Remote sensing of aerosol plumes: a semianalytical model Alexandre Alakian,1,* Rodolphe Marion,1 and Xavier Briottet2 1
Commissariat à l’Energie Atomique, BP 12, 91680 Bruyères-le-Châtel, France
2
Office National d’Etudes et de Recherches Aérospatiales, 2 rue Edouard Belin, 31055 Toulouse, France *Corresponding author:
[email protected] Received 6 August 2007; revised 18 December 2007; accepted 13 February 2008; posted 22 February 2008 (Doc. ID 86104); published 4 April 2008
A semianalytical model, named APOM (aerosol plume optical model) and predicting the radiative effects of aerosol plumes in the spectral range ½0:4; 2:5 μm, is presented in the case of nadir viewing. It is devoted to the analysis of plumes arising from single strong emission events (high optical depths) such as fires or industrial discharges. The scene is represented by a standard atmosphere (molecules and natural aerosols) on which a plume layer is added at the bottom. The estimated at-sensor reflectance depends on the atmosphere without plume, the solar zenith angle, the plume optical properties (optical depth, singlescattering albedo, and asymmetry parameter), the ground reflectance, and the wavelength. Its mathematical expression as well as its numerical coefficients are derived from MODTRAN4 radiative transfer simulations. The DISORT option is used with 16 fluxes to provide a sufficiently accurate calculation of multiple scattering effects that are important for dense smokes. Model accuracy is assessed by using a set of simulations performed in the case of biomass burning and industrial plumes. APOM proves to be accurate and robust for solar zenith angles between 0° and 60° whatever the sensor altitude, the standard atmosphere, for plume phase functions defined from urban and rural models, and for plume locations that extend from the ground to a height below 3 km. The modeling errors in the at-sensor reflectance are on average below 0.002. They can reach values of 0.01 but correspond to low relative errors then (below 3% on average). This model can be used for forward modeling (quick simulations of multi/hyperspectral images and help in sensor design) as well as for the retrieval of the plume optical properties from remotely sensed images. © 2008 Optical Society of America OCIS codes: 280.1100, 290.1090.
1. Introduction
Understanding the effect of aerosols on climate and on human health is a field of research that still requires new developments [1]. Aerosols are liquid and solid particles suspended in the atmosphere that originate from natural or man-made sources of emission. They can reflect and absorb solar radiation (the aerosol direct effect) and modify cloud properties (the aerosol indirect effect) [2]. These phenomena are very variable depending on sources of emission, which makes difficult the assessment of the aerosol effect on the radiative budget [2]. Aerosol particles 0003-6935/08/111851-16$15.00/0 © 2008 Optical Society of America
can originate from various sources. For example, carbonaceous substances are generally emitted by biomass burning and industrial incomplete combustions, sea-salt particles by the ocean, and ashes and sulfuric acid particles by volcanic eruptions. Windblown mineral particles gather desert dust, sulfate, and nitrate aerosols resulting from gas to particles conversion [1]. These particles generally remain in the boundary layer (typically between 0 and 2 km) or can be raised to higher altitudes during their transport [1]. For example, smoke from large fires can be emitted from near the ground to 7 − 8 km above the ground, depending on the fire intensity. Aerosol particles are characterized by their shape, their size, their chemical composition, and their amount. These properties determine their radiative 10 April 2008 / Vol. 47, No. 11 / APPLIED OPTICS
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characteristics. Assuming sphericity and homogeneity of particles as well as their complex refraction index and their size distribution, Mie theory [3] allows one to compute the optical properties of aerosols. Whereas molecules have localized spectral features that allow their assessment, aerosols exhibit slow spectral variations and then they have a radiative effect in a wide range of wavelengths. Remote sensing methods are well suited for aerosol characterization. They generally make use of a few wavelengths [4,5] and multiangle [6,7] and polarization information [8]. Active remote sensing can also be used (lidars [9]). In order to better understand the global effect of aerosols, it is necessary to characterize the local sources of emission [10]. In this study, we focus on smoke plumes that result from single local high emission events, like biomass burning plumes and industrial plumes, observed by sensors with high spatial resolution (e.g., tens of meters). Studies on pollution and climate are using models on particles circulation htat are valid at a global scale [10]. These models require local data as references. This paper aims at characterizing the local sources of aerosols and then contributes to providing such data. Moreover, it would help in understanding the spatial distribution of natural and anthropogenic aerosols. The paper is devoted to the mathematical parametrization (semianalytical model) of the spectral signal (at-sensor reflectance) above plumes in the spectral range ½0:4; 2:5 μm. Multiple scattering plays a great role in the case of dense plumes. It can be accurately computed by using algorithms like DISORT [11] (discrete ordinates radiative transfer), but unfortunately this operation is very time consuming and cannot be applied for satellite operational algorithms. Disposing of a semianalytical model that would take into account multiple scattering would then be a fruitful advance in the purpose of forward modeling but also in the purpose of solving the inverse problem. Indeed, usual techniques mainly rely on a lookup table scheme involving complex numerical procedures and a huge amount of data. The size of the database can be in principle be reduced using the polynomial approach [12] or neural networks techniques [13]. The main shortcoming of these methods is, however, inflexibility. For instance, the change of the position, width, or number of spectral channels forces us to construct a new database or neural network training. This does not allow us to use algorithms developed for a given spectrometer or radiometer to interpret data obtained from different instruments on board multiple satellite platforms using the same database constructed. Similar problems arise also in our particular case if one needs to feed the algorithm with new or updated aerosol microphysical information. Our goal is then to develop a simple, explicitly invertable, semianalytical formula for remote sensing of aerosol plumes from the MODTRAN4 [14] numerical forward model (not explicitly invertable). Note that 1852
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such an approach has already been conducted for littoral [15], vegetation [16], and cloud [17] studies. The outline of this paper is the following. In a first step, the plume signal is modeled as a function of the optical properties of aerosols (optical depth, singlescattering albedo, asymmetry parameter), the viewing conditions (solar zenith angle, sensor is at nadir) and the ground reflectance. It yields a semianalytical model named APOM (aerosol plume optical model). The obtained formula allows one to explicitly compute the at-sensor signal from the parameters mentioned above. In a second step, the accuracy and the robustness of the model are assessed by using optical properties computed from a data set of biomass burning and industrial particles. The possible extended uses of the model in forward and inverse purposes are detailed and discussed at the end. 2.
