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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 7 — Mar. 26, 2012
  • pp: 7973–7993
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MOCRA: a Monte Carlo code for the simulation of radiative transfer in the atmosphere

Margherita Premuda, Elisa Palazzi, Fabrizio Ravegnani, Daniele Bortoli, Samuele Masieri, and Giorgio Giovanelli  »View Author Affiliations


Optics Express, Vol. 20, Issue 7, pp. 7973-7993 (2012)
http://dx.doi.org/10.1364/OE.20.007973


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Abstract

This paper describes the radiative transfer model (RTM) MOCRA (MOnte Carlo Radiance Analysis), developed in the frame of DOAS (Differential Optical Absorption Spectroscopy) to correctly interpret remote sensing measurements of trace gas amounts in the atmosphere through the calculation of the Air Mass Factor. Besides the DOAS-related quantities, the MOCRA code yields: 1- the atmospheric transmittance in the vertical and sun directions, 2- the direct and global irradiance, 3- the single- and multiple- scattered radiance for a detector with assigned position, line of sight and field of view. Sample calculations of the main radiometric quantities calculated with MOCRA are presented and compared with the output of another RTM (MODTRAN4). A further comparison is presented between the NO2 slant column densities (SCDs) measured with DOAS at Evora (Portugal) and the ones simulated with MOCRA. Both comparisons (MOCRA-MODTRAN4 and MOCRA-observations) gave more than satisfactory results, and overall make MOCRA a versatile tool for atmospheric radiative transfer simulations and interpretation of remote sensing measurements.

© 2012 OSA

1. Introduction

Propagation of light in the atmosphere can be mathematically described by the Radiative Transfer Equation (RTE), whose solution can be obtained both analytically [1

1. J. Lenoble, ed., “Standard Procedures to Compute Atmospheric Radiative Transfer in a Scattering Atmosphere - Volume I, International Association of Meteorology and Atmospheric Physics (IAMAP), ” Boulder, Colorado, USA (1977).

,2

2. Y. Fouquart, W. M. Irvine, and J. Lenoble, eds., “Standard Procedures to Compute Atmospheric Radiative Transfer in a Scattering Atmosphere - Volume II, International Association of Meteorology and Atmospheric Physics (IAMAP),” Boulder, Colorado, USA (1980).

] and numerically [3

3. K. Stamnes, S.-C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27(12), 2502–2509 (1988). [CrossRef] [PubMed]

]. The atmosphere is a non homogeneous medium where absorption, scattering and polarization effects combine themselves and make the system extremely difficult for analytical treatment. The Monte Carlo (MC) method is a powerful numerical tool for performing radiative transfer simulations including absorption, polarization and multiple scattering. The method requires the knowledge of the atmospheric physical properties, such as the interaction coefficients and scattering phase functions, obtained by means of different theoretical models according to the type of atmospheric constituents (gases or aerosol particles). Several examples of MC studies of radiative transfer in scattering and absorbing media can be found in the literature [4

4. G. N. Plass and G. W. Kattawar, “Monte carlo calculations of light scattering from clouds,” Appl. Opt. 7(3), 415–419 (1968). [CrossRef] [PubMed]

6

6. E. Tinet, S. Avrillier, and J. M. Tualle, “Fast semianalytical Monte Carlo simulation for time-resolved light propagation in turbid media,” J. Opt. Soc. Am. A 13(9), 1903–1915 (1996). [CrossRef]

].

The Monte Carlo approach for atmospheric radiative transfer consists in using probabilistic methods to trace the trajectories of individual photons, which are subject to random events. A random number, for example, can be used to decide whether a photon gets scattered or absorbed on its way, and to fix the flight direction in case of scattering. The strict Monte Carlo experiment consists in firing photons out of a radiation source along an initial direction and in following their path through the atmosphere where they are scattered, by which compound and in which direction, till they hit the detector or they are absorbed. The problem in this “forward” procedure is that simulating realistic dimensions and aperture angle of a detector can result in a very low probability for one single photon to get recorded by the modeled receiving apparatus. This would require a very high number of photon histories to be processed to obtain statistically significant results. To overcome this problem, because of linearity of the RTE and reciprocity relationship between the Green’s function and its adjoint [7

7. A. De Matteis and R. Simonini, “A New Monte Carlo Approach to the Adjoint Boltzmann Equation,” Nucl. Sci. Eng. 65, 93–105 (1978).

,8

8. A. De Matteis and R. Simonini, “A Monte Carlo Biasing Scheme for the Adjoint Photon Transport,” Nucl. Sci. Eng. 67, 309–316 (1978).

], it is possible to write an adjoint RTE and to perform a “backward” Monte Carlo simulation. Each photon history is traced backward through the atmosphere as being generated by the detector in the line of sight direction, with a photon “weight” proportional to the source intensity, and coming out of the atmosphere towards the sun. Examples of such a simulation scheme can be found in references [9

9. D. G. Collins, W. G. Blättner, M. B. Wells, and H. G. Horak, “Backward monte carlo calculations of the polarization characteristics of the radiation emerging from spherical-shell atmospheres,” Appl. Opt. 11(11), 2684–2696 (1972). [CrossRef] [PubMed]

, 10

10. W. G. Blättner, H. G. Horak, D. G. Collins, and M. B. Wells, “Monte carlo studies of the sky radiation at twilight,” Appl. Opt. 13(3), 534–547 (1974). [CrossRef] [PubMed]

], while in references [4

4. G. N. Plass and G. W. Kattawar, “Monte carlo calculations of light scattering from clouds,” Appl. Opt. 7(3), 415–419 (1968). [CrossRef] [PubMed]

6

6. E. Tinet, S. Avrillier, and J. M. Tualle, “Fast semianalytical Monte Carlo simulation for time-resolved light propagation in turbid media,” J. Opt. Soc. Am. A 13(9), 1903–1915 (1996). [CrossRef]

] the forward scheme is followed. It is worth pointing out that, in case of atmospheric simulations, a backward Monte Carlo scheme is even simpler than in other contexts, as for instance photon transport in shielding calculations for nuclear reactors. Photons interacting with atmospheric constituents, in fact, do not undergo energy changes, which would require specific and careful treatment [8

8. A. De Matteis and R. Simonini, “A Monte Carlo Biasing Scheme for the Adjoint Photon Transport,” Nucl. Sci. Eng. 67, 309–316 (1978).

]. It has been demonstrated that the backward Monte Carlo technique is superior to the forward method in real case applications to atmospheric problems, by greatly reducing the computational demands [11

11. B. Mayer, S. W. Hoch, and C. D. Whiteman, “Validating the MYSTIC three-dimensional radiative transfer model with observations from the complex topography of Arizona's Meteor Crater,” Atmos. Chem. Phys. Discuss. 10(5), 13373–13405 (2010), doi:. [CrossRef]

].

A certain number of backward Monte Carlo RTMs have been developed to simulate remote sensing observations of trace gas amounts measured through the Differential Optical Absorption Spectroscopy (DOAS) technique [12

12. J. F. Noxon, “Nitrogen dioxide in the stratosphere and troposphere measured by ground-based absorption spectroscopy,” Science 189(4202), 547–549 (1975). [CrossRef] [PubMed]

16

16. A. Petritoli, G. Giovanelli, I. Kostadinov, F. Ravegnani, D. Bortoli, P. Bonasoni, F. Evangelisti, U. Bonafe, and F. Calzolari, “Tropospheric and stratospheric NO2 amount deduced by slant column measurements at Mt. Cimone Station,” Adv. Space Res. 29(11), 1691–1695 (2002). [CrossRef]

]. DOAS is an established remote sensing method which identifies and quantifies the trace gases in the atmosphere, taking advantage of their absorption structures in the near UV and visible wavelengths of the solar spectrum (DOAS measuring solar or other natural radiation is called “passive DOAS”). The molecular absorption is analyzed to obtain the concentration of trace gases integrated along the optical path (or Slant Column Density, SCD) between the sun and the receiver. In the case of a vertical looking ground-based sensor (“zenith” configuration), radiative transfer models are used to convert the measured SCD into Vertical Column Density (the concentration of the absorber integrated along the vertical, VCD), via the calculation of the so-called Air Mass Factor (AMF). Perliski and Solomon [17

17. L. M. Perliski and S. Solomon, “On the Evaluation of Air Mass Factors for Atmospheric Near-Ultraviolet and Visible Absorptiion Spectroscopy,” J. Geophys. Res. 98(D6), 10363–10374 (1993). [CrossRef]

], for instance, developed a multiple scattering backward Monte Carlo RTM to calculate the AMF of their target trace gases from DOAS observations performed with an upward-looking detector. They compared their results with those obtained with a single scattering model [18

18. S. Solomon, A. Schmeltekopf, and R. W. Sanders, “On the interpretation of zenith sky absorption measurements,” J. Geophys. Res. 92(D7), 8311–8319 (1987). [CrossRef]

], and concluded that multiple scattering significantly affects the measurements at twilight. The TRACY (Trace gas RAdiative Transfer Monte Carlo Y(I)implementation) series models [19

19. J. Pukıte, S. Kühl, T. Deutschmann, W. Wilms-Grabe, C. Friedeburg, U. Platt, and T. Wagner, “Retrieval of stratospheric trace gases from SCIAMACHY limb measurements,” Proceedings of the First Atmospheric Science Conference, 8–12 May, ESA/ESRIN, Frascati, Italy, ESA SP-628, (2006). http://earth.esa.int/workshops/atmos2006/participants/1148/paper proc Frasc 2.pdf

,20

20. S. Kühl, J. Puķīte, T. Deutschmann, U. Platt, and T. Wagner, “SCIAMACHY Limb Measurements of NO2, BrO and OClO, Retrieval of vertical profiles: Algorithm, first results, sensitivity and comparison studies,” Adv. Space Res. 42(10), 1747–1764 (2008). [CrossRef]

] solve the radiative transfer equation using the backward Monte Carlo method and derive so-called “box air mass factors” which describe the sensitivity of the measurements as a function of atmospheric layer altitude. The TRACY-II version [21

21. J. Puķīte, S. Kühl, T. Deutschmann, U. Platt, and T. Wagner, “Accounting for the effect of horizontal gradients in limb measurements of scattered sunlight,” Atmos. Chem. Phys. 8(12), 3045–3060 (2008). [CrossRef]

], in particular, has been used to retrieve the vertical profiles of the absorbing trace gases from limb-viewing satellite measurements.

