Monte Carlo Radiative Transfer Model Research Articles

Two-dimensional Monte Carlo radiative transfer modelling of the disc-shaped secondary of Epsilon Aurigae

We present two dimensional Monte-Carlo radiative transfer models for the disk of the eclipsing binary $\epsilon$ Aur by fitting its spectral energy distribution from optical to the far-IR wavelengths. We also report new observations of $\epsilon$ Aur made by AKARI in its five mid and far-IR photometric bands and were used to construct our SED. The disk is optically thick and has flared disk geometry containing gas and dust with a gas to dust mass ratio of 100. We have taken the primary of the binary to be a F0Iae-type post-AGB star and the disk is heated by a B5V hot star with a temperature of 15,000 K at the center of the disk. We take the radius of the disk to be 3.8 AU for our models as constrained from the IR interferometric imaging observations of the eclipsing disk. Our models imply that the disk contains grains which are much bigger than the ISM grains (grain sizes 10$\mu$ to 100$\mu$). The grain chemistry of the disk is carbonaceous and our models show that silicate and ISM dust chemistry do not reproduce the slope of the observed SED in the mid-IR to far-IR regions. This implies that the formation of the disk shaped secondary in $\epsilon$ Aur system could be the result of accretion of matter and or mass transfer from the primary which is now a F0Iae post-AGB star. It is not a proto-planetary disk. The disk is seen nearly edge on with an inclination angle larger than 85$^{o}$. We propose from our radiative transfer modeling that the disk is not solid and have a void of 2AU radius at the center within which no grains are present making the region nearly transparent. The disk is not massive, its mass is derived to be less than 0.005M$_{\odot}$.

THIRTY YEARS OF SN 1980K: EVIDENCE FOR LIGHT ECHOES

We report optical and mid-infrared photometry of SN 1980K between 2004 and 2010, which show slow monotonic fading consistent with previous spectroscopic and photometric observations made 8 to 17 years after outburst. The slow rate-of-change over two decades suggests that this evolution may result from scattered and thermal light echoes off of extended circumstellar material. We present a semi- analytic dust radiative-transfer model that uses an empirically corrected effective optical depth to provide a fast and robust alternative to full Monte-Carlo radiative transfer modeling for homogenous dust at low to intermediate optical depths. We find that unresolved echoes from a thin circumstellar shell 14-15 lt-yr from the progenitor, and containing about 0.02 Msun of carbon-rich dust, can explain the broadband spectral and temporal evolution. The size, mass and dust composition are in good agreement with the contact discontinuity observed in scattered echoes around SN 1987A. The origin of slowly-changing high-velocity [O I] and Halpha lines is also considered. We propose an origin in shocked high-velocity metal-rich clumps of ejecta, rather than arising in the impact of ejecta on slowly-moving circumstellar material, as is the case with hot spots in SN 1987A.

The NIMO Monte Carlo model for box-air-mass factor and radiance calculations

A new fully spherical multiple scattering Monte Carlo radiative transfer model named NIMO (NIWA Monte Carlo model) is presented. The ray tracing algorithm is described in detail along with the treatment of scattering and absorption, and the simulation of backward adjoint trajectories. The primary application of NIMO is the calculation of box-air-mass factors (box-AMFs), which are used to convert slant column densities (SCDs) of trace gases, derived from UV–visible multiple axis Differential Optical Absorption Spectroscopy (MAX-DOAS) measurements, into vertical column densities (VCDs). Box-AMFs are also employed as weighting functions for optimal estimation retrievals of vertical trace gas profiles from SCDs. Monte Carlo models are well suited to AMF calculations at high solar zenith angles (SZA) and at low viewing elevation angles where multiple scattering is important. Additionally, the object-oriented structure of NIMO makes it easily extensible to new applications by plugging in objects for new absorbing or scattering species. Box-AMFs and radiances, calculated for various wavelengths, SZAs, viewing elevation and azimuth angles and aerosol scenarios, are compared with results from nine other models using a set of exercises from a recent radiative transfer model intercomparison. NIMO results for these simulations are well within the range of variability of the other models.

