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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 6, JUNE 2008

McCART: Monte Carlo Code for Atmospheric Radiative Transfer Vanni Nardino, Fabrizio Martelli, Piero Bruscaglioni, Giovanni Zaccanti, Samuele Del Bianco, Donatella Guzzi, Paolo Marcoionni, and Ivan Pippi

Abstract—McCART is a numerical procedure to solve the radiative transfer equation for light propagation through the atmosphere from visible to near-infrared wavelengths. The procedure has been developed to study the effect of the atmosphere in the remote sensing of the Earth, using aerospace imaging spectrometers. The simulation is run for a reference layered plane nonabsorbing atmosphere and a plane ground with uniform reflectance. For a given distribution of ground reflectance and for a specific profile of scattering and absorption properties of the atmosphere, the spectral response of the sensor is obtained in a short time from the results of the Monte Carlo simulation by using scaling relationships and symmetry properties. The procedure also includes an accurate analysis of the adjacency and trapping effects due to multiple scattering of photons coming from neighboring pixels. McCART can generate synthetic images of the Earth’s surface for arbitrary viewing conditions. The results can be used to establish the limits of applicability of approximate algorithms for the processing and analysis of hyperspectral images acquired by imaging spectrometers. In addition, the algorithm can be used to develop procedures for atmospheric correction for the accurate retrieval of the spectral ground reflectance. Index Terms—Aerosols, atmospheric propagation, multilayered media, remote sensing, scattering.

I. I NTRODUCTION

M

EASUREMENTS of remote sensing of the Earth at visible and near-infrared wavelengths are greatly influenced by the turbidity of the atmosphere. Scattering and absorption by aerosol and gases highly complicate light propagation compared with the idealized case of a perfectly transparent atmosphere. As a consequence, the relationship between the measured signal and the physical parameters of interest becomes very complicated, and the relevant information can be retrieved only if a suitable inversion procedure is available. The first step in developing a reliable inversion procedure is the availability of an efficient solver of the forward problem, i.e., an algorithm that, given the characteristics of the light source and of the receiver and the optical properties of the ground and of the atmosphere, enables us to determine the response of the receiver. Light propagation through turbid media is described by the radiative transfer equation (RTE) [1]–[3]. Several radiative transfer codes are used to find solutions of the RTE. These Manuscript received July 3, 2007; revised November 19, 2007. V. Nardino, F. Martelli, P. Bruscaglioni, and G. Zaccanti are with the Dipartimento di Fisica, Università degli Studi di Firenze, 50019 Sesto Fiorentino, Italy. S. Del Bianco, D. Guzzi, P. Marcoionni, and I. Pippi are with the Istituto di Fisica Applicata “Nello Carrara,” Consiglio Nazionale delle Ricerche, 50019 Sesto Fiorentino, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TGRS.2008.916464

codes are based on an explicit solution of the RTE or on a Monte Carlo (MC) approach to the radiative transfer problem. The first ones are currently used in atmospheric radiation applications like MODTRAN4 [4], [5] which uses the DISORT [6] radiative transfer code as a subroutine for the implementation of the azimuth dependence of multiple scattering, 6S which solves the RTE in an iterative way using successive orders of scattering [7], COMANCHE [8] which uses an analytical formulation of the upwelling radiance, SHARM-3-D [9], [10] which uses the method of spherical harmonics [11]–[13], and SHDOM which uses the spherical harmonics discrete ordinate method [14], [15]. These codes use different levels of sophistication when modeling complex effects, like radiation trapping between the ground and atmosphere, and neighboring pixel adjacency effects. For example, MODTRAN uses a single spatially homogeneous spectral reflectance to model the surface surrounding the imaged pixel. Both MODTRAN and 6S are used by several model-based methods (ATREM [16], [17], FLAASH [18], and ATCOR [19]) to generate lookup tables for atmospheric-effect correction. Similarly, McCART uses MODTRAN to generate molecular absorption profiles. Models based on an MC approach [20]–[23] solve the RTE, using probabilistic modeling of the associated radiative transfer processes. These methods are flexible, and their selfconsistency may be predicted by examining the variance between estimates made from subsets of the simulation. The main goal of this paper is to describe a methodology based on MC simulations, the McCART procedure, which, making use of scaling relations that greatly reduce the computation time, enables us to evaluate in a reasonable time the spectral at-sensor radiance for a specific distribution of reflectance of the Earth’s surface. The McCART procedure has been developed to evaluate the effect of a user-defined atmosphere on aerospace multispectral and hyperspectral images of the Earth. The direct problem is solved: the procedure is able to obtain the at-sensor radiance, given the spectral surface reflectance distribution [Lambertian or characterized by a bidirectional reflectance distribution function (BRDF)]. The model, described in Section II, is based on a numerical MC simulation carried out for a nonabsorbing reference atmosphere with layered plane-parallel structure and for a nonabsorbing plane Earth surface. Taking advantage of the translational symmetry of the problem and of the weak dependence of the at-sensor radiance on the viewing direction, a large number of useful trajectories are drawn and stored in a reasonable time

0196-2892/$25.00 © 2008 IEEE

NARDINO et al.: McCART: MONTE CARLO CODE FOR ATMOSPHERIC RADIATIVE TRANSFER

Fig. 1. (a) Geometrical scheme and symbols. (b) Examples of trajectories of photons contributing to the at-sensor radiance.

even with a personal computer. From the stored trajectories, the effect of a specific profile of absorption and scattering properties of the atmosphere on the response is evaluated in a short time, making use of Beer’s law scaling relationships. The response is divided into six contributions according to the number of reflections at the Earth’s surface and of scattering events in the atmosphere. Finally, for a specific distribution of spectral reflectance of the ground surface, the response is obtained, properly weighting each contribution according to the reflectance of each pixel. Therefore, the response includes an accurate analysis of adjacency and trapping effects with respect to radiative transfer codes like MODTRAN and 6S. Because McCART does not include thermal emission, it can be applied from visible to short-wave infrared wavelengths. The range of applicability is discussed in Section III, whereas examples of results predicted by the procedure are reported in Section IV, where discussions of the obtained results are given. Conclusions are drawn in Section V. II. D ESCRIPTION OF THE A LGORITHM A. General Remarks The goal of the McCART procedure is to simulate the response of a pixel of a spectral image collected by an airborne system observing the Earth’s surface that is illuminated by the sun. More specifically, we are interested in evaluating the response of the pixel R(λ, rP ) = KL(λ, rP , sˆP )ΩP cos θP

(1)

from which the at-sensor radiance L(λ, rP , sˆP ) can be obtained. In (1), λ denotes the wavelength; K is the relative instrumental spectral response; rP , sˆP , and ΩP are the position, the viewing direction, and the acceptance solid angle of the sP · n ˆ R ; and n ˆ R is the normal pixel, respectively; cos θP = −ˆ to the surface of the optical receiver [Fig. 1(a)]. For a perfectly transparent atmosphere and a plane Earth surface, the at-sensor radiance would be given by L(λ, rP , sˆP ) = S(λ)

cos θS ρ(λ, rE )pE (λ, sˆS , sˆP ) cos θP

(2)

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sS · where S(λ) is the spectral solar irradiance, cos θS = −ˆ ˆ E is the n ˆ E , sˆS is the direction of the solar incident light, n normal to the Earth’s surface, ρ(λ, rE ) is the spectral reflectance of the observed pixel at rE , and pE (λ, sˆ , sˆ) is the distribution function, normalized to one over the solid angle 2π, describing the angular distribution of the light reflected by the ground that is illuminated by light in direction sˆ related to the BRDF. For example, for a Lambertian plane ˆ E /π. In this idealized consurface, it is pE (λ, sˆ , sˆ) = sˆ · n dition, L(λ, rP , sˆP ) only depends on the reflectance of the observed pixel and on the geometry of observation. As shown in Fig. 1(b), the problem is highly complicated by the turbidity of the atmosphere because, due to manifold scattering events in the atmosphere and/or to ground reflections, L(λ, rP , sˆP ) becomes a very complicated function both of the optical properties of the atmosphere and of the distribution of reflectance on a large surface around the observed pixel. The procedure we have developed to evaluate the response of the pixel is based on an MC simulation. We point out that the elementary MC procedure, i.e., a numerical simulation of the trajectories of the photons that illuminate the top of the atmosphere, which propagate through the atmosphere as far as they are absorbed (by the ground or by the atmosphere), they escape from the atmosphere, or they are eventually received, would be inapplicable due to the prohibitively long computation time. In fact, it would be necessary to process an extremely large number of photons to simulate the illumination of the vast surface around the pixel of interest from which, due to multiple scattering, photons can reach the receiver. Furthermore, also due to the small solid angle subtended by the pixel, the probability that the photon is received is extremely small, and because the optical properties of both the atmosphere and ground depend on the wavelength, it would be necessary to repeat the simulation for each value of λ. To speed up the convergence of the MC procedure, we need to introduce some simplifications in order to make possible the use of symmetry properties and scaling relationships. We have therefore assumed a plane Earth surface and a plane layered atmosphere uniformly illuminated by a plane wave. The McCART procedure can be summarized as follows. 1) An elementary MC simulation is run only once for a nonabsorbing horizontally homogeneous reference atmosphere and a ground with constant reflectance ρ = 1. With these assumptions, the response only depends on the height at which the sensor is placed and on the solar and viewing geometry, and translational properties can be used to reduce the computation time. All the trajectories of received photons (typically 105 ) are stored on a file, together with the number of launched photons NL . 2) Making use of scaling relationships, from the stored information, the effect of the atmosphere with a specific profile of scattering and absorption on the response of the pixel is first evaluated for a ground with uniform reflectance ρ = 1. 3) Received photons are divided into six contributions according to the number of scattering events in the atmosphere and of ground reflections. Moreover, photons

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are also classified according to the distance of the point of ground reflection from the center of the observed pixel to evaluate radial distribution functions. 4) With the information from points 2) and 3), the at-sensor radiance for a specific distribution of reflectance ρ(λ, rE ) can easily be obtained. Steps 2), 3), and 4) take short computation time and are repeated for each wavelength of interest. The longer computation time is taken by the MC simulation. In fact, the acceptance solid angle of the pixel is very small (specifically for spacecraft systems), so the probability of finding a useful trajectory is extremely small and a very large number of trajectories should be simulated to obtain statistically reliable results. However, because the radiance L(λ, rP , sˆP ) for a uniform surface is usually a slowly varying function of sˆP , the MC simulation for the reference atmosphere can be carried out for a large acceptance solid angle ΩP , and the response for the solid angle ΩP can be obtained, scaling the results for ΩP . B. Elementary MC Simulation The elementary MC simulation is carried out for a nonabsorbing plane-parallel layered atmosphere and for a plane ground with uniform reflectance ρ = 1. The kth layer is characterized by the scattering coefficient µsk and by the scalar scattering function pk (θ). Instead of directly simulating the atmosphere uniformly illuminated by the sun (infinitely extended source) and a very small receiver (a pointlike detector), for which the probability of receiving a photon is prohibitively small, we first simulate the complementary problem, i.e., a pointlike source and an infinitely extended receiver. In this case, the probability of receiving a photon is much higher because all photons intersecting the plane z = zP at which the receiver is placed, within the acceptance solid angle of the pixel, are received. The modeled geometry is shown in Fig. 1(a). All photons are injected at the top of the atmosphere at the point P0 = (0, 0, zT OA ) along the direction sˆS of the solar radiation. The photon path is traced as a series of discrete steps. Each step is determined stochastically by using standard MC rules starting from random numbers ξ between zero and unity: The length li of the path between two subsequent scattering events Pi , Pi+1 , the scattering angle θi , and the azimuth angle ϕi are calculated as follows: li (ξ) = z (zi+1 − zi )/ cos αi with zii+1 µs (z)/ cos αi dz = − ln(ξ) and αi angle between the direction of the photon and the z-axis, θ (ξ) 2π 0 i p(θ ) sin θ dθ = ξ, and ϕi (ξ) = 2πξ. User-defined aerosol models are utilized for the evaluation of the scattering properties of the atmosphere. The scattering functions are evaluated starting from the size distribution of aerosols. At each step, the following checking steps are made. 1) If the trajectory has crossed the Earth’s surface (plane z = 0), Pi+1 is posed equal to the point of intersection and the trajectory gone on drawing a new direction and a new path length. The new direction (θE , ϕE ) is drawn s , sˆ). Examples according to the distribution function pE (ˆ reported in Section III refer to a Lambertian surface,

ˆ E /π. In this case, for which pE (λ, sˆ , sˆ) = sˆ · n √ θE and ϕE are obtained as follows: cos θE (ξ) = 1 − ξ and ϕE = 2πξ. 2) If the trajectory has crossed the plane z = zP at which the sensor is placed, the angle between the direction of the photon and the viewing direction sˆP is calculated. If the angle is within the acceptance solid angle of the pixel, the intersection Pout = (xout , yout , zP ) with the plane z = zP is calculated, the trajectory P0 , P1 , . . . , Pout , after a translation, is stored, and the counter of received photons is increased. Subsequently, the trajectory goes on from the point Pi+1 until the photon leaves the atmosphere. Each emitted photon can therefore yield more than one contribution, according to the possibility for a photon of contributing several times to the radiance in different points on the plane z = zP . This point is insignificant when McCART is used to evaluate the atsensor radiance measured by a sensor with a small acceptance angle, but it becomes crucial when McCART is used, for instance, to evaluate the irradiance measured by a solar spectral irradiometer (an example is shown in Section IV). 3) If the trajectory has crossed the top of the atmosphere (plane z = zTOA ), the photon is given up, and a new photon is launched. The simulated trajectories refer to the pencil light beam and to the infinitely extended receiver. Due to the symmetry of the problem, these trajectories can also be used to solve the problem of an infinitely extended source and of a pointlike detector. In fact, because the optical properties of the atmosphere and of the ground are assumed independent of the x and y coordinates, two parallel trajectories have the same probability. Thus, for each simulated received trajectory P0 , P1 , . . . , Pout with P0 = (0, 0, zTOA ) and Pout = (xout , yout , zP ), there is a parallel , obtained trajectory with the same probability, P0 , P1 , . . . , Pout with the translation x = x − xout and y = y − yout , for which = (0, 0, zP ). With this P0 = (−xout , −yout , zTOA ) and Pout translation applied to each received trajectory, we obtain the corresponding received trajectories for the infinitely extended source and the pointlike receiver. Thanks to this symmetry, we obtain a statistically representative set of received trajectories in a reasonable computation time. In principle, the simulation should be carried out for each pixel within the field of view of the receiver. However, because, for a surface with uniform reflectance, the radiance is a slowly varying function of the viewing direction (see Section III-C), the MC results obtained for a specific viewing direction sˆP are also applicable to the pixels within a cone of several degrees around sˆP . C. Scaling Relationships for µa and µs To evaluate the effect of the atmosphere on the response of the pixel when the profile of scattering and absorption for a specific wavelength λ is considered, we first calculate the probability P (λ) that the photon is received for the ground with ρ = 1. For the nonabsorbing reference atmosphere, the probability is simply given by Pref = NR /NL with a relative error

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√ εr = 1/ NR , where NR is the number of received photons and NL is the number of launched photons. For an atmosphere with scattering and absorption coefficients µsk (λ) and µak (λ), the probability changes because the probability of each trajectory changes. To evaluate the new probability, we used two scaling relationships previously used to study light propagation through the atmosphere [24] and through biological tissues [25]. To each trajectory, a weight wj (λ) is given, equal to the ratio between the probability of following the trajectory after and before changing the optical properties of the atmosphere, and P (λ) is obtained as P (λ) =

NR

wj (λ)/NL .

