Remote Sensing Amartis V2: 3d Radiative Transfer Code in the [0.4; 2.5 Μm] Spectral Domain Dedicated to Urban Areas

The availability of new very high spatial resolution sensors has for the past few years allowed a precise description of urban areas, and thus the settlement of specific ground or atmosphere characterization methods. However, in order to develop such techniques, a radiative transfer tool dedicated to such an area is necessary. AMARTIS v2 is a new radiative transfer code derived from the radiative transfer code AMARTIS specifically dedicated to urban areas. It allows to simulate airborne and spaceborne multiangular observations of 3D scenes in the [0.4; 2.5µm] domain with the ground's geometry, urban materials optical properties, atmospheric modeling and sensor characteristics entirely defined by the user. After a general presentation of AMARTIS v2 and a description of the performed calculations, results of radiometric intercomparisons with other radiative transfer codes are presented and the new offered potentials are illustrated with four realistic examples, representative of current issues in urban areas remote sensing.

Remote Sens

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Unfortunately, it is limited by the description of the landscape, defined by a 2D profile which is infinitely reproduced in the third dimension.Secondly, each urban surface, like roofs, walls and roads, is considered as uniform with unique optical properties.Furthermore, the aerosol distribution can only be described by a Junge law [27] whereas Thomas et al. [28] has shown the high variability of urban aerosols radiative properties.Therefore, in order to overcome these limitations a new radiative transfer code derived from AMARTIS v1 has been developed, taking into account a real 3D description of the landscape, each scene element being defined by its spectral bidirectional reflectance.The atmosphere modeling has also been improved with the addition of new aerosol models.This tool is also now able to perform the simulation and the visualization of synthetic remote sensing images.
This paper aims to present this new tool: AMARTIS v2.In a first step, a description of the code is given (Section 2), then a comparison of its radiometric performances with other radiative transfer codes is detailed (Section 3).Finally, its new potentialities are illustrated with four examples: study of the radiative transfer for an urban canyon (Section 4.1), analysis of the impact of highly reflective windows in a shadowed canyon street (Section 4.2), comparison of the signal coming from an irradiated area and those from a shaded area in the case of a complex landscape (case of a crossroad, Section 4.3), and finally, evaluation of the directional effects induced on the at-sensor level signal by an urban canyon when using broader resolution at a street scale (Section 4.4).Conclusions and perspectives are then discussed (Section 5).

General Description
AMARTIS v2 [29] aims to remove the drawbacks of the previous code AMARTIS v1.Its main functionalities, i.e., scene description, sensor and atmosphere characteristics and radiative transfer modeling, are discussed below.
This code performs monochromatic radiative transfer computations in the [0.4; 2.5 µm] spectral domain.Its inputs are the geometry and the materials optical properties of a 3D scene, the atmospheric properties and the viewing and irradiating conditions.It computes all the radiative components of the signal, both at ground level (irradiance) and at sensor level (radiance) at every point of the landscape.Thus, it can simulate the radiance image of the scene acquired by the sensor.The atmospheric radiative properties are modeled in AMARTIS v2 thanks to the radiative transfer code 6S [19].The aerosols can be modeled by the standard models of 6S, by their physical properties (with notably Junge or multimodal distributions) or directly by their optical properties (spectral variation of the optical thicknesses, single scattering albedos and phase functions).The gaseous atmosphere can be modeled by the standard models of 6S or by the ozone and water vapor contents.

Sensor
AMARTIS v2 allows the simulation of airborne or satellite sensors.The sensor is defined by the following parameters: its zenith and azimuth viewing angles defined by the optical axis orientation pointed at the centre of the scene, its pixels matrix (number of pixels by rows and columns and pixel size), its spatial resolution, the wavelengths of observation (AMARTIS v2 simulates monochromatic observations), and the focal length of the instrument.The altitude of the sensor is deduced from the previous geometrical parameters.The modeled instrument has a perfect signal-to-noise ratio corresponding to no instrumental noise.

