

Infrared emission (500–2000 cm −1) of laboratory ice clouds

The spectral infrared emission (500–2000 cm −1 ) of laboratory ice clouds has been measured with a Fourier transform infrared spectrometer. A cloudscope (a novel microscope equipped with a video camera) captured cloud ice crystals for microphysical characterization of the cloud. Theoretical emission calculations based on modified Mie theory, observed crystal size distributions, and an approximation for IR radiative transfer agree reasonably well with measurements. Model sensitivity to anomalous diffraction theory, size distribution, and particle shape are explored. Atmospheric applications are that the single-scattering approximation is dubious; that the zero-scattering approximation is accurate to with 20% in the atmospheric window region; and that recent observations of strong spectral variations in the upwelling IR emission from some cirrus is largely due to contributions by small ice crystals. The ratio of visible optical depth to IR optical depth at 10.85 μm (the “LIRAD” ratio) was 2.4, and is at the lower end of reported values from tropical cirrus despite the relatively small laboratory crystal sizes observed.

Improved application of exponential sum fitting transmissions to inhomogeneous atmosphere

An improved method of applying exponential sum fitting transmissions (ESFT) to inhomogeneous atmospheres (referred to as inhomogeneous ESFT or IESFT) is presented, and its use in a broadband radiative transfer model is documented. Instead of using a scaling function to represent the pressure and temperature dependence of the absorption coefficient we apply the ESFT to all pressures and temperatures based on weight terms determined at reference conditions. The exponential terms corresponding to all pressures and temperatures form a look‐up table which can be used to interpolate to the corresponding value at the real pressure and temperature. The IESFT is performed using a nonlinear least squares routine with some constraints to ensure that the ranking of the exponential terms is maintained. The method is tested against line‐by‐line calculations, and the results show that the use of IESFT in the two stream approximation of radiative transfer can reduce the cooling rate error by a factor of 2 relative to the use of a scaling function, particularly in the upper atmosphere. An application of the treatment of overlapping gaseous absorption developed by Mlawer et al. [1997] to ESFT is described. The basic idea of this method is to treat the mixture of two absorbing gases as a single gas by introducing a binary interpolation parameter which reflects the relative abundance of the two designated gases. Tests against line‐by‐line calculations show that this method is not only accurate but also computationally efficient.

Colors of RR Lyrae Stars

ABSTRACTThe methods of Davis & Cox are applied to a series of models described by Bono & Stellingwerf to determine the colors of RR Lyrae stars. Convection is ignored, and the radiation flow is treated by a complete variable Eddington, multifrequency‐dependent radiative transfer approximation.

S-curves of X-ray heated accretion discs and the black hole soft X-ray transients

It has recently been suggested that the thermal stability of accretion discs in the high M˙ regime may be influenced by X-rays that irradiate the disc. We investigate this problem by calculating the vertical structure of externally X-ray irradiated accretion discs using a two-stream bi-grey approximation for radiative transfer. The effects of X-ray heating depend on the hardness of the incident X-ray spectrum, and on the magnitude of the X-ray flux relative to the viscously generated flux. For heating by soft X-rays, a temperature inversion occurs in the upper atmosphere of the disc when X-ray heating dominates. For heating by hard X-rays, no significant X-ray inversion occurs. In general, the S-curves become increasingly distorted as the irradiation flux increases. The location of the upper critical point of the S-curve, and the corresponding critical mass transfer rate M˙crit above which the disc is thermally stable, is significantly reduced if the heating is predominantly by soft X-rays. Our results are used to discuss the stability of accretion discs in black hole (BH) soft X-ray transients (SXTs). In particular, we show that the BH-SXTs lie below the X-ray irradiated instability lines, and that the BH-SXT GRO1655-40 is in the regime where the disc instability is first triggered close to the outer edge of the disc.