(3)

j=1

The weight can be expressed as wj = waj wsj , with waj and wsj weights being due to changes of absorption and scattering, respectively. For absorption, according to the Lambert–Beer law, the weight of the jth trajectory is K L µak ljk (4) waj = exp − k=1

where µak is the absorption coefficient of the kth layer, ljk is the total length followed by the jth photon into the kth layer, and KL is the number of layers into which the atmosphere has been divided. For a fixed wavelength grid, the absorption coefficients are calculated using a MODTRAN4-based band model derived directly from the spectral line parameters of the HITRAN96 database [26]. If the absorption coefficient has small variations within the wavelength interval, the weight can be evaluated using, in (4), the average value for µak . On the contrary, if strong variations occur, due, for instance, to a narrow absorption band, the interval must be properly sampled, and the average value of the weight must be calculated. The exact knowledge of the path length followed by each received photon allows us to include the effect of absorption. The weight wsj is obtained, remembering that, in a nonabsorbing medium, the probability for a photon of undergoing a scattering event within a volume element subtended by dl at distance l from the previous one, and being scattered with an angle θ within the solid angle dω, is given by µs dl exp(−µs l)p(θ)dω. Therefore, if the scattering coefficient of the layers is changed from µsk to µsk , without changing the scattering functions, the weight of the photon is given by KL njk µsk exp [−(µsk − µsk ) ljk ] (5) wsj = µsk k=1

where njk is the number of scattering events undergone by the photon inside the kth layer. Equations (4) and (5) are exact scaling relationships, provided that the scattering function remains constant when the scattering coefficient is changed. However, the statistical fluctuations due to the finite number of simulated trajectories limit their range of applicability. The range of applicability will be discussed in Section III.

Fig. 2. Classification of received photons. Six different types of trajectories are distinguished. For each type, examples of trajectories are shown.

D. Contributions to the Response and Distribution Functions To evaluate the effect of the ground on the response of the pixel when a specific distribution of reflectance is considered, the weight of each trajectory wj should be multiplied by the reflectance of the points in which the photon has been reflected. This could be made, but if many pixels should be processed, as is the case for simulating spectral images of a ground with known reflectance, or for processing images to reconstruct the map of spectral ground reflectance, a long computation time would be necessary. The response of the pixel can be obtained in a shorter time using average information instead of the information carried by each trajectory separately. For this purpose, the contribution to the response calculated for ρ = 1 should be divided according to the type of trajectory followed by the photon, and the radial distribution functions describing the probability that received photons have been reflected at distance d from the center of the observed pixel should be evaluated. When the trajectories are processed to apply the scaling relationships, their weights wj ’s are accumulated separately according to the type of trajectory. As shown in Fig. 2, we have distinguished six different types of trajectories, and for each type, we have evaluated the fractional contributions Cn ’s(n = 1, 6)to the probability P (λ). C1 is the fraction of photons received without ground reflections. Contributions C2 − C5 refer to photons reflected only once: C2 is the fraction of photons received without scattering events in the atmosphere; C3 is the fraction of scattered photons that illuminate the observed pixel and are received without scattering events between the ground and the receiver; C4 is the fraction of received photons that, after the ground reflection, have been scattered only once; and C5 is the fraction of photons that have undergone multiple scattering after the ground reflection. Contribution C6 is the fraction of photons received after multiple ground reflections. This contribution is further divided according to the number of reflections: C6,m is the fraction of photons reflected m times. We point out that each photon is classified in one and only one contribution. Contributions C2 and C3 produce a clear image of the observed pixel. The other contributions blur the image: contribution C1 comes from the brightness of the atmosphere, whereas contributions C4 , C5 , and C6 come from the

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surrounding pixels and are responsible for the adjacency effects. Contributions C4 and C5 have been distinguished in order to make possible the use of an empirical scaling relationship for the angular field of view (see Section III-C). The distribution functions Dn (d)’s describing the probability that received photons have been reflected at distance d from the center of the observed pixel have been evaluated separately for contributions Cn ’s, with n = 4, 5, and 6. The plane z = 0 has been divided into rings concentric with the observed pixel, and for each contribution, a histogram has been obtained, adding the weights of photons reflected within each ring. The distribution function has been obtained as j wji (6) Dn (di ) = Cn π [(di + ∆d/2)2 − (di − ∆d/2)2 ] where the summation is on photons reflected at distance (di − ∆d/2) ≤ d < (di + ∆d/2). For the contribution C6 , referring to photons with multiple reflections, we have evaluated only one distribution. Photons have been classified using, for each trajectory, the average distance dj . In principle, the distribution function for each m of C6,m could be evaluated, but a significant increase in the computation time would be necessary to process a sufficiently large number of photons in order to obtain statistically reliable results. E. At-Sensor Radiance If the ground is divided into pixels and ρij is the reflectance of the ij-pixel, the probability Pij that photons are received when the ij-pixel is observed can be obtained as Pij = P C1 + (C2 + C3 )ρij + C4 D4 (di j )ρi j AP i j

+C5

D5 (di j )ρi j AP +

i j

m

C6m ρ

(7)

m

cos θS 1 . cos θP ΩP

A. Tests on the Drawn Trajectories The algorithm to draw the trajectories has been checked by comparisons with exact analytical and numerical results. In particular, to check the correctness of the algorithm to draw the scattering points, we have run the MC code for a homogeneous infinite atmosphere, and we have evaluated the first and second moments of the coordinates of the points at which the different orders of scattering occur. The numerical results have been compared with exact analytic relationships [27], and we have obtained an excellent agreement: Discrepancies were within the standard deviation of MC results even when a large number of trajectories (108 ) were simulated and the error was as small as 0.01%. To check the correctness of the algorithm used for the ground reflectance, we have considered a transparent atmosphere and compared the MC results with (2) for a different solar and viewing geometry. Other tests have been made to check that, as expected by the RTE, for a nonabsorbing atmosphere with the scattering function that is independent of the height, and a ground with ρ = 1, the probability P is independent of the profile of scattering coefficients, provided that the optical thickness z TOA

τs =

µs (z)dz 0

where di j is the distance between the ij- and the i j -pixels, AP is the area of the pixel, and ρ is the average reflectance over an area around the pixel from which photons with multiple reflections are received. The area is estimated from the distribution function D6 (d). We point out that P , Cn , Dn (d), ρij , ρ, and, thus, Pij are functions of λ. Once the probability Pij has been evaluated for the wavelengths of interest, the at-sensor radiance for the ij-pixel can be obtained as Lij (λ, sˆP ) = S(λ)Pij (λ)

In fact, with the MC method, the RTE can be solved without the need of simplifying assumptions, and the accuracy of the results is only limited by the statistical fluctuations. However, developing an MC code in order to simulate the response of hyperspectral sensors for remote sensing is not trivial at all, and checks should be made to control its correctness. This is particularly important for the McCART procedure in which scaling relationships and symmetry properties are widely used.

(8)

III. T ESTS ON THE M C CART P ROCEDURE AND A S CALING R ELATIONSHIP FOR THE A NGULAR F IELD OF V IEW The results of MC simulations are often used as a gold standard to check the accuracy of approximate solutions of the RTE.

zP τsP =

µs (z)dz

(9)

0

remain constant. Furthermore, comparisons have been made for the distribution function D4 (d) referring to photons reflected by the ground, which have been received after a single scattering event. Because the ground is uniformly illuminated, the reference distribution can be easily calculated with a numerical integration [an example of comparisons will be shown in Fig. 6(e)]. All the comparisons gave excellent agreement. B. Applicability Range of the Scaling Relationships Until the scattering functions can be considered independent of µs and µa , the scaling relationships presented in Section II are exact relationships, but their range of applicability is limited by statistical noise due to the finite number of simulated trajectories and needs to be checked. In fact, starting from a set of useful trajectories predetermined by the elementary MC simulation, the relative error εr on the results increases when the range of values that the weighting factors can assume increases. When the absorption coefficient is varied, the weight of the photon only depends on the length of the trajectory inside

NARDINO et al.: McCART: MONTE CARLO CODE FOR ATMOSPHERIC RADIATIVE TRANSFER

Fig. 3. Scattering properties of the atmosphere used for numerical simulations. (a) Reference profile of scattering coefficient. (b) Scattering functions.

each layer. In any case, 0 ≤ waj ≤ 1, and reliable results can be obtained for a large range of conditions. When the scattering coefficient is varied, the weight strongly depends also on the number of scattering events. This involves larger variations of the weighting factor, and the noise on the scaled results can rapidly increase, particularly for µsk > µsk . The range of applicability of the scaling relationships has been investigated, with simulations carried out for idealized models of atmosphere. To explore the scaling relationship for the scattering coefficient, we have started from a profile of µs , which is typical of a rural midlatitude summer atmosphere with relative humidity of 70% and visibility at 23 km, at λ = 450 nm [Fig. 3(a)] [4], [5]. The optical thickness due to scattering is τs = 0.71. The MC simulations have been repeated for 11 profiles of scattering coefficient obtained, multiplying the one of Fig. 3(a) by constant factors. We have considered values of τs ranging from 0 to 3. The scattering functions that we have used are shown in Fig. 3(b): we have used a scattering function that is typical of a clear atmosphere [28] for the lower layers (z < 12 km), of stratospheric aerosols for 12 < z < 35 km [29], [30], and the Rayleigh scattering function for the higher layers (z > 35 km). The scattering functions have been obtained from Mie theory starting from the size distribution of the aerosols for the wavelength λ = 450 nm. The atmosphere was divided into 50 layers. We considered a nonabsorbing atmosphere and a sensor at zp = 1.5 km looking downward along the z-axis. We simulated ˆE the response for the pixel with viewing direction sˆP = n (Fig. 1) and angular field-of-view semiaperture α = 10 mrad. The sun was at a zenith angle of 30◦ . Fig. 4(a) shows the comparison between the results obtained for the probability P from the 11 direct MC simulations (symbols) and the results obtained from the scaling relationship (5) starting from the MC results for τs = 1.5 (continuous lines). For the scaled results, the error bars are also reported (for the direct simulations, the error was smaller than the symbols). Discrepancies between the scaled results and the results of direct simulations are larger than one standard deviation only for τs > 2.5. Fig. 4(b) shows the comparison for the corresponding fractional contributions Cn ’s(n = 1, 6). Also, for Cn , there is an excellent agreement for τs < 2.5. Examples of results for the distribution functions Dn (d)’s are shown in Fig. 4(c). We focused our attention on the distributions D4 and D5 , the ones explicitly used in (7) to evaluate the response Rij . The figures show the comparisons between the results obtained from the direct MC simulations

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at τs = 0.9 and those obtained scaling from τs = 1.5. Comparisons show that the scaled results are in excellent agreement with the results of the direct MC simulations. Anyway, the MC results showed that the distribution functions slightly depend on τs . To better visualize the extension of the region around the observed pixel that influences the measured at-sensor radiance, in Fig. 4(d), we have shown the integrated distributions d 2π 0 Dn (d )d dd as a function of d. This figure shows that the integrated distributions reach the value of 0.8 for d ∼ = 1, 5, and 12 km for D4 , D5 , and D6 , respectively, indicating that ∼ =20% of the contributions C4 , C5 , and C6 comes from pixels at distances d > 1, 5, and 12 km, respectively, from the observed pixel. The size of the pixel is 30 m. The results of Fig. 4 show that, from a single MC simulation, the scaling relationship for the scattering coefficient enables us to obtain reliable results on a wide range of values of τs that probably include the values of practical interest for remote sensing of the Earth’s surface. In fact, in Fig. 4(b), we note that, for τs = 2.5, the contributions C2 and C3 (those of photons that are useful for the image) are less than 2% and 20%, respectively, of total received photons. We want to point out that the results of Fig. 4 refer to an idealized nonabsorbing atmosphere with scattering functions that are independent of the optical thickness τs and to a Lambertian surface with uniform reflectance ρ = 1. To investigate the range of applicability of the scaling relationship for the absorption coefficient, we have started from the MC simulation carried out for the nonabsorbing atmosphere with the profile of scattering coefficient and the scattering functions shown in Fig. 3. The reference profile of absorption that we considered is shown in Fig. 5(a). It is a profile, which is typical of a rural midlatitude summer atmosphere with relative humidity of 70%, at λ = 2500 nm [4], [5]. The optical thickness due to absorption is τa = 0.73. The scaling relationship for absorption has been used to simulate the response for values of τa between 0 and 10. The profiles of absorption have been obtained, multiplying the one of Fig. 5(a) by constant factors. The geometry is the same as used in the simulation of Fig. 4. Examples of results are shown in Fig. 5. Fig. 5(b) and (c) shows the probability P and the fractional contributions Cn ’s as a function of τa (the error bars are not shown because the error was smaller than the symbols apart from the results for τa > 5). The results of Fig. 5(c) show that when τa increases, the contribution of multiple reflections, C6 , rapidly decreases, and particularly, the contribution of photons received without ground reflections, C1 , rapidly increases: For τa > 10 is C1 ∼ = 1, indicating that received photons are only due to atmospheric backscattering and that no useful information on the ground reflectance can be obtained. Fig. 5(d) shows examples of results for the distribution function D4 (d). As τa increases, the distribution function becomes sharper, indicating that the surface around the observed pixel that affects the at-sensor radiance significantly decreases. A significantly larger effect has been observed for D5 and D6 . The results of Figs. 4 and 5 show that, from a single MC simulation, the results for a wide range of values of scattering and absorption coefficients can be obtained by using the scaling relationships. We can therefore expect, from a single

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Fig. 4. Comparisons between the results of direct MC simulations and the results from the scaling relationship for the scattering coefficient. (a) Probability P and (b) fractional contributions Cn ’s as a function of τs . Symbols: Results of direct MC simulations. Continuous lines: Results from the scaling relationship d Dn (d )d dd as a function of d. applied to the MC results for τs = 1.5. (c) Distribution functions D4 and D5 for τs = 0.9. (d) Integrated distributions 2π 0

Fig. 5. Examples of results obtained with the scaling relationship for the absorption coefficient. (a) Reference profile of absorption coefficient used. (b) and (c) Probability P and fractional contributions Cn ’s as a function of τa . (d) Examples of results for the distribution function D4 (d). The geometry and the scattering properties are the same as for Fig. 4. τs = 0.9.