Description of the Aerosol Plume Optical Model
In this section, we first present the theoretical description of APOM. Afterward, the algorithm and data used to derive its numerical coefficients are presented. A.
Analytical Formulation
Considering a plane-parallel atmosphere, the atsensor radiance is the result of photons coming from the landscape and the atmosphere. The at-sensor reflectance ρsensor is defined here as ρsensor ¼
πLsensor ; μs Es
ð1Þ
where Lsensor is the at-sensor radiance, μs is the cosine of the solar zenith angle θs , and Es is the extraterrestrial solar irradiance. Considering reflectance rather than radiance allows us to overcome variations of Sun irradiance and Sun geometry. This notation is used whatever the altitude of the sensor, which can be either airborne or satelliteborne. Assuming a homogeneous Lambertian surface with a reflectance ρ, ρsensor can be expressed as [18] ρsensor ¼ ρatm þ
T atm ρ ; 1 − Satm ρ
ð2Þ
where ρatm is the upwelling atmospheric reflectance, and the second term is the additional contribution due to the surface. Wavelength dependency of each term has been omitted for clarity. According to Refs. [19,20], the total transmittance T atm of the atmosphere along the Sun–ground–sensor path may be interpreted as the product of the transmittance (direct plus diffuse) in the solar direction for uniform illumination of the atmosphere from above by the transmittance (direct plus diffuse) in the sensor direction for uniform illumination of the atmosphere from below. Satm is the atmospheric spherical albedo for uniform illumination of the atmosphere from below. The distinction between illumination from above and below is crucial in our case because
the atmosphere is vertically inhomogeneous (see Ref. [21] for a detailed discussion on this subject). Note that in this study, all numerical computations of the atmospheric terms ρatm , T atm , and Satm have been made using MODTRAN4, so this distinction is fully accounted for. Now, let us consider a homogeneous absorbing aerosol plume added in a layer just above the ground. The atmospheric terms ρatm , T atm , and Satm are then modified by the presence of the plume. The goal of APOM is to derive a mathematical formula that models the variation of these terms in the presence of the plume. For this, APOM assumes that each atmospheric term can be expressed as the combination of two terms: a first one that is the corresponding atmospheric term for the same scene without plume and a second one due to the plume (see Fig. 1). APOM proposes to model ρatm , T atm , and Satm for a nadir viewing sensor as functions of the plume optical properties (fully characterized by the optical depth τ, the single-scattering albedo ω0 , and the asymmetry parameter g), and the solar zenith angle. At a given wavelength, ρatm , T atm , and Satm are modeled with the following expressions: atm ρatm ðτ; ω0 ; g; μs Þ ¼ ρatm 0 ðμs Þ þ ρplume ðτ; ω0 ; g; μs Þ; ð3Þ
atm T atm ðτ; ω0 ; g; μs Þ ¼ T atm 0 ðμs ÞT plume ðτ; ω0 ; g; μs Þ;
þ Satm Satm ðτ; ω0 ; gÞ ¼ Satm 0 plume ðτ; ω0 ; gÞ;
takes into account gaseous and natural aerosol profiles from ground to sensor and will be referred as standard in the following. By natural aerosols, we point out the aerosols that are present in the scene without plume, i.e., the background aerosols. atm atm ρatm plume , T plume , and Splume are the corresponding atmospheric terms that are due to the plume. They include the effect of the plume itself, the coupling between plume particles and atmospheric gases, and, especially for ρatm plume, the transmission loss between the top of the plume and the sensor. The coupling between plume particles and natural aerosols is neglected (this assumption will be discussed in Subsection 3.D). The plume terms are expressed by ρatm plume ðτ; ω0 ; g; μs Þ 2
0 13 Nτ X αk ðω0 ; g; μs Þτk A5; ¼ α0 ðω0 ; gÞ41 − exp@
ð6Þ
k¼1
0 @ T atm plume ðτ; ω0 ; g; μs Þ ¼ exp
Nτ X
1 γ k ðω0 ; g; μs Þτk A;
ð7Þ
k¼1
ð4Þ ð5Þ
atm atm are, respectively, the upwhere ρatm 0 , T 0 , and S0 welling atmospheric reflectance, the total transmittance (direct plus diffuse), and the atmospheric spherical albedo in the case of a given atmosphere without plume. This atmosphere without plume
Satm plume ðτ; ω0 ; gÞ 2
0 13 Nτ X βk ðω0 ; gÞτk A5; ¼ β0 ðω0 ; gÞ41 − exp@
ð8Þ
k¼1
where N τ is the degree of the polynomial in τ, α0 and β0 are functions of ω0 and g, ∀ k ∈ f1; :::; N τ g, αk and γ k are functions of ω0, g, and μs , and βk are
Fig. 1. APOM principle. When adding the plume, changes in photons paths are taken into account by the equations of ρatm, T atm , and Satm . atm At-sensor atmospheric terms (ρatm , T atm , Satm ) are modeled as combinations of standard atmospheric terms (ρatm , T atm 0 , S0 ) and plume atm , Satm ). Standard terms are computed by considering the scene without plume and by taking into account , T atmospheric terms (ρatm plume plume plume gaseous and natural aerosol profiles from ground to sensor. Plume terms are computed from APOM coefficients. 10 April 2008 / Vol. 47, No. 11 / APPLIED OPTICS
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Fig. 2. Evolution of ρatm − ρatm as a function of τ (ω0 and g being fixed). (a) g ¼ 0:45, ω0 ∈ f0:45; 0:60; 0:69; 0:77; 0:84; 0:90; 0:95; 0:98g (the 0 higher ω0, the upper the curve), (b) ω0 ¼ 0:84, g ∈ f0:01; 0:10; 0:20; 0:30; 0:45; 0:60; 0:75; 0:90g (the higher g, the lower the curve). In these simulations, nadir viewing sensor altitude is 20 km, standard atmosphere is 1976 U.S. Standard with no natural aerosol, solar zenith angle θs is 15°, and aerosol plume phase function type is urban (as defined in MODTRAN4). Dependency on τ can be roughly modeled with a law in the form αð1 − eβτÞ, α < 0, β < 0.