More recently, DOAS has been deployed also in “off-axis” configuration [22

22. G. Hönninger, C. von Friedeburg, and U. Platt, “Multi axis differential optical absorption spectroscopy (MAX-DOAS),” Atmos. Chem. Phys. 4(1), 231–254 (2004), http://www.atmos-chem-phys.net/4/231/2004/acp-4-231-2004.pdf. [CrossRef]

24

24. G. Giovanelli, E. Palazzi, A. Petritoli, D. Bortoli, I. Kostadinov, F. Margelli, S. Pagnutti, M. Premuda, F. Ravegnani, and G. Trivellone, “Perspectives of 2D and 3D mapping of Atmospheric Pollutants over Urban Areas by means of airborne DOAS Spectrometer,” Ann. Geophys. 49(1), 133–142 (2006).

], i.e. ground-based instrument with viewing direction other than vertical one, for a better characterization of tropospheric absorbers. A very recent application of the off-axis configuration is the FRE-DOAS (Flow Rate Emission with DOAS) technique, which was used by Premuda et al. [25

25. M. Premuda, S. Masieri, D. Bortoli, I. Kostadinov, A. Petritoli, and G. Giovanelli, “Evaluation of vessel emissions in a lagoon area with ground based Multi axis DOAS measurements,” Atmos. Environ. 45(29), 5212–5219 (2011), doi:, http://dx.doi.org/10.1016/j.atmosenv.2011.05.067. [CrossRef]

] to calculate gaseous pollutant flow rates from moving sources like ships. The AMF concept can be generalized to off-axis configuration measurements and appropriate RTMs have to be used for data interpretation. Palazzi et al. [26

26. E. Palazzi, A. Petritoli, G. Giovanelli, I. Kostadinov, D. Bortoli, F. Ravegnani, and S. S. Sackey, “PROMSAR: A backward Monte Carlo spherical RTM for the analysis of DOAS remote sensing measurements,” Adv. Space Res. 36(5), 1007–1014 (2005). [CrossRef]

] developed the PROMSAR (PROcessing of Multi-Scattered Atmospheric Radiation) code to perform backward Monte Carlo simulations of DOAS observations for vertical and off-axis looking detectors. PROMSAR Box AMFs and radiances were validated through comparison with a series of state-of-the-art UV/visible RTMs [27

27. T. Wagner, J. P. Burrows, T. Deutschmann, B. Dix, C. von Friedeburg, U. Frieß, F. Hendrick, K.-P. Heue, H. Irie, H. Iwabuchi, Y. Kanaya, J. Keller, C. A. McLinden, H. Oetjen, E. Palazzi, A. Petritoli, U. Platt, O. Postylyakov, J. Pukite, A. Richter, M. van Roozendael, A. Rozanov, V. Rozanov, R. Sinreich, S. Sanghavi, and F. Wittrock, “Comparison of box-air-mass-factors and radiances for Multiple-Axis Differential Optical Absorption Spectroscopy (MAX-DOAS) geometries calculated from different UV/visible radiative transfer models,” Atmos. Chem. Phys. 7(7), 1809–1833 (2007). [CrossRef]

].

In this paper, the backward Monte Carlo model MOCRA (MOnte Carlo Radiance Analysis) is described. MOCRA is an up-to-date version of the PROMSAR model [26

26. E. Palazzi, A. Petritoli, G. Giovanelli, I. Kostadinov, D. Bortoli, F. Ravegnani, and S. S. Sackey, “PROMSAR: A backward Monte Carlo spherical RTM for the analysis of DOAS remote sensing measurements,” Adv. Space Res. 36(5), 1007–1014 (2005). [CrossRef]

] derived from the PREMAR code [28

28. E. Cupini, M. G. Borgia and M. Premuda, Il codice PREMAR per la simulazione Montecarlo del trasporto della radiazione nell’atmosfera. (RT/INN/97/5. 1997).

,29

29. E. Cupini, G. Ferro, N. Ferrari, Monte Carlo Analysis of Radiative Transport in Oceanographic Lidar Measurements. (ENEA/RT/INN/2001/7, 2001).

]. MOCRA performs radiative transfer calculations in the UV/Visible spectral range, with the possibility to simulate complex topography, surface albedos and a highly inhomogeneous atmosphere, both vertically and horizontally. The model is suitable for the simulation of different observational set-up, including satellite-limb and nadir viewing geometries, which were not foreseen in the PROMSAR model.

2. Monte Carlo solution of the radiative transfer equation

The radiative transfer equation (RTE) for time independent problems can be written in the integral form as in Eq. (1) [35

35. G. C. Pomraning, Radiation Hydrodynamics, Los Alamos National Laboratory Radiation Hydrodynamics Short Course Attendees, (1982), http://www.osti.gov/energycitations/servlets/purl/656708-zO5SF0/webviewable/656708.pdf

]
I(r,Ω)=Γ(rs,Ω)exp[0|rrs|dsk(rsΩ)]+0|rrs|dsQ(rsΩ,Ω)exp[0sdsk(rsΩ)]
(1)
where I(r,Ω) is the specific intensity of radiation (or radiance) received in the direction Ω at the observer point r, defined as the product of the photons distribution function by the quantity chν, with c the velocity of light in a vacuum, h the Planck’s constant and ν the radiation frequency. Γ(rs,Ω) is the specific intensity of the radiation source, k is the extinction coefficient, accounting for both scattering and absorption contributions. Q is the source term defined by Eq. (2)

Q(r,Ω)=S(r)+4πdΩks(r,ΩΩ)I(r,Ω)l.
(2)

The quantity S in Eq. (2) can be defined as S(r)=kaB; where B is the Planck’s function [30

30. A. Berk, L. S. Bernstein, and D. C. Robertson, “MODTRAN: a moderate resolution model for LOWTRAN7,” Geophysics Laboratory, Air Force Systems Command, United States Air Force, Hanscom AFB, MA (1989).

], and ka'is derived from the absorption coefficient ka by means of Eq. (3)
ka=ka[1exp(hν/KT)]
(3)
where K is the Boltzmann constant and T is the absolute temperature.

A more detailed description of the radiative transfer equation can be found in references [36

36. R. L. Liboff, Kinetic Theory: Classical, Quantum and Relativistic Descriptions. (Prentice Hall, 1989).

,37

37. S. Chandrasekhar, Radiative Transfer, (Dover Publications, Inc., 1960).

] which define the Stokes parameters to describe the effects of polarization.

As it can be seen in Eq. (1), the specific intensity of radiation is determined by two contributions. The first one is the specific intensity hitting the system’s surface at point rs. The second term accounts for emitted and scattered photons from each path element ds between rs and r along Ω direction. Each term is exponentially attenuated by scattering and absorption along the path between the source point and the observer location. The photonic interpretation of radiation allows applying the models usually adopted for simulating particles transport to solve the RTE in the atmosphere; many such models rely on the Monte Carlo approach. The Monte Carlo modeling involves a free optical path (or optical depth) to be first simulated. The optical depth is the exponent of the attenuation factors in the RTE (Eq. (1)):

τ(r,rs)=0|rrs|dsk(rsΩ)
(4)

The Monte Carlo approach to the solution of atmospheric radiative transfer consists of simulating the trajectories and occurring events, that together we call “histories”, of each individual photon emitted by the radiation source. Being the histories independent of one another by definition, the Monte Carlo method allows for the straightforward parallelization and the optimization of the computing resources.

The adjoint transport equation is very similar to Eq. (1). The source terms Γ(rs,Ω) and S(r), in fact, are evaluated at the actual position of the source, but in the integration intervals the quantity rs is replaced by rd, the detector position, and the incoming and outgoing directions Ω and Ω are exchanged each other. Due to isotropy of the medium, however, the scattering kernel depends on these directions only through their scalar product, so that exchanging the order of directions does not change the value of scattering kernel.

3. Description of MOCRA

MOCRA is a FORTRAN code which, given a monochromatic electro-magnetic radiation in the UV-near Infrared frequency range, performs a Monte Carlo simulation of the radiative transfer according to the physical and geometrical data of the assigned atmospheric environment. It is suitable to perform simulations of passive remote sensing measurements, as for instance DOAS observations. In this case a backward Monte Carlo scheme is adopted to simulate the UV-Visible radiance collected by a detector. The code simulates the atmosphere as a spherical multi-layer environment, taking into account also horizontal variations of physical parameters through three-dimensional multi-region rectangular and spherical geometries. The surface albedo is taken into account, assuming a Lambertian surface. Refraction is simulated according to Snell’s law.

In order to improve the performance of MOCRA, semi-analytical estimates of specific quantities (e.g., the absorption rates) and variance reducing techniques such as forced collision and expected leakage, are foreseen.

To evaluate the effects of small perturbations in the physical components of the atmosphere, a reference environment and several perturbed scenarios can simultaneously be simulated in a single run of the code. For each environment of interest in a simulation, the external library of physical data (interaction coefficients, phase functions and refractivity indexes) is required.