MC simulation, that we can reconstruct the at-sensor spectral response for a wide range of wavelengths. However, when the scaling relationships are used to evaluate the spectral response, we must point out that the scaling relationship for scattering, (5), is valid, provided that the scattering function does not appreciably vary when µs is changed. Because both µs and p(θ) vary with the wavelength, it is necessary to study how the response changes with the scattering function. This point has been investigated, repeating MC simulations for the scattering functions obtained using the same size distributions for the aerosols assumed for Fig. 3 but different values of λ between 450 and 2500 nm. Comparisons showed that sim-

ulations performed for a small number of wavelengths, for which the scattering functions can be considered constant for each step of calculation, are sufficient to reconstruct the spectral response without any interpolation procedure and with an excellent accuracy over the whole range of wavelengths. For the considered cases, characterized by smooth aerosol phase functions, only three wavelengths (450, 600, and 800 nm) proved to be sufficient (simulation at 450 nm scaled between 400 and 550 nm, simulation at 600 nm scaled between 550 and 750 nm, whereas simulation at 800 nm scaled between 750 and 2500 nm). Absorption and scattering coefficients are referred to a user-defined grid. The results obtained from a

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Fig. 6. Dependence of MC results on the angular field-of-view semiaperture α. (a) and (b) Averaged probability P/(ˆ sP · n ˆ R Ωp ) and fractional contributions Cn ’s as a function of α. (c) and (d) Distribution functions D4 and D5 for some values of α. (e) Examples of comparisons reported between MC predictions and results from numerical integration for D4 . (f) Examples of integrated distributions D4 and D5 for some values of α. The geometry and the scattering properties are the same as for Fig. 4. τs = 0.9 and τa = 0.

single MC simulation at 450 nm are more approximated but probably sufficiently accurate for many applications. C. Scaling Relationship for the Angular Field of View The longer computation time in using the McCART procedure is taken by the MC simulation for the reference atmosphere. As an example, the simulation for τs = 1.5 and α = 10 mrad used for Fig. 4 has taken about 1 h on a 3.2-GHz Pentium 4 personal computer. The simulation referred to a sensor at zp = 1.5 km. The angular field-of-view semiaperture of the pixel was α = 10 mrad, and the size of the corresponding ground pixel was 30 m. The computation time strongly depends on α. In fact, because, for the uniform surface with reflectance ρ = 1 for which the elementary MC simulation is carried out, the radiance is a slowly varying function of the viewing direction sˆP , the probability P comes about proportional to the acceptance solid angle ΩP ∼ = 2πα2 /2 and the computation time approximately proportional to 1/α2 . Therefore, the computation time to simulate a sensor with a very small value of α, as for spacecraft sensors for which α can be a few tens of microradians, becomes prohibitively long. We therefore explored the possibility of obtaining the response, scaling the results of MC simulations carried out for a large value of α. For this, we studied how the probability P , the

fractional contributions Cn ’s, and the distribution functions Dn ’s vary with α. Examples of results are shown in Fig. 6. We considered a nonabsorbing atmosphere with the scattering properties of Fig. 3 and a uniform Earth surface with ρ = 1. The geometry is the same as for Fig. 4. Fig. 6(a) and (b) shows the averaged probability P/(cos θP Ωp ) (the average of cos θP is calculated over the solid angle Ωp ) and the fractional contributions Cn ’s as a function of α. Fig. 6(c) and (d) shows the distribution functions D4 and D5 for some values of α. Fig. 6(a) shows that P/(cos θP Ωp ) remains almost constant over a large range of values of α, indicating that the radiance is a slowly varying function of the direction sˆP . Furthermore, Fig. 6(b) shows that also the fractional contributions Cn ’s are almost independent of α. This is true even for the contributions C1 and C4 , the ones that are more strictly related to the shape of the scattering function. These results indicate that the at-sensor radiance for a small angular aperture can be obtained from an MC simulation for a large value of α. The distribution functions D4 and D5 have a significantly different dependence on the angular field of view: D5 weakly depends on α (curves for α ≤ 100 mrad are almost indistinguishable). On the contrary, D4 is strongly affected: It is nearly constant for d < zP tan α, i.e., over the area of the observed pixel, and has a high peak for α → 0. This dependence should be taken into account when data simulated for a certain value of α are used to predict

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Fig. 7. Simulated spectral response of a sensor for measuring the radiance. The geometry is the same as for Fig. 4. (a) Spectral transmittance that we used; the contributions of scattering and absorption are reported separately. (b) Probability Pij (λ) for a ground with constant reflectance ρ(λ) = 0.5 and for a grassland with the spectral reflectance shown in (c). (c) Spectral reflectance for grassland and spectrum of the solar irradiance used to evaluate the (d) at-sensor radiance.

the results for different values of α. Fortunately, as mentioned in Section III-A, the distribution function D4 can be easily calculated with a numerical integration. In Fig. 6(e), examples of comparison between results obtained from MC simulations and by numerical integration are shown. The excellent agreement also gives a test on the correctness of the MC procedure. Finally, we observe that, in spite of the strong differences observed for small values of d, particularly for the distribution function D4 , small differences are observed when the integrated distributions are considered, indicating that the angular aperture that subtends the observed pixel has a little influence on the extension of the surrounding region that affects the at-sensor radiance. Examples of comparisons are shown in Fig. 6(f) for both D4 and D5 . To investigate the effect of the scattering function on the range of applicability of the scaling relationship for the angular field of view, we repeated the calculations using, for the lower layers of the atmosphere (z < 12 km), a scattering function with a significantly higher forward peak with respect to the one of Fig. 3(b): we used the scattering function of the C1 cloud model [31] at 450 nm for which p(θ = 0) = 324 and the asymmetry factor is g = 0.86. The results that we obtained showed that the scattering function significantly affects the redistribution of photons among different fractional contributions Cn ’s, but again, the fractional contributions are almost independent of the angular field of view over a wide range of values of α. Similar results have been obtained for the distribution functions Dn ’s: they are significantly affected by the scattering function (as the forward peak increases, the distributions, particularly D4 and D5 , become sharper, indicating that the surface around the observed pixel that affects the at-sensor radiance significantly decreases), but the dependence of D5 and D6 on α remains weak. In conclusion, the at-sensor radiance can be precisely evaluated, even for extremely small values of the angular field of

view, by using the results of an MC simulation carried out for a value of α as large as 100 mrad, provided that the distribution function D4 is evaluated by numerical integration. For α = 100 mrad and τs = 1.5, 105 useful trajectories can be obtained with a computation time of about 1 min. The weak dependence of the radiance on the viewing direction enables us to use the MC results obtained for the viewing direction sˆP to evaluate the response for the pixels within a cone of several degrees around sˆP . A way to speed up the convergence of the MC simulation may be the use of a semianalytic MC procedure [32], [33]. With the semianalytic approach, trajectories are simulated by using the same rules as for the elementary MC simulation, and each time an atmospheric or surface scattering event occurs, the probability that it would result in a photon entering the receiver is evaluated. Thus, for the geometry investigated, every scattering event below the altitude of the receiver contributes and provides information for the specific viewing geometry. With the elementary MC simulation, the response is obtained, summing up the few contributions of the received photons, all with the same weight. On the contrary, with the semianalytic approach, the response is obtained, summing up many small contributions with different weights. Because the probability is proportional to the product of the scattering function toward the sensor by the transmittance to the sensor, the range of values that it can assume increases as the optical thickness of the atmosphere increases or the scattering function becomes more forward peaked. To obtain the same convergence, it is therefore necessary to sum up a significantly larger number of contributions. For a small value of α and a moderate optical thickness, the convergence of a semianalytic MC simulation is significantly faster with respect to an elementary MC simulation. However, to apply the scaling relationships to the results of a semianalytic MC simulation, it would be necessary to change the weight of each probability. It would be therefore

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Fig. 8. Simulated response of a solar spectral irradiometer. (a) and (b) Probability Pij (λ) and spectral irradiance for two different ground reflectances. (c) Fractional contributions as a function of λ. (d) Distribution functions D4 , D5 , and D6 and their integrals as a function of d for λ = 450 nm.

necessary to store a significantly larger number of trajectories and a significantly longer computation time. IV. E XAMPLES OF R ESULTS As examples of results obtained with McCART, we report the simulated response of a sensor for measuring the radiance (Fig. 7) and the irradiance (Fig. 8). For Fig. 7, we have considered the same geometry of Fig. 4: we simulated the ˆ E and response for the pixel with viewing direction sˆP = n angular field-of-view semiaperture α = 10 mrad for a sensor at zp = 1.5 km and the sun at a zenith angle of 30◦ . Also, the size distributions used for evaluating the scattering functions are the same as those used for Fig. 4. The spectral transmittance that we used is shown in Fig. 7(a). Contributions due z to scattering, Ts(λ) = exp(− 0 TOA µs (λ)dz), and absorption, zTOA µa (λ)dz), are reported separately. Ta (λ) = exp(− 0 Fig. 7(b) shows the probability Pij (λ) (7) for a ground with constant reflectance ρ(λ) = 0.5 and for a grassland with the spectrum of reflectance shown in Fig. 7(c) [34]. Fig. 7(d) shows the corresponding at-sensor radiance obtained with the solar spectrum [35] shown in Fig. 7(c). Fig. 8 refers to a solar spectral irradiometer measuring the downward irradiance at ground level. The spectral transmittance of the atmosphere and the solar zenith angle are the same as those used in the simulation of Fig. 7. Fig. 8(a) and (b) shows the probability Pij (λ) and the spectral irradiance for two different ground reflectances, respectively: we considered ρ(λ) = 1 and the spectral reflectance of the grassland shown in Fig. 7(c). The results are significantly different for the two cases, showing that measurements of irradiance are significantly affected by scattered photons received after ground reflections. Differences are larger at shorter wavelengths where the turbidity of the atmosphere is higher. For the perfectly reflecting ground, ρ(λ) = 1, it is interesting to note that, due to contributions of

ground reflections, it results Pij (λ) > 1, even though part of the solar radiation is backscattered before reaching the Earth’s surface. Fig. 8(c) shows the fractional contributions Cn ’s, evaluated for the ground with ρ(λ) = 1, as a function of λ. For the spectral irradiometer, the contributions C2 and C3 are equal to zero. For C1 , the fraction of photons received without ground reflections, we distinguished the fractional contributions of photons received with and without scattering events (C1a and C1b , respectively). Fig. 8(d) shows the distribution functions D4 , D5 , and D6 (noisy curves) and their integrals as a function of d for λ = 450 nm. At this wavelength, τs = 0.9 and τa = 0. Comparison of Fig. 8(d) with Figs. 4(d) and 6(f) shows that the region that affects the measure of spectral irradiance is significantly more extended with respect to measurements of at-sensor radiance. Finally, we show an example of images generated by the McCART procedure. We considered a surface made up of alternately black and white pixels with reflectances (independent of the wavelength) ρ = 0 and ρ = 1, respectively, observed by a spacecraft sensor at zp = 600 km looking downward along the z-axis. The pixel was subtended by an angular field-of-view semiaperture α = 0.01 mrad, and the sun was at a zenith angle of 30◦ . The profiles of scattering and absorption coefficients that we assumed are similar to those shown in Figs. 3(a) and 5(a), and the scattering functions are those of Fig. 3(b). Fig. 9 shows six images simulated at λ = 450, 591, 760, 817, 1260, and 2054 nm. At these wavelengths, the optical thicknesses of the atmosphere due to scattering and absorption were τs = 2.5 and τa = 0, τs = 1.53 and τa = 0.01, τs = 0.98 and τa = 1.71, τs = 0.86 and τa = 0.049, τs = 0.38 and τa = 0.095, and τs = 0.16 and τa = 0.27, respectively. For a comparison, the figure also shows the image obtained for a transparent atmosphere (τs = τa = 0). The figures have been obtained, plotting with a linear grayscale the probability Pij that photons are received when

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Fig. 9. Example of images generated by the McCART procedure. A surface made up of alternately black and white pixels was observed by a spacecraft sensor at zp = 600 km looking downward along the z-axis with the sun at a zenith angle of 30◦ . The image expected for a transparent atmosphere (τs = τa = 0) has also been reported.

the ij-pixel is observed. For each figure, we used the same values for the minimum (black corresponds to zero) and the maximum (white corresponds to the response evaluated for the pixel with ρ = 1 when a transparent atmosphere is considered). The images at different wavelengths are significantly different: at λ = 450 nm, the image is dominated by photons received after multiple scattering in the atmosphere or multiple ground reflections (contributions C1 , C5 , and C6 ), and the response for the black pixel is only 10% lower than for the white pixel. As the wavelength increases, due to the significant decrease of τs , the contributions of photons received from the observed pixel without scattering events (contributions C2 and C3 ) become more and more important, and the contrast increases: as an example, at λ = 2054 nm, the response for the black pixel is only 10% of that for the white pixel. However, at some wavelengths, due to the strong absorption, the response is greatly reduced, and the image becomes dark. As an example, at 760 nm, the response is reduced by a factor ≈40 with respect to λ = 450 nm. V. D ISCUSSION AND C ONCLUSION McCART is a numerical procedure devoted to the evaluation of the effect of the atmosphere on hyperspectral images of the Earth’s surface and on measurements of solar radiation. The procedure is based on MC simulations and on scaling relationships. For MC simulations, the atmosphere, illuminated by the sun, is modeled as a nonabsorbing plane-parallel layered medium, and the Earth’s surface is modeled as a plane with uniform reflectance. The symmetry of the problem and the weak angular dependence of the radiance enable us to speed up the MC simulations: simulations can be carried out for an infinitely extended receiver with a large angular field of view. Thanks to scaling relationships from a single MC simulation, it is possible, in a short time, to reconstruct the spectral response on a wide range of wavelengths for a specific profile of optical

properties of the atmosphere and for a specific distribution of ground reflectance. The method allows one to estimate the a priori error on the various contributions to the received radiance, when the properties of the atmosphere are specified. The main advantage of the McCART procedure is the accuracy: McCART provides a solution of the RTE that suffers for the statistical fluctuations that are intrinsic to MC simulations. The simulated response includes the effects of adjacent pixels and trapping. Therefore, the code could be used to verify the accuracy of radiative transfer codes which are not able to accurately simulate trapping and/or adjacency effects when a user-defined reflectance map is given. In Section IV, we have shown examples of simulated spectral responses both for a sensor for remote measurements of reflectance of the Earth’s surface and for a solar irradiometer. Examples of simulated images have also been shown. The simulated images can be used to check the reliability of inversion procedures devoted to the retrieval of the spectral reflectance of the Earth’s surface. Another advantage of the McCART procedure is the possibility of obtaining detailed information on photon migration. As shown in Section IV, it is possible, for example, to evaluate separately the contributions due to different types of trajectories or the extension of the ground that affects the response. McCART can therefore be used as a tool for a deeper investigation of the effects of the atmosphere on measurements of solar radiation. The symmetries, utilized to speed up the MC simulation, are applicable only to a plane geometry and therefore represent a limit of the McCART procedure. The code does not consider the horizontal variability of some atmospheric gas constituents like water vapor. Only 2-D terrains can be considered; surface topography is not taken into account. Moreover, based on MC simulations, the main limit of the McCART procedure is the computation time. However, the use of scaling relationships and symmetry properties enabled us to drastically reduce the