Pomraning–Eddington approximation for radiative transfer in a homogeneous solid cylinder

The radiative transfer in a solid cylinder containing a homogeneous turbid medium with anisotropic scattering is considered. The medium has a diffuse and specular reflecting boundary illuminated by an external incidence and contains an internal energy source. This general problem can be solved in terms of the solution of the corresponding source-free problem with a specular reflecting boundary and isotropic external incidence. The Pomraning–Eddington approximation is used to solve the source-free problem. Three different weight functions are used to verify the boundary condition to find the constants of the solution. The partial flux, the irradiance and the net flux at the boundary for the general problem are calculated.

Cirrus cloud optical properties in far infrared

Abstract A new parameterization of cloud optical properties in the Far Infrared (FIR) (10–200 μ) has been carried out. The forward and scattered transmissivity and the scattered reflectivity are all included in the emissivity expression. Results are compared with the ones calculated by other authors and with Mie theory. The parameterization developed has been implemented in a two-stream approximation radiative transfer code and the TOA upward flux variations as function of cirrus clouds optical and physical parameters are discussed. Results show that the scattering effects added generally increase the optical depth of the cloud and decreases the OLR, with most relevant effects on medium optical depth clouds.

The Jeans Mass Constraint and the Fragmentation of Molecular Cloud Cores

The collapse and fragmentation of molecular cloud cores into binary and multiple protostar systems is a demanding computational problem, in part because of the large range of length scales involved. Truelove et al. proposed that the computational cell size Δx must be smaller than 1/4 of the Jeans length, λJ, if artificial (numerical) fragmentation is to be avoided. For a uniform Cartesian grid, Truelove et al.'s Jeans condition is equivalent to ensuring that the mass of a cell never exceeds ~1/64 of a Jeans mass. It is shown here that for a nonuniform spherical grid, artificial fragmentation can be avoided provided that the cell size of a cube with approximately the same volume as the spherical coordinate cell [Δx=(ΔxrΔxθΔx)1/3] is less than λJ/4 (i.e., that the mass inside each cell is much less than a Jeans mass), even if one of the three cell lengths (Δxr, Δxθ, or Δx) exceeds λJ/4. For a nonuniform grid, resolving a small fraction of a Jeans mass is less restrictive than resolving 1/4 of a Jeans length in each coordinate direction; resolving all three Jeans lengths is desirable but not necessary in order to avoid artificial fragmentation. The well-resolved collapse of an initially Gaussian-profile cloud is then followed with both isothermal and nonisothermal (Eddington approximation radiative transfer) thermodynamics and is shown to lead to fragmentation into a binary protostar system in both cases. The Jeans mass constraint appears to be a valuable indicator of physically realistic fragmentation.

Intercomparison of models representing direct shortwave radiative forcing by sulfate aerosols

The importance of aerosols as agents of climate change has recently been highlighted. However, the magnitude of aerosol forcing by scattering of shortwave radiation (direct forcing) is still very uncertain even for the relatively well characterized sulfate aerosol. A potential source of uncertainty is in the model representation of aerosol optical properties and aerosol influences on radiative transfer in the atmosphere. Although radiative transfer methods and codes have been compared in the past, these comparisons have not focused on aerosol forcing (change in net radiative flux at the top of the atmosphere). Here we report results of a project involving 12 groups using 15 models to examine radiative forcing by sulfate aerosol for a wide range of values of particle radius, aerosol optical depth, surface albedo, and solar zenith angle. Among the models that were employed were high and low spectral resolution models incorporating a variety of radiative transfer approximations as well as a line‐by‐line model. The normalized forcings (forcing per sulfate column burden) obtained with the several radiative transfer models were examined, and the discrepancies were characterized. All models simulate forcings of comparable amplitude and exhibit a similar dependence on input parameters. As expected for a non‐light‐absorbing aerosol, forcings were negative (cooling influence) except at high surface albedo combined with small solar zenith angle. The relative standard deviation of the zenith‐angle‐averaged normalized broadband forcing for 15 models was 8% for particle radius near the maximum in this forcing (∼0.2 μm) and at low surface albedo. Somewhat greater model‐to‐model discrepancies were exhibited at specific solar zenith angles. Still greater discrepancies were exhibited at small particle radii, and much greater discrepancies were exhibited at high surface albedos, at which the forcing changes sign; in these situations, however, the normalized forcing is quite small. Discrepancies among the models arise from inaccuracies in Mie calculations, differing treatment of the angular scattering phase function, differing wavelength and angular resolution, and differing treatment of multiple scattering. These results imply the need for standardized radiative transfer methods tailored to the direct aerosol forcing problem. However, the relatively small spread in these results suggests that the uncertainty in forcing arising from the treatment of radiative forcing of a well‐characterized aerosol at well‐specified surface albedo is smaller than some of the other sources of uncertainty in estimates of direct forcing by anthropogenic sulfate aerosols and anthropogenic aerosols generally.