Method Description
In remote sensing, a flat ground assumption is usually made to model the signal at ground and sensor levels.However, in cities, at very high spatial resolution, this hypothesis is no longer valid because of the complexity introduced by the relief which induced specific radiative effects.Thus, it becomes necessary to use a new formalism adapted to those areas.The signal at ground and at sensor levels is the result of several radiative components as described in Figure 3 ( [24,25]).
The irradiance at ground level (I tot ) is the sum of four components (Figure 3(a)): the direct irradiance (I dir ), the scattered irradiance (I scat ), the Earth-atmosphere coupling irradiance (I coup ) and the downward reflected irradiance (I refl ).The radiance at sensor level (R tot ) is the sum of three components (Figure 3(b)): the direct radiance (R dir ), the environment radiance (R env ) and the atmospheric radiance (R atm ).
The formalism and the radiative calculations performed in AMARTIS v2 are the same as in AMARTIS v1, except for I scat and R atm .
Their expressions are now detailed but to avoid complications, the wavelength dependence has been omitted in the formulations.
The direct irradiance corresponds to the photons directly coming from the sun, and is defined for a point P at ground level by [33] to the photons directly transmitted to the ground after reflections from the neighborhood.The Earth-atmosphere coupling irradiance cannot be expressed by an analytical formulation because of the complex geometry of the scene.This is the reason why this component is computed by means of Monte Carlo methods [25].The downward reflected irradiance might be calculated by an analytical method.This has not been selected as the Earth-atmosphere coupling computation simultaneously calculates the downward reflected irradiance.This Monte Carlo principle consists of modeling the radiometric flux by analyzing the propagation of a high number of photons (usually ~10 9 to 10 10 per simulation) in the Earth-atmosphere system.The phenomena of scattering and absorption by the atmosphere and of reflection and absorption by the ground are modeled by statistical laws [25].As the Earth-atmosphere coupling irradiance and the downward reflected irradiance result from interactions with the ground, the neighborhood of the scene has an impact on the signal.As it is not modeled by the 3D scene, this impact is calculated by duplicating virtually the original scene.A comparable approach was implemented in AMARTIS v1 [33].As the ray-tracing was not implemented, a method by photon propagation was used.Now, the introduction of the ray-tracer RayBooster (http://www.hpc-sa.com)has considerably improved the computation time.
The direct radiance corresponds to the photons reflected by the surface directly toward the sensor.It is defined by: where dir dir R , scat dir R , and refl coup dir R _ are respectively the contributions to the direct radiance resulting from the direct irradiance, the scattered irradiance and the coupling and reflected irradiances; i Ω is the solid angle corresponding to the field of view of the pixel i; ω d S is the surface at ground level seen by this pixel in the direction of ω d ; ) , ( ϕ θ is the radiance scattered by the sky in the direction defined by the zenith angle θ and azimuth angleϕ (in W.m −2 .µm−1 .sr−1 ); is the direct transmission of the atmosphere between ω d S and the sensor; dd ρ is the bidirectional reflectance of the corresponding material; and hd ρ its hemispheric-directional reflectance.
The environment radiance, R env , corresponds to the photons coming from the surface and scattered by the atmosphere.As for the Earth-atmosphere coupling irradiance, this term cannot be modeled by an analytic expression and is also solved with Monte Carlo calculations [25].
The atmospheric radiance, R atm , corresponds to the photons that have been scattered by the atmosphere without reaching the ground and is directly calculated thanks to the radiative transfer code 6S.Thus, AMARTIS v2 allows the computation of the incident irradiances to the ground as described above (I dir , I scat , I coup and I refl ) and of the sum of those components (I tot ) for each facet of the scene.Whereas irradiances are calculated for each facet, radiances are calculated for each pixel.In order to represent these different irradiances at the sensor resolution and to model the at sensor radiances, the different irradiances incident to a given facet must be aggregated.Each irradiance type incident to a ground pixel is the sum of the corresponding irradiance type incident to each facet included in this ground pixel and weighted by the solid angle defined by this facet.The sum of these solid angles equals the instantaneous field of view of a pixel.
The radiances at sensor level (R dir , R atm and R env ) are directly calculated for each pixel and their sum will simulate the synthetic image acquired by the sensor (R tot ).Therefore, all the radiative components of the signal can be represented on different images.
To perform these calculations, the following atmospheric radiative terms are directly derived from