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computation time compared to direct simulation MC approaches. When running the McCART code on a common personal computer (for example, a 3.2-GHz Pentium 4) to evaluate the at-sensor radiance, a few minutes is sufficient to run the MC simulation to obtain a sufficiently accurate result (all the results reported in Sections III and IV, referring to a 50-layer atmosphere, have been obtained with 105 useful trajectories). A few seconds is sufficient to process the trajectories to obtain the probability P , the fractional contributions Cn ’s, and the distribution functions Dn (d)’s. At present, the longer computation time to simulate an image is taken by the summations in (7), necessary to include the effects of adjacent pixels, which should be evaluated on an extremely large number of pixels. We expect a significant reduction of the computation time by developing dedicated software and using parallel processing. This would also make possible the use of the McCART procedure to develop algorithms of atmospheric compensation for an accurate retrieval of the spectral reflectance of the Earth’s surface. McCART provides a model that can rapidly estimate the effects of adjacency; therefore, it can be applied to retrieved hyperspectral surface reflectance data, showing the effect of adjacency modeling. McCART does not include thermal emission. Therefore, it can be applied from visible to short-wave infrared wavelengths. R EFERENCES [1] S. Chandrasekhar, Radiative Transfer. New York: Dover, 1960. [2] A. Ishimaru, Wave Propagation and Scattering in Random Media. New York: Academic, 1978. [3] E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image Transfer Through a Scattering Medium. Berlin, Germany: Springer-Verlag, 1991. [4] F. X. Kneizys, E. P. Shettle, L. W. Abreau, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, and S. A. Clough, “Users guide to LOWTRAN 7,” Air Force Geophys. Lab., Bedford, MA, Tech. Rep. AFGL-TR-8-0177, 1988. [5] A. Berk, G. P. Anderson, L. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, Jr., S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, and M. L. Hoke, “MODTRAN4 radiative transfer modeling for atmospheric correction,” in Proc. SPIE—Opt. Spectrosc. Tech. Instrum. Atmos. Space Res. III, M. Larar, Ed., 1999, vol. 3756, pp. 348–353. [6] K. Stamses, 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., vol. 27, no. 12, pp. 2502–2509, 1998. [7] E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, and J.-J. Morcette, “Second simulation of the satellite signal in the solar spectrum, 6S: An overview,” IEEE Trans. Geosci. Remote Sens., vol. 35, no. 3, pp. 675– 686, May 1997. [8] C. Miesch, L. Poutier, V. Achard, X. Briottet, X. Lenot, and Y. Boucher, “Direct and inverse radiative transfer solutions for visible and nearinfrared hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens., vol. 43, no. 7, pp. 1552–1562, Jul. 2005. [9] A. I. Lyapustin and Y. Wang, “Parameterized code SHARM-3D for radiative transfer over inhomogeneous surfaces,” Appl. Opt., vol. 44, no. 35, pp. 7602–7610, Dec. 2005. [10] A. I. Lyapustin, “Radiative transfer code SHARM for atmospheric and terrestrial applications,” Appl. Opt., vol. 44, no. 36, pp. 7764–7772, Dec. 2005. [11] T. Z. Muldashev, A. I. Lyapustin, and U. M. Sultangazin, “Spherical harmonics method in the problem of radiative transfer in the atmosphere–surface system,” J. Quant. Spectrosc. Radiat. Transf., vol. 61, no. 3, pp. 393–404, Feb. 1999. [12] A. Lyapustin and Y. Knyazikhin, “Green’s function method for the radiative transfer problem. I. Homogeneous non-Lambertian surface,” Appl. Opt., vol. 40, no. 21, pp. 3495–3501, Jul. 2001.

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[13] A. Lyapustin and Y. Knyazikhin, “Green’s function method in the radiative transfer problem. II. Spatially heterogeneous anisotropic surface,” Appl. Opt., vol. 41, no. 27, pp. 5600–5606, Sep. 2002. [14] K. F. Evans, “The spherical harmonics discrete ordinate method for threedimensional atmospheric radiative transfer,” J. Atmos. Sci, vol. 55, no. 3, pp. 429–446, Feb. 1998. [15] K. F. Evans and W. J. Wiscombe, “Improvements to the SHDOM radiative transfer modelling package,” in Proc. 13th ARM Sci. Team Meeting, Broomfield, CO, Mar. 31–Apr. 4 2003. [16] W. Boardman, “Post-ATREM polishing of AVIRIS apparent reflectance data using EFFORT: A lesson in accuracy versus precision,” in Proc. Summaries 7th JPL Airborne Earth Sci. Workshop, 1998, vol. 1, pp. 53–54. JPL Publication 97-21. [17] Atmosphere Removal Program (ATREM) User’s Guide, Version 3.1, Univ. Colorado, CSES, Boulder, CO, 1999. [18] M. W. Matthew, S. M. Adler-Golden, A. Berk, G. Felde, G. P. Anderson, D. Gorodestzky, S. Paswaters, and M. Shippert, “Atmospheric correction of spectral imagery: Evaluation of the FLAASH algorithm with AVIRIS data,” in Proc. SPIE—Algorithms Technol. Multispectral, Hyperspectral, Ultraspectral Imagery IX, S. S. Shen and P. E. Lewis, Eds., 2003, vol. 5093, pp. 474–482. [19] P. M. Teillet, “Surface reflectance retrieval using atmospheric correction algorithms,” in Proc. IGARSS 12th Can. Symp. Remote Sens., Vancouver, BC, Canada, 1989, pp. 864–867. [20] S. C. Richtsmeier, A. Berk, L. S. Bernstein, and S. M. Adler-Golden, “A 3-dimensional radiative-transfer hyperspectral image simulator for algorithm validation,” in Proc. Int. Symp. Spectr. Sens. Res., Quebec, QC, Canada, Jun. 10–15, 2001, pp. 537–545. [21] O. V. Postylyakov, “Linearized vector radiative transfer model MCC++ for a spherical atmosphere,” J. Quant. Spectrosc. Radiat. Transf., vol. 88, no. 1–3, pp. 297–317, Sep. 2004. [22] B. Mayer, “I3RC phase 1 results from the MYSTIC Monte Carlo model,” in Proc. I3RC: Abstract 1st/2nd Int. Workshops, Tucson, AZ, Nov. 17–19, 1999, pp. 49–54. [23] F. Spada, M. C. Krol, and P. Stammes, “McSCIA: Application of the equivalence theorem in a Monte Carlo radiative transfer model for spherical shell atmospheres,” Atmos. Chem. Phys., vol. 6, no. 12, pp. 4823–4842, 2006. [24] A. Battistelli, P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, “Use of two scaling relations in the study of multiple-scattering effects on the transmittance of light beams through a turbid atmosphere,” J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 2, no. 6, pp. 903–912, 1985. [25] A. Sassaroli, C. Blumetti, F. Martelli, L. Alianelli, D. Contini, A. Ismaelli, and G. Zaccanti, “Monte Carlo procedure for investigating light propagation and imaging of highly scattering media,” Appl. Opt., vol. 37, no. 31, pp. 7392–7400, 1998. [26] W. M. Irvine, “The formation of absorption bands and the distribution of photon optical paths in a scattering atmosphere,” Bull. Astron. Inst. Netherlands, vol. 17, pp. 266–279, 1963. [27] G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, “Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt., vol. 3, no. 5, pp. 897–905, Sep. 1994. [28] K. T. Whitby and B. Cantrel, “Atmospheric aerosols: Characteristics and measurement,” in Proc. Int. Conf. Environ. Sens. Assessment, Las Vegas, NV, Sep. 14–19, 1975, vol. 2, pp. 29-1–29-6. [29] D. Guzzi, M. Morandi, V. Santacesaria, L. Stefanutti, P. Agostini, B. Liley, and J. P. Wolf, “Four years of stratospheric aerosol measurements in the northern and southern hemispheres,” Geophys. Res. Lett., vol. 26, no. 14, pp. 2199–2202, 1999. [30] T. Deshler, M. E. Hervig, D. J. Hofmann, J. M. Rosen, and J. B. Liley, “Thirty years of in situ stratospheric aerosol size distribution measurements from Laramie, Wyoming(41◦ N), using balloon-borne instruments,” J. Geophys. Res.—Atmos., vol. 108, no. D5, pp. 4167–4178, 2003. [31] D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions. New York: Elsevier, 1969. [32] L. R. Poole, D. D. Venable, and J. W. Campbell, “Semianalytic Monte Carlo radiative transfer model for oceanographic lidar systems,” Appl. Opt., vol. 20, no. 20, pp. 3253–3656, 1981. [33] P. Bruscaglioni, G. Zaccanti, L. Pantani, and L. Stefanutti, “An approximate procedure to isolate single scattering contribution to lidar returns from fogs,” Int. J. Remote Sens., vol. 4, no. 2, pp. 399–417, 1983. [34] John Hopkins University Spectral Library, Vegetation Database, 2005. [Online]. Available: http://speclib.jpl.nasa.gov/archive/JHU/becknic/ vegetation/vegetation.txt [35] W. Zissis, The Infrared Handbook, W. Wolfe and J. Zissis, Eds. Washington, DC: Dept. Navy, Office Naval Res., 1978, ch. 3.

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Vanni Nardino received the B.S. degree in physics from the Università degli Studi di Firenze, Firenze, Italy, in 2004. He continued working in the research group of G. Zaccanti in the Physic Department of Florence, developing Monte Carlo algorithms for simulating the contribution of the atmosphere on hyperspectral remote sensing images. He is currently pursuing the Ph.D. degree at Pisa University, Pisa, Italy, with research about UV and visible ground irradiance estimation from satellite images. In 2005, he was with the research group of I. Pippi in the Istituto di Fisica Applicata “Nello Carrara,” Firenze, developing reflectance retrieval procedures based on numerical simulation of satellite images. He is currently working in the aerospace field, developing algorithms for satellite data processing, atmospheric parameter retrieval, and sensor calibration procedures.

Fabrizio Martelli received the Ph.D. degree from the University of Electro-communications (UEC), Tokyo, Japan in 2002, with a thesis on photon migration through homogeneous and layered diffusive media. He is currently with the Dipartimento di Fisica, Università degli Studi di Firenze, Firenze, Italy. His current major research interests include the modeling of light propagation through diffusive media and the development of inversion procedures to retrieve the optical properties of biological tissues and turbid media.

Piero Bruscaglioni received the Laurea degree in physics from the Università degli Studi di Firenze, Firenze, Italy, in 1956. He was an Assistant Professor in 1962 and an Associate Professor in 1987–2002 with the Università degli Studi di Firenze, teaching courses in electromagnetic wave theory and in physics such as basic acoustics. He is currently with the Dipartimento di Fisica, Università degli Studi di Firenze, conducting research on antenna models and propagation of light beams in turbid media, with models of multiple scattering and applications to lidar returns from atmospheric structures and to atmospheric blurring. In collaboration with the Dipartimento di Elettronica e Telecomunicazioni, Università degli Studi di Firenze, he has carried out studies on the analysis of human voice with regard to infant cries, and the effect of vocal tract diseases among infants, and to particular features of singing voices. He is currently responsible for the participation of the department at a European project on lidar returns from clouds, with particular regard to the presence of nonspherical scatterers, such as ice particles in the high troposphere.

Giovanni Zaccanti is currently a Researcher with the Dipartimento di Fisica, Università degli Studi di Firenze, Firenze, Italy. His research activity has been related to the field of light propagation through turbid media. He has developed models for light propagation both through the atmosphere and through biological tissues. He has also developed methodologies for measuring the optical properties of turbid media.

Samuele Del Bianco received the B.S. and M.S. degrees in physics from the Università degli Studi di Firenze, Firenze, Italy, where he is working toward the Ph.D. degree, studying several problems of photon migration through diffusive media both with a theoretical and with an experimental approach. He is currently with the Istituto di Fisica Applicata “Nello Carrara,” Consiglio Nazionale delle Ricerche, Firenze. His major research interest is related to the retrieval procedures of nonlinear problems.

Donatella Guzzi was born in Firenze, Italy, on May 2, 1963. She received the Laurea degree in physics from the Università degli Studi di Firenze, Firenze, in 1990. From 1991, she was with IROE-CNR for four years, working on fiber-optic sensors for environmental monitoring. From 1995, she worked on the scattering of light by particulate in atmosphere and in the analysis of lidar data at IROE-CNR with the LIDAR group of the institute. Her activities included analysis of the scattering properties of the atmospheric aerosols and clouds and their characterization. Since January 2001, she has been with the Aerospace High Resolution Optical Sensor Group, Istituto di Fisica Applicata “Nello Carrara,” Consiglio Nazionale delle Ricerche, Firenze. Her main activities include the implementation and development of algorithm for atmospheric corrections of remotely sensed data, the study of the propagation of radiation in the atmosphere, the development and calibration of aerospace high-resolution optical sensor, the validation of remotely sensed data by means of in-field measurements, and the integration of remotely sensed data with modeling and in situ canopy reflectance measurements for the carbon balance estimates in vegetation.

Paolo Marcoionni was born in Prato, Italy, in 1973. He received the Laurea degree in physics from the Università degli Studi di Firenze, Firenze, Italy, in 1999, and the Ph.D. degree in earth science from the University of Parma, Parma, Italy, in 2006. His research interests include hyperspectral remote sensing, inverse modeling of remotely sensed data, digital image processing, high-resolution interferometric imaging, and sensor characterization. Since 2006, he has been with ICL—Integrated Color Line srl, Italy, where he is involved in development of robots for industrial automation and quality-control spectrophotometric systems. He collaborates with the Istituto di Fisica Applicata “Nello Carrara,” Consiglio Nazionale delle Ricerche, where he participates in several research projects devoted to high-resolution remote sensing by aerospace imaging spectrometers.

Ivan Pippi was born in Firenze, Italy, in 1949. He received the Diploma in electronics from the Technical High School, Firenze, in 1968. From 1969 to 1970, he was with the Dipartimento di Fisica, Università degli Studi di Firenze, Firenze. Since 1970, he has been with the Istituto di Ricerca sulle Onde Elettromagnetiche, Consiglio Nazionale delle Ricerche (CNR), Firenze, first dealing with astrophysics research and then, since 1976, dealing with remote sensing techniques. He has been the Leader of the High Resolution Aerospace Optical Sensors Group, IFAC, CNR, since 1986, managing several national and international research projects mainly supported by the Italian and European Space Agencies. His research interest in remote sensing was first focused on laserradar development for meteorological studies and Earth observation. Then, he started studying the applications to environment monitoring of aerospace optical sensors operating in visible and infrared wavelengths. He has been participating in developing and characterizing several imaging spectrometers and interferometers and in their data calibration and validation through remote sensing campaigns performed on equipped test sites.

