delta-Eddington Approximation Research Articles

Towards Quantitative Evaluation of Tissue Absorption Coefficients Using Light Fluence Correction in Optoacoustic Tomography.

Optoacoustic tomography is a fast developing imaging modality, combining the high contrast available from optical excitation of tissue with the high resolution and penetration depth of ultrasound detection. Light is subject to both absorption and scattering when traveling through tissue; adequate knowledge of tissue optical properties and hence the spatial fluence distribution is required to create an optoacoustic image that is directly proportional to chromophore concentrations at all depths. Using data from a commercial multispectral optoacoustic tomography (MSOT) system, we implemented an iterative optimization for fluence correction based on a finite-element implementation of the delta-Eddington approximation to the Radiative Transfer Equation (RTE). We demonstrate a linear relationship between the image intensity and absorption coefficients across multiple wavelengths and depths in phantoms. We also demonstrate improved feature visibility and spectral recovery at depth in phantoms and with in vivo measurements, suggesting our approach could in the future enable quantitative extraction of tissue absorption coefficients in biological tissue.

Fast linear solver for radiative transport equation with multiple right hand sides in diffuse optical tomography

It is well known that radiative transfer equation (RTE) provides more accurate tomographic results than its diffusion approximation (DA). However, RTE-based tomographic reconstruction codes have limited applicability in practice due to their high computational cost. In this article, we propose a new efficient method for solving the RTE forward problem with multiple light sources in an all-at-once manner instead of solving it for each source separately. To this end, we introduce here a novel linear solver called block biconjugate gradient stabilized method (block BiCGStab) that makes full use of the shared information between different right hand sides to accelerate solution convergence. Two parallelized block BiCGStab methods are proposed for additional acceleration under limited threads situation. We evaluate the performance of this algorithm with numerical simulation studies involving the Delta–Eddington approximation to the scattering phase function. The results show that the single threading block RTE solver proposed here reduces computation time by a factor of 1.5–3 as compared to the traditional sequential solution method and the parallel block solver by a factor of 1.5 as compared to the traditional parallel sequential method. This block linear solver is, moreover, independent of discretization schemes and preconditioners used; thus further acceleration and higher accuracy can be expected when combined with other existing discretization schemes or preconditioners.

Microwave Radiometric Remote Sensing of Volcanic Ash Clouds From Space: Model and Data Analysis

The potential of satellite passive microwave sensors to provide quantitative information about near-source volcanic ash cloud parameters is assessed. To this aim, ground-based microwave weather radar and spaceborne microwave radiometer observations are used together with forward-model simulations. The latter are based on 2-D simulations with the numerical plume model Active Tracer High-Resolution Atmospheric Model (ATHAM), in conjunction with the radiative transfer model Satellite Data Simulator Unit (SDSU) that is based on the deltaEddington approximation and includes Mie scattering. The study area is the Icelandic subglacial volcanic region. The analyzed case study is that of the Grimsvotn eruption in May 2011. ATHAM input parameters are adjusted using available ground data, and sensitivity tests are conducted to investigate the observed brightness temperatures and their variance. The tests are based on the variation of environmental conditions like the terrain emissivity, water vapor, and ice in the volcanic plume. Quantitative correlation analysis between ATHAM/SDSU forward-model columnar content simulations and available microwave radiometric brightness temperature measurements, derived from the Special Sensor Microwave Imager/Sounder (SSMIS), are encouraging in terms of both dynamic range and correlation coefficient. The correlation coefficients are found to vary from -0.37 to -0.63 for SSMIS channels from 91 to 183 ± 1 GHz, respectively. The larger sensitivity of the brightness temperature at 183 ± 1 GHz to the columnar content, with respect to other channels, allowed us to consider this channel as the basis for a model-based polynomial relationship of volcanic plume height as a function of the measured SSMIS brightness temperature.

Regolith grain sizes of Saturn’s rings inferred from Cassini–CIRS far-infrared spectra

We analyze far-infrared (10–650cm−1) emissivity spectra of Saturn’s main rings obtained by the Cassini Composite Infrared Spectrometer (CIRS). In modeling of the spectra, the single scattering albedos of regolith grains are calculated using the Mie theory, diffraction is removed with the delta-Eddington approximation, and the hemispherical emissivities of macroscopic free-floating ring particles are calculated using the Hapke’s isotropic scattering model. Only pure crystalline water ice is considered and the size distribution of regolith grains is estimated. We find that good fits are obtained if the size distribution is broad ranging from 1μm to 1–10cm with a power law index of ∼3. This means that the largest regolith grains are comparable to the smallest free-floating particles in size and that the power law indices for both free-floating particles and regolith grains are similar to each other. The apparent relative abundance of small grains increases with decreasing solar phase angle (or increasing mean temperature). This trend is particularly strong for the C ring and is probably caused by eclipse cooling in Saturn’s shadow, which relatively suppresses warming up of grains larger than the thermal skin depth (∼1mm) under subsequent solar illumination.