Comparison of AMARTIS v2 with Existing Radiative Transfer Codes
The potentialities of AMARTIS v2 are unique.Indeed, it allows both the use of complex 3D scenes and a large modeling of the atmosphere.This is the reason why the comparison between AMARTIS v2 and existing radiative transfer codes should be limited.Nevertheless, major efforts have been done in order to estimate its performances.A comparison has been performed, first with AMARTIS v1 to assess they provide the same results on simple cases, and then with 6S to validate the atmospheric modeling.

Comparison with AMARTIS v1
AMARTIS v2 is first compared with AMARTIS v1 to check the coherence of those two codes.As AMARTIS v1 can only take into account simple 3D scenes and is limited in atmospheric modeling, the comparison procedure is restrained to conditions AMARTIS v1 can simulate.
This comparison is performed for the different radiative components of the signal.However, different atmospheric scattering models are used in AMARTIS v1 and v2.Indeed, in AMARTIS v1, a Gauss-Seidel model [34] is used whereas in AMARTIS v2, the 6S kernel used performs different analytical calculations based on the successive orders of scattering [19].Thus, the scattered irradiance and the atmospheric radiance are not compared.
Two different synthetic landscapes are chosen (Figure 4): an urban canyon and a mountainous scene.The modeled grounds are considered as lambertian with spectrally constant reflectances ρ.For each scene, two different reflectance configurations are defined (Figure 4).Those scenes are observed in nadir viewing by a sensor onboard an aircraft flying at 2.25 km altitude with a spatial resolution of 10 cm.The sun has a zenith angle of 30° and is located in a plan perpendicular to the street and the valley axis.The simulations are performed at 440, 870 and 1,600 nm.The atmosphere is modeled by the standard "mid-latitude summer" model of 6S for the molecules and by a Junge distribution [27] for the aerosols.This distribution is defined by particles of radius ranging from 0.01 to 10 µm, with a refractive index of 1.35 + 0.007i, spectrally constant in the [0.4; 2.5 µm] domain and with a Junge parameter of 3.32.Their abundances are defined from the following visibilities: 5 km and 23 km.As can be noted in Table 1 and in Figure 5, these results are in very good agreement.The values of the direct irradiances are equal because their analytical expressions used in the two codes are identical.The direct and environment radiances show some small discrepancies that are directly due to the difference of total irradiance at ground level resulting from the different atmospheric scattering models: mean difference of 3% between AMARTIS v2 and AMARTIS v1 for those three terms.
The Earth-atmosphere coupling irradiances and the downward reflected irradiances exhibit higher discrepancies, up to 8.9% in the case of I coup .Those differences are due to two reasons.First, as previously, a part of those discrepancies are due to the difference of scattered irradiance at ground level.This can have a huge impact on the calculation of I coup and I refl notably in the street cases where the presence of shadows (areas where the direct irradiance is null) increases the relative contribution of the scattered irradiance.However, this is not the only reason.As can be seen in Figure 5 and especially in Figure 6, the other reason is the noisy signal resulting from AMARTIS v1 simulations.This noisy signal is due to the use of fewer photons for the Monte Carlo calculations in AMARTIS v1 than in AMARTIS v2.Indeed, AMARTIS v2 allows obtaining a very good convergence of the components calculated with Monte Carlo techniques by simulating the propagation of numerous photons (~10 10 per simulation) in the atmosphere.This is possible thanks to the efficient ray-tracing tool that allows obtaining reasonable computation times (~1 h per simulation).The Monte Carlo methods of AMARTIS v1 are implemented in a way that makes the computations very slow compared to AMARTIS v2.Thus, it has not been possible to simulate the propagation of as many photons (less than 10 9 per simulation) because of the long computation times (up to 12 h per simulation in this case) and therefore we have obtained a less good convergence of I coup , I refl and R env .Finally, it is interesting to note in Figures 5 and 6 that the components of the signal have the same spatial variations in AMARTIS v2 simulations as in AMARTIS v1 simulations, thanks to the precise ray tracing software used to handle the geometry of the code.