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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 6, JUNE 2008

McCART: Monte Carlo Code for Atmospheric Radiative Transfer Vanni Nardino, Fabrizio Martelli, Piero Bruscaglioni, Giovanni Zaccanti, Samuele Del Bianco, Donatella Guzzi, Paolo Marcoionni, and Ivan Pippi

Abstract—McCART is a numerical procedure to solve the radiative transfer equation for light propagation through the atmosphere from visible to near-infrared wavelengths. The procedure has been developed to study the effect of the atmosphere in the remote sensing of the Earth, using aerospace imaging spectrometers. The simulation is run for a reference layered plane nonabsorbing atmosphere and a plane ground with uniform reflectance. For a given distribution of ground reflectance and for a specific profile of scattering and absorption properties of the atmosphere, the spectral response of the sensor is obtained in a short time from the results of the Monte Carlo simulation by using scaling relationships and symmetry properties. The procedure also includes an accurate analysis of the adjacency and trapping effects due to multiple scattering of photons coming from neighboring pixels. McCART can generate synthetic images of the Earth’s surface for arbitrary viewing conditions. The results can be used to establish the limits of applicability of approximate algorithms for the processing and analysis of hyperspectral images acquired by imaging spectrometers. In addition, the algorithm can be used to develop procedures for atmospheric correction for the accurate retrieval of the spectral ground reflectance. Index Terms—Aerosols, atmospheric propagation, multilayered media, remote sensing, scattering.

I. I NTRODUCTION

M

EASUREMENTS of remote sensing of the Earth at visible and near-infrared wavelengths are greatly influenced by the turbidity of the atmosphere. Scattering and absorption by aerosol and gases highly complicate light propagation compared with the idealized case of a perfectly transparent atmosphere. As a consequence, the relationship between the measured signal and the physical parameters of interest becomes very complicated, and the relevant information can be retrieved only if a suitable inversion procedure is available. The first step in developing a reliable inversion procedure is the availability of an efficient solver of the forward problem, i.e., an algorithm that, given the characteristics of the light source and of the receiver and the optical properties of the ground and of the atmosphere, enables us to determine the response of the receiver. Light propagation through turbid media is described by the radiative transfer equation (RTE) [1]–[3]. Several radiative transfer codes are used to find solutions of the RTE. These Manuscript received July 3, 2007; revised November 19, 2007. V. Nardino, F. Martelli, P. Bruscaglioni, and G. Zaccanti are with the Dipartimento di Fisica, Università degli Studi di Firenze, 50019 Sesto Fiorentino, Italy. S. Del Bianco, D. Guzzi, P. Marcoionni, and I. Pippi are with the Istituto di Fisica Applicata “Nello Carrara,” Consiglio Nazionale delle Ricerche, 50019 Sesto Fiorentino, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TGRS.2008.916464

codes are based on an explicit solution of the RTE or on a Monte Carlo (MC) approach to the radiative transfer problem. The first ones are currently used in atmospheric radiation applications like MODTRAN4 [4], [5] which uses the DISORT [6] radiative transfer code as a subroutine for the implementation of the azimuth dependence of multiple scattering, 6S which solves the RTE in an iterative way using successive orders of scattering [7], COMANCHE [8] which uses an analytical formulation of the upwelling radiance, SHARM-3-D [9], [10] which uses the method of spherical harmonics [11]–[13], and SHDOM which uses the spherical harmonics discrete ordinate method [14], [15]. These codes use different levels of sophistication when modeling complex effects, like radiation trapping between the ground and atmosphere, and neighboring pixel adjacency effects. For example, MODTRAN uses a single spatially homogeneous spectral reflectance to model the surface surrounding the imaged pixel. Both MODTRAN and 6S are used by several model-based methods (ATREM [16], [17], FLAASH [18], and ATCOR [19]) to generate lookup tables for atmospheric-effect correction. Similarly, McCART uses MODTRAN to generate molecular absorption profiles. Models based on an MC approach [20]–[23] solve the RTE, using probabilistic modeling of the associated radiative transfer processes. These methods are flexible, and their selfconsistency may be predicted by examining the variance between estimates made from subsets of the simulation. The main goal of this paper is to describe a methodology based on MC simulations, the McCART procedure, which, making use of scaling relations that greatly reduce the computation time, enables us to evaluate in a reasonable time the spectral at-sensor radiance for a specific distribution of reflectance of the Earth’s surface. The McCART procedure has been developed to evaluate the effect of a user-defined atmosphere on aerospace multispectral and hyperspectral images of the Earth. The direct problem is solved: the procedure is able to obtain the at-sensor radiance, given the spectral surface reflectance distribution [Lambertian or characterized by a bidirectional reflectance distribution function (BRDF)]. The model, described in Section II, is based on a numerical MC simulation carried out for a nonabsorbing reference atmosphere with layered plane-parallel structure and for a nonabsorbing plane Earth surface. Taking advantage of the translational symmetry of the problem and of the weak dependence of the at-sensor radiance on the viewing direction, a large number of useful trajectories are drawn and stored in a reasonable time

0196-2892/$25.00 © 2008 IEEE

NARDINO et al.: McCART: MONTE CARLO CODE FOR ATMOSPHERIC RADIATIVE TRANSFER

Fig. 1. (a) Geometrical scheme and symbols. (b) Examples of trajectories of photons contributing to the at-sensor radiance.

even with a personal computer. From the stored trajectories, the effect of a specific profile of absorption and scattering properties of the atmosphere on the response is evaluated in a short time, making use of Beer’s law scaling relationships. The response is divided into six contributions according to the number of reflections at the Earth’s surface and of scattering events in the atmosphere. Finally, for a specific distribution of spectral reflectance of the ground surface, the response is obtained, properly weighting each contribution according to the reflectance of each pixel. Therefore, the response includes an accurate analysis of adjacency and trapping effects with respect to radiative transfer codes like MODTRAN and 6S. Because McCART does not include thermal emission, it can be applied from visible to short-wave infrared wavelengths. The range of applicability is discussed in Section III, whereas examples of results predicted by the procedure are reported in Section IV, where discussions of the obtained results are given. Conclusions are drawn in Section V. II. D ESCRIPTION OF THE A LGORITHM A. General Remarks The goal of the McCART procedure is to simulate the response of a pixel of a spectral image collected by an airborne system observing the Earth’s surface that is illuminated by the sun. More specifically, we are interested in evaluating the response of the pixel R(λ, rP ) = KL(λ, rP , sˆP )ΩP cos θP

(1)

from which the at-sensor radiance L(λ, rP , sˆP ) can be obtained. In (1), λ denotes the wavelength; K is the relative instrumental spectral response; rP , sˆP , and ΩP are the position, the viewing direction, and the acceptance solid angle of the sP · n ˆ R ; and n ˆ R is the normal pixel, respectively; cos θP = −ˆ to the surface of the optical receiver [Fig. 1(a)]. For a perfectly transparent atmosphere and a plane Earth surface, the at-sensor radiance would be given by L(λ, rP , sˆP ) = S(λ)

cos θS ρ(λ, rE )pE (λ, sˆS , sˆP ) cos θP

(2)

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sS · where S(λ) is the spectral solar irradiance, cos θS = −ˆ ˆ E is the n ˆ E , sˆS is the direction of the solar incident light, n normal to the Earth’s surface, ρ(λ, rE ) is the spectral reflectance of the observed pixel at rE , and pE (λ, sˆ , sˆ) is the distribution function, normalized to one over the solid angle 2π, describing the angular distribution of the light reflected by the ground that is illuminated by light in direction sˆ related to the BRDF. For example, for a Lambertian plane ˆ E /π. In this idealized consurface, it is pE (λ, sˆ , sˆ) = sˆ · n dition, L(λ, rP , sˆP ) only depends on the reflectance of the observed pixel and on the geometry of observation. As shown in Fig. 1(b), the problem is highly complicated by the turbidity of the atmosphere because, due to manifold scattering events in the atmosphere and/or to ground reflections, L(λ, rP , sˆP ) becomes a very complicated function both of the optical properties of the atmosphere and of the distribution of reflectance on a large surface around the observed pixel. The procedure we have developed to evaluate the response of the pixel is based on an MC simulation. We point out that the elementary MC procedure, i.e., a numerical simulation of the trajectories of the photons that illuminate the top of the atmosphere, which propagate through the atmosphere as far as they are absorbed (by the ground or by the atmosphere), they escape from the atmosphere, or they are eventually received, would be inapplicable due to the prohibitively long computation time. In fact, it would be necessary to process an extremely large number of photons to simulate the illumination of the vast surface around the pixel of interest from which, due to multiple scattering, photons can reach the receiver. Furthermore, also due to the small solid angle subtended by the pixel, the probability that the photon is received is extremely small, and because the optical properties of both the atmosphere and ground depend on the wavelength, it would be necessary to repeat the simulation for each value of λ. To speed up the convergence of the MC procedure, we need to introduce some simplifications in order to make possible the use of symmetry properties and scaling relationships. We have therefore assumed a plane Earth surface and a plane layered atmosphere uniformly illuminated by a plane wave. The McCART procedure can be summarized as follows. 1) An elementary MC simulation is run only once for a nonabsorbing horizontally homogeneous reference atmosphere and a ground with constant reflectance ρ = 1. With these assumptions, the response only depends on the height at which the sensor is placed and on the solar and viewing geometry, and translational properties can be used to reduce the computation time. All the trajectories of received photons (typically 105 ) are stored on a file, together with the number of launched photons NL . 2) Making use of scaling relationships, from the stored information, the effect of the atmosphere with a specific profile of scattering and absorption on the response of the pixel is first evaluated for a ground with uniform reflectance ρ = 1. 3) Received photons are divided into six contributions according to the number of scattering events in the atmosphere and of ground reflections. Moreover, photons

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are also classified according to the distance of the point of ground reflection from the center of the observed pixel to evaluate radial distribution functions. 4) With the information from points 2) and 3), the at-sensor radiance for a specific distribution of reflectance ρ(λ, rE ) can easily be obtained. Steps 2), 3), and 4) take short computation time and are repeated for each wavelength of interest. The longer computation time is taken by the MC simulation. In fact, the acceptance solid angle of the pixel is very small (specifically for spacecraft systems), so the probability of finding a useful trajectory is extremely small and a very large number of trajectories should be simulated to obtain statistically reliable results. However, because the radiance L(λ, rP , sˆP ) for a uniform surface is usually a slowly varying function of sˆP , the MC simulation for the reference atmosphere can be carried out for a large acceptance solid angle ΩP , and the response for the solid angle ΩP can be obtained, scaling the results for ΩP . B. Elementary MC Simulation The elementary MC simulation is carried out for a nonabsorbing plane-parallel layered atmosphere and for a plane ground with uniform reflectance ρ = 1. The kth layer is characterized by the scattering coefficient µsk and by the scalar scattering function pk (θ). Instead of directly simulating the atmosphere uniformly illuminated by the sun (infinitely extended source) and a very small receiver (a pointlike detector), for which the probability of receiving a photon is prohibitively small, we first simulate the complementary problem, i.e., a pointlike source and an infinitely extended receiver. In this case, the probability of receiving a photon is much higher because all photons intersecting the plane z = zP at which the receiver is placed, within the acceptance solid angle of the pixel, are received. The modeled geometry is shown in Fig. 1(a). All photons are injected at the top of the atmosphere at the point P0 = (0, 0, zT OA ) along the direction sˆS of the solar radiation. The photon path is traced as a series of discrete steps. Each step is determined stochastically by using standard MC rules starting from random numbers ξ between zero and unity: The length li of the path between two subsequent scattering events Pi , Pi+1 , the scattering angle θi , and the azimuth angle ϕi are calculated as follows: li (ξ) = z (zi+1 − zi )/ cos αi with zii+1 µs (z)/ cos αi dz = − ln(ξ) and αi angle between the direction of the photon and the z-axis, θ (ξ) 2π 0 i p(θ ) sin θ dθ = ξ, and ϕi (ξ) = 2πξ. User-defined aerosol models are utilized for the evaluation of the scattering properties of the atmosphere. The scattering functions are evaluated starting from the size distribution of aerosols. At each step, the following checking steps are made. 1) If the trajectory has crossed the Earth’s surface (plane z = 0), Pi+1 is posed equal to the point of intersection and the trajectory gone on drawing a new direction and a new path length. The new direction (θE , ϕE ) is drawn s , sˆ). Examples according to the distribution function pE (ˆ reported in Section III refer to a Lambertian surface,

ˆ E /π. In this case, for which pE (λ, sˆ , sˆ) = sˆ · n √ θE and ϕE are obtained as follows: cos θE (ξ) = 1 − ξ and ϕE = 2πξ. 2) If the trajectory has crossed the plane z = zP at which the sensor is placed, the angle between the direction of the photon and the viewing direction sˆP is calculated. If the angle is within the acceptance solid angle of the pixel, the intersection Pout = (xout , yout , zP ) with the plane z = zP is calculated, the trajectory P0 , P1 , . . . , Pout , after a translation, is stored, and the counter of received photons is increased. Subsequently, the trajectory goes on from the point Pi+1 until the photon leaves the atmosphere. Each emitted photon can therefore yield more than one contribution, according to the possibility for a photon of contributing several times to the radiance in different points on the plane z = zP . This point is insignificant when McCART is used to evaluate the atsensor radiance measured by a sensor with a small acceptance angle, but it becomes crucial when McCART is used, for instance, to evaluate the irradiance measured by a solar spectral irradiometer (an example is shown in Section IV). 3) If the trajectory has crossed the top of the atmosphere (plane z = zTOA ), the photon is given up, and a new photon is launched. The simulated trajectories refer to the pencil light beam and to the infinitely extended receiver. Due to the symmetry of the problem, these trajectories can also be used to solve the problem of an infinitely extended source and of a pointlike detector. In fact, because the optical properties of the atmosphere and of the ground are assumed independent of the x and y coordinates, two parallel trajectories have the same probability. Thus, for each simulated received trajectory P0 , P1 , . . . , Pout with P0 = (0, 0, zTOA ) and Pout = (xout , yout , zP ), there is a parallel , obtained trajectory with the same probability, P0 , P1 , . . . , Pout with the translation x = x − xout and y = y − yout , for which = (0, 0, zP ). With this P0 = (−xout , −yout , zTOA ) and Pout translation applied to each received trajectory, we obtain the corresponding received trajectories for the infinitely extended source and the pointlike receiver. Thanks to this symmetry, we obtain a statistically representative set of received trajectories in a reasonable computation time. In principle, the simulation should be carried out for each pixel within the field of view of the receiver. However, because, for a surface with uniform reflectance, the radiance is a slowly varying function of the viewing direction (see Section III-C), the MC results obtained for a specific viewing direction sˆP are also applicable to the pixels within a cone of several degrees around sˆP . C. Scaling Relationships for µa and µs To evaluate the effect of the atmosphere on the response of the pixel when the profile of scattering and absorption for a specific wavelength λ is considered, we first calculate the probability P (λ) that the photon is received for the ground with ρ = 1. For the nonabsorbing reference atmosphere, the probability is simply given by Pref = NR /NL with a relative error

NARDINO et al.: McCART: MONTE CARLO CODE FOR ATMOSPHERIC RADIATIVE TRANSFER

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√ εr = 1/ NR , where NR is the number of received photons and NL is the number of launched photons. For an atmosphere with scattering and absorption coefficients µsk (λ) and µak (λ), the probability changes because the probability of each trajectory changes. To evaluate the new probability, we used two scaling relationships previously used to study light propagation through the atmosphere [24] and through biological tissues [25]. To each trajectory, a weight wj (λ) is given, equal to the ratio between the probability of following the trajectory after and before changing the optical properties of the atmosphere, and P (λ) is obtained as P (λ) =

NR

wj (λ)/NL .