functions of ω0 and g. The dependency on wavelength cannot be easily modeled (see below), which is why each wavelength is considered separately. We will now describe how these expressions have been established. Let us consider a given solar zenith angle θs . When there is no plume (i.e., τ ¼ 0), ρatm equals the standard upwelling atmospheric reflecatm varies. As the tance ρatm 0 . When τ increases, ρ plume is assumed to be lying at the bottom of the atmosphere and to have a limited vertical extension (typically a few hundreds meters, i.e., far lower than the total atmospheric column), we consider that adding aerosols can only have an increasing effect on ρatm . Computations show that the error in ρatm begotten by this assumption is below 0.001 in the spectral must be modeled range ½0:4; 2:5 μm. Then ρatm − ρatm 0 with a positive function. The solution should tend to-
ward 0 when τ tends toward 0 whatever the values of ω0 and g. Another condition is that the solution should converge toward a limit value for the high values of τ. Indeed, when the plume is very dense, the photons cannot pass through a wall of aerosols, and then adding more particles would not change is reanything. The dependency on τ of ρatm − ρatm 0 presented in Fig. 2 for various couples ðω0 ; gÞ. The shape of the dependency can pretty well be fitted with a law in the form αð1 − eβτ Þ, α > 0, β < 0. This form checks the limit conditions defined before. Generalizing the τ-dependency as a polynomial in τ (with a constant part equaling zero) in the exponential allows a better fit of ρatm and then its expression becomes the one given by Eqs. (3) and (6). As the functions α0 ðω0 ; gÞ and αk ðω0 ; g; μs Þ have a smooth dependency on ω0 and g (see Fig. 3 for α0 ),
Fig. 3. Surface α0 ðω0 ; gÞ. The smooth behavior along ω0 and g allows a fit using a polynomial approach.
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they can be modeled by performing a polynomial approach (for a given μs ): ∀ k ∈ f0; …; N τ g;
αk ðω0 ; gÞ ¼
X
ηijk ωi0 gj ; ð9Þ
0≤i≤N ω 0≤j≤N g
where ηijk are the surface fitting parameters, and N ω and N g are, respectively, the degrees of the polynomials in ω0 and g. We can notice that if ω0 equals 0 (i.e., no scattering), then ρatm should be the same whatever the value of g: as no photons are scattered, the direction of scattering does not matter. Then every ηijk should equal 0 for i ¼ 0; j ≥ 1; ∀ k. Along the Sun–ground path, the photons that cross a horizontal plume (optical depth τ) with a zenith angle θs travel along an effective optical depth τμ−1 s . Along the ground–sensor path, the effective optical depth is τ because the sensor is at nadir. Then, the dependency on the solar zenith angle θs can be included by replacing ηijk by ðbijk þ cijk μ−k s Þ. bijk characterizes the ground–sensor path, whereas cijk characterizes the Sun–ground path. Then, ρatm plume can be written as follows: X
ρatm plume ðτ; ω0 ; g; μs Þ ¼ 2
0
× 41 − exp@
ði;jÞ∈I×J
X ði;j;kÞ∈I×J×K
aij ωi0 gj 13
k i j A5 ; ð10Þ ðbijk þ cijk μ−k s Þτ ω0 g
where I ¼ f0; …; N ω g, J ¼ f0; …; N g g, K ¼ f1; …; N τ g, and aij ¼ bijk ¼ cijk ¼ 0 for i ¼ 0; j ≥ 1; ∀ k. These considerations have been made for one wavelength. Modifying the wavelength does not change the shape of ρatm plume , but the fitting parameters aij , bijk , and cijk are different. This is due to a coupling [22] between the aerosol plume and the gases from the standard atmosphere. This coupling is due to Rayleigh scattering (important for the shorter wavelengths) and to the numerous gaseous absorption lines along the spectrum. Then, the fitting parameters are calculated for each wavelength separately. Unfortunately, it seems that it is not possible to model each parameter as a simple function of wavelength. Note that if only τ, ω0 , g, and μs intervene in the expression of ρatm plume , the coefficients aij , bijk , and cijk take into account the coupling between the plume and atmospheric gases (changing the standard atmosphere, and then the coupling leads to low modeling errors; see Subsection 3.E). Similarly, the same study has been conducted for atm atm Satm plume and T plume . The behavior of Splume can be modatm eled exactly in the same way as ρplume . However, we can note two differences. The first one is that Satm plume does not depend on solar zenith angle because by definition it is computed from an integration over all angles. The second difference is that Satm plume can take negative values for low values of ω0. Indeed,
adding dark absorbing particles in the atmosphere decreases its scattering effects and then decreases Satm . We obtain Satm plume ðτ; ω0 ; gÞ ¼
X ði;jÞ∈I×J
2
0
dij ωi0 gj 41 − exp@
X
ði;j;kÞ∈I×J×K
13 f ijk τk ωi0 gj A5; ð11Þ
where dij ¼ f ijk ¼ 0 for i ¼ 0; j ≥ 1; ∀ k. For T atm plume, we have the following expression: T atm plume ðτ; ω0 ; g; μs Þ 0 X ¼ exp@ ði;j;kÞ∈I×J×K
1 k i jA ; ðuijk þ vijk μ−k s Þτ ω0 g
ð12Þ
where uijk ¼ vijk ¼ 0 for i ¼ 0; j ≥ 1; ∀ k. B.