Comments on “Four-Stream Spherical Harmonic Expansion Approximation for Solar Radiative Transfer”

Comments on : four-stream spherical harmonic expansion approximation for solar radiative transfer.

Four-Stream Isosector Approximation for Solar Radiative Transfer

Abstract For radiative transfer in a thin atmosphere, an analytical four-stream isosector approximation for solar radiative transfer is presented. This approximation method is based on the assumption of four spherical sectors of isotropic intensities. Calculations show that the four-stream isosector approximation model substantially improves the accuracy in reflection, transmission, and absorption with respect to the Coakley–Chýlek model. For an optical thickness less than unity, the four-stream isosector approximation has errors mostly under 5%, in contrast to errors up to 20% or higher for the Coakley–Chýlek model. This four-stream isosector approximation can be applied to atmospheric aerosol layers or thin cirrus clouds.

Multiple Scattering Parameterization in Thermal Infrared Radiative Transfer

A systematic formulation of various radiative transfer parameterizations is presented, including the absorption approximation (AA), δ-two-stream approximation (D2S), δ-four-stream approximation (D4S), and δ-two- and four-stream combination approximation (D2/4S), in a consistent manner for thermal infrared flux calculations. The D2/4S scheme uses a source function from the δ-two-stream approximation and evaluates intensities in the four-stream directions. A wide range of accuracy checks for monochromatic emissivity of a homogeneous layer and broadband heating rates and fluxes in nonhomogeneous atmospheres is performed with respect to the “exact” results computed from the δ-128-stream scheme for radiative transfer. The computer time required for the calculations using different radiative transfer parameterizations is compared. The results pertaining to the accuracy and efficiency of various radiative transfer approximations can be utilized to decide which approximate method is most appropriate for a particular application. In view of its overall high accuracy and computational economy, it is recommended that the D2/4S scheme is well suited for GCM and climate modeling applications.

Aerosol Optical Depth over Oceans: High Space- and Time-Resolution Retrieval and Error Budget from Satellite Radiometry

A method to retrieve aerosol vertical optical depth at 0.64 mm from satellite observations of cloud-free scenes over oceans with high spatial resolution (;18) and instantaneous temporal resolution is described and evaluated. The observed radiance is treated as the linear sum of contributions to path radiance by different scattering processes in the atmosphere‐ocean system. This treatment allows examination of errors in the retrieved vertical aerosol optical depth contributed by each process and approximation. Random error in retrieved aerosol optical depth is typically 0.03. The systematic error due to absolute calibration uncertainty in the measured radiance is 0.01. The largest errors and biases are due to radiative transfer approximations (122%) and assumptions regarding aerosol microphysical and optical properties (220%). The latter errors, which are due to the optical properties (e.g., phase function), vary systematically with latitude and season because of the variation of the mean observing geometry. This method is applied to Advanced Very High Resolution Radiometer global area coverage data, and example maps of aerosol optical depth are presented for specific dates in July and October 1986. The aerosol optical depth derived from the satellite data is suitable for examining large aerosol signatures by instantaneous comparison of the amplitude and location of aerosol plumes with model predictions based on meteorological conditions at and preceding the time of observation.