Comparison of light scattering models for diffuse optical tomography

We reconstruct images of the absorption and the scattering coefficients for diffuse optical tomography using five different models for light propagation in tissues: (1) the radiative transport equation, (2) the delta-Eddington approximation, (3) the Fokker-Planck approximation, (4) the Fokker-Planck-Eddington approximation and (5) the generalized Fokker-Planck-Eddington approximation. The last four models listed are approximations of the radiative transport equation that take into account forward-peaked scattering analytically. Using simulated data from the numerical solution of radiative transport equation, we solve the inverse problem for the absorption and scattering coefficients using the transport-backtransport method. Through comparison of the numerical results, we show that all of these light scattering models produce good image reconstructions. In addition, we show that these approximations afford considerable computational savings over solving the radiative transport equation. However, all of the models exhibit significant "cross-talk" between absorption and scattering coefficient images. Among the approximations, we have found that the generalized Fokker-Planck-Eddington equation produced the best image reconstructions in comparison with the image reconstructions produced by the radiative transport equation.

On the finite volume method and the discrete ordinates method regarding radiative heat transfer in acute forward anisotropic scattering media

The ability of the finite volume method (FVM) and the discrete ordinates method (DOM) to model radiative heat transfer in acute forward anisotropic scattering media has been investigated. The test case involves a purely scattering medium in a cubic enclosure, irradiated by one boundary with diffuse emission. Four phase functions have been considered: three of the Henyey–Greenstein type with respective asymmetry factors of 0.2, 0.8 and 0.93, and a Mie phase function with a strong forward scattering peak (computed for a size parameter of 245 and corresponding to an asymmetry factor of 0.93). Results obtained with the FVM are in good agreement with Monte Carlo reference solutions, whatever the level of acute anisotropic scattering (for asymmetry factors up to 0.93). The DOM combined with the renormalization procedures of the phase function proposed by Kim and Lee (Effect of anisotropic scattering on radiative heat transfer in two-dimensional rectangular enclosures. Int J Heat Mass Transfer 1988;31:1711–21. [1]) and Wiscombe (On initialization error and flux conservation in the doubling method. JQSRT 1976;18:637–58. [2]) provides accurate results only for the smallest asymmetry factor. As the asymmetry factor increases, the renormalization procedures induce strong modifications in the values of the discretized phase function resulting in an underestimation of the effective attenuation by scattering. This error has been found to increase with optical thickness. In fact, when using the DOM, results would be more accurate combining this method with a Delta-Eddington approximation of the phase function, instead of using the actual phase function which is altered too much by renormalization.

Saharan dust in nighttime thermal imagery: Detection and reduction of related biases in retrieved sea surface temperature

Guided by radiative transfer modelling of the effects of Saharan dust (aerosol) on brightness temperatures at 3.9, 8.7, 11 and 12 μm measured by the Spinning Enhanced Visible and Infrared Imager (SEVIRI), we propose an indicator of the presence of Saharan dust in nighttime (infrared-only) SEVIRI imagery. Radiative transfer modelling is performed using a fast atmospheric transmittance model and a delta-Eddington approximation for scattering. Using aerosol single scattering properties appropriate to simulation of Saharan dust gives qualitatively good simulations of dust-affected brightness temperatures. From these simulations, we show that aerosol-free brightness temperatures are tightly clustered around the principal axis (explaining 99% of variance) in a 3-dimensional brightness-temperature difference space (the axes being 3.9 μm minus 8.7 μm, 3.9 μm minus 12 μm, and 11 μm minus 12 μm); in contrast, brightness temperatures affected by aerosol are simulated to be significantly off-axis when transformed into this space. Although the detailed effects of aerosol on brightness temperatures are not quantitatively reproduced by our simulations, this off-axis characteristic is empirically found to be also true of Saharan-dust-affected brightness temperatures in observations. The second principal component of the brightness-temperature difference space is identified as a useful index for Saharan dust. Comparisons with independent satellite measurements of aerosol optical depth (AOD, at 0.55 μm) show that the new Saharan dust index (SDI) is loosely correlated with AOD ( r = 0.34). The correlation is loose because the SDI is also sensitive to the height distribution of aerosol, which affects the aerosol's thermal emission. The SDI is used to develop an empirical correction scheme for SST retrievals affected by Saharan dust. Application of the SDI correction scheme removes an independent error of 0.2 K from SEVIRI SSTs validated against buoys from all latitudes.