Estimation of the Model Coefficients
We have described the general formulation of the atmospheric terms in the presence of a plume. We now present the algorithm and data used to establish the model. The radiative transfer code used to simulate ρsensor is MODTRAN4. The DISORT option is used with 16 fluxes to accurately account for multiple scattering effects. The establishment of the model requires us to fix some parameters, which are the sensor altitude, the atmosphere without plume that is referred as the standard atmosphere, the type of the aerosol plume phase function, and the plume location. They will be referred to in the following as the fixed parameters. The atmosphere is defined by the 1976 U.S. Standard profile, without natural aerosols (the effect of their addition to the scene is assessed in Subsection 3.D), with an added plume located between the ground and an altitude of half a kilometer. The remote sensor is located at an altitude of twenty kilometers, looking at the nadir direction. The aerosol phase function of the plume is a key parameter that quantifies the probability of the scattering directions. In the case of dense plumes, for which multiple scattering is quite important, the mean behavior of this function may be sufficient to describe its whole effect, and then the asymmetry parameter g is used for this purpose. However, an asymmetry parameter must be connected to an associated phase function that then needs to be chosen. Indeed, g only characterizes the average behavior of the phase function. Two phase functions that have the same average are not necessarily equal. As the aerosol phase function depends on the type of particle and as the model presented is general (in τ, ω0 , and g), the phase function corresponding to urban particles as described in MODTRAN4 has been chosen. Urban particles composition (by number) is 20% of soot, 56% of water-soluble substance, and 24% of dustlike aerosols [23]. This choice is 10 April 2008 / Vol. 47, No. 11 / APPLIED OPTICS
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motivated by the fact that biomass burning particles and industrial particles are very often carbonaceous [24–26] and then must have a similar behavior to urban particles. The urban phase function is computed from Mie theory, and then is preferred to the analytical phase function of Henyey–Greenstein [27], which is generally less accurate. The effect of this choice will be discussed in Subsection 3.E. The simulations of ρatm, Satm , T atm , and ρsensor by MODTRAN4 are, respectively, referred to as atm atm sensor ρatm MODTRAN , SMODTRAN , T MODTRAN , and ρMODTRAN . APOM simulations are referred to as ρatm APOM , atm sensor Satm , T , and ρ . APOM APOM APOM The method used for the determination of the atm model coefficients is the same for ρatm APOM, SAPOM , atm atm and T APOM ;then it is only described for ρAPOM. For a given wavelength λ0 , the simulations with the full sets for τ, ω0 , g, and θs are considered. The estimates ^ijk ðλ0 Þ, and ^cijk ðλ0 Þ of the fitting parameters ^ ij ðλ0 Þ, b a aij ðλ0 Þ, bijk ðλ0 Þ, and cijk ðλ0 Þ are computed simultaneously by ^ijk ðλ0 Þ; ^cijk ðλ0 Þi ¼ Arg min Δρðaij ; bijk ; cijk Þ; h^ aij ðλ0 Þ; b aij ;bijk ;cijk
ð13Þ where Δρðaij ; bijk ; cijk Þ ¼
X τ0 ∈I τ ω00 ∈I ω
0 0 0 0 ðρatm MODTRAN ðτ ; ω0 ; g ; μs Þ
g0 ∈I g θ0s ∈I θ
− f ρ ðaij ; bijk ; cijk ; τ0 ; ω00 ; g0 ; μ0s ÞÞ2 ; ð14Þ and where f ρ ðaij ; bijk ; cijk ; τ; ω0 ; g; μs Þ is defined by the expression of ρatm APOM ðτ; ω0 ; g; μs Þ given in Eqs. (3) and (10), and I τ , I ω , I g , and I θ are, respectively, the variation ranges of τ, ω0 , g, and θs . We use a generalized reduced gradient method [28] to solve Eq. (13). The values of aij, bijk , and cijk have been constrained between −8 and 8 in order to accelerate the algorithm convergence and to reduce the number of local minima of Δρ. Their initial values are computed randomly between −0:8 and 0.1 (the numerical values have been obtained by trial and error). As the function Δρ may have many local minima, the algorithm is run several times with different random initial inputs. The optimal coefficients are chosen as the ones that minimize Δρ. This process is conducted for every wavelength. Finally, we obtain the optimal expressions of aij ðλÞ, bijk ðλÞ, and cijk ðλÞ in the least mean squares sense. N τ , N ω , and N g must be chosen carefully. Too low values will lead to a poor modeling, whereas too high values can lead to overfitting phenomena and increase considerably the computing time required for solving Eq. (13). We found that choosing N τ ¼ 3, N ω ¼ 4, and N g ¼ 3 was a good trade-off between 1856
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model accuracy and computation time (see Section 3 for details on modeling errors). For each wavelength, 119 non-null coefficients are needed to model ρatm , 68 non-null coefficients for Satm , and 102 non-null coefficients for T atm. Then ρsensor is modeled with 289 coefficients at each wavelength. APOM coefficients can be obtained from the author upon request. The presented model has been computed with a spectral resolution of 15 cm−1. Such a resolution is sufficient for a future use with hyperspectral sensors that have resolutions of about 10 nm in the spectral range ½0:4; 2:5 μm. If one is required to use the model for a particular sensor, coefficients that are adapted to its spectral bands are required. Then, for a given sensor, all the computations performed so far with MODTRAN4 have to be convolved with the normalized spectral response. The method described before is then applied to determine the model coefficients. In the example of the hyperspectral sensor AVIRIS (Airborne Visible/InfraRed Imaging Spectrometer [29], 224 bands), the plume signal measured by this sensor can be modeled with 64,736 coefficients (224 bands × 289 coefficients). Note that once the database of simulations has been computed at 15 cm−1 , the model coefficients corresponding to a given sensor are determined fast and can be used for all images from this sensor. The APOM coefficients computed over all the wavelengths of interest allow us to generate quasiinstantaneously ρsensor within ½0:4; 2:5 μm, whatever the plume optical properties τ, ω0 , and g, and whatever the solar zenith angle θs (note that a full MODTRAN4 run with DISORT 16 fluxes is very time consuming; see the conclusion section at the end for a discussion). The set of simulations has been conducted by varying the optical properties of the plume, the wavelength, and the solar zenith angle. These parameters will be referred to in the following as the varying parameters (by opposition to the fixed parameters described before). The optical depth τ can take values in the set I τ ¼ f0:0; 0:1; 0:2; 0:4; 0:6; 0:8; 1:0; 1:3; 1:6; 2:0; 2:4; 2:8; 3:2g. The single-scattering albedo ω0 takes its values in the set I ω ¼ f0:00; 0:45; 0:60; 0:69; 0:77; 0:84; 0:90; 0:95; 0:98g. Values of the asymmetry parameter g belong to I g ¼ f0:01; 0:10; 0:20; 0:30; 0:45; 0:60; 0:75; 0:90g. Ranges are defined from available data on plumes (see Subsection. 3.B for details). g can theoretically take negative values, but the simulations of optical properties as done in Subsection 3.B show that it never happens in the considered cases. These sets have been established by studying the effect of each parameter: simulations show that this effect increases faster and faster as ω0 increases and τ and g decrease, which explains the variable steps used for each set. For wavelength, as said above, all the computations are made on MODTRAN4 wavelength grid 15 cm−1 in the range ½0:4; 2:5 μm. The solar zenith angles θs are chosen in the set