Comparison with 6S
AMARTIS v2 has been compared with 6S for flat grounds to check both the correct use of 6S in AMARTIS v2 as a radiative transfer tool above canopy level and the atmospheric modeling.
In AMARTIS v2, I dir , I scat , R dir and R atm are calculated in the same way as in 6S or result from 6S calculations.So, the two codes should give similar results for those radiative components.However, it is necessary to check it.Then, as the I coup , I refl and R env components are calculated with different methods, a comparison has to be done.
To do so, a scene of lambertian reflectance 0.2 (spectrally constant) is used.A nadir satellite viewing is simulated at 440, 870 and 1,600 nm with a solar zenith angle of 30° and a molecular atmosphere modeled with the "mid-latitude summer" model.Four aerosols distributions are used, defined by spectrally constant single scattering albedos and asymmetry factors.The values of those parameters are presented in Table 2.The phase functions of aerosols distributions are defined by the Henyey-Greenstein function [35].The concentrations of those particles are described by two visibilities: 5 km and 23 km.Another case is also computed corresponding to an atmosphere without aerosols.

Table 2.
Single scattering albedos and asymmetry factors of the 4 aerosols models used for the comparison between AMARTIS v2 and 6S.

Aerosols models
Single scattering albedo Asymmetry factor M1 0.6 0.6 M2 0.6 0.9 M3 0.9 0.6 M4 0.9 0.9 In all, 27 simulations were performed both with AMARTIS v2 and 6S.The mean value of the absolute differences and the corresponding standard deviations obtained with 6S and for the central pixel of the images computed by AMARTIS v2 were calculated and presented in Table 3.To be able to assess the impact of the discrepancies on the total signal at ground and sensor levels, the mean differences and their standard deviation normalized by I tot for the irradiances and by R tot for the radiances are also presented.Finally, in order to have the order of magnitude of the different components of the signal, the minimum, mean and maximum values obtained with the AMARTIS v2 simulations have been added to this table.
As expected, the results obtained for analytical calculations (I dir , I scat , R dir and R atm ) are identical with mean discrepancies compared to the total signal of 0.0% except for R dir where a slight difference of 0.2% is obtained.This is explained by the difference obtained for the total signal at ground level due to the discrepancy on the Earth-atmosphere coupling irradiance.For the environment radiance, the mean difference compared to the total radiance is 0.8%, corresponding to a mean absolute error of 0.5 W.m −2 .µm−1 .sr−1 .For the Earth-atmosphere coupling irradiance, the mean difference compared to the total irradiance is 0.4%, corresponding to a mean absolute error of 3.3 W.m −2 .µm−1 .Those components are the ones that show the maximum discrepancies in absolute level with 6S because of the different calculation methods (analytical calculations for 6S, Monte Carlo calculations for AMARTIS v2).However, as can be noticed in Table 3, they are also the components that contribute the least to the total signal at ground and sensor levels and those differences have very little impact on the signal modeling.Finally, good agreements were obtained between 6S and AMARTIS v2 for flat ground simulations, proven by the very low difference for the total signals at sensor level, with a mean value of 0.4 W.m −2 .µm−1 .sr−1 corresponding to a relative value of 0.7%.

Illustrations of the Improvements Brought by AMARTIS v2
As explained previously, AMARTIS v2 is a radiative transfer tool that allows the calculation of all the components of the signal with a good accuracy (cf.previous section) and their representation.It notably allows the simulation of the radiance image of the scene acquired by the sensor.
To illustrate the gain brought by this code compared to AMARTIS v1 and to present its new potentialities, four examples are presented.First, a simulation of the observation of an urban canyon is detailed.Then, the use of materials exhibiting high directional reflection behaviors is presented with the example of the specular reflection of light by window panes.A radiative analysis of the signal in sunny and shady areas in the case of a crossroad follows.Finally, the aggregation process over a heterogeneous landscape is presented in the case of a street pattern and the directional effects induced by this aggregation are evaluated.
Note that the geometric accuracy of AMARTIS v2 is a major issue for such complex 3D scenes.However, this accuracy is directly linked to the precision of the ray-tracing tool used to handle the 3D.