In the following subsections the main features of the MOCRA code are described: section 3.1 describes the radiation source and detectors representation; in section 3.2 the geometric and physical treatment of the atmosphere is depicted; section 3.3 concerns the photon history tracking, the end of history options and the variance reducing techniques; section 3.4 deals with perturbative analysis and section 3.5 gives an outlook of the quantities calculated by the model.

3.1 Radiation source and detectors

MOCRA assumes the Sun as the monochromatic source of radiation. The incoming photon directions are given by means of direction cosines (us, vs, ws), which can be either assigned by the user, or calculated on the basis of the astronomical parameters [44

44. M. Iqbal, An Introduction to Solar Radiation (Academic Press, 1983).

]. Through a basic knowledge of the standard meridian longitude with respect to the Greenwich meridian together with the longitude and latitude of the observer, the solar zenith and azimuth angles can be computed once the day of the year and the standard solar time are assigned. In this regard, the observer local time can be either assigned by the user or computed on the basis of the standard solar time, standard longitudes and the equation of time, accounting for the non-uniform terrestrial motion.

The radiation receivers handled by MOCRA can be schematized as point or disk detectors. A special treatment is foreseen for upward looking point detectors at ground level, in order to better take into account the refraction phenomenon at twilight when performing single scattering radiance calculations. In the following, the main receiver types allowed by MOCRA and the way in which collection of solar radiation is formally treated are described.

Vertical upward looking point detectors. Let I(z) be the radiance intensity due to a solar ray reaching the detector from the z-height level. Integrating the contributions from all z levels from the ground to the top boundary of the atmosphere gives the total intensity IT, as in Eq. (5):
IT=I(z)dz
(5)
or equivalently in Eq. (6):
IT=iΔziΔziI(z)(dz/Δzi)
(6)
where the entire vertical range is subdivided into N intervals of thickness Δzi (i = 1,…,N) and the integrand is changed accordingly.

In the Monte Carlo simulation, a stratified sampling procedure can be performed, that consists in randomly choosing a set of z points uniformly distributed within each height subinterval Δzi, and in estimating the corresponding I(z) value. The average intensity value over the points belonging to the i-th subinterval times the vertical layer thickness Δzi gives the contribution of that layer to the total intensity, the latter accounting for both single and multiple scattering radiance terms. Single scattering radiance (IS) is calculated using a forward Monte Carlo simulation of the photon path from the solar source coordinate to the altitude of each selected arrival point on the z-axis taking into account refraction. The effect of refraction is particularly important for SZA near or greater than 90° (horizon) and for solar rays crossing the lower atmospheric layers. Multiple scattering radiance (IM) is calculated using a backward Monte Carlo simulation from the selected vertical points to the sun. The albedo phenomenon is foreseen assuming a Lambertian surface, refraction is not accounted for along the path from the last scattering point to the sun.

Point detectors with assigned position and line of sight. In a O(x,y,z) reference system of coordinates, let Pd = (xd, yd, zd) be the point detector spatial coordinates and (ud, vd, wd) the direction cosines of its line of sight. Starting from Pd, a backward Monte Carlo simulation of a photon history with initial flight direction (ud, vd, wd) is performed, taking into account both single and total scattering radiance.

Disk detectors with assigned diameter, Field Of View (FOV), position and orientation. Let Pd = (xd, yd, zd) be the centre of the disk and (ud, vd, wd) the direction cosines of the outward direction normal to the disk surface. By assuming D (≥ 0) the diameter of the disk and α (>0) its FOV, a fictitious cone with semi-amplitude α and the disk as a basis is generated. The vertex point Pv, ≡ (xv, yv, zv) (possibly located outside the geometrical system) is assumed to be the virtual source point in a backward Monte Carlo simulation, with initial motion direction Ω for each photon uniformly distributed inside the cone. For each Ω direction, the crossing point with the detector disk gives the “true” starting point for the photon. Let ΔΩ be the solid angle amplitude and I(Ω) the contribution to radiance intensity collected by the receiver from the Ω direction. The total intensity IT over all possible directions is assumed to be:

IT=ΔΩI(Ω)dΩ=ΔΩΔΩI(Ω)dΩ/ΔΩ
(7)

Equation (7) can be simulated through a standard backward Monte Carlo estimating procedure for IT. If D = 0, analogous considerations can be made, with the true starting point coinciding with the virtual one. If α = 0, the starting point is chosen uniformly on the disk and the initial direction coincides with the detector axis.

3.2 Geometrical and physical description of atmospheric environments

The atmosphere is modeled as a set of either (chemically) homogeneous or non homogeneous horizontal layers. In both cases, the albedo phenomenon is accounted for whenever the ground surface is hit. In case of non homogeneous horizontal layers, refraction is foreseen whenever the boundaries between contiguous regions are crossed. The physical characteristics of each region are provided to MOCRA with a data set contained in an external library. The various kinds of geometrical environments that are foreseen by the code are described in the following.

  • 1) Horizontally homogeneous atmosphere. Starting from the ground, the atmosphere is divided into a series of horizontally homogeneous spherical shells, each one with its own physical properties. Two reference systems are simultaneously considered for the photon history tracking: O(x,y,z) and (R,φ), where R is the radius from the collision point to the earth centre and φ is the angle between the direction of vector R and the motion direction. The system based on (R,φ) coordinates is more suitable with respect to the Cartesian system to compute the escape probability and the leakage optical path for the travelling photon.
  • 2) Horizontally non homogeneous atmosphere. Horizontal variations of the atmospheric parameters and different topography scenarios are simulated by the code using a multi-region rectangular or spherical geometry:
    • a. Multi-region rectangular geometry. Given a rectangular domain (xmin, xmax, ymin, ymax), a set of sub-regions is generated by dividing the interval (xmin, xmax) in a number of subintervals. For each of them a subdivision of the interval (ymin, ymax) is performed. Each such generated sub-region has its own atmospheric environment and surface characteristics.
    • b. Multi-region spherical geometry (Fig. 1
      Fig. 1 Multi-region 3D spherical geometry: a) a set of three cones with the vertex at the Erth centre; b) vertical half planes having the z axis as intersection line with azimuth angle ψi and ψj defined in (x,y) plane; c) a horizontal section of a possible geometry with three cones and 5, 7 half planes for the second and third cone, respectively; d) a vertical section with three cones of aperture θ1, θ2, θ3 and ground heights h1, h2 and h3.
      ). Let O(x,y,z) be a reference system with origin on the Earth centre and the z-coordinate axis oriented according to the zenith of the observer on the Earth surface. This geometry model foresees a set of “N” cones, each one with the vertex located at the centre of the Earth, the axis coincident with the z-coordinate axis, and the bisector of the vertex angle θ (< 90°) coincident with the zenith angle with respect to the vertical axis (Fig. 1a). It is assumed that θ1 < … < θN. A set of “M” vertical half planes can be associated to each cone, except that to the first one: the cones axis (the z-coordinate axis) is the intersection of all half planes. As shown in Fig. 1b, for each vertical half plane, starting from the x-axis in the (x,y) coordinate plane, the azimuth angle ψj (0 ≤ ψ1 < … < ψM = ψ1 + 360°) is defined, being ψk + 1 - ψk < 180° (k = 1, …, M - 1). In this way, the i-th geometrical space (i > 1) between the i-th and (i + 1)-th contiguous cones is subdivided in a number of (Mi - 1) regions. Figure 1c shows the horizontal section of a possible geometry with 3 cones and 5, 7 half planes for the second and third cone respectively. Each region generated according to the previous procedure, has its own vertical subdivision as previously described for the spherical multi-layer geometry, so that different ground levels can belong to different zones. Figure 1d shows a vertical section of conic regions with bisectors of vertex angles θ1, θ2, θ3 and ground heights h1, h2 and h3. A simple application of this type of geometry in MOCRA has already been described in reference [45

      45. M. Premuda, S. Masieri, D. Bortoli, F. Margelli, F. Ravegnani, A. Petritoli, I. Kostadinov, G. Giovanelli, and E. Cupini, “A Monte Carlo simulation of radiative transfer in the atmosphere applied to ToTaL-DOAS,” Proc. SPIE 7475, 74751A, 74751A-12 (2009). [CrossRef]

      ].
      Fig. 1Multi-region 3D spherical geometry: a) a set of three cones with the vertex at the Erth centre; b) vertical half planes having the z axis as intersection line with azimuth angle ψi and ψj defined in (x,y) plane; c) a horizontal section of a possible geometry with three cones and 5, 7 half planes for the second and third cone, respectively; d) a vertical section with three cones of aperture θ1, θ2, θ3 and ground heights h1, h2 and h3.
    Fig. 1Multi-region 3D spherical geometry: a) a set of three cones with the vertex at the Erth centre; b) vertical half planes having the z axis as intersection line with azimuth angle ψi and ψj defined in (x,y) plane; c) a horizontal section of a possible geometry with three cones and 5, 7 half planes for the second and third cone, respectively; d) a vertical section with three cones of aperture θ1, θ2, θ3 and ground heights h1, h2 and h3.

3.3 Path tracking and variance reducing techniques

History tracking in Monte Carlo simulations is based on sampling random variables distributed according to appropriate, discrete or continuous, pdfs which describe the random walk of photons and their interactions with atmospheric constituents. As mentioned above, the present work uses sampling via inversion of cdfs.

Batches of pre-assigned number of photon histories are generally run in a MOCRA simulation; each batch is assumed as a single numerical experiment. Besides being more appropriate than a simulation of all photon histories in a unique model run as regard the variance calculation, this procedure allows high number of histories (as it should be necessary) to be run without significant precision loss when collecting the results coming from the simulation [49

49. E. Palazzi, “Retrieval of trace gases vertical profile in the lower atmosphere combining Differential Optical Absorption Spectroscopy with radiative transfer models,” PhD thesis (Bologna University, Italy, 2008), http://amsdottorato.cib.unibo.it/983/1/Tesi_Palazzi_Elisa.pdf

]. A restart option is also foreseen, to make the results of an assigned number of batches available for a subsequent continuation of the simulation. To make the comparison between slightly different calculations (in the physical or geometrical parameters) sufficiently robust, a strategy on the random number generation is adopted for both batches and histories inside a batch so that, in each calculation, corresponding histories begin with the same random number. This guarantees that unaltered corresponding histories in the two calculations give the same results for the same quantity.