I θ ¼ f15°; 30°; 45°; 60°g. Note that the model only depends on τ, ω0 , and g;then the model is fully general (for aerosol plumes with g > 0). 3. Accuracy and Robustness of Aerosol Plume Optical Model A. Principles of Aerosol Plume Optical Model Performances Evaluation
APOM performances are now estimated by considering MODTRAN4 as a benchmark. Its accuracy is assessed by varying the plume optical properties τ, ω0 , and g as well as the solar zenith angle θs and wavelength (i.e., the varying parameters). Afterward, natural aerosols are added to the atmosphere, and their effects on modeling errors are evaluated. Finally, the robustness of the model is assessed by varying the sensor altitude, the standard atmospheric model, the plume phase function, and the plume location (i.e., the fixed parameters). The validation strategy is detailed below. In a first step, a large set of spectral optical properties is computed with Mie theory in the spectral range ½0:4; 2:5 μm from a wide range of microphysical properties that can be observed in biomass burning and industrial plumes (see Subsection 3.B and Table 1). These optical properties are used to assess the modeling errors. In a second step, the atmospheric terms ρatm, T atm , atm S , and ρsensor associated with these optical properties are computed by using MODTRAN4 and APOM and by considering the original fixed parameters used to establish the model (nadir viewing sensor altitude of 20 km, U.S. 1976 standard atmosphere, and urban plume phase function), and no natural aerosol. As the model has been established for solar zenith angles belonging to the set f15°; 30°; 45°; 60°g, its accuracy needs to be estimated for a wider range of angles. Then the set f0°; 10°; 20°; 30°; 40°; 50°; 60°; 70°g is considered for the assessment. MODTRAN4 and model simulations are compared in Subsection 3.C in terms of values (full spectrum) and absolute spectral errors that are defined for each wavelength as atm Δρatm ¼ jρatm ΔT atm ¼ jT atm MODTRAN − ρAPOM j, MODTRAN − atm atm atm T APOM j, ΔS ¼ jSMODTRAN − Satm j, and Δρsensor ¼ APOM sensor sensor jρMODTRAN − ρAPOM j (wavelength dependency is omitted for clarity). In a third step, the effect of adding natural aerosols in the atmosphere is estimated. For this particular case, assessment is performed by setting θs to 20° and by using a set of 16 types of plume particles randomly selected from Table 1 that correspond to Table 1.
various spectral behaviors and by varying the particles concentration (i.e., the amplitude of optical depth). The same error analysis between MODTRAN4 and model simulations is led by considering each condition separately (see Subsection 3.D). In a fourth step, the sensor altitude, the standard atmospheric model, the aerosol phase function types, and the plume location are varied to assess the model robustness. The sets of optical properties defined from Table 1 are used, and θs is varied in the set f10°; 30°; 50°g. In a fifth step, the respective effect of modeling errors in ρatm , T atm , and Satm on ρsensor is assessed (see Subsection 3.F). Furthermore, the effect of uncertainties on ground reflectance is also discussed. The flowchart in Fig. 4 illustrates the main lines of the validation strategy. For every simulation, a gray ground reflectance is considered. A mean value of 0.3 is used so the joint effects of the atmospheric terms on the sensor signal can be observed. B.
Microphysical Properties of Aerosol Plumes
Biomass burning and industrial particles are considered to validate the model. Note that other types of particles (e.g., volcanic particles) could also have been considered because the model is general in τ, ω0 , and g. The aerosol optical properties are defined from their microphysical properties which are the concentration, the refractive index (composition), and the size distribution. They will be detailed below. Biomass burning particles are described with a coated sphere where a black carbon (BC) core is surrounded by a nonabsorbing organic shell (OC) [30]. This is the BC content of the aerosol particles that determines their absorbing behavior. Then the lower the BC, the higher ω0 and the higher the scattering. The relative amount of BC usually ranges from 2% to 30% [24]. The wavelength-dependent refractive index of particles has to be assumed. For BC, the values from Fenn et al. [31] for soot are used. For wavelengths above 2:0 μm, the values of Sutherland and Khanna [32] are taken for OC. For other wavelengths, very limited information about the refractive index of OC exists. Then by inspiration from the work of Trentmann et al. [30], the index of ammonium sulfate was chosen for wavelengths in the range ½0:4; 2:0 μm because it has the same spectral behavior. The refractive index of the internal mixture is computed using the Maxwell–Garnett mixing rule [33], assuming the same density for BC and OC, as performed in Ref. [30]. As the humidification factor is generally low for biomass burning particles [34],
Aerosol Microphysical Properties Used to Model Optical Properties of Biomass Burning Particles and Industrial Particles
Microphysical Property
Biomass Burning Particles
Industrial Particles
Composition Size distribution modal radius rm ðμmÞ Size distribution standard deviation σ m τ550
2%BC 10%BC 30%BC 0:05 0:10 0:15 1:35 1:65 1:90 0:4 0:8 1:3 2:4
40% BC AS 0:05 0:15 1:60 1:90 0:4 0:8 1:3
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Fig. 4. Flowchart describing assessment method of APOM performances. For accuracy assessment, τ, ω0 , g, and θs are varying. For robustness assessment, the same parameters are varying as well as standard atmosphere, sensor altitude, plume location, and type of aerosol plume phase function.