THE THREE-DIMENSIONAL DISTRIBUTION OF DUST IN NGC 891

We produce three-dimensional Monte-Carlo radiative transfer models of the edge-on spiral galaxy NGC 891, a fast-rotating galaxy thought to be an analogue to the Milky Way. The models contain realistic spiral arms and a fractal distribution of clumpy dust. We fit our models to Hubble Space Telescope images corresponding to the B and I bands, using shapelet analysis and a genetic algorithm to generate 30 statistically best-fitting models. These models have a strong preference for spirality and clumpiness, with average face-on attenuation decreasing from 0.24(0.16) to 0.03(0.03) mag in the B(I) band between 0.5 and 2 radial scale-lengths. Most of the attenuation comes from small high-density clumps with low (<10%) filling factors. The fraction of dust in clumps is broadly consistent with results from fitting NGC 891's spectral energy distribution. Because of scattering effects and the intermixed nature of the dust and starlight, attenuation is smaller and less wavelength-dependent than the integrated dust column-density. Our clumpy models typically have higher attenuation at low inclinations than previous radiative transfer models using smooth distributions of stars and dust, but similar attenuation at inclinations above 70 degrees. At all inclinations most clumpy models have less attenuation than expected from previous estimates based on minimizing scatter in the Tully-Fisher relation. Mass-to-light ratios are higher and the intrinsic scatter in the Tully-Fisher relation is larger than previously expected for galaxies similar to NGC 891. The attenuation curve changes as a function of inclination, with R_(B,B-I)=A_(B)/E(B-I) increasing by ~0.75 from face-on to near-edge-on orientations.

基于蒙特卡罗方法的卫星光学波段闪电辐射观测模拟

基于蒙特卡罗方法的卫星光学波段闪电辐射观测模拟

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.

Shoot scattering phase function for Scots pine and its effect on canopy reflectance

Spectral and directional reflectance properties of coniferous forests are known to differ from those of broadleaf forests. Many reasons have been proposed for this, including differences in the optical properties of leaves and shoots, the latter being considered the basic unit in radiative transfer modeling of a coniferous canopy. Unfortunately, very little empirical data is available on the spectrodirectional scattering properties of shoots. Here, we present results of angular measurements (using an ASD FieldSpec 3 spectroradiometer mounted on LAGOS) of ten Scots pine shoots in the spectral range 400–2000 nm. The shoots were found to scatter anisotropically with most of the radiation reflected back into the hemisphere where the radiation source was positioned. To describe the measured directional scattering pattern, we propose a phase function consisting of isotropic and Lambertian scattering components. Next, we used the proposed scattering phase function in a Monte Carlo radiative transfer model. Angular reflectance of a modeled horizontally homogeneous shoot canopy has, due to shoot scattering anisotropy, an enhanced “dark spot” as compared with a canopy composed of isotropic scatterers and a quantitatively similar leaf canopy.

Cloud Impact on Surface Altimetry From a Spaceborne 532-nm Micropulse Photon-Counting Lidar: System Modeling for Cloudy and Clear Atmospheres

This paper establishes a framework that simulates the behavior of a spaceborne 532-nm micropulse photon-counting lidar in cloudy and clear atmospheres in support of the ICESat-2 mission. Adopted by the current mission design, the photon-counting system will be used to obtain surface altimetry for ICESat-2. To investigate how clouds affect surface elevation retrievals, a 3-D Monte Carlo radiative transfer model is used to simulate the photon path distribution and the Poisson distribution is adopted for the number of photon returns. Since the photon-counting system only registers the time of the first arriving photon within the detector “dead time,” the retrieved average surface elevation tends to bias toward higher values. This is known as the first photon bias. With the scenarios simulated here, the first photon bias for clear sky is about 6.5 cm. Clouds affect surface altimetry in two ways: 1) Cloud attenuation lowers the average number of arriving photons and hence reduces the first photon bias, and 2) cloud forward scattering increases the photon path length and makes the surface appear further away from the satellite. Compared with that for clear skies, the average surface elevation detected by the photon-counting system for cloudy skies with optical depth of 1.0 is 4.0-6.0 cm lower for the simulations conducted. The effect of surface roughness on the accuracy of elevation retrievals is also discussed.