Analytical estimates of specific quantities can be performed in the simulation to reduce as much as possible the variance of the calculation. Absorption rates, for instance, are estimated semi-analytically through the distance estimator described above. Another example of analytical estimators is the leakage estimator which is evaluated when a collision occurs. At each collision point the optical distance τe from that point to the outer atmospheric boundary along the motion direction is computed. The escaping weight we=weτe is the analytical contribution to leakage from the collision point. The sum over all collisions of these contributions gives the analytical estimate of leakage for that history. The colliding weight wc=w(1eτe) is assigned to the photon and used for evaluating the collision estimator in the following part of the history.

It should be noticed that, despite the variance is a function of the number of histories, N, its reduction cannot be achieved simply increasing N, because this would increase the computational time. To reduce the variance of a factor 1/F, in fact, it is necessary to multiply the number of histories by a factor F2 [38

38. S. Chatigny, M. Morin, D. Asselin, Y. Painchaud, and P. Beaudry, “Hybrid Monte Carlo for photon transport through optically thick scattering media,” Appl. Opt. 38(28), 6075–6086 (1999). [CrossRef] [PubMed]

]. Therefore, in addition to analytical evaluations, that certainly reduce the computing time, variance reducing techniques (vrt) are adopted. They have the property of somewhat altering the natural sequence of the events a photon would undergo, in order to increase the photon statistics saving also computing time, giving rise to a more efficient contribution to the required estimates. Probably the most common vrt, also used in MOCRA, is forced collision: for motion directions that do not hit a reflecting boundary, the leakage optical distance τe is computed and the probabilities for a photon to escape from the system (eτe) and to collide inside it (1eτe) are computed. In this way, a travelling photon with weight we=weτe will escape from the system, while a travelling photon with weight wc=w(1eτe) will collide. It is worth pointing out that both photons fates are considered when the absorption distance estimator is calculated. When a forced collision is simulated, the optical path before collision is randomly selected from the pdf f(x)=ex1eτe (truncated exponential distribution) rather than from the simply exponential distribution, and the photon travels with weight w=wc. End of history criterion 1 is adopted in this case and, as a precaution, a maximum number of collisions per history is either assigned by the user or assumed equal to 1000. Forced collision, however, is a time consuming technique and may cause considerable changes in the travelling photon weight which can negatively affect the variance of the calculation. Therefore, it should only be used as far as information for a specific layer (or region) of the atmosphere is not correctly achievable using a standard simulation (for layers with small optical thickness, for instance, for which an unbiased simulation would give rise to small interaction probabilities). A Figure Of Merit (FOM) can be defined to simply measure the efficiency of the simulation process:
FOM=1tσμ2
(12)
where t is the computing time and σμ is the standard deviation of the estimated parameter.

A more comprehensive review of variance reducing techniques can be found in references [28

28. E. Cupini, M. G. Borgia and M. Premuda, Il codice PREMAR per la simulazione Montecarlo del trasporto della radiazione nell’atmosfera. (RT/INN/97/5. 1997).

,50

50. E. Cupini, PREMAR-2 A Monte Carlo code for radiative transport simulation in atmospheric environments. (RT/INN/98/20. 1998).

].

3.4 Perturbative calculations

Small perturbations in the physical atmospheric components, with respect to a reference atmosphere, and their effect on radiation transport, can be accounted for in MOCRA. To do so, photon histories are traced sampling the intervening events from the pdfs describing both the unperturbed and the perturbed situation. To this aim an “importance sampling” Monte Carlo technique [42

42. M. Premuda, “Montecarlo simulation of Radiative transfer in Atmospheric environments for problems arising from remote sensing measurements,” in Theory and Applications of Monte Carlo Simulations, S. Mordechai, ed. (INTECH, 2011).

], which allows defining proper weight factors for the photons travelling in the perturbed and unperturbed environments is employed. In the radiation transport simulation, let k and k be the extinction coefficients in a given region for the unperturbed and the perturbed environment, respectively. When crossing a boundary of an atmospheric region in which the perturbative calculation is performed, a weight factor equal to exp((kk)d) has to be associated to travelling particle in the perturbed environment, being d the distance from the starting point to the boundary, since the probability to leave the region is given by exp(kd) in the perturbed environment and exp(kd) in the unperturbed one. If a collision occurs in the region, the weight factor will be (k/k)exp((kk)d), where d is the distance traveled in the region. kexp(kd) is the probability for the photon to collide after a distance d is travelled in the perturbed environment and kexp(kd) is the analogous quantity in the unperturbed environment. Therefore, when perturbative calculations are carried out, at least 2 weights are attached to each photon during a simulation, one for each situation involved.

Estimators of the differential effects of the perturbation on certain quantities are computed by collecting, during the history tracking, the differences in the quantities of interest between the perturbed and unperturbed environments, so that the effects of small perturbations, difficult to evaluate with separate calculations, are computed in a single run of the code. The maximum number of perturbed environments that can be handled by the code is limited only by memory usage. It must be remarked that, according to this technique, the leading photon history is governed by the unperturbed system, so that an end of history in this system will cause the end of histories also in the perturbed environments. Perturbations are not foreseen to affect the angular distributions. When horizontally non homogeneous atmospheric conditions are considered and perturbations are foreseen in different regions, the unperturbed environment must be the same for all of them.

3.5 Computed quantities

The first quantities computed by the MOCRA code are the atmospheric optical depth τ, defined by Eq. (4), the atmospheric transmittance, defined as Tr=eτ, and the spatial distribution of single and total scattering irradiance reaching the ground surface. Besides that, the single and total scattering radiance and the ground reflected radiance, received by a simulated detector are evaluated. Option is left to avoid scattered radiance or reflected radiance calculation. The aerosol and molecular contributions to scattering are separately accounted for. The spatial distribution of the single and total scattering contribution to the radiance is calculated along the detector axis, according to the geometrical subdivision assigned by the user along the detector axis. Radiance calculations using a source other than the sun, placed at an assigned distance from a point detector, can be performed. For each molecules of interest, absorption vertical optical depths, intensity weighted optical paths for the single and total scattering radiance and for the reflected radiance, and the AMF are calculated. The AMF of a specific trace gas, in particular, is used to convert the SCD measured with DOAS instruments into a VCD. Several types of RTMs have been developed for such an aim taking into account either single (e.g., the RTM AMEFCO [23

23. A. Petritoli, F. Ravegnani, G. Giovanelli, D. Bortoli, U. Bonafè, I. Kostadinov, and A. Oulanovsky, “Off-axis measurements of atmospheric trace gases by use of an airborne ultraviolet-visible spectrometer,” Appl. Opt. 41(27), 5593–5599 (2002). [CrossRef] [PubMed]

]), or multiple (e.g., the PROMSAR RTM [26

26. E. Palazzi, A. Petritoli, G. Giovanelli, I. Kostadinov, D. Bortoli, F. Ravegnani, and S. S. Sackey, “PROMSAR: A backward Monte Carlo spherical RTM for the analysis of DOAS remote sensing measurements,” Adv. Space Res. 36(5), 1007–1014 (2005). [CrossRef]

]) scattering. Being I and I* the radiance (single or total scattering) detected by a receiver with and without the trace gas of interest in the atmosphere, respectively, the AMF of the species can be calculated as in Eq. (13) [51

51. A. Sarkissian, H. K. Roscoe, and D. J. Fish, “Ozone Measurement by Zenith-Sky Spectrometers: an Evaluation of Errors in Air-Mass Factors calculated by Radiative Transfer Models,” J. Quant. Spectrosc. Radiat. Transf. 54(3), 471–480 (1995). [CrossRef]

]:
AMF=ln(II*)δa
(13)
where δa is the vertical absorption optical depth of the molecular species of interest. Equation (13) is a general expression for the AMF of a given trace gas, which holds especially in case of strong absorbers and will be thus referred to also as AMFSA. For weak absorbers an approximating formula for the AMF (AMFWA) is given by Eq. (14) (see also reference [51

51. A. Sarkissian, H. K. Roscoe, and D. J. Fish, “Ozone Measurement by Zenith-Sky Spectrometers: an Evaluation of Errors in Air-Mass Factors calculated by Radiative Transfer Models,” J. Quant. Spectrosc. Radiat. Transf. 54(3), 471–480 (1995). [CrossRef]

]):
AMFWA=δOPTδa
(14)
where δOPT is the intensity weighted absorption optical path from the sun to the detector defined as
δOPT=iDiIi*iIi*
(15)
where Di is the distance traveled by photons in the i-th layer, Ii* is the radiance collected by the detector from the i-th layer in the absence of the examined gas.

Due to the possibility for the MOCRA code to perform simultaneously radiative calculations with and without the molecular particle of interest (perturbative calculation), the AMF can be calculated as in Eq. (13). The approximated value of the AMF can altogether be calculated (Eq. (14), and compared with the former to check the hypothesis of weak absorption. An example of such a comparison is shown in Fig. 2
Fig. 2 AMFs for O3 (left) and for NO2 (right) calculated as in Eq. (13) (AMFSA, red line) and Eq. (14) (AMFWA, blue line). The red line for NO2 has been shifted upward of 0.5 to make it distinguishable from the blue line.
for the O3 and NO2 AMFs at 310 nm, for a ground based vertically looking detector.