the effect of humidity on the optical properties is neglected in this study. The size distribution of biomass burning aerosols is modeled as a monomodal lognormal law. Indeed, even if several modes exist, one has generally much more radiative effect than others [24]. The range of modal radii and standard deviations varies between fresh smoke and aged smoke. Collection of data shows modal radii between 0:05 μm for fresh smoke and 0:15 μm for aged smoke, with a medium value of 0:10 μm and standard deviations between 1.35 and 1.90, with a medium value of 1.65 [24]. Plume particles concentrations are highly variable [34] in space and in time. In our study, they have been defined from the optical depth at λ ¼ 550 nm, which is referred to as τ550. From available measurements [34], τ550 typically varies from 0.4 to 2.4, which is representative of observations and applies to thick plumes. A few pieces of information have been gathered so far concerning industrial particles. The analysis of the measurements performed during the ESCOMPTE (Expérience sur Site pour Contraindre les Modèles de Pollution atmosphérique et de Transport d’Emissions) campaign [26] shows that industrial particles are generally composed of carbonaceous particles as we can find in biomass burning particles but also ammonium sulfate particles, noted AS. The composition, the size distribution, and the concentrations used to characterize industrial particles are also summarized in Table 1. 1858
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Figure 5 presents the optical properties computed from Mie theory for two types of particles: the first one corresponds to a biomass burning plume, the second one to an industrial plume. These two examples show that each optical property may have very variable spectral behaviors. The large database of optical properties that have been computed from Table 1 may allow us to cover most of the encountered behaviors. By considering every wavelength, this database gathers optical depths varying between 0 and 3.5, single-scattering albedos varying between 0.03 and 1, and asymmetry parameters varying between 0.04 and 0.88. Note that every triplet of optical properties [τðλÞ, ω0 ðλÞ, gðλÞ] in which τðλÞ could exceed 3.5 has been excluded from the database. C.
Model Accuracy
As an illustration, Fig. 6 compares ρsensor as computed by MODTRAN4 and by the model. The optical properties that are used are those represented in Fig. 5 and correspond to biomass burning and industrial plumes. The highest absolute error [about 0.005 at λ ¼ 0:8 μm; see Fig. 6(b)] corresponds to a relative error of 2%. APOM fits accurately MODTRAN4 simulation at every wavelength in both cases that correspond to very different spectral optical properties. As the simulations corresponding to biomass burning and industrial plumes present the same characteristics in terms of values and absolute errors, they will be represented in the same figures without distinguishing them. Indeed, the r2 correlation
Fig. 5. Aerosol optical properties for (a) biomass burning plume (BC content of 10%, rm ¼ 0:15 μm, σ m ¼ 1:65 and τ550 ¼ 0:8) and (b) industrial plume (AS particles, rm ¼ 0:05 μm, σ m ¼ 1:60, and τ550 ¼ 0:8).
Fig. 6. Comparison between MODTRAN4 and APOM simulations at θs ¼ 30° for ρsensor in case of (a) biomass burning plume and (b) industrial plume (associated optical properties are represented in Fig. 5). For clarity, the represented spectral errors are multiplied by 10. In both cases, modeling errors are on average below 0.002. They reach a maximum of about 0.005 in (b), which corresponds to a relative error of 2%.
coefficients between MODTRAN4 and APOM are about the same in both cases for ρatm, T atm , Satm , and ρsensor (r2 ≥ 0:999 for each case). Figure 7 presents ρatm , T atm , Satm , and ρsensor (for ρ ¼ 0:3) values as calculated by MODTRAN4 and by APOM over ½0:4; 2:5 μm for θs varying between 0° and 60°. Figure 8 shows the corresponding spectral errors Δρatm , ΔT atm , ΔSatm , and Δρsensor . Each atmospheric term is analyzed separately. Δρatm does not exceed 0.012 (mean error below 0.002), as seen in Fig. 8(a). Note that the highest absolute errors correspond to low relative errors below 3% (for example, Δρatm ¼ 0:012 corresponds to ρatm ¼ 0:40). ΔT atm remains below 0.008 over ½0:4; 2:5 μm (mean error around 0.002). The highest errors in T atm mainly correspond to relative errors around 2% (r2 > 0:999). Satm is accurately modeled in the full spectrum (average error below 0.002, r2 ¼ 0:999) but can reach values of 0.02 inside strong gaseous absorption bands (and 0.01 outside them) as seen in Figs. 7(c)
and 8(c). Fortunately, such errors have little effect on ρsensor (more details are given in Subsection 3.F). Errors in ρatm , T atm , and Satm may compensate. Figures 8(a) and 8(d) show that the spectral behavior of Δρsensor is about the same as Δρatm , which is quite accurate. This behavior is explained in Subsection 3.F. The results of the sensitivity study on ρatm , T atm , Satm , and ρsensor by considering each solar zenith angle separately over the spectral fields ½0:4; 0:5 μm, ½0:5; 0:8 μm, and ½0:8; 2:5 μm outside strong gaseous absorption bands are gathered in Table 2. The standard deviations associated with the mean errors approximately equal them. For angles between 0° and 60°, the mean modeling errors are very low and remain below 0.002 in ½0:4; 2:5 μm. The model has been established for angles varying between 15° and 60°, and then the extrapolation between 0° and 15° remains accurate. On the contrary, for an angle of 70°, extrapolation is far less accurate 10 April 2008 / Vol. 47, No. 11 / APPLIED OPTICS
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Fig. 7. Comparison of (a) ρatm , (b) T atm , (c) Satm , and (d) ρsensor (for ρ ¼ 0:3) values as computed by MODTRAN4 and by APOM for biomass burning and industrial plumes and for θs between 0° and 60°. Computations performed outside and inside strong gaseous absorption bands are respectively represented in dark gray and light gray. Correlation coefficient is higher than 0.999 for every case.