The effects of dust on the optical and infrared evolution of SN 2004et

We present an analysis of multi-epoch observations of the Type II-P supernova SN 2004et. New and archival optical spectra of SN 2004et are used to study the evolution of the Halpha and [O I] 6300A line profiles between days 259 and 646. Mid-infrared imaging was carried out between 2004 to 2010. We include Spitzer `warm' mission photometry at 3.6 and 4.5um obtained on days 1779, 1931 and 2151, along with ground-based and HST optical and near-infrared observations obtained between days 79 and 1803. Multi-wavelength light curves are presented, as well as optical-infrared spectral energy distributions (SEDs) for multiple epochs. Starting from about day 300, the optical light curves provide evidence for an increasing amount of circumstellar extinction attributable to newly formed dust, with the additional extinction reaching 0.8-1.5 magnitudes in the V-band by day 690. The overall SEDs were fitted with multiple blackbody components, in order to investigate the luminosity evolution of the supernova, and then with Monte Carlo radiative transfer models using smooth or clumpy dust distributions, in order to estimate how much new dust condensed in the ejecta. The luminosity evolution was consistent with the decay of 56Co in the ejecta up until about day 690, after which an additional emission source is required, in agreement with the findings of Kotak et al. (2009). Clumped dust density distributions consisting of 20% amorphous carbons and 80% silicates by mass were able to match the observed optical and infrared SEDs, with dust masses that increased from 8x10^{-5} Msun on day 300 to 1.5x10^{-3} Msun on day 690, still significantly lower than the values needed for core collapse supernovae to make a significant contribution to the dust enrichment of galaxies.

Effect of sastrugi on snow bidirectional reflectance and its application to MODIS data

[1] Snow surface roughness such as sastrugi on the Antarctic ice sheet can be a cause of error for remote sensing of snow parameters. The effect of sastrugi on snow bidirectional reflectance was assessed by a field experiment, model simulations, and satellite measurements. The hemispherical-directional reflectance factor (HDRF) of artificial sastrugi-like linear ridges measured at Nakasatsunai, Hokkaido, Japan, exhibited different patterns from that of a flat surface, with the difference of more than ±50% for some geometries. A 3-D Monte Carlo radiative transfer model (MC model) reproduced both the HDRF measurements for the artificial ideal sastrugi and previous measurements for natural sastrugi at the South Pole. Furthermore, the sastrugi effect was applied to remote sensing. Failure to include the surface roughness in models for developing snow-grain-size lookup tables can lead to order-of-magnitude retrieval errors. Using the MC model and multiangle data derived from the Moderate Resolution Imaging Spectrometer over the South Pole during the 2003–2004 summer, the sastrugi and snow parameters were retrieved. The height-to-width ratio of sastrugi reduced from 0.1 to 0.02, whereas the azimuth angle was nearly constant within the range of 0°–30° during the summer. The snow grain size showed a seasonal variation, which depended on the spectral channel. These retrieved parameters were consistent with existing ground measurements. The results suggest that a combination of multiangle data and a 3-D radiative transfer model can be used to quantitatively estimate surface roughness, along with snow grain size, on ice sheets.

The 3-dimensional structure of NGC 891 and M51

AbstractWe investigate the three-dimensional structure of the nearby edge-on spiral galaxy NGC 891 using 3D Monte Carlo radiative transfer models, with realistic spiral structure and fractally clumped dust. Using the spiral and clumpiness parameters found from recently completed scattered light models we produce lower resolution SED models which reproduce the spatially-integrated UV-to-FIR SED of NGC 891. Our models contain a color gradient across the major axis of the galaxy - similar to what is seen in images of the NGC 891. With minor adjustment our SED models are able to match the SED of M51, a similar galaxy at a near face-on inclination.