Transient Coupled Heat Transfer in Multilayer Non-gray Semitransparent Media with Reflective Foils

In this paper, transient coupled radiative and conductive heat transfer in multilayer thermal insulation (MTI), which consists of non-gray semitransparent materials and reflective foils, is investigated for re-entry aerodynamic heating. The governing equation for combined radiation and conduction heat transfer in MTI is solved using a finite volume numerical method, whereas the radiative transfer equation (RTE) is solved using a finite difference method, in which the spectral scattering and absorption coefficients are determined using the Mie theory and the scattering phase function is modeled by the Delta-Eddington approximation. A comparison is carried out between numerical results for coupled heat transfer in non-gray MTI and those obtained by the two-flux model in which the medium is assumed to be gray. Finally, the numerical optimization of MTI is discussed.

Modelling the ultraviolet environment at the surface of Mars and design of the Beagle 2 UV sensor

This thesis describes a multi-layer radiative transfer UV model which was used to aid in the design of the UV sensor on Beagle 2, which will soon provide the first ever in situ measurement of UV flux at the martian surface. The model uses the delta-Eddington approximation for diffuse flux and new low temperature gas absorption cross-sections and aerosol optical properties. Dust, H2O clouds and morning fogs are found to modify the martian surface UV spectrum. Dust storms have been shown to attenuate the surface UV flux by more than an order of magnitude, though some UV persists even at extremely high optical depths. The seasonal variation of surface UV irradiance was found to produce maximum exposure areas highly dependent upon dust activity over the martian year. Dust activity is also shown to distort the annual latitudinal dose, with high dust loading in the southern hemisphere resulting in a higher annual dose than in the north. The introduction of O3 abundances of 1.64 x 1017cm-2 into the model resulted in only partial protection for micro-organisms, since wavelengths shorter than 230 nm still penetrate to the surface. DNA-weighting of a martian UV spectrum shows the surface UV environment to be 103 times more damaging than on Earth. UV detection signatures of atmospheric phenomena and anticipated events for the case of Beagle 2 are presented: a dust devil encounter creates a double minimum profile in the UV flux, and solar eclipses by Phobos produce a single minimum. Clouds increase the diffuse/direct UV ratio, and fogs create a distinct dip in the morning profile when normalised to clear days.

Effects of aerosol UV extinction on the formation of ozone and secondary particulate matter

Abstract A radiative transfer model based on the Delta-Eddington approximation is coupled into the framework of a three-dimensional air quality model that represents airborne particles as a source-oriented external mixture. Model simulations performed for Southern California on 25 September 1996 show that ultra-violet (UV) irradiance calculated by the model is in good agreement with measured ground level UV at central Los Angeles. Ozone performance statistics show that the use of the fully coupled radiative transfer calculation within the air quality models leads to more realistic predictions in regions with significant concentration gradients of airborne particulate matter and absorbing gases. The new air quality model with fully coupled radiative transfer calculations predicts lower ozone concentrations (10–22% reduction) and lower PM2.5 concentrations (2.3–8.5% reduction) relative to air quality models that use decoupled radiative transfer calculations or surface UV observations from sparse measurement networks. The greatest ozone reduction predicted by the fully coupled model occurs in polluted regions where photolysis rates are greatly reduced. The feedback effect on ozone concentrations decreased when VOC emissions increased but was insensitive to temperature. The greatest PM2.5 reduction occurs in regions with high gas-phase ammonia and particle-phase ammonium nitrate concentrations. The representation of airborne particulate matter as an internal mixture vs. source-oriented external mixture and homogenous particle vs. core-and-shell configuration did not have a significant effect on the relationship between UV feedback and secondary pollutant concentrations.

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.