As expected, significant differences in the AMF of O3 (left panel) between the two formulas can be observed, due to the fact that ozone is a strong absorber in the near UV. The difference between the exact and approximated AMF is even greater than 50% for solar zenith angles (SZA) near or greater than 90°. On the other hand, NO2 is a much weaker absorber than ozone at 310 nm, leading to differences always less than 1% (0.1%) between the two AMF calculations in case of single (multiple) scattering simulations (the red line for NO2 AMF in the figure is shifted of 0.5 upward).

A very useful, quite recent concept is that of box-AMF [27

27. T. Wagner, J. P. Burrows, T. Deutschmann, B. Dix, C. von Friedeburg, U. Frieß, F. Hendrick, K.-P. Heue, H. Irie, H. Iwabuchi, Y. Kanaya, J. Keller, C. A. McLinden, H. Oetjen, E. Palazzi, A. Petritoli, U. Platt, O. Postylyakov, J. Pukite, A. Richter, M. van Roozendael, A. Rozanov, V. Rozanov, R. Sinreich, S. Sanghavi, and F. Wittrock, “Comparison of box-air-mass-factors and radiances for Multiple-Axis Differential Optical Absorption Spectroscopy (MAX-DOAS) geometries calculated from different UV/visible radiative transfer models,” Atmos. Chem. Phys. 7(7), 1809–1833 (2007). [CrossRef]

]. The box-AMF for a certain atmospheric “box” can be calculated as the derivative of the SCD within that box with respect to the VCD [21

21. J. Puķīte, S. Kühl, T. Deutschmann, U. Platt, and T. Wagner, “Accounting for the effect of horizontal gradients in limb measurements of scattered sunlight,” Atmos. Chem. Phys. 8(12), 3045–3060 (2008). [CrossRef]

]. This is equivalent to calculate the derivative of the logarithm of the intensity with respect to the number density nb of the gas in the box or, more precisely, the absorption coefficient βb, as indicated in Eq. (16).

AMFb=dSCDgdVCDb=1hbσbdlnIgdnb=1hbdlnIgdβb.
(16)

It can be noted that Eq. (16) reduces to Eq. (13), provided that δa is substituted by its value δa=hbβb for the box. As for the AMF, the MOCRA code calculates the box-AMF and its approximated expression for weak absorbers (or little-explored regions), which gives a reasonable estimate of this factor with less computational demand. The approximated box-AMF for a specific trace gas is given by Eq. (14), provided that δa and δOPT are calculated for the box, so that δa=hbβb and in Eq. (15) Di is supposed equal to zero if the layer is outside the box.

When perturbative calculations are not performed, the approximated expression for the box-AMF can still be applied, provided that Ii* in Eq. (15) is replaced by Ii, i.e. the radiance detected in the presence of the gas of interest. Several boxes can be considered in a single simulation. The assumption of monochromatic photons and the energy dependence of absorption are probably the reasons why the ozone AMF in the visible wavelength range (the so-called “Chappuis band”, see section 4) presents a strong variability with wavelength. This gives rise to some problems in using these results for the interpretation of passive DOAS measurements, which would be the subject of future investigations.

4. Results, applications and comparison exercises

This section gives an outlook of possible applications of the MOCRA model, especially in the frame of DOAS. However, to verify the reliability of the code in solving radiance problems, two comparison exercises between MOCRA and MODTRAN4 for a variety of reciprocal positions between the instrument line of sight (ud, vd, wd) and the solar direction (us, vs, ws) and different atmospheric visibility values, are first presented. Some applications of MOCRA in the frame of DOAS will be subsequently shown, for which a deterministic model, like MODTRAN4, should not obtain sufficiently accurate results due to multiple scattering effects and geometric requirements. The first exercise consists of simulating a point detector, located at an altitude of 1 km above the ground (as if it was flying on board a research aircraft) with lines of sight α=180° and α=150° (with respect to the vertical upward direction). The solar zenith angles are θ=30° and θ=60°, while azimuth angles are ψ=0° and ψ=90° between the vertical plane containing the detector line of sight and the sun direction. In Tables 1

Table 1. Point Detector: MOCRA - MODTRAN-4 Comparison for vis = 5 Km (for wavelength and atmospheric conditions see text)*

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and 2

Table 2. Point Detector: MOCRA - MODTRAN-4 Comparison for vis = 23 Kma

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the obtained results are quoted for a visibility of 5 and 23 km, respectively. The considered atmospheric scenario chosen for this simulation is that of MODTRAN4 for mid-latitude winter conditions, for a wavelength of 550 nm. The solar constant for that wavelength is calculated with MODTRAN4 as 6.1134 10−6 Watt/(cm2 cm−1), a typical value for January. The mean surface albedo (0.3) was assumed.

For each combination of α,ψ,θ and the two visibility values, the compared radiometric quantities are single scattering and total scattering radiance, reflected radiance at ground, along with the sum of total and reflected radiances. For each quantity the percent difference between the MOCRA and MOTRAN4 estimates is reported. Discussing the details of the comparison as a function of the different values of the input parameters is outside the scope of the paper. It is important to stress that a satisfactory agreement between MOCRA and MODTRAN4 for both scattered and reflected radiances has been found. The difference between the two codes is less than 1% for single scattering and less than 3% for multiple scattering. Monte Carlo methods have proven to be particularly suitable to simulate radiative transfer problems with marked multiple scattering anisotropy [42

42. M. Premuda, “Montecarlo simulation of Radiative transfer in Atmospheric environments for problems arising from remote sensing measurements,” in Theory and Applications of Monte Carlo Simulations, S. Mordechai, ed. (INTECH, 2011).

] and geometrically confined sources with narrow fields of view, that can give rise to troubles known as “ray effect” and “false scattering” [52

52. J. C. Chai, H. S. Lee, and S. V. Patankar, “Ray Effect and False Scattering in Discrete Ordinates Method,” Numer. Heat Transfer, Part B 24(4), 373- 389 (1993), http://www.informaworld.com/10.1080/10407799308955899

], if not properly treated.

A second comparison exercise between MOCRA and MODTRAN4 was performed assuming a horizontal looking ground-based point detector. In this comparison exercise we focus on single scattering calculations only. In this way, the effects of using different methods to solve the RTE when multiple scattering is included are not accounted for and do not contribute to possible discrepancies in the results. A mid-latitude winter model for the atmosphere was used in this exercise. The considered radiation wavelengths were 310 nm, 435 nm, 550 nm and 700 nm. Seven visibility values, chosen by considerations about the relationship between visibility and aerosol load, (5 km, 7.6 km, 9.5 km, 13 km, 18 km, 23 km and 35 km), solar zenith angles of 30° and 60°, and solar azimuth angles of 0° and 90° have been considered. Being too long to show all results, they are summarized in Table 3

Table 3. Horizontal Looking Point Detector Radiance, MOCRA-MODTRAN 4 Comparison, Mid Latitude Winter Model, Urban Extinction. Units: watt/ (cm2-ster-cm−1)

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as the average values over all visibilities, solar zenith angles and azimuth angles. The table shows, for each radiation wavelength and the respective value of the solar constant calculated with MODTRAN4, the average radiance values computed with MOCRA and MODTRAN4, along with their mean and standard deviation differences. The mean difference between the two codes is relatively low, less than 3%, with maximum standard deviation of about 5% at 310 nm, suggesting a really good agreement between the two models in calculating the single scattering radiance reaching such a simulated detector in a cloud-free atmosphere [25

25. M. Premuda, S. Masieri, D. Bortoli, I. Kostadinov, A. Petritoli, and G. Giovanelli, “Evaluation of vessel emissions in a lagoon area with ground based Multi axis DOAS measurements,” Atmos. Environ. 45(29), 5212–5219 (2011), doi:, http://dx.doi.org/10.1016/j.atmosenv.2011.05.067. [CrossRef]

].

In what follows, we move to some practical uses of the MOCRA model with particular emphasis on DOAS applications: calculation of the AMF and simulation of measured SCDs.

Figure 3
Fig. 3 AMFs computed for ozone and nitrogen dioxide at three different wavelengths: 310 nm (blue), 435 nm (green), 550 nm (red). The atmospheric model used was that of MODTRAN4 for mid latitude winter, with urban aerosol extinction, visibility = 35 Km.
shows the O3 (left) and NO2 (right) AMFs as a function of the SZA at 310 nm (blue), 435 nm (green), and 550 nm (red), calculated with MOCRA for a vertical looking detector at ground.

The choice between spectral intervals in which DOAS measurements of a specific gas are performed is usually driven by considerations about the absorption cross section of the gas. It is obviously important to perform measurements of a gas in a spectral range in which its absorption cross section is quite high (strong absorption). This requirement is not sufficient, however: it is also required that the absorption features of the trace gases are easily recognizable by the dedicated elaboration software. Figure 4
Fig. 4 Absolute absorption cross sections for O3 and NO2 in the 250-650 nm spectral range at a 273 °K temperature.
shows the absolute absorption cross sections (ACS) of O3 and NO2, plotted together in the 250-650 nm spectral range. The absorption cross section of O3 is higher in the near UV (Hartley-Huggins band) than in the visible (Chappuis band). Chappuis band, however, is usually chosen for ozone retrieval mainly because the O3 ACS is less temperature dependent and the AMF is less wavelength dependent here than in the UV. The spectral range around 430 nm is normally chosen for DOAS NO2 measurements.