Remote Sens
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Conclusions and Perspectives
In this paper, a new radiative transfer code is presented: AMARTIS v2.It constitutes a powerful tool to study radiative transfer in urban areas and to simulate images of urban scenes acquired by high or very high spatial resolution sensors onboard aircrafts or satellites, in the [0.4; 2.5µm] domain.It allows the simulation of all the components of the signal at ground and at sensor levels for diverse atmospheric conditions, and for various solar and viewing configurations.Complex geometries and realistic materials radiative properties (using bidirectional reflection functions, like the window pane bidirectional reflection function) can be taken into account.
This code also allows performing numerous phenomenological studies in urban areas.Indeed, as shown in this paper, this radiative transfer tool can for instance study the contribution of every component of the signal both at ground and at sensor levels.It also allows the comparison of the signal in the sunny and in the shady area of a landscape.The fourth case illustrates the aggregation process over a heterogeneous landscape.Thus, the evaluation of the introduced non-linearity in the mixing process can be evaluated which will help in the development of new non-linear unmixing methods.
Beyond those examples, this tool is very useful and efficient to develop new remote sensing methods.First, it allows the comprehension of radiative phenomena in such media [36].Then, it can be used to compute realistic synthetic remote sensing images, whose total parameters are defined by the user, in order to test new processing methods like atmospheric characterisation, atmospheric compensation, anomaly detection or classification.This tool can obviously be used for multispectral sensors, but the monochromatic calculations performed by AMARTIS v2 makes this radiative transfer code an appropriate tool to develop applications for hyperspectral sensors.It is currently used to test a new characterization procedure of urban aerosols radiative properties based on the transitions between sunny and shady areas [17,18].It will also be used in the near future to test the new version of ICARE (atmospheric compensation code over urban areas from hyperspectral acquisitions [5]) able to process spectral and multiangular acquisitions which might give access to walls classification and also to new unmixing methods as proposed by Zeng [49].
This code has nevertheless two drawbacks.First, it requires many input parameters to completely describe the scene, notably the complex geometry and the radiometric properties of the ground that can make the modeling of realistic scenes difficult.However, the recent efforts performed by the remote sensing community on the merging of LIDAR and hyperspectal remote sensing data allow obtaining complete characterizations of 3D complex scenes with the assessment of their spatial and spectral properties.Then, such a code is time consuming.Indeed, with the Onera's cluster dedicated to scientific calculations (64 bits cluster with 108 CPU), the AMARTIS v2 simulations last between a few minutes and a few hours.The duration of those calculations, performed without parallel processing, depends on the number of photons used for the Monte Carlo methods and on the number of facets describing the scene.For instance, the simulation of the observation of a scene of 500 m × 500 m with a spatial resolution of 1 m lasts between two and three hours.
Several points are considered to improve the performances of AMARTIS v2.First, new efforts must be done to pursue its validation.As no other radiative transfer code can both decompose the signal at ground and sensor levels and offer the modeling of complex 3D scenes, it will be necessary to validate AMARTIS v2 only on the total radiance entering the instrument.To do so, it will be possible to use Figure simula mount

Figure 6 .
Figure 6.Comparison of simulation results obtained with AMARTIS v1 (thin line) and AMARTIS v2 (thick line) in the street case No. 1 (cf.Figure 4) at 440 nm with a visibility of 5 km.

Figure 4 )
Figure 6.Comparison of simulation results obtained with AMARTIS v1 (thin line) and AMARTIS v2 (thick line) in the street case No. 1 (cf.Figure 4) at 440 nm with a visibility of 5 km.
Figure scene

Table 3 .
Comparison of the results obtained between AMARTIS v2 and 6S on the 27 simulation cases.