(3)

j=1

The weight can be expressed as wj = waj wsj , with waj and wsj weights being due to changes of absorption and scattering, respectively. For absorption, according to the Lambert–Beer law, the weight of the jth trajectory is K L µak ljk (4) waj = exp − k=1

where µak is the absorption coefficient of the kth layer, ljk is the total length followed by the jth photon into the kth layer, and KL is the number of layers into which the atmosphere has been divided. For a fixed wavelength grid, the absorption coefficients are calculated using a MODTRAN4-based band model derived directly from the spectral line parameters of the HITRAN96 database [26]. If the absorption coefficient has small variations within the wavelength interval, the weight can be evaluated using, in (4), the average value for µak . On the contrary, if strong variations occur, due, for instance, to a narrow absorption band, the interval must be properly sampled, and the average value of the weight must be calculated. The exact knowledge of the path length followed by each received photon allows us to include the effect of absorption. The weight wsj is obtained, remembering that, in a nonabsorbing medium, the probability for a photon of undergoing a scattering event within a volume element subtended by dl at distance l from the previous one, and being scattered with an angle θ within the solid angle dω, is given by µs dl exp(−µs l)p(θ)dω. Therefore, if the scattering coefficient of the layers is changed from µsk to µsk , without changing the scattering functions, the weight of the photon is given by KL njk µsk exp [−(µsk − µsk ) ljk ] (5) wsj = µsk k=1

where njk is the number of scattering events undergone by the photon inside the kth layer. Equations (4) and (5) are exact scaling relationships, provided that the scattering function remains constant when the scattering coefficient is changed. However, the statistical fluctuations due to the finite number of simulated trajectories limit their range of applicability. The range of applicability will be discussed in Section III.

Fig. 2. Classification of received photons. Six different types of trajectories are distinguished. For each type, examples of trajectories are shown.

D. Contributions to the Response and Distribution Functions To evaluate the effect of the ground on the response of the pixel when a specific distribution of reflectance is considered, the weight of each trajectory wj should be multiplied by the reflectance of the points in which the photon has been reflected. This could be made, but if many pixels should be processed, as is the case for simulating spectral images of a ground with known reflectance, or for processing images to reconstruct the map of spectral ground reflectance, a long computation time would be necessary. The response of the pixel can be obtained in a shorter time using average information instead of the information carried by each trajectory separately. For this purpose, the contribution to the response calculated for ρ = 1 should be divided according to the type of trajectory followed by the photon, and the radial distribution functions describing the probability that received photons have been reflected at distance d from the center of the observed pixel should be evaluated. When the trajectories are processed to apply the scaling relationships, their weights wj ’s are accumulated separately according to the type of trajectory. As shown in Fig. 2, we have distinguished six different types of trajectories, and for each type, we have evaluated the fractional contributions Cn ’s(n = 1, 6)to the probability P (λ). C1 is the fraction of photons received without ground reflections. Contributions C2 − C5 refer to photons reflected only once: C2 is the fraction of photons received without scattering events in the atmosphere; C3 is the fraction of scattered photons that illuminate the observed pixel and are received without scattering events between the ground and the receiver; C4 is the fraction of received photons that, after the ground reflection, have been scattered only once; and C5 is the fraction of photons that have undergone multiple scattering after the ground reflection. Contribution C6 is the fraction of photons received after multiple ground reflections. This contribution is further divided according to the number of reflections: C6,m is the fraction of photons reflected m times. We point out that each photon is classified in one and only one contribution. Contributions C2 and C3 produce a clear image of the observed pixel. The other contributions blur the image: contribution C1 comes from the brightness of the atmosphere, whereas contributions C4 , C5 , and C6 come from the

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surrounding pixels and are responsible for the adjacency effects. Contributions C4 and C5 have been distinguished in order to make possible the use of an empirical scaling relationship for the angular field of view (see Section III-C). The distribution functions Dn (d)’s describing the probability that received photons have been reflected at distance d from the center of the observed pixel have been evaluated separately for contributions Cn ’s, with n = 4, 5, and 6. The plane z = 0 has been divided into rings concentric with the observed pixel, and for each contribution, a histogram has been obtained, adding the weights of photons reflected within each ring. The distribution function has been obtained as j wji (6) Dn (di ) = Cn π [(di + ∆d/2)2 − (di − ∆d/2)2 ] where the summation is on photons reflected at distance (di − ∆d/2) ≤ d < (di + ∆d/2). For the contribution C6 , referring to photons with multiple reflections, we have evaluated only one distribution. Photons have been classified using, for each trajectory, the average distance dj . In principle, the distribution function for each m of C6,m could be evaluated, but a significant increase in the computation time would be necessary to process a sufficiently large number of photons in order to obtain statistically reliable results. E. At-Sensor Radiance If the ground is divided into pixels and ρij is the reflectance of the ij-pixel, the probability Pij that photons are received when the ij-pixel is observed can be obtained as Pij = P C1 + (C2 + C3 )ρij + C4 D4 (di j )ρi j AP i j

+C5

D5 (di j )ρi j AP +

i j

m

C6m ρ

(7)

m

cos θS 1 . cos θP ΩP

A. Tests on the Drawn Trajectories The algorithm to draw the trajectories has been checked by comparisons with exact analytical and numerical results. In particular, to check the correctness of the algorithm to draw the scattering points, we have run the MC code for a homogeneous infinite atmosphere, and we have evaluated the first and second moments of the coordinates of the points at which the different orders of scattering occur. The numerical results have been compared with exact analytic relationships [27], and we have obtained an excellent agreement: Discrepancies were within the standard deviation of MC results even when a large number of trajectories (108 ) were simulated and the error was as small as 0.01%. To check the correctness of the algorithm used for the ground reflectance, we have considered a transparent atmosphere and compared the MC results with (2) for a different solar and viewing geometry. Other tests have been made to check that, as expected by the RTE, for a nonabsorbing atmosphere with the scattering function that is independent of the height, and a ground with ρ = 1, the probability P is independent of the profile of scattering coefficients, provided that the optical thickness z TOA

τs =

µs (z)dz 0

where di j is the distance between the ij- and the i j -pixels, AP is the area of the pixel, and ρ is the average reflectance over an area around the pixel from which photons with multiple reflections are received. The area is estimated from the distribution function D6 (d). We point out that P , Cn , Dn (d), ρij , ρ, and, thus, Pij are functions of λ. Once the probability Pij has been evaluated for the wavelengths of interest, the at-sensor radiance for the ij-pixel can be obtained as Lij (λ, sˆP ) = S(λ)Pij (λ)

In fact, with the MC method, the RTE can be solved without the need of simplifying assumptions, and the accuracy of the results is only limited by the statistical fluctuations. However, developing an MC code in order to simulate the response of hyperspectral sensors for remote sensing is not trivial at all, and checks should be made to control its correctness. This is particularly important for the McCART procedure in which scaling relationships and symmetry properties are widely used.

(8)

III. T ESTS ON THE M C CART P ROCEDURE AND A S CALING R ELATIONSHIP FOR THE A NGULAR F IELD OF V IEW The results of MC simulations are often used as a gold standard to check the accuracy of approximate solutions of the RTE.

zP τsP =

µs (z)dz

(9)

0

remain constant. Furthermore, comparisons have been made for the distribution function D4 (d) referring to photons reflected by the ground, which have been received after a single scattering event. Because the ground is uniformly illuminated, the reference distribution can be easily calculated with a numerical integration [an example of comparisons will be shown in Fig. 6(e)]. All the comparisons gave excellent agreement. B. Applicability Range of the Scaling Relationships Until the scattering functions can be considered independent of µs and µa , the scaling relationships presented in Section II are exact relationships, but their range of applicability is limited by statistical noise due to the finite number of simulated trajectories and needs to be checked. In fact, starting from a set of useful trajectories predetermined by the elementary MC simulation, the relative error εr on the results increases when the range of values that the weighting factors can assume increases. When the absorption coefficient is varied, the weight of the photon only depends on the length of the trajectory inside

NARDINO et al.: McCART: MONTE CARLO CODE FOR ATMOSPHERIC RADIATIVE TRANSFER

Fig. 3. Scattering properties of the atmosphere used for numerical simulations. (a) Reference profile of scattering coefficient. (b) Scattering functions.

each layer. In any case, 0 ≤ waj ≤ 1, and reliable results can be obtained for a large range of conditions. When the scattering coefficient is varied, the weight strongly depends also on the number of scattering events. This involves larger variations of the weighting factor, and the noise on the scaled results can rapidly increase, particularly for µsk > µsk . The range of applicability of the scaling relationships has been investigated, with simulations carried out for idealized models of atmosphere. To explore the scaling relationship for the scattering coefficient, we have started from a profile of µs , which is typical of a rural midlatitude summer atmosphere with relative humidity of 70% and visibility at 23 km, at λ = 450 nm [Fig. 3(a)] [4], [5]. The optical thickness due to scattering is τs = 0.71. The MC simulations have been repeated for 11 profiles of scattering coefficient obtained, multiplying the one of Fig. 3(a) by constant factors. We have considered values of τs ranging from 0 to 3. The scattering functions that we have used are shown in Fig. 3(b): we have used a scattering function that is typical of a clear atmosphere [28] for the lower layers (z < 12 km), of stratospheric aerosols for 12 < z < 35 km [29], [30], and the Rayleigh scattering function for the higher layers (z > 35 km). The scattering functions have been obtained from Mie theory starting from the size distribution of the aerosols for the wavelength λ = 450 nm. The atmosphere was divided into 50 layers. We considered a nonabsorbing atmosphere and a sensor at zp = 1.5 km looking downward along the z-axis. We simulated ˆE the response for the pixel with viewing direction sˆP = n (Fig. 1) and angular field-of-view semiaperture α = 10 mrad. The sun was at a zenith angle of 30◦ . Fig. 4(a) shows the comparison between the results obtained for the probability P from the 11 direct MC simulations (symbols) and the results obtained from the scaling relationship (5) starting from the MC results for τs = 1.5 (continuous lines). For the scaled results, the error bars are also reported (for the direct simulations, the error was smaller than the symbols). Discrepancies between the scaled results and the results of direct simulations are larger than one standard deviation only for τs > 2.5. Fig. 4(b) shows the comparison for the corresponding fractional contributions Cn ’s(n = 1, 6). Also, for Cn , there is an excellent agreement for τs < 2.5. Examples of results for the distribution functions Dn (d)’s are shown in Fig. 4(c). We focused our attention on the distributions D4 and D5 , the ones explicitly used in (7) to evaluate the response Rij . The figures show the comparisons between the results obtained from the direct MC simulations

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at τs = 0.9 and those obtained scaling from τs = 1.5. Comparisons show that the scaled results are in excellent agreement with the results of the direct MC simulations. Anyway, the MC results showed that the distribution functions slightly depend on τs . To better visualize the extension of the region around the observed pixel that influences the measured at-sensor radiance, in Fig. 4(d), we have shown the integrated distributions d 2π 0 Dn (d )d dd as a function of d. This figure shows that the integrated distributions reach the value of 0.8 for d ∼ = 1, 5, and 12 km for D4 , D5 , and D6 , respectively, indicating that ∼ =20% of the contributions C4 , C5 , and C6 comes from pixels at distances d > 1, 5, and 12 km, respectively, from the observed pixel. The size of the pixel is 30 m. The results of Fig. 4 show that, from a single MC simulation, the scaling relationship for the scattering coefficient enables us to obtain reliable results on a wide range of values of τs that probably include the values of practical interest for remote sensing of the Earth’s surface. In fact, in Fig. 4(b), we note that, for τs = 2.5, the contributions C2 and C3 (those of photons that are useful for the image) are less than 2% and 20%, respectively, of total received photons. We want to point out that the results of Fig. 4 refer to an idealized nonabsorbing atmosphere with scattering functions that are independent of the optical thickness τs and to a Lambertian surface with uniform reflectance ρ = 1. To investigate the range of applicability of the scaling relationship for the absorption coefficient, we have started from the MC simulation carried out for the nonabsorbing atmosphere with the profile of scattering coefficient and the scattering functions shown in Fig. 3. The reference profile of absorption that we considered is shown in Fig. 5(a). It is a profile, which is typical of a rural midlatitude summer atmosphere with relative humidity of 70%, at λ = 2500 nm [4], [5]. The optical thickness due to absorption is τa = 0.73. The scaling relationship for absorption has been used to simulate the response for values of τa between 0 and 10. The profiles of absorption have been obtained, multiplying the one of Fig. 5(a) by constant factors. The geometry is the same as used in the simulation of Fig. 4. Examples of results are shown in Fig. 5. Fig. 5(b) and (c) shows the probability P and the fractional contributions Cn ’s as a function of τa (the error bars are not shown because the error was smaller than the symbols apart from the results for τa > 5). The results of Fig. 5(c) show that when τa increases, the contribution of multiple reflections, C6 , rapidly decreases, and particularly, the contribution of photons received without ground reflections, C1 , rapidly increases: For τa > 10 is C1 ∼ = 1, indicating that received photons are only due to atmospheric backscattering and that no useful information on the ground reflectance can be obtained. Fig. 5(d) shows examples of results for the distribution function D4 (d). As τa increases, the distribution function becomes sharper, indicating that the surface around the observed pixel that affects the at-sensor radiance significantly decreases. A significantly larger effect has been observed for D5 and D6 . The results of Figs. 4 and 5 show that, from a single MC simulation, the results for a wide range of values of scattering and absorption coefficients can be obtained by using the scaling relationships. We can therefore expect, from a single

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Fig. 4. Comparisons between the results of direct MC simulations and the results from the scaling relationship for the scattering coefficient. (a) Probability P and (b) fractional contributions Cn ’s as a function of τs . Symbols: Results of direct MC simulations. Continuous lines: Results from the scaling relationship d Dn (d )d dd as a function of d. applied to the MC results for τs = 1.5. (c) Distribution functions D4 and D5 for τs = 0.9. (d) Integrated distributions 2π 0

Fig. 5. Examples of results obtained with the scaling relationship for the absorption coefficient. (a) Reference profile of absorption coefficient used. (b) and (c) Probability P and fractional contributions Cn ’s as a function of τa . (d) Examples of results for the distribution function D4 (d). The geometry and the scattering properties are the same as for Fig. 4. τs = 0.9.