(modeling errors are between three and six times higher than for other angles) but still acceptable (mean errors in ρsensor remain below 0.01). On the whole, for solar zenith angles between 0° and 60°, the model reproduces accurately the simulations performed with MODTRAN4 using the DISORT 16 fluxes option: modeling errors are in general lower than 0.002. Moreover, high absolute errors correspond to low relative errors (below 3% in general). D. Impact Evaluation of Natural Aerosols
The presence of natural aerosols in the scene has not been considered during the computation of APOM coefficients. APOM assumes that their effect can atm be fully included in the computation of ρatm 0 , T0 , atm and S0 for a scene without plume. The coupling between natural aerosols and plume aerosols is then assumed to be negligible. This hypothesis is assessed now. Natural aerosols located above and below the plume have little interaction with it (computations show that they do not affect APOM performances), but it is not the case for the ones located inside the plume. Indeed, natural and plume aerosols are mixed externally, which modifies the global plume optical properties. As this phenomenon is not taken 1860
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into account in the model, its effect needs to be estimated. For a given type of plume particle and a given type of natural aerosol, the analysis is performed as follows. On one hand, the optical properties of the external mixture between plume aerosols and natural aerosols are computed. MODTRAN4 uses these optical properties as inputs to simulate ρatm , T atm , Satm , and ρsensor . On the other hand, APOM simulations are performed by using as inputs the plume optical properties (which do not take into account the atm atm natural aerosols) as well as ρatm com0 , T 0 , and S0 puted with MODTRAN4 for a scene without a plume and containing a layer of natural aerosols located between the ground and an altitude of half a kilometer (i.e., the plume location). MODTRAN4 and APOM simulations are then compared in terms of absolute spectral errors. Three types of natural aerosols defined in MODTRAN4 are considered in this study: rural, maritime, and urban. MODTRAN4 atmospheric profiles show that the natural aerosol column located inside the plume (0–500 m) represents generally less than 25% of the natural aerosol column over all the atmosphere. Then, by considering visibilities between 10 km and 20 km at 550 nm over all the atmosphere
Fig. 8. Spectral errors between MODTRAN4 and APOM for (a) ρatm , (b) T atm , (c) Satm , and (d) ρsensor (for ρ ¼ 0:3) for biomass burning and industrial plumes and for θs between 0° and 60°. Mean values are represented by black dots.
Table 2.
Mean Errors in ρatm , T atm , Satm and ρsensor for Different Solar Zenith Angles and Over Three Spectral Intervalsa and Outside Strong Gaseous Absorption Bandsb
103 Δρatm
103 ΔT atm
103 ΔSatm
103 Δρsensor
Bands
Λ1
Λ2
Λ3
Λ1
Λ2
Λ3
Λ1
Λ2
Λ3
Λ1
Λ2
Λ3
0° 10° 20° 30° 40° 50° 60° 0°–60°
1.5 1.9 1.3 0.8 1.3 1.4 1.0 1.3
0.9 1.1 0.9 0.9 1.3 1.4 1.3 1.1
0.7 0.7 0.7 0.8 1.0 1.2 1.3 0.9
2.2 2.0 1.6 1.7 2.4 2.7 1.5 2.0
2.2 2.1 1.9 1.7 1.7 1.8 1.6 1.9
2.0 2.0 2.0 1.9 1.9 1.7 1.7 1.9
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8
1.7 1.7 1.7 1.7 1.7 1.7 1.7 1.7
1.5 1.7 1.3 1.0 1.3 1.3 1.0 1.3
1.3 1.2 1.0 0.9 1.2 1.3 1.5 1.2
1.1 1.0 1.0 0.9 1.0 1.2 1.5 1.1
70°
8.8
6.8
2.7
14
7.8
3.8
1.0
0.8
1.7
7.4
6.9
3.5
Λ1 ¼ ½0:4; 0:5 μm, Λ2 ¼ ½0:5; 0:8 μm, Λ3 ¼ ½0:8; 2:5 μm. 0°60° gives the mean errors over all angles between 0° and 60°. The standard deviations associated with the mean errors approximately equal them. a
b
(i.e., τnatural varying between 0.2 and 0.4), the natural 550 aerosol column inside the plume τnatural varies be550 tween 0.05 and 0.10. Both values have been used for the assessment. Simulations show that absolute errors have the same level whatever the plume optical depth. For each natural aerosol type, Table 3 gathers the absolute spectral errors over the set of simulations dedicated to this study by making a distinction between
values of τnatural (see Subsection 3.A for details). As 550 the number of simulations performed is not as high as for the study of the other parameters, a direct comparison cannot be reasonably done. However, it clearly appears that errors in ρatm , T atm , and ρsensor are almost as low for rural and maritime natural aerosols as the errors obtained in Table 2. Errors are higher when urban natural aerosols are considered, especially for τnatural ¼ 0:1. This can be 550 10 April 2008 / Vol. 47, No. 11 / APPLIED OPTICS
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Table 3.