The second example of application of the MOCRA model discussed in this paper is the simulation of the slant column densities of a given trace gas, and the comparison with real observations. Such a comparison is useful to make assumptions on the vertical profile of the trace gas whose SCD is simulated by the model, as well as the atmospheric conditions at the time of measurements (such as the aerosol load, the atmospheric visibility, the presence of clouds, etc., see e.g [25

25. M. Premuda, S. Masieri, D. Bortoli, I. Kostadinov, A. Petritoli, and G. Giovanelli, “Evaluation of vessel emissions in a lagoon area with ground based Multi axis DOAS measurements,” Atmos. Environ. 45(29), 5212–5219 (2011), doi:, http://dx.doi.org/10.1016/j.atmosenv.2011.05.067. [CrossRef]

,51

51. A. Sarkissian, H. K. Roscoe, and D. J. Fish, “Ozone Measurement by Zenith-Sky Spectrometers: an Evaluation of Errors in Air-Mass Factors calculated by Radiative Transfer Models,” J. Quant. Spectrosc. Radiat. Transf. 54(3), 471–480 (1995). [CrossRef]

].). In the example reported on here, the modeled SCD for NO2 are compared to experimental values obtained at the observatory of the Geophysics centre of Evora (38°33′36”N, 7°54′ W), Portugal, on 22 August 2009 in the p.m. part of the day. The atmospheric scenario for the MOCRA simulation was that of mid-latitude summer conditions with the visibility value set according to the measurements performed by a nephelometer placed in the measurement site. Figure 5
Fig. 5 Comparison between simulated (blue) and measured (cyan) NO2 slant column densities. The measurements were carried out on 22 august 2009 at Evora (38°33′36”N, 7°54′ W), Portugal.
shows the comparison of simulated (blue symbols) and measured (cyan symbols) NO2 SCDs. The percent error is on average less than 20% for solar zenith angles less the 56°, and increases for greater solar zenith angles, though keeping the agreement between the model and observations reasonably good.

5. Summary and conclusions

In this paper, the radiative transfer model MOCRA has been described. It is a FORTRAN code which performs a Monte Carlo simulation of radiative transfer in a very general three-dimensional atmospheric scenario. When simulating passive remote sensing measurements, such those performed by a DOAS spectrometer, the MOCRA model adopts a backward Monte Carlo scheme, which is a useful tool to overcome problems arising from the narrow field of view of the detector. The main advantages of this code can be resumed in the following items:

  • 1) the possibility to take into account not only vertical, but also horizontal variations of the physical and chemical atmospheric parameters, along with topographic features of the surface, through the implementation of three dimensional multi region rectangular and spherical geometries;
  • 2) the possibility to evaluate the effects of small atmospheric perturbations in a single model run (perturbative calculations). In this way, small differences in some key parameters that influence the radiation transport and photons interactions with the atmosphere can be accounted for. This has direct consequence when the model is used to interpret DOAS remote sensing observations, since the AMF (and box-AMF) of a target trace gas can be evaluated through the exact definition formula. A number of different atmospheric perturbations can be accounted for simultaneously in a single run simply defining, for each of them, appropriate perturbed environments with respect to a reference scenario.

Other interesting features of MOCRA are the realistic treatment of multiple scattering, which relies on the employed Monte Carlo approach, and the use of variance reducing techniques to obtain statistically significant results with reasonable computing times. The choice of the particular variance reducing technique to be implemented depends on the problem at hand and strongly relies on the user experience. Forced collision is mostly used in MOCRA.

In the present paper, MOCRA has been first compared to the state-of-the-art model MODTRAN4, giving overall a very satisfactory agreement in the radiative quantities we have focused on, such as the single and total scattering radiance and the ground reflected radiance, for a variety of atmospheric scenarios and measurement configurations. For the single and multiple scattering radiance, for instance, the percent differences between the two models are less than 1% and less than 3% respectively. Multiple scattering is differently implemented in MOCRA with respect to MODTRAN4, leading to higher, though still small, differences between the two models in the multiple than single scattering radiance calculations.

In the frame of DOAS, the MOCRA model provides the main input values required to retrieve trace gases VCDs, AMFs, box-AMFs and vertical profiles by means of appropriate algorithms applied to the SCDs directly retrieved from the measurements of diffuse solar radiation. The paper has discussed some examples of application of this model to typical DOAS configurations, simulating measurements that could be performed with ground-based zenith or horizontal looking detectors, or detectors placed at a certain atmospheric height with respect to the sea level, or directly comparing simulations with SCD observations.

Due to the 3D spherical geometry, moreover, MOCRA has already given preliminary interesting results in computing the NO2 box-AMFs for limb viewing measurements performed with the SCIAMACHY (Scanning Imaging Absorption Spectrometer for Atmospheric CHartographY) instrument onboard ENVISAT [53

53. E. Papandrea, E. Arnone, E. Castelli, and M. Premuda, “Sounding the Upper Troposphere-Lower Stratosphere, with Satellite measurements,” Proc. “ESA Living Planet Symposium,” Bergen, Norway, 28 June- 2 July 2010 (ESA-SP-686, December 2010).

]. These preliminary results deserve further investigations for their important application to limb-viewing satellite measurements in target atmospheric regions like the (still under-sampled) Upper Troposphere-Lower Stratosphere (UT-LS), a challenge region for most satellite measurements.

Acknowledgments

This work was partially funded by ENVIREN (Environmental Regional Network) laboratory and TRIL-ICTP (Training in Italian Laboratories-International Centre of Theoretical Physics) Trieste.

The paper was also financially supported through FEDER (Programa Operacional Factores de Competitividade – COMPETE) and National funding through FCT – Fundação para a Ciência e a Tecnologia in the framework of project FCOMP-01-0124-FEDER-014024 (Refª. FCT PTDC/AAC-CLI/114031/2009).

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S. Solomon, A. Schmeltekopf, and R. W. Sanders, “On the interpretation of zenith sky absorption measurements,” J. Geophys. Res. 92(D7), 8311–8319 (1987). [CrossRef]

19.

J. Pukıte, S. Kühl, T. Deutschmann, W. Wilms-Grabe, C. Friedeburg, U. Platt, and T. Wagner, “Retrieval of stratospheric trace gases from SCIAMACHY limb measurements,” Proceedings of the First Atmospheric Science Conference, 8–12 May, ESA/ESRIN, Frascati, Italy, ESA SP-628, (2006). http://earth.esa.int/workshops/atmos2006/participants/1148/paper proc Frasc 2.pdf

20.

S. Kühl, J. Puķīte, T. Deutschmann, U. Platt, and T. Wagner, “SCIAMACHY Limb Measurements of NO2, BrO and OClO, Retrieval of vertical profiles: Algorithm, first results, sensitivity and comparison studies,” Adv. Space Res. 42(10), 1747–1764 (2008). [CrossRef]

21.

J. Puķīte, S. Kühl, T. Deutschmann, U. Platt, and T. Wagner, “Accounting for the effect of horizontal gradients in limb measurements of scattered sunlight,” Atmos. Chem. Phys. 8(12), 3045–3060 (2008). [CrossRef]

22.

G. Hönninger, C. von Friedeburg, and U. Platt, “Multi axis differential optical absorption spectroscopy (MAX-DOAS),” Atmos. Chem. Phys. 4(1), 231–254 (2004), http://www.atmos-chem-phys.net/4/231/2004/acp-4-231-2004.pdf. [CrossRef]

23.

A. Petritoli, F. Ravegnani, G. Giovanelli, D. Bortoli, U. Bonafè, I. Kostadinov, and A. Oulanovsky, “Off-axis measurements of atmospheric trace gases by use of an airborne ultraviolet-visible spectrometer,” Appl. Opt. 41(27), 5593–5599 (2002). [CrossRef] [PubMed]

24.

G. Giovanelli, E. Palazzi, A. Petritoli, D. Bortoli, I. Kostadinov, F. Margelli, S. Pagnutti, M. Premuda, F. Ravegnani, and G. Trivellone, “Perspectives of 2D and 3D mapping of Atmospheric Pollutants over Urban Areas by means of airborne DOAS Spectrometer,” Ann. Geophys. 49(1), 133–142 (2006).

25.

M. Premuda, S. Masieri, D. Bortoli, I. Kostadinov, A. Petritoli, and G. Giovanelli, “Evaluation of vessel emissions in a lagoon area with ground based Multi axis DOAS measurements,” Atmos. Environ. 45(29), 5212–5219 (2011), doi:, http://dx.doi.org/10.1016/j.atmosenv.2011.05.067. [CrossRef]

26.

E. Palazzi, A. Petritoli, G. Giovanelli, I. Kostadinov, D. Bortoli, F. Ravegnani, and S. S. Sackey, “PROMSAR: A backward Monte Carlo spherical RTM for the analysis of DOAS remote sensing measurements,” Adv. Space Res. 36(5), 1007–1014 (2005). [CrossRef]

27.

T. Wagner, J. P. Burrows, T. Deutschmann, B. Dix, C. von Friedeburg, U. Frieß, F. Hendrick, K.-P. Heue, H. Irie, H. Iwabuchi, Y. Kanaya, J. Keller, C. A. McLinden, H. Oetjen, E. Palazzi, A. Petritoli, U. Platt, O. Postylyakov, J. Pukite, A. Richter, M. van Roozendael, A. Rozanov, V. Rozanov, R. Sinreich, S. Sanghavi, and F. Wittrock, “Comparison of box-air-mass-factors and radiances for Multiple-Axis Differential Optical Absorption Spectroscopy (MAX-DOAS) geometries calculated from different UV/visible radiative transfer models,” Atmos. Chem. Phys. 7(7), 1809–1833 (2007). [CrossRef]

28.

E. Cupini, M. G. Borgia and M. Premuda, Il codice PREMAR per la simulazione Montecarlo del trasporto della radiazione nell’atmosfera. (RT/INN/97/5. 1997).

29.

E. Cupini, G. Ferro, N. Ferrari, Monte Carlo Analysis of Radiative Transport in Oceanographic Lidar Measurements. (ENEA/RT/INN/2001/7, 2001).