Modified Delta-Eddington Approximation for Solar Reflection, Transmission, and Absorption Calculations

Abstract The fractional factor f of δ-function scaling in the δ-Eddington approximation modifies the fractional scattering into the forward peak. As shown in this paper, reasonably choosing the factor f can yield a great improvement of transmission, reflection, and absorption calculations in the condition of the optical depth τ ⩽ 1. Based on this fact, a modified δ-Eddington approximation is empirically and mathematically developed using a parameterization model of the factor f that mainly depends on asymmetry factor g0, total optical depth τ, single scattering albedo ϖ, (ground) surface reflectance A, and cosine of solar zenith angle μ0. There are 69 120 sets of comparative numerical tests, covering seven aerosol and two cloud size distributions, as well as three Henyey–Greenstein phase functions. Among the exiting two-stream approximations, δ-Eddington generally has better transmission, reflection, and absorption accuracy as τ ⩽ 1. In an average sense, in the condition of A ⩽ 0.6, τ ⩽ 1, 0.1 ⩽ μ0 ⩽ 1.0,...

Estimation of actinic flux and photolysis rate constant of NO 2 from aerosol size data

Aerosol mass measurements made using Andersen cascade impactor at different locations in an urban area were used to study the variation in power-law exponent in the Junge-type number distribution. The discrete mass size data were inverted by modified Twomey algorithm to obtain continuous mass size distribution from which number distributions were derived. These were used for evaluating actinic fluxes and hence NO 2 photolysis rate constants employing a radiative transfer model with delta-Eddington approximation. Results of the spatial variation of Junge power-law exponent over the urban region are presented. The relative magnitude of photolysis rate constants over regions with differing aerosol emission characteristics are illustrated. Parametric studies indicated the importance of considering fractional elemental carbon content of the aerosols in the atmosphere. Results of a limited validation study of the model are also presented using quartz crystal microbalance impactor data at one of the sites.

Impact of Shortwave Radiative Effects of Dust Aerosols on the Summer Season Heat Low over Saudi Arabia

Abstract A two-stream scattering scheme based on the delta-Eddington approximation is incorporated into the Florida State University Limited Area Model for computing the shortwave radiative fluxes due to dust aerosols over the Saudi Arabian region and to study their impact on synoptic-scale systems and the diurnal cycle over the region. The radiative properties of dust corresponding to different categories of dustiness are determined from the results of field experiments. Satellite imagery and visibility are used to determine the intensity and extent of the dust layer. Two parallel simulations, one including the radiative effects of dust aerosols and the other without them, were made over a 6-day period starting with 1200 UTC 25 June 1979 using First GARP (Global Atmospheric Research Program) Global Experiment IIIb data analyses from ECMWF. A comparison of the two experiments shows that the dust aerosol radiative heating strengthens the heat low over Saudi Arabia. Furthermore, the radiative heating of the...

3D Radiative Transfer in Weakly Inhomogeneous Medium. Part I: Diffusive Approximation

The solution of the three-dimensional radiative transfer equation in weakly horizontally inhomogeneous medium has been obtained in the diffusion approximation using the expansion of the three-dimensional delta-Eddington approximation. The solution approach, referred as the gradient correction (GC) method, expands the horizontal fluxes and the source function in terms of the horizontal gradient of the extinction coefficient and/or the cloud-top boundary. In the transfer equation, only the zeroth- and first-order gradient terms are retained and hence the following limitations apply. First, the length of the horizontal variations of optical properties of the medium should be large in comparison to the mean radiative transport length. Second, the ratio of the vertical to horizontal scales should be small enough so that fluxes from boundaries may be neglected. Since there are no restrictions to the amplitude of the optical properties variations, this method may even be applicable to a medium with strong horizontal variations of optical properties, as long as scales of the variations are large enough in comparison to the radiative transport length. The analytical solutions are in excellent agreement with the more accurate numerical solutions. The solution also shows the solar zenith angle dependence of the albedo, similar to that observed in analyses of satellite imagery. The GC approach may be useful as a fast and computationally inexpensive method both for the correction of the independent pixel approximation used for extraction of cloud fields from satellite imagery and possibly for the calculation of the radiation fluxes in climate models.

Generalized four-flux radiative transfer model

General solutions for a four-flux radiative transfer model, derivedfrom the radiative transfer equation and based on Lorenz-Miescattering and absorption parameters, have been obtained. Forwardand backward average path-length parameters have been considered aswell as forward-scattering ratios for diffuse anisotropic radiationgoing into the forward and the backward hemispheres. The reportedsolutions are generalizations of those obtained by Maheu et al. [Appl. Opt. 23, 3353-3362(1984)]. Compared with the generalized solutions, numericalcalculations indicate that the delta-Eddington approximation and thestandard four-flux model of Maheu et al. overestimate thecollimated-diffuse reflectance of particulate coatings, whereas thesemodels give similar results in the case of collimated-diffusetransmittance.