MC simulation, that we can reconstruct the at-sensor spectral response for a wide range of wavelengths. However, when the scaling relationships are used to evaluate the spectral response, we must point out that the scaling relationship for scattering, (5), is valid, provided that the scattering function does not appreciably vary when µs is changed. Because both µs and p(θ) vary with the wavelength, it is necessary to study how the response changes with the scattering function. This point has been investigated, repeating MC simulations for the scattering functions obtained using the same size distributions for the aerosols assumed for Fig. 3 but different values of λ between 450 and 2500 nm. Comparisons showed that sim-

ulations performed for a small number of wavelengths, for which the scattering functions can be considered constant for each step of calculation, are sufficient to reconstruct the spectral response without any interpolation procedure and with an excellent accuracy over the whole range of wavelengths. For the considered cases, characterized by smooth aerosol phase functions, only three wavelengths (450, 600, and 800 nm) proved to be sufficient (simulation at 450 nm scaled between 400 and 550 nm, simulation at 600 nm scaled between 550 and 750 nm, whereas simulation at 800 nm scaled between 750 and 2500 nm). Absorption and scattering coefficients are referred to a user-defined grid. The results obtained from a

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Fig. 6. Dependence of MC results on the angular field-of-view semiaperture α. (a) and (b) Averaged probability P/(ˆ sP · n ˆ R Ωp ) and fractional contributions Cn ’s as a function of α. (c) and (d) Distribution functions D4 and D5 for some values of α. (e) Examples of comparisons reported between MC predictions and results from numerical integration for D4 . (f) Examples of integrated distributions D4 and D5 for some values of α. The geometry and the scattering properties are the same as for Fig. 4. τs = 0.9 and τa = 0.

single MC simulation at 450 nm are more approximated but probably sufficiently accurate for many applications. C. Scaling Relationship for the Angular Field of View The longer computation time in using the McCART procedure is taken by the MC simulation for the reference atmosphere. As an example, the simulation for τs = 1.5 and α = 10 mrad used for Fig. 4 has taken about 1 h on a 3.2-GHz Pentium 4 personal computer. The simulation referred to a sensor at zp = 1.5 km. The angular field-of-view semiaperture of the pixel was α = 10 mrad, and the size of the corresponding ground pixel was 30 m. The computation time strongly depends on α. In fact, because, for the uniform surface with reflectance ρ = 1 for which the elementary MC simulation is carried out, the radiance is a slowly varying function of the viewing direction sˆP , the probability P comes about proportional to the acceptance solid angle ΩP ∼ = 2πα2 /2 and the computation time approximately proportional to 1/α2 . Therefore, the computation time to simulate a sensor with a very small value of α, as for spacecraft sensors for which α can be a few tens of microradians, becomes prohibitively long. We therefore explored the possibility of obtaining the response, scaling the results of MC simulations carried out for a large value of α. For this, we studied how the probability P , the

fractional contributions Cn ’s, and the distribution functions Dn ’s vary with α. Examples of results are shown in Fig. 6. We considered a nonabsorbing atmosphere with the scattering properties of Fig. 3 and a uniform Earth surface with ρ = 1. The geometry is the same as for Fig. 4. Fig. 6(a) and (b) shows the averaged probability P/(cos θP Ωp ) (the average of cos θP is calculated over the solid angle Ωp ) and the fractional contributions Cn ’s as a function of α. Fig. 6(c) and (d) shows the distribution functions D4 and D5 for some values of α. Fig. 6(a) shows that P/(cos θP Ωp ) remains almost constant over a large range of values of α, indicating that the radiance is a slowly varying function of the direction sˆP . Furthermore, Fig. 6(b) shows that also the fractional contributions Cn ’s are almost independent of α. This is true even for the contributions C1 and C4 , the ones that are more strictly related to the shape of the scattering function. These results indicate that the at-sensor radiance for a small angular aperture can be obtained from an MC simulation for a large value of α. The distribution functions D4 and D5 have a significantly different dependence on the angular field of view: D5 weakly depends on α (curves for α ≤ 100 mrad are almost indistinguishable). On the contrary, D4 is strongly affected: It is nearly constant for d < zP tan α, i.e., over the area of the observed pixel, and has a high peak for α → 0. This dependence should be taken into account when data simulated for a certain value of α are used to predict

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Fig. 7. Simulated spectral response of a sensor for measuring the radiance. The geometry is the same as for Fig. 4. (a) Spectral transmittance that we used; the contributions of scattering and absorption are reported separately. (b) Probability Pij (λ) for a ground with constant reflectance ρ(λ) = 0.5 and for a grassland with the spectral reflectance shown in (c). (c) Spectral reflectance for grassland and spectrum of the solar irradiance used to evaluate the (d) at-sensor radiance.

the results for different values of α. Fortunately, as mentioned in Section III-A, the distribution function D4 can be easily calculated with a numerical integration. In Fig. 6(e), examples of comparison between results obtained from MC simulations and by numerical integration are shown. The excellent agreement also gives a test on the correctness of the MC procedure. Finally, we observe that, in spite of the strong differences observed for small values of d, particularly for the distribution function D4 , small differences are observed when the integrated distributions are considered, indicating that the angular aperture that subtends the observed pixel has a little influence on the extension of the surrounding region that affects the at-sensor radiance. Examples of comparisons are shown in Fig. 6(f) for both D4 and D5 . To investigate the effect of the scattering function on the range of applicability of the scaling relationship for the angular field of view, we repeated the calculations using, for the lower layers of the atmosphere (z < 12 km), a scattering function with a significantly higher forward peak with respect to the one of Fig. 3(b): we used the scattering function of the C1 cloud model [31] at 450 nm for which p(θ = 0) = 324 and the asymmetry factor is g = 0.86. The results that we obtained showed that the scattering function significantly affects the redistribution of photons among different fractional contributions Cn ’s, but again, the fractional contributions are almost independent of the angular field of view over a wide range of values of α. Similar results have been obtained for the distribution functions Dn ’s: they are significantly affected by the scattering function (as the forward peak increases, the distributions, particularly D4 and D5 , become sharper, indicating that the surface around the observed pixel that affects the at-sensor radiance significantly decreases), but the dependence of D5 and D6 on α remains weak. In conclusion, the at-sensor radiance can be precisely evaluated, even for extremely small values of the angular field of

view, by using the results of an MC simulation carried out for a value of α as large as 100 mrad, provided that the distribution function D4 is evaluated by numerical integration. For α = 100 mrad and τs = 1.5, 105 useful trajectories can be obtained with a computation time of about 1 min. The weak dependence of the radiance on the viewing direction enables us to use the MC results obtained for the viewing direction sˆP to evaluate the response for the pixels within a cone of several degrees around sˆP . A way to speed up the convergence of the MC simulation may be the use of a semianalytic MC procedure [32], [33]. With the semianalytic approach, trajectories are simulated by using the same rules as for the elementary MC simulation, and each time an atmospheric or surface scattering event occurs, the probability that it would result in a photon entering the receiver is evaluated. Thus, for the geometry investigated, every scattering event below the altitude of the receiver contributes and provides information for the specific viewing geometry. With the elementary MC simulation, the response is obtained, summing up the few contributions of the received photons, all with the same weight. On the contrary, with the semianalytic approach, the response is obtained, summing up many small contributions with different weights. Because the probability is proportional to the product of the scattering function toward the sensor by the transmittance to the sensor, the range of values that it can assume increases as the optical thickness of the atmosphere increases or the scattering function becomes more forward peaked. To obtain the same convergence, it is therefore necessary to sum up a significantly larger number of contributions. For a small value of α and a moderate optical thickness, the convergence of a semianalytic MC simulation is significantly faster with respect to an elementary MC simulation. However, to apply the scaling relationships to the results of a semianalytic MC simulation, it would be necessary to change the weight of each probability. It would be therefore

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Fig. 8. Simulated response of a solar spectral irradiometer. (a) and (b) Probability Pij (λ) and spectral irradiance for two different ground reflectances. (c) Fractional contributions as a function of λ. (d) Distribution functions D4 , D5 , and D6 and their integrals as a function of d for λ = 450 nm.

necessary to store a significantly larger number of trajectories and a significantly longer computation time. IV. E XAMPLES OF R ESULTS As examples of results obtained with McCART, we report the simulated response of a sensor for measuring the radiance (Fig. 7) and the irradiance (Fig. 8). For Fig. 7, we have considered the same geometry of Fig. 4: we simulated the ˆ E and response for the pixel with viewing direction sˆP = n angular field-of-view semiaperture α = 10 mrad for a sensor at zp = 1.5 km and the sun at a zenith angle of 30◦ . Also, the size distributions used for evaluating the scattering functions are the same as those used for Fig. 4. The spectral transmittance that we used is shown in Fig. 7(a). Contributions due z to scattering, Ts(λ) = exp(− 0 TOA µs (λ)dz), and absorption, zTOA µa (λ)dz), are reported separately. Ta (λ) = exp(− 0 Fig. 7(b) shows the probability Pij (λ) (7) for a ground with constant reflectance ρ(λ) = 0.5 and for a grassland with the spectrum of reflectance shown in Fig. 7(c) [34]. Fig. 7(d) shows the corresponding at-sensor radiance obtained with the solar spectrum [35] shown in Fig. 7(c). Fig. 8 refers to a solar spectral irradiometer measuring the downward irradiance at ground level. The spectral transmittance of the atmosphere and the solar zenith angle are the same as those used in the simulation of Fig. 7. Fig. 8(a) and (b) shows the probability Pij (λ) and the spectral irradiance for two different ground reflectances, respectively: we considered ρ(λ) = 1 and the spectral reflectance of the grassland shown in Fig. 7(c). The results are significantly different for the two cases, showing that measurements of irradiance are significantly affected by scattered photons received after ground reflections. Differences are larger at shorter wavelengths where the turbidity of the atmosphere is higher. For the perfectly reflecting ground, ρ(λ) = 1, it is interesting to note that, due to contributions of

ground reflections, it results Pij (λ) > 1, even though part of the solar radiation is backscattered before reaching the Earth’s surface. Fig. 8(c) shows the fractional contributions Cn ’s, evaluated for the ground with ρ(λ) = 1, as a function of λ. For the spectral irradiometer, the contributions C2 and C3 are equal to zero. For C1 , the fraction of photons received without ground reflections, we distinguished the fractional contributions of photons received with and without scattering events (C1a and C1b , respectively). Fig. 8(d) shows the distribution functions D4 , D5 , and D6 (noisy curves) and their integrals as a function of d for λ = 450 nm. At this wavelength, τs = 0.9 and τa = 0. Comparison of Fig. 8(d) with Figs. 4(d) and 6(f) shows that the region that affects the measure of spectral irradiance is significantly more extended with respect to measurements of at-sensor radiance. Finally, we show an example of images generated by the McCART procedure. We considered a surface made up of alternately black and white pixels with reflectances (independent of the wavelength) ρ = 0 and ρ = 1, respectively, observed by a spacecraft sensor at zp = 600 km looking downward along the z-axis. The pixel was subtended by an angular field-of-view semiaperture α = 0.01 mrad, and the sun was at a zenith angle of 30◦ . The profiles of scattering and absorption coefficients that we assumed are similar to those shown in Figs. 3(a) and 5(a), and the scattering functions are those of Fig. 3(b). Fig. 9 shows six images simulated at λ = 450, 591, 760, 817, 1260, and 2054 nm. At these wavelengths, the optical thicknesses of the atmosphere due to scattering and absorption were τs = 2.5 and τa = 0, τs = 1.53 and τa = 0.01, τs = 0.98 and τa = 1.71, τs = 0.86 and τa = 0.049, τs = 0.38 and τa = 0.095, and τs = 0.16 and τa = 0.27, respectively. For a comparison, the figure also shows the image obtained for a transparent atmosphere (τs = τa = 0). The figures have been obtained, plotting with a linear grayscale the probability Pij that photons are received when

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Fig. 9. Example of images generated by the McCART procedure. A surface made up of alternately black and white pixels was observed by a spacecraft sensor at zp = 600 km looking downward along the z-axis with the sun at a zenith angle of 30◦ . The image expected for a transparent atmosphere (τs = τa = 0) has also been reported.

the ij-pixel is observed. For each figure, we used the same values for the minimum (black corresponds to zero) and the maximum (white corresponds to the response evaluated for the pixel with ρ = 1 when a transparent atmosphere is considered). The images at different wavelengths are significantly different: at λ = 450 nm, the image is dominated by photons received after multiple scattering in the atmosphere or multiple ground reflections (contributions C1 , C5 , and C6 ), and the response for the black pixel is only 10% lower than for the white pixel. As the wavelength increases, due to the significant decrease of τs , the contributions of photons received from the observed pixel without scattering events (contributions C2 and C3 ) become more and more important, and the contrast increases: as an example, at λ = 2054 nm, the response for the black pixel is only 10% of that for the white pixel. However, at some wavelengths, due to the strong absorption, the response is greatly reduced, and the image becomes dark. As an example, at 760 nm, the response is reduced by a factor ≈40 with respect to λ = 450 nm. V. D ISCUSSION AND C ONCLUSION McCART is a numerical procedure devoted to the evaluation of the effect of the atmosphere on hyperspectral images of the Earth’s surface and on measurements of solar radiation. The procedure is based on MC simulations and on scaling relationships. For MC simulations, the atmosphere, illuminated by the sun, is modeled as a nonabsorbing plane-parallel layered medium, and the Earth’s surface is modeled as a plane with uniform reflectance. The symmetry of the problem and the weak angular dependence of the radiance enable us to speed up the MC simulations: simulations can be carried out for an infinitely extended receiver with a large angular field of view. Thanks to scaling relationships from a single MC simulation, it is possible, in a short time, to reconstruct the spectral response on a wide range of wavelengths for a specific profile of optical

properties of the atmosphere and for a specific distribution of ground reflectance. The method allows one to estimate the a priori error on the various contributions to the received radiance, when the properties of the atmosphere are specified. The main advantage of the McCART procedure is the accuracy: McCART provides a solution of the RTE that suffers for the statistical fluctuations that are intrinsic to MC simulations. The simulated response includes the effects of adjacent pixels and trapping. Therefore, the code could be used to verify the accuracy of radiative transfer codes which are not able to accurately simulate trapping and/or adjacency effects when a user-defined reflectance map is given. In Section IV, we have shown examples of simulated spectral responses both for a sensor for remote measurements of reflectance of the Earth’s surface and for a solar irradiometer. Examples of simulated images have also been shown. The simulated images can be used to check the reliability of inversion procedures devoted to the retrieval of the spectral reflectance of the Earth’s surface. Another advantage of the McCART procedure is the possibility of obtaining detailed information on photon migration. As shown in Section IV, it is possible, for example, to evaluate separately the contributions due to different types of trajectories or the extension of the ground that affects the response. McCART can therefore be used as a tool for a deeper investigation of the effects of the atmosphere on measurements of solar radiation. The symmetries, utilized to speed up the MC simulation, are applicable only to a plane geometry and therefore represent a limit of the McCART procedure. The code does not consider the horizontal variability of some atmospheric gas constituents like water vapor. Only 2-D terrains can be considered; surface topography is not taken into account. Moreover, based on MC simulations, the main limit of the McCART procedure is the computation time. However, the use of scaling relationships and symmetry properties enabled us to drastically reduce the