Mean Errors in ρatm , T atm , Satm , and ρsensor over Three Spectral Intervalsa outside Strong Gaseous Absorption Bands for Different Natural Aerosolsb
103 Δρatm Aerosol Maritime Rural Urban
Bands τnatural 550 τnatural 550 τnatural 550 τnatural 550 τnatural 550 τnatural 550
¼ 0:05 ¼ 0:10 ¼ 0:05 ¼ 0:10 ¼ 0:05 ¼ 0:10
103 ΔT atm
103 ΔSatm
103 Δρsensor
Λ1
Λ2
Λ3
Λ1
Λ2
Λ3
Λ1
Λ2
Λ3
Λ1
Λ2
Λ3
2.2 2.6 2.6 3.2 4.8 7.4
0.9 1.3 0.9 1.2 1.7 2.4
0.5 0.4 0.5 0.5 0.6 0.6
1.7 1.3 1.7 1.9 3.9 6.0
2.1 1.4 2.0 1.5 2.3 3.0
1.7 1.6 1.7 1.5 1.9 1.7
5.8 10 6.0 11 7.0 13
7.9 14 7.6 14 6.3 12
2.7 4.6 1.9 3.2 2.0 3.8
2.3 3.1 2.8 4.0 5.4 9.1
0.7 1.1 0.8 1.1 2.1 3.8
0.4 0.4 0.5 0.4 0.5 0.6
a Λ1 ¼ ½0:4; 0:5 μm, Λ2 ¼ ½0:5; 0:8 μm, Λ3 ¼ ½0:8; 2:5 μm.bτnatural is the optical depth of natural aerosols inside the plume (0–0:5 km). The 550 total optical depth of natural aerosols over all the atmosphere is given by τnatural ¼ 4τnatural 550 550
explained by the fact that such absorbing particles have low values of ω0 (below 0.65 at every wavelength) and then reduce the global ω0 in the case of nonabsorbing plumes (AS): a value of ω0 that changes from 1 to 0.97 may have a significant effect on the atmospheric terms. In the worst case (AS plume particles with τ550 ¼ 1:3, urban natural aero¼ 0:1), Δρsensor (for ρ ¼ 0:3) can sols with τnatural 550 reach values of 0.03 for wavelengths close to 0:4 μm, which corresponds to a relative error of about 8%. Errors in Satm can reach values as high as 0.03 but have little effect in ρsensor , as it will be shown in Subsection 3.F It is possible to remove the modeling errors due to natural aerosols by doing as follows. As APOM is a model that depends only on optical properties, it can directly take into account the presence of natural aerosols inside the plume. Indeed, it is just needed to compute the optical properties of the external mixture between plume and natural aerosols and to introduce them into APOM rather than the plume optical properties alone. A rigorous application atm atm would require then to compute ρatm 0 , T 0 , and S0 by considering only the natural aerosols located above the top of the plume (otherwise, the contribution of the natural aerosols inside the plume would be taken into account twice). E. Model Robustness
APOM should be used in a general purpose. Then, it is necessary to assess its robustness by changing its fixed parameters, which are the sensor altitude, the standard atmospheric model in the scene, the type of the aerosol phase function, and the plume location. Each parameter is studied separately. The sensor altitude takes values of 2 km, 5 km, 10 km, 20 km (APOM), and 99 km. The standard atmospheric model and the aerosol phase function types are being varied with models defined in MODTRAN4. The standard atmospheric model is chosen as tropical, midlatitude summer (Mid-L sum), midlatitude winter (Mid-L win), subarctic summer (Sub-A sum), subarctic winter (Sub-A win), and 1976 U.S. standard. The plume location is defined by the altitudes of its bottom and top (zmin − zmax ) which are 0–0:5 km (APOM), 0–1 km, 0–2 km, 0–3 km, 0–4 km, 0–6 km, 0:5–1 km, 2–3 km, 3–5 km, and 6–8 km. 1862
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As explained in Subsection 2.B, the asymmetry parameter g is connected to a plume phase function which is typical of urban particles as defined in MODTRAN4. To assess the effect of this choice, simulations are performed by connecting g to other plume phase function types defined in MODTRAN4, which are the rural type and the maritime type. The rural type is defined with particles composed of 70% of water-soluble substance and 30% of dustlike aerosols. The maritime type is defined by a mixing of sea salts and water [23]. The model requires standard atmospheric terms that characterize the scene without plume. Then, atm atm are computed for each configuraρatm 0 , T 0 , and S0 tion corresponding to different sensor altitudes and different standard atmospheric models. Afterward, they are introduced as inputs in the model. The results of the sensitivity study at MODTRAN4 resolution have been gathered in Table 4. It appears that the model is not sensitive to the sensor altitude. Indeed, the modeling errors for a sensor altitude of 20 km are about the same as for other altitudes. The same conclusion can be made by comparing the errors from the U.S. Standard atmosphere to the other standard atmospheres outside strong gaseous absorption bands. However, inside the strong absorption bands the modeling errors are increased by a factor between 1.5 and 3, as seen in Fig. 9. The different standard atmospheres do not present the same concentrations of gases, which is why the deviation with the U.S. Standard atmosphere is mostly located in the absorption bands. However, Δρatm and ΔT atm do not exceed 0.020 (on average below 0.004). High values of ΔSatm around 0.04 do not have much effect on ρsensor (see Subsection 3.F). Then the model can be used over the whole spectrum, whatever the type of the standard atmosphere. Concerning the type of the plume phase function, rural and maritime types provide the same level of errors for T atm and Satm as the urban type (benchmark). For ρatm and ρsensor , the rural type leads to quite accurate results (mean error below 0.005), whereas the maritime type is less accurate (mean error around 0.01) but is still acceptable. As the aerosol plume types are close to the urban or the rural ones (rural type is actually the urban type without
Table 4.
Mean Errors in ρatm , T atm , Satm , and ρsensor over Three Spectral Intervalsa outside Strong Gaseous Absorption Bands for Different Sensor Altitudes, Standard Atmospheric Models (No Natural Aerosol), Plume Phase