Confirmation of the magnetic oblique rotator model for the Of?p star HD 191612★

This paper reports high-precision Stokes V spectra of HD 191612 acquired using the ESPaDOnS spectropolarimeter at the Canada-France-Hawaii Telescope, in the context of the Magnetism in Massive stars (MiMeS) Project. Using measurements of the equivalent width of the Halpha line and radial velocities of various metallic lines, we have updated both the spectroscopic and orbital ephemerides of this star. We confirm the presence of a strong magnetic field in the photosphere of HD 191612, and detect its variability. We establish that the longitudinal field varies in a manner consistent with the spectroscopic period of 537.6 d, in an approximately sinusoidal fashion. This demonstrates a firm connection between the magnetic field and the processes responsible for the line and continuum variability. Interpreting the variation of the longitudinal magnetic field within the context of the dipole oblique rotator model we obtain a best-fit surface magnetic field model with obliquity beta=67\pm 5 deg and polar strength Bd=2450\pm 400 G . The inferred magnetic field strength implies an equatorial wind magnetic confinement parameter eta*~50, supporting a picture in which the Halpha emission and photometric variability have their origin in an oblique, rigidly rotating magnetospheric structure resulting from a magnetically channeled wind. This interpretation is supported by our successful Monte Carlo radiative transfer modeling of the photometric variation, which assumes the enhanced plasma densities in the magnetic equatorial plane above the star implied by such a picture. Predictions of the continuum linear polarisation resulting from Thompson scattering from the magnetospheric material indicate that the Stokes Q and U variations are highly sensitive to the magnetospheric geometry, and that expected amplitudes are in the range of current instrumentation. (abridged)

THE DUST PROPERTIES OF TWO HOT R CORONAE BOREALIS STARS AND A WOLF-RAYET CENTRAL STAR OF A PLANETARY NEBULA: IN SEARCH OF A POSSIBLE LINK

We present new Spitzer/IRS spectra of two hot R Coronae Borealis (RCB) stars, one in the Galaxy,V348 Sgr, and one lying in the LMC, HV 2671. These two objects may constitute a link between the RCB stars and the late Wolf-Rayet ([WCL]) class of central stars of planetary nebula (CSPNe) such as CPD -56 8032 that has little or no hydrogen in their atmospheres. HV 2671 and V348 Sgr are members of a rare subclass that has significantly higher effective temperatures than most RCB stars, but sharing the traits of hydrogen deficiency and dust formation that define the cooler RCB stars. The [WC] CSPNe star, CPD -56 8032, displays evidence for dual-dust chemistry showing both PAHs and crystalline silicates in its mid-IR spectrum. HV 2671 shows strong PAH emission but shows no sign of having crystalline silicates. The spectrum of V348 Sgr is very different from those of CPD -56 8032 and HV 2671. The PAH emission seen strongly in the other two stars is not present. Instead, the spectrum is dominated by a broad emission centered at about 8.2 micron. The mid-IR spectrum of CPD -56 8032 shows emission features that may be associated with C60. The other two stars do not show evidence for C60. HV 2671 has also been detected by Herschel/PACS and SPIRE. V348 Sgr and CPD -56 8032 have been detected by AKARI/FIS. These data were combined with Spitzer, IRAS, 2MASS and other photometry to produce their spectral energy distributions from the visible to the far-IR. Monte Carlo radiative transfer modeling was used to study the circumstellar dust around these stars. HV 2671 and CPD -56 8032 require both a flared inner disk with warm dust and an extended diffuse envelope with cold dust to to fit their SEDs. The SED of V348 Sgr can be fit with a much smaller disk and envelope.

Wind modelling of very massive stars up to 300 solar masses

The stellar upper-mass limit is highly uncertain. Some studies have claimed there is a universal upper limit of ~150 M⊙. A factor that is often overlooked is that there might be a significant difference between the present-day and the initial masses of the most massive stars – as a result of mass loss. The upper-mass limit may easily supersede ~200 M⊙. For these reasons, we present new mass-loss predictions from Monte Carlo radiative transfer models for very massive stars (VMS) in the mass range 40–300 M⊙, and with very high luminosities 6.0 ≤ log (L ⋆ /L⊙) ≤ 7.03, corresponding to large Eddington factors Γ. Using our new dynamical approach, we find an upturn or “kink” in the mass-loss versus Γ dependence, at the point where the model winds become optically thick. This coincides with the location where our wind efficiency numbers surpass the single-scattering limit of η = 1, reaching values up to η ≃ 2.5. In all, our modelling suggests a transition from common O-type winds to Wolf-Rayet characteristics at the point where the winds become optically thick. This transitional behaviour is also revealed with respect to the wind acceleration parameter, β, which starts at values below 1 for the optically thin O-stars, and naturally reaches values as high as 1.5–2 for the optically thick Wolf-Rayet models. An additional finding concerns the transition in spectral morphology of the Of and WN characteristic He ii line at 4686 Å. When we express our mass-loss predictions as a function of the electron scattering Eddington factor Γe ~ L ⋆ /M ⋆ alone, we obtain an Ṁ vs. Γe dependence that is consistent with a previously reported power law Ṁ (Vink 2006) that was based on our previous semi-empirical modelling approach. When we express Ṁ in terms of both Γe and stellar mass, we find optically thin winds and Ṁ ∝ for the Γe range 0.4 ≲ Γe ≲ 0.7, and mass-loss rates that agree with the standard Vink et al. recipe for normal O stars. For higher Γe values, the winds are optically thick and, as pointed out, the dependence is much steeper, Ṁ ∝ . Finally, we confirm that the effect of Γ on the predicted mass-loss rates is much stronger than for the increased helium abundance, calling for a fundamental revision in the way stellar mass loss is incorporated in evolutionary models for the most massive stars.

Non-line-of-sight single-scatter propagation model for noncoplanar geometries

In this paper, a geometrical propagation model is developed that generalizes the classical single-scatter model under the assumption of first-order scattering and non-line-of-sight (NLOS) communication. The generalized model considers the case of a noncoplanar geometry, where it overcomes the restriction that the transmitter and the receiver cone axes lie in the same plane. To verify the model, a Monte Carlo (MC) radiative transfer model based on a photon transport algorithm is constructed. Numerical examples for a wavelength of 266 nm are illustrated, which corresponds to a solar-blind NLOS UV communication system. A comparison of the temporal responses of the generalized model and the MC simulation results shows close agreement. Path loss and delay spread are also shown for different pointing directions.

Solar radiative transfer simulations in Saharan dust plumes: particle shapes and 3-D effect

Radiative fields of three-dimensional inhomogeneous Saharan dust clouds have been calculated at solar wavelength (0.6μm) by means of a Monte Carlo radiative transfer model. Scattering properties are taken from measurements in the SAMUM campaigns, from light scattering calculations for spheroids based on the MIESCHKA code, from Mie theory for spheres and from the geometric optics method assuming irregular shaped particles. Optical properties of different projected area equivalent shapes are compared. Large differences in optical properties are found especially in the phase functions.Results of radiative transfer calculations based on the Monte Carlo method are shown exemplarily for one dust cloud simulated by the cloud resolving atmospheric circulation model LM-MUSCAT-DES. Shape-induced differences in the radiation fluxes are pronounced, for example, the domain averaged normalized radiance is about 30% lower in the case of a dust plume consisting of spheroids or irregular particles compared to spheres. The effect of net horizontal photon transport (3-D effect) on the reflected radiance fields is only notable at the largest gradients in optical thickness. For example, the reflectance at low sun position differs locally about 15% when horizontal photon transport is accounted for. ‘Sharp edges’ due to 1-D calculations are smoothed out in the 3-D case.

A sensitivity analysis of Ring effect to aerosol properties and comparison to satellite observations

Abstract. In this study we explore the sensitivity of satellite observations of the Ring effect (at various wavelengths) to atmospheric aerosol properties. Compared to clouds, aerosols have a rather weak influence on the Ring effect, thus the requirements on the accuracy of the measurements and the radiative transfer simulations are high. In this study, we show that for moderate and high aerosol optical depth (AOD), Ring effect observations are sensitive enough to yield information not only on the AOD, but also on the absorbing properties of aerosols and the aerosol layer height. The latter two quantities are especially important for the determination of the radiative effects of aerosols. Our investigations are based on observations by the satellite instrument SCIAMACHY on ENVISAT (2004–2008) and on model simulations using the Monte-Carlo radiative transfer model McArtim. In addition to the Ring effect we investigate the impact of aerosols on the absorptions of the oxygen molecule (O2) and dimer (O4) as well as the radiance. In general good consistency between measured and simulated quantities is found. In some cases also systematic differences occurred, which are probably mainly related to the strong polarisation sensitivity of the SCIAMACHY instrument. Our study indicates that Ring effect observations have important advantages for aerosol retrievals: they can be analysed with high accuracy in various wavelength ranges; and depending on the wavelength range, they show different sensitivities on aerosol properties like single scattering albedo, optical depth or layer height. The results of this study are of particular interest for future aerosol inversion algorithms for satellite instruments with reduced polarisation sensitivity and smaller ground pixels, capable of measuring the Ring effect with higher accuracy.

HERSCHELOBSERVATIONS OF A NEWLY DISCOVERED UX Ori STAR IN THE LARGE MAGELLANIC CLOUD

The LMC star, SSTISAGE1C J050756.44-703453.9, was first noticed during a survey of EROS-2 lightcurves for stars with large irregular brightness variations typical of the R Coronae Borealis (RCB) class. However, the visible spectrum showing emission lines including the Balmer and Paschen series as well as many Fe II lines is emphatically not that of an RCB star. This star has all of the characteristics of a typical UX Ori star. It has a spectral type of approximately A2 and has excited an H II region in its vicinity. However, if it is an LMC member, then it is very luminous for a Herbig Ae/Be star. It shows irregular drops in brightness of up to 2 mag, and displays the reddening and "blueing" typical of this class of stars. Its spectrum, showing a combination of emission and absorption lines, is typical of a UX Ori star that is in a decline caused by obscuration from the circumstellar dust. SSTISAGE1C J050756.44-703453.9 has a strong IR excess and significant emission is present out to 500 micron. Monte Carlo radiative transfer modeling of the SED requires that SSTISAGE1C J050756.44-703453.9 has both a dusty disk as well as a large extended diffuse envelope to fit both the mid- and far-IR dust emission. This star is a new member of the UX Ori subclass of the Herbig Ae/Be stars and only the second such star to be discovered in the LMC.

Validating the MYSTIC three-dimensional radiative transfer model with observations from the complex topography of Arizona's Meteor Crater

Abstract. The MYSTIC three-dimensional Monte-Carlo radiative transfer model has been extended to simulate solar and thermal irradiances with a rigorous consideration of topography. Forward as well as backward Monte Carlo simulations are possible for arbitrarily oriented surfaces and we demonstrate that the backward Monte Carlo technique is superior to the forward method for applications involving topography, by greatly reducing the computational demands. MYSTIC is used to simulate the short- and longwave radiation fields during a clear day and night in and around Arizona's Meteor Crater, a bowl-shaped, 165-m-deep basin with a diameter of 1200 m. The simulations are made over a 4 by 4 km2 domain using a 10-m horizontal resolution digital elevation model and meteorological input data collected during the METCRAX (Meteor Crater Experiment) field experiment in 2006. Irradiance (or radiative flux) measurements at multiple locations inside the crater are then used to evaluate the simulations. MYSTIC is shown to realistically model the complex interactions between topography and the radiative field, resolving the effects of terrain shading, terrain exposure, and longwave surface emissions. The effects of surface temperature variations and of temperature stratification within the crater atmosphere on the near-surface longwave irradiance are then evaluated with additional simulations.

Cloud 3D effects on broadband heating rate profiles: I. Model simulation

Solar broadband heating directly drives the atmospheric and ocean circulations, and is largely determined by cloud spatial 3-diminesional (3D) structures. To study the cloud 3D effects on radiation, a 3D broadband Monte-Carlo radiative transfer model, along with an Independent Pixel/Column Approximation (IPA) method, is used to simulate radiation and heating rate of three typical cloud fields generated by cloud resolving models (CRM). A quantitative and statistical estimation of cloud 3D effects has been developed to investigate the impact of cloud 3D structures on both heating rate strength, STD_Bias, and vertical distribution, CorrCoef. The cloud 3D structures affect some clouds more in heating rate strength and others more in vertical distribution. It is crucial to use the combination of CorrCoef and STD_Bias for better quantitative evaluation of the 3D effects. Furthermore, there is no simple way to define a critical resolution (or average radius), within which the IPA heating rate profiles closely represent the true 3D heating rate profiles. The critical radius (or resolution) strongly depends on solar incident angle as well as cloud vertical distribution. Also, the critical radii for clear-sky columns are larger than for cloudy columns, although the corresponding STD_Bias for clear-sky columns are smaller than for cloudy columns. Analysis based on two different statistical average methods illustrates that the cloud 3D effects due to the dimensionality difference between the 3D clouds (circle average) and 2D clouds (line average) significantly impact on the heating rate profiles.