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computation time compared to direct simulation MC approaches. When running the McCART code on a common personal computer (for example, a 3.2-GHz Pentium 4) to evaluate the at-sensor radiance, a few minutes is sufficient to run the MC simulation to obtain a sufficiently accurate result (all the results reported in Sections III and IV, referring to a 50-layer atmosphere, have been obtained with 105 useful trajectories). A few seconds is sufficient to process the trajectories to obtain the probability P , the fractional contributions Cn ’s, and the distribution functions Dn (d)’s. At present, the longer computation time to simulate an image is taken by the summations in (7), necessary to include the effects of adjacent pixels, which should be evaluated on an extremely large number of pixels. We expect a significant reduction of the computation time by developing dedicated software and using parallel processing. This would also make possible the use of the McCART procedure to develop algorithms of atmospheric compensation for an accurate retrieval of the spectral reflectance of the Earth’s surface. McCART provides a model that can rapidly estimate the effects of adjacency; therefore, it can be applied to retrieved hyperspectral surface reflectance data, showing the effect of adjacency modeling. McCART does not include thermal emission. Therefore, it can be applied from visible to short-wave infrared wavelengths. R EFERENCES [1] S. Chandrasekhar, Radiative Transfer. New York: Dover, 1960. [2] A. Ishimaru, Wave Propagation and Scattering in Random Media. New York: Academic, 1978. [3] E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image Transfer Through a Scattering Medium. Berlin, Germany: Springer-Verlag, 1991. [4] F. X. Kneizys, E. P. Shettle, L. W. Abreau, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, and S. A. Clough, “Users guide to LOWTRAN 7,” Air Force Geophys. Lab., Bedford, MA, Tech. Rep. AFGL-TR-8-0177, 1988. [5] A. Berk, G. P. Anderson, L. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, Jr., S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, and M. L. Hoke, “MODTRAN4 radiative transfer modeling for atmospheric correction,” in Proc. SPIE—Opt. Spectrosc. Tech. Instrum. Atmos. Space Res. III, M. Larar, Ed., 1999, vol. 3756, pp. 348–353. [6] K. Stamses, 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., vol. 27, no. 12, pp. 2502–2509, 1998. [7] E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, and J.-J. Morcette, “Second simulation of the satellite signal in the solar spectrum, 6S: An overview,” IEEE Trans. Geosci. Remote Sens., vol. 35, no. 3, pp. 675– 686, May 1997. [8] C. Miesch, L. Poutier, V. Achard, X. Briottet, X. Lenot, and Y. Boucher, “Direct and inverse radiative transfer solutions for visible and nearinfrared hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens., vol. 43, no. 7, pp. 1552–1562, Jul. 2005. [9] A. I. Lyapustin and Y. Wang, “Parameterized code SHARM-3D for radiative transfer over inhomogeneous surfaces,” Appl. Opt., vol. 44, no. 35, pp. 7602–7610, Dec. 2005. [10] A. I. Lyapustin, “Radiative transfer code SHARM for atmospheric and terrestrial applications,” Appl. Opt., vol. 44, no. 36, pp. 7764–7772, Dec. 2005. [11] T. Z. Muldashev, A. I. Lyapustin, and U. M. Sultangazin, “Spherical harmonics method in the problem of radiative transfer in the atmosphere–surface system,” J. Quant. Spectrosc. Radiat. Transf., vol. 61, no. 3, pp. 393–404, Feb. 1999. [12] A. Lyapustin and Y. Knyazikhin, “Green’s function method for the radiative transfer problem. I. Homogeneous non-Lambertian surface,” Appl. Opt., vol. 40, no. 21, pp. 3495–3501, Jul. 2001.

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[13] A. Lyapustin and Y. Knyazikhin, “Green’s function method in the radiative transfer problem. II. Spatially heterogeneous anisotropic surface,” Appl. Opt., vol. 41, no. 27, pp. 5600–5606, Sep. 2002. [14] K. F. Evans, “The spherical harmonics discrete ordinate method for threedimensional atmospheric radiative transfer,” J. Atmos. Sci, vol. 55, no. 3, pp. 429–446, Feb. 1998. [15] K. F. Evans and W. J. Wiscombe, “Improvements to the SHDOM radiative transfer modelling package,” in Proc. 13th ARM Sci. Team Meeting, Broomfield, CO, Mar. 31–Apr. 4 2003. [16] W. Boardman, “Post-ATREM polishing of AVIRIS apparent reflectance data using EFFORT: A lesson in accuracy versus precision,” in Proc. Summaries 7th JPL Airborne Earth Sci. Workshop, 1998, vol. 1, pp. 53–54. JPL Publication 97-21. [17] Atmosphere Removal Program (ATREM) User’s Guide, Version 3.1, Univ. Colorado, CSES, Boulder, CO, 1999. [18] M. W. Matthew, S. M. Adler-Golden, A. Berk, G. Felde, G. P. Anderson, D. Gorodestzky, S. Paswaters, and M. Shippert, “Atmospheric correction of spectral imagery: Evaluation of the FLAASH algorithm with AVIRIS data,” in Proc. SPIE—Algorithms Technol. Multispectral, Hyperspectral, Ultraspectral Imagery IX, S. S. Shen and P. E. Lewis, Eds., 2003, vol. 5093, pp. 474–482. [19] P. M. Teillet, “Surface reflectance retrieval using atmospheric correction algorithms,” in Proc. IGARSS 12th Can. Symp. Remote Sens., Vancouver, BC, Canada, 1989, pp. 864–867. [20] S. C. Richtsmeier, A. Berk, L. S. Bernstein, and S. M. Adler-Golden, “A 3-dimensional radiative-transfer hyperspectral image simulator for algorithm validation,” in Proc. Int. Symp. Spectr. Sens. Res., Quebec, QC, Canada, Jun. 10–15, 2001, pp. 537–545. [21] O. V. Postylyakov, “Linearized vector radiative transfer model MCC++ for a spherical atmosphere,” J. Quant. Spectrosc. Radiat. Transf., vol. 88, no. 1–3, pp. 297–317, Sep. 2004. [22] B. Mayer, “I3RC phase 1 results from the MYSTIC Monte Carlo model,” in Proc. I3RC: Abstract 1st/2nd Int. Workshops, Tucson, AZ, Nov. 17–19, 1999, pp. 49–54. [23] F. Spada, M. C. Krol, and P. Stammes, “McSCIA: Application of the equivalence theorem in a Monte Carlo radiative transfer model for spherical shell atmospheres,” Atmos. Chem. Phys., vol. 6, no. 12, pp. 4823–4842, 2006. [24] A. Battistelli, P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, “Use of two scaling relations in the study of multiple-scattering effects on the transmittance of light beams through a turbid atmosphere,” J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 2, no. 6, pp. 903–912, 1985. [25] A. Sassaroli, C. Blumetti, F. Martelli, L. Alianelli, D. Contini, A. Ismaelli, and G. Zaccanti, “Monte Carlo procedure for investigating light propagation and imaging of highly scattering media,” Appl. Opt., vol. 37, no. 31, pp. 7392–7400, 1998. [26] W. M. Irvine, “The formation of absorption bands and the distribution of photon optical paths in a scattering atmosphere,” Bull. Astron. Inst. Netherlands, vol. 17, pp. 266–279, 1963. [27] G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, “Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt., vol. 3, no. 5, pp. 897–905, Sep. 1994. [28] K. T. Whitby and B. Cantrel, “Atmospheric aerosols: Characteristics and measurement,” in Proc. Int. Conf. Environ. Sens. Assessment, Las Vegas, NV, Sep. 14–19, 1975, vol. 2, pp. 29-1–29-6. [29] D. Guzzi, M. Morandi, V. Santacesaria, L. Stefanutti, P. Agostini, B. Liley, and J. P. Wolf, “Four years of stratospheric aerosol measurements in the northern and southern hemispheres,” Geophys. Res. Lett., vol. 26, no. 14, pp. 2199–2202, 1999. [30] T. Deshler, M. E. Hervig, D. J. Hofmann, J. M. Rosen, and J. B. Liley, “Thirty years of in situ stratospheric aerosol size distribution measurements from Laramie, Wyoming(41◦ N), using balloon-borne instruments,” J. Geophys. Res.—Atmos., vol. 108, no. D5, pp. 4167–4178, 2003. [31] D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions. New York: Elsevier, 1969. [32] L. R. Poole, D. D. Venable, and J. W. Campbell, “Semianalytic Monte Carlo radiative transfer model for oceanographic lidar systems,” Appl. Opt., vol. 20, no. 20, pp. 3253–3656, 1981. [33] P. Bruscaglioni, G. Zaccanti, L. Pantani, and L. Stefanutti, “An approximate procedure to isolate single scattering contribution to lidar returns from fogs,” Int. J. Remote Sens., vol. 4, no. 2, pp. 399–417, 1983. [34] John Hopkins University Spectral Library, Vegetation Database, 2005. [Online]. Available: http://speclib.jpl.nasa.gov/archive/JHU/becknic/ vegetation/vegetation.txt [35] W. Zissis, The Infrared Handbook, W. Wolfe and J. Zissis, Eds. Washington, DC: Dept. Navy, Office Naval Res., 1978, ch. 3.

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Vanni Nardino received the B.S. degree in physics from the Università degli Studi di Firenze, Firenze, Italy, in 2004. He continued working in the research group of G. Zaccanti in the Physic Department of Florence, developing Monte Carlo algorithms for simulating the contribution of the atmosphere on hyperspectral remote sensing images. He is currently pursuing the Ph.D. degree at Pisa University, Pisa, Italy, with research about UV and visible ground irradiance estimation from satellite images. In 2005, he was with the research group of I. Pippi in the Istituto di Fisica Applicata “Nello Carrara,” Firenze, developing reflectance retrieval procedures based on numerical simulation of satellite images. He is currently working in the aerospace field, developing algorithms for satellite data processing, atmospheric parameter retrieval, and sensor calibration procedures.

Fabrizio Martelli received the Ph.D. degree from the University of Electro-communications (UEC), Tokyo, Japan in 2002, with a thesis on photon migration through homogeneous and layered diffusive media. He is currently with the Dipartimento di Fisica, Università degli Studi di Firenze, Firenze, Italy. His current major research interests include the modeling of light propagation through diffusive media and the development of inversion procedures to retrieve the optical properties of biological tissues and turbid media.

Piero Bruscaglioni received the Laurea degree in physics from the Università degli Studi di Firenze, Firenze, Italy, in 1956. He was an Assistant Professor in 1962 and an Associate Professor in 1987–2002 with the Università degli Studi di Firenze, teaching courses in electromagnetic wave theory and in physics such as basic acoustics. He is currently with the Dipartimento di Fisica, Università degli Studi di Firenze, conducting research on antenna models and propagation of light beams in turbid media, with models of multiple scattering and applications to lidar returns from atmospheric structures and to atmospheric blurring. In collaboration with the Dipartimento di Elettronica e Telecomunicazioni, Università degli Studi di Firenze, he has carried out studies on the analysis of human voice with regard to infant cries, and the effect of vocal tract diseases among infants, and to particular features of singing voices. He is currently responsible for the participation of the department at a European project on lidar returns from clouds, with particular regard to the presence of nonspherical scatterers, such as ice particles in the high troposphere.

Giovanni Zaccanti is currently a Researcher with the Dipartimento di Fisica, Università degli Studi di Firenze, Firenze, Italy. His research activity has been related to the field of light propagation through turbid media. He has developed models for light propagation both through the atmosphere and through biological tissues. He has also developed methodologies for measuring the optical properties of turbid media.

Samuele Del Bianco received the B.S. and M.S. degrees in physics from the Università degli Studi di Firenze, Firenze, Italy, where he is working toward the Ph.D. degree, studying several problems of photon migration through diffusive media both with a theoretical and with an experimental approach. He is currently with the Istituto di Fisica Applicata “Nello Carrara,” Consiglio Nazionale delle Ricerche, Firenze. His major research interest is related to the retrieval procedures of nonlinear problems.

Donatella Guzzi was born in Firenze, Italy, on May 2, 1963. She received the Laurea degree in physics from the Università degli Studi di Firenze, Firenze, in 1990. From 1991, she was with IROE-CNR for four years, working on fiber-optic sensors for environmental monitoring. From 1995, she worked on the scattering of light by particulate in atmosphere and in the analysis of lidar data at IROE-CNR with the LIDAR group of the institute. Her activities included analysis of the scattering properties of the atmospheric aerosols and clouds and their characterization. Since January 2001, she has been with the Aerospace High Resolution Optical Sensor Group, Istituto di Fisica Applicata “Nello Carrara,” Consiglio Nazionale delle Ricerche, Firenze. Her main activities include the implementation and development of algorithm for atmospheric corrections of remotely sensed data, the study of the propagation of radiation in the atmosphere, the development and calibration of aerospace high-resolution optical sensor, the validation of remotely sensed data by means of in-field measurements, and the integration of remotely sensed data with modeling and in situ canopy reflectance measurements for the carbon balance estimates in vegetation.

Paolo Marcoionni was born in Prato, Italy, in 1973. He received the Laurea degree in physics from the Università degli Studi di Firenze, Firenze, Italy, in 1999, and the Ph.D. degree in earth science from the University of Parma, Parma, Italy, in 2006. His research interests include hyperspectral remote sensing, inverse modeling of remotely sensed data, digital image processing, high-resolution interferometric imaging, and sensor characterization. Since 2006, he has been with ICL—Integrated Color Line srl, Italy, where he is involved in development of robots for industrial automation and quality-control spectrophotometric systems. He collaborates with the Istituto di Fisica Applicata “Nello Carrara,” Consiglio Nazionale delle Ricerche, where he participates in several research projects devoted to high-resolution remote sensing by aerospace imaging spectrometers.

Ivan Pippi was born in Firenze, Italy, in 1949. He received the Diploma in electronics from the Technical High School, Firenze, in 1968. From 1969 to 1970, he was with the Dipartimento di Fisica, Università degli Studi di Firenze, Firenze. Since 1970, he has been with the Istituto di Ricerca sulle Onde Elettromagnetiche, Consiglio Nazionale delle Ricerche (CNR), Firenze, first dealing with astrophysics research and then, since 1976, dealing with remote sensing techniques. He has been the Leader of the High Resolution Aerospace Optical Sensors Group, IFAC, CNR, since 1986, managing several national and international research projects mainly supported by the Italian and European Space Agencies. His research interest in remote sensing was first focused on laserradar development for meteorological studies and Earth observation. Then, he started studying the applications to environment monitoring of aerospace optical sensors operating in visible and infrared wavelengths. He has been participating in developing and characterizing several imaging spectrometers and interferometers and in their data calibration and validation through remote sensing campaigns performed on equipped test sites.

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