30.

A. Berk, L. S. Bernstein, and D. C. Robertson, “MODTRAN: a moderate resolution model for LOWTRAN7,” Geophysics Laboratory, Air Force Systems Command, United States Air Force, Hanscom AFB, MA (1989).

31.

F. Kneizys, D. Robertson, L. W. Abreu, P. Acharya, G. P. Anderson, L. S. Rothman, J. H. Chetwynd, J. E. A. Selby, E. P. Shettle, W. O. Gallery, A. Berk, S. A. Clough, and L. S. Bernstein, “The MODTRAN 2/3 Report and LOWTRAN7 MODEL,” Phillips Laboratory, Geophysics Directorate, Hanscom AFB, MA (1996).

32.

P. Acharya, S. M. Adler-Golden, G. P. Anderson, A. Berk, L. S. Bernstein, J. H. Chetwynd, et al., “Modtran version 3.7/4.0 user’s manual,” Air Force Research Laboratory, Space Vehicles Directorate, Air Force MATERIEL Command, Hanscom AFB, MA (1998).

33.

A. Berk, G. P. Anderson, P. K. Acharya, J. H. Chetwynd, L. Bernstein, and E. P. Shettle, “MODTRAN4 user’s manual,” Air Force Research Laboratory, Space Vehicles Directorate, Air Force Materiel Command, Hanscom AFB, MA (1999).

34.

D. Nowak, L. Vuilleumier, C. N. Long, and A. Ohmura, “Solar irradiance computations compared with observations at the Baseline Surface Radiation Network Payerne site,” J. Geophys. Res. 113(D14), D14206 (2008), doi:. [CrossRef]

35.

G. C. Pomraning, Radiation Hydrodynamics, Los Alamos National Laboratory Radiation Hydrodynamics Short Course Attendees, (1982), http://www.osti.gov/energycitations/servlets/purl/656708-zO5SF0/webviewable/656708.pdf

36.

R. L. Liboff, Kinetic Theory: Classical, Quantum and Relativistic Descriptions. (Prentice Hall, 1989).

37.

S. Chandrasekhar, Radiative Transfer, (Dover Publications, Inc., 1960).

38.

S. Chatigny, M. Morin, D. Asselin, Y. Painchaud, and P. Beaudry, “Hybrid Monte Carlo for photon transport through optically thick scattering media,” Appl. Opt. 38(28), 6075–6086 (1999). [CrossRef] [PubMed]

39.

C. P. Robert and G. Casella, Monte Carlo Statistical Methods (Springer, 2004).

40.

A. De Matteis and S. Pagnutti, “Long-range Correlation-analysis of the Wichmann-Hill random number generator,” Stat. Comput. 3(2), 67–70 (1993). [CrossRef]

41.

A. De Matteis and S. Pagnutti, “Controlling correlations in parallel Monte Carlo,” Parallel Comput. 21(1), 73–84 (1995). [CrossRef]

42.

M. Premuda, “Montecarlo simulation of Radiative transfer in Atmospheric environments for problems arising from remote sensing measurements,” in Theory and Applications of Monte Carlo Simulations, S. Mordechai, ed. (INTECH, 2011).

43.

M. H. Kalos and P. A. Whitlock, Monte Carlo Methods. Volume I: Basics (John Wiley & Sons, 1986).

44.

M. Iqbal, An Introduction to Solar Radiation (Academic Press, 1983).

45.

M. Premuda, S. Masieri, D. Bortoli, F. Margelli, F. Ravegnani, A. Petritoli, I. Kostadinov, G. Giovanelli, and E. Cupini, “A Monte Carlo simulation of radiative transfer in the atmosphere applied to ToTaL-DOAS,” Proc. SPIE 7475, 74751A, 74751A-12 (2009). [CrossRef]

46.

K. Y. Kondratyev, Radiation in the Atmosphere (Academic Press, 1969).

47.

F. X. Kneizys, S. A. Clough, E. P. Shettle, L. S. Rothmann and R. W. Fenn, “Linear Absorption and Scattering of Laser Beams,” Air Force Geophysical Laboratory, AFGL-TR-84 0265 (1984).

48.

F. X. Kneizys, E. P. Shettle, W. O. Gallery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, S. A. Clough and R. W. Fenn, “Atmospheric Transmittance/Radiance: Computer Code LOWTRAN6,” AFGL-TR-83–0187, (1983).

49.

E. Palazzi, “Retrieval of trace gases vertical profile in the lower atmosphere combining Differential Optical Absorption Spectroscopy with radiative transfer models,” PhD thesis (Bologna University, Italy, 2008), http://amsdottorato.cib.unibo.it/983/1/Tesi_Palazzi_Elisa.pdf

50.

E. Cupini, PREMAR-2 A Monte Carlo code for radiative transport simulation in atmospheric environments. (RT/INN/98/20. 1998).

51.

A. Sarkissian, H. K. Roscoe, and D. J. Fish, “Ozone Measurement by Zenith-Sky Spectrometers: an Evaluation of Errors in Air-Mass Factors calculated by Radiative Transfer Models,” J. Quant. Spectrosc. Radiat. Transf. 54(3), 471–480 (1995). [CrossRef]

52.

J. C. Chai, H. S. Lee, and S. V. Patankar, “Ray Effect and False Scattering in Discrete Ordinates Method,” Numer. Heat Transfer, Part B 24(4), 373- 389 (1993), http://www.informaworld.com/10.1080/10407799308955899

53.

E. Papandrea, E. Arnone, E. Castelli, and M. Premuda, “Sounding the Upper Troposphere-Lower Stratosphere, with Satellite measurements,” Proc. “ESA Living Planet Symposium,” Bergen, Norway, 28 June- 2 July 2010 (ESA-SP-686, December 2010).

OCIS Codes
(000.5490) General : Probability theory, stochastic processes, and statistics
(010.5620) Atmospheric and oceanic optics : Radiative transfer

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: August 1, 2011
Revised Manuscript: January 2, 2012
Manuscript Accepted: January 2, 2012
Published: March 22, 2012

Citation
Margherita Premuda, Elisa Palazzi, Fabrizio Ravegnani, Daniele Bortoli, Samuele Masieri, and Giorgio Giovanelli, "MOCRA: a Monte Carlo code for the simulation of radiative transfer in the atmosphere," Opt. Express 20, 7973-7993 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-7973