Sensitivity analysis of a UV radiation transfer model and experimental photolysis rates of NO 2 in the atmosphere of Mexico City

Photolysis of key species such as nitrogen dioxide, ozone and aldehydes, are the elementary reactions leading to the formation of photochemical smog. Calculations and experimental measurements (in particular, nitrogen dioxide) of these key reactions are reported, as well as, the sensitivity of them to the ozone column, local albedo and properties of the urban aerosol layer in Mexico City. A radiation transfer model (RTM ), based on the delta-Eddington approximation, was used in the calculations. The results show the importance of providing local resolution for photolysis rates by considering different local conditions within the modeling domain of air quality models.

Approximations of Phase Function in Calculating the Spectral Albedo of Snow Surface with Multiple Scattering.

Four types of approximations of the Mie phase function were studied in calculating multiple scattering by snow particles with the doubling method. These involve the two renormalizations of Hansen and Grant, the delta-M method and direct truncation. These four approximations were compared for snow surface albedo with effective grain radii of 50, 200 and 1000μm in a wavelength region from 0.3 to 3.0μm with the delta-Eddington approximation as a reference. In the Hansen's renormalization, the maximum albedo error exceeds 0.1 for snow with an effective radius of 1000μm at small solar zenith angles. The delta-M method overestimates snow albedos at all solar zenith angles in a wavelength region smaller than 1.4μm for snow with effective radius of 1000μm. This is due to insufficient angle resolution (0.1° in a scattering angle region less than 2°) in the forward peak region of the look-up table of the Mie phase function. It has been shown that even with ten times higher resolution in the scattering angle region less than 10° a sufficient accuracy could not be obtained for an effective radius of 1000μm in a wavelength region smaller than 0.6μm. Reasonable results were obtained by the Grant's renormalization and direct truncation approximation for all cases of effective grain radii studied. It was also found that these methods save computation time and memory because sufficient accuracy is obtained even with a low angle resolution of 0.1° in the forward peak region of phase function. In direct truncation, the result was not sensitive to the choice of a truncation angle between 5° and 20°.

Scattering of thermal radiation in the freeboard of a 1 MWt fluidized bed combustion with coal and limestone feeding

The effect of the scattering of thermal radiation on the emissivity of the particle holdup in the freeboard of a 1-MW t fluidized-bed combustor in conjunction with coal, and with coal and limestone feeding is compared. The cloud absorption and scattering coefficients and the scattering phase function are those obtained by Mie theory on the basis of the physical and chemical characterization of solids samples taken at the outlet from the freeboard. The cloud emissivity has been calculated by a discrete ordinate four-flux method for cylindrical axisymmetric coordinates using Delta-Eddington approximation of the phase function to take into account both the near isotropic scattering by small particles and the pronounced anisotropic scattering by large particles with respects to thermal wavelengths. Large particles make an essential contribution to the absorption thickness of the cloud, whereas their contribution to the effective scattering is minor compared to that of small particles, because of the increasing forward stretching of the phase function with increasing particle size parameter. The scattering of thermal radiation lowers the emissivity of the particle cloud to a larger extent in conjunction with limestone feeding by the combined effect of the increased albedo due to the low absorption index of limestone, and of the increased isotropy of scattering due to the large proportion of small particles in the particle size distribution. The change of absorption thickness of the particle could at the outlet from the freeboard between the two cases is small, in spite of the near threefold increase of solids concentration measured when limestone was added to the coal feed.

Parameterization of the shortwave radiative properties of water clouds for use in GCMs

The influence of the micro- and macrophysics of water clouds on the scattering and radiative properties of clouds is investigated using versatile cloud drop size distributions (DSDs) and Mie theory for single scattering and the delta-Eddington approximation for multiple scattering. A new parameterization scheme for the shortwave radiative properties of water clouds is presented. As for single- scattering properties, a new parameterization for cloud optical thickness (τ) is proposed. This is based upon the seperation of the dependence ofτ on the total number of DSDs, the cloud thickness, and the liquid water content, combined with equivalent radius. The cloud bulk radiative properties, i.e., the flux reflectance, transmittance, and absorptance, are uniquely fitted by a dimensionless parameter (X) or the optical thicknessτ. The parameterization is compared with other schemes. Finally, the features and potential use of the scheme are discussed.