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References

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  2. Y. Fouquart, W. M. Irvine, and J. Lenoble, eds., “Standard Procedures to Compute Atmospheric Radiative Transfer in a Scattering Atmosphere - Volume II, International Association of Meteorology and Atmospheric Physics (IAMAP),” Boulder, Colorado, USA (1980).
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  9. D. G. Collins, W. G. Blättner, M. B. Wells, H. G. Horak, “Backward monte carlo calculations of the polarization characteristics of the radiation emerging from spherical-shell atmospheres,” Appl. Opt. 11(11), 2684–2696 (1972). [CrossRef] [PubMed]
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  11. B. Mayer, S. W. Hoch, C. D. Whiteman, “Validating the MYSTIC three-dimensional radiative transfer model with observations from the complex topography of Arizona's Meteor Crater,” Atmos. Chem. Phys. Discuss. 10(5), 13373–13405 (2010), doi:. [CrossRef]
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  13. U. Platt, D. Perner, H. W. Patz, “Simultaneous measurement of atmospheric CH2O, O3 e NO2 by differential optical absorption,” J. Geophys. Res. 84(C10), 6329–6335 (1979). [CrossRef]
  14. U. Platt, D. Perner, “Direct measurement of atmospheric CH2O, HNO2, O3, NO2, and SO2 by differential optical absorption in the near UV,” J. Geophys. Res. 85(C12), 7453–7458 (1980). [CrossRef]
  15. H. K. Roscoe, P. V. Johnston, M. Van Roozendael, A. Richter, A. Sarkissian, J. Roscoe, K. E. Preston, J.-C. Lambert, C. Hermans, W. DeCuyper, S. Dzienus, T. Winterrath, J. Burrows, F. Goutail, J.-P. Pommereau, E. D’Almeida, J. Hottier, C. Coureul, R. Didier, I. Pundt, L. M. Bartlett, C. T. McElroy, J. E. Kerr, A. Elokhov, G. Giovanelli, F. Ravegnani, M. Premuda, I. Kostadinov, F. Erle, T. Wagner, K. Pfeilsticker, M. Kenntner, L. C. Marquard, M. Gil, O. Puentedura, M. Yela, D. W. Arlander, B. A. Kastad Hoiskar, C. W. Tellefsen, K. Karlsen Tornkvist, B. Heese, R. L. Jones, S. R. Aliwell, R. A. Freshwater, “Slant column measurements of O3 and NO2 during NDSC intercomparison of zenith-sky UV-visible spectrometer in June 1996,” J. Atmos. Chem. 32(2), 281–314 (1999).
  16. A. Petritoli, G. Giovanelli, I. Kostadinov, F. Ravegnani, D. Bortoli, P. Bonasoni, F. Evangelisti, U. Bonafe, F. Calzolari, “Tropospheric and stratospheric NO2 amount deduced by slant column measurements at Mt. Cimone Station,” Adv. Space Res. 29(11), 1691–1695 (2002). [CrossRef]
  17. L. M. Perliski, S. Solomon, “On the Evaluation of Air Mass Factors for Atmospheric Near-Ultraviolet and Visible Absorptiion Spectroscopy,” J. Geophys. Res. 98(D6), 10363–10374 (1993). [CrossRef]
  18. S. Solomon, A. Schmeltekopf, R. W. Sanders, “On the interpretation of zenith sky absorption measurements,” J. Geophys. Res. 92(D7), 8311–8319 (1987). [CrossRef]
  19. J. Pukıte, S. Kühl, T. Deutschmann, W. Wilms-Grabe, C. Friedeburg, U. Platt, and T. Wagner, “Retrieval of stratospheric trace gases from SCIAMACHY limb measurements,” Proceedings of the First Atmospheric Science Conference, 8–12 May, ESA/ESRIN, Frascati, Italy, ESA SP-628, (2006). http://earth.esa.int/workshops/atmos2006/participants/1148/paper proc Frasc 2.pdf
  20. S. Kühl, J. Puķīte, T. Deutschmann, U. Platt, T. Wagner, “SCIAMACHY Limb Measurements of NO2, BrO and OClO, Retrieval of vertical profiles: Algorithm, first results, sensitivity and comparison studies,” Adv. Space Res. 42(10), 1747–1764 (2008). [CrossRef]
  21. J. Puķīte, S. Kühl, T. Deutschmann, U. Platt, T. Wagner, “Accounting for the effect of horizontal gradients in limb measurements of scattered sunlight,” Atmos. Chem. Phys. 8(12), 3045–3060 (2008). [CrossRef]
  22. G. Hönninger, C. von Friedeburg, U. Platt, “Multi axis differential optical absorption spectroscopy (MAX-DOAS),” Atmos. Chem. Phys. 4(1), 231–254 (2004), http://www.atmos-chem-phys.net/4/231/2004/acp-4-231-2004.pdf . [CrossRef]
  23. A. Petritoli, F. Ravegnani, G. Giovanelli, D. Bortoli, U. Bonafè, I. Kostadinov, A. Oulanovsky, “Off-axis measurements of atmospheric trace gases by use of an airborne ultraviolet-visible spectrometer,” Appl. Opt. 41(27), 5593–5599 (2002). [CrossRef] [PubMed]
  24. G. Giovanelli, E. Palazzi, A. Petritoli, D. Bortoli, I. Kostadinov, F. Margelli, S. Pagnutti, M. Premuda, F. Ravegnani, G. Trivellone, “Perspectives of 2D and 3D mapping of Atmospheric Pollutants over Urban Areas by means of airborne DOAS Spectrometer,” Ann. Geophys. 49(1), 133–142 (2006).
  25. M. Premuda, S. Masieri, D. Bortoli, I. Kostadinov, A. Petritoli, G. Giovanelli, “Evaluation of vessel emissions in a lagoon area with ground based Multi axis DOAS measurements,” Atmos. Environ. 45(29), 5212–5219 (2011), doi:, http://dx.doi.org/10.1016/j.atmosenv.2011.05.067 . [CrossRef]
  26. E. Palazzi, A. Petritoli, G. Giovanelli, I. Kostadinov, D. Bortoli, F. Ravegnani, S. S. Sackey, “PROMSAR: A backward Monte Carlo spherical RTM for the analysis of DOAS remote sensing measurements,” Adv. Space Res. 36(5), 1007–1014 (2005). [CrossRef]
  27. T. Wagner, J. P. Burrows, T. Deutschmann, B. Dix, C. von Friedeburg, U. Frieß, F. Hendrick, K.-P. Heue, H. Irie, H. Iwabuchi, Y. Kanaya, J. Keller, C. A. McLinden, H. Oetjen, E. Palazzi, A. Petritoli, U. Platt, O. Postylyakov, J. Pukite, A. Richter, M. van Roozendael, A. Rozanov, V. Rozanov, R. Sinreich, S. Sanghavi, F. Wittrock, “Comparison of box-air-mass-factors and radiances for Multiple-Axis Differential Optical Absorption Spectroscopy (MAX-DOAS) geometries calculated from different UV/visible radiative transfer models,” Atmos. Chem. Phys. 7(7), 1809–1833 (2007). [CrossRef]
  28. E. Cupini, M. G. Borgia and M. Premuda, Il codice PREMAR per la simulazione Montecarlo del trasporto della radiazione nell’atmosfera. (RT/INN/97/5. 1997).
  29. E. Cupini, G. Ferro, N. Ferrari, Monte Carlo Analysis of Radiative Transport in Oceanographic Lidar Measurements. (ENEA/RT/INN/2001/7, 2001).
  30. A. Berk, L. S. Bernstein, and D. C. Robertson, “MODTRAN: a moderate resolution model for LOWTRAN7,” Geophysics Laboratory, Air Force Systems Command, United States Air Force, Hanscom AFB, MA (1989).
  31. F. Kneizys, D. Robertson, L. W. Abreu, P. Acharya, G. P. Anderson, L. S. Rothman, J. H. Chetwynd, J. E. A. Selby, E. P. Shettle, W. O. Gallery, A. Berk, S. A. Clough, and L. S. Bernstein, “The MODTRAN 2/3 Report and LOWTRAN7 MODEL,” Phillips Laboratory, Geophysics Directorate, Hanscom AFB, MA (1996).
  32. P. Acharya, S. M. Adler-Golden, G. P. Anderson, A. Berk, L. S. Bernstein, J. H. Chetwynd, et al., “Modtran version 3.7/4.0 user’s manual,” Air Force Research Laboratory, Space Vehicles Directorate, Air Force MATERIEL Command, Hanscom AFB, MA (1998).
  33. A. Berk, G. P. Anderson, P. K. Acharya, J. H. Chetwynd, L. Bernstein, and E. P. Shettle, “MODTRAN4 user’s manual,” Air Force Research Laboratory, Space Vehicles Directorate, Air Force Materiel Command, Hanscom AFB, MA (1999).
  34. D. Nowak, L. Vuilleumier, C. N. Long, A. Ohmura, “Solar irradiance computations compared with observations at the Baseline Surface Radiation Network Payerne site,” J. Geophys. Res. 113(D14), D14206 (2008), doi:. [CrossRef]
  35. G. C. Pomraning, Radiation Hydrodynamics, Los Alamos National Laboratory Radiation Hydrodynamics Short Course Attendees, (1982), http://www.osti.gov/energycitations/servlets/purl/656708-zO5SF0/webviewable/656708.pdf
  36. R. L. Liboff, Kinetic Theory: Classical, Quantum and Relativistic Descriptions. (Prentice Hall, 1989).
  37. S. Chandrasekhar, Radiative Transfer, (Dover Publications, Inc., 1960).
  38. S. Chatigny, M. Morin, D. Asselin, Y. Painchaud, P. Beaudry, “Hybrid Monte Carlo for photon transport through optically thick scattering media,” Appl. Opt. 38(28), 6075–6086 (1999). [CrossRef] [PubMed]
  39. C. P. Robert and G. Casella, Monte Carlo Statistical Methods (Springer, 2004).
  40. A. De Matteis, S. Pagnutti, “Long-range Correlation-analysis of the Wichmann-Hill random number generator,” Stat. Comput. 3(2), 67–70 (1993). [CrossRef]
  41. A. De Matteis, S. Pagnutti, “Controlling correlations in parallel Monte Carlo,” Parallel Comput. 21(1), 73–84 (1995). [CrossRef]
  42. M. Premuda, “Montecarlo simulation of Radiative transfer in Atmospheric environments for problems arising from remote sensing measurements,” in Theory and Applications of Monte Carlo Simulations, S. Mordechai, ed. (INTECH, 2011).
  43. M. H. Kalos and P. A. Whitlock, Monte Carlo Methods. Volume I: Basics (John Wiley & Sons, 1986).
  44. M. Iqbal, An Introduction to Solar Radiation (Academic Press, 1983).
  45. M. Premuda, S. Masieri, D. Bortoli, F. Margelli, F. Ravegnani, A. Petritoli, I. Kostadinov, G. Giovanelli, E. Cupini, “A Monte Carlo simulation of radiative transfer in the atmosphere applied to ToTaL-DOAS,” Proc. SPIE 7475, 74751A, 74751A-12 (2009). [CrossRef]
  46. K. Y. Kondratyev, Radiation in the Atmosphere (Academic Press, 1969).
  47. F. X. Kneizys, S. A. Clough, E. P. Shettle, L. S. Rothmann and R. W. Fenn, “Linear Absorption and Scattering of Laser Beams,” Air Force Geophysical Laboratory, AFGL-TR-84 0265 (1984).
  48. F. X. Kneizys, E. P. Shettle, W. O. Gallery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, S. A. Clough and R. W. Fenn, “Atmospheric Transmittance/Radiance: Computer Code LOWTRAN6,” AFGL-TR-83–0187, (1983).
  49. E. Palazzi, “Retrieval of trace gases vertical profile in the lower atmosphere combining Differential Optical Absorption Spectroscopy with radiative transfer models,” PhD thesis (Bologna University, Italy, 2008), http://amsdottorato.cib.unibo.it/983/1/Tesi_Palazzi_Elisa.pdf
  50. E. Cupini, PREMAR-2 A Monte Carlo code for radiative transport simulation in atmospheric environments. (RT/INN/98/20. 1998).
  51. A. Sarkissian, H. K. Roscoe, D. J. Fish, “Ozone Measurement by Zenith-Sky Spectrometers: an Evaluation of Errors in Air-Mass Factors calculated by Radiative Transfer Models,” J. Quant. Spectrosc. Radiat. Transf. 54(3), 471–480 (1995). [CrossRef]
  52. J. C. Chai, H. S. Lee, and S. V. Patankar, “Ray Effect and False Scattering in Discrete Ordinates Method,” Numer. Heat Transfer, Part B 24(4), 373- 389 (1993), http://www.informaworld.com/10.1080/10407799308955899
  53. E. Papandrea, E. Arnone, E. Castelli, and M. Premuda, “Sounding the Upper Troposphere-Lower Stratosphere, with Satellite measurements,” Proc. “ESA Living Planet Symposium,” Bergen, Norway, 28 June- 2 July 2010 (ESA-SP-686, December 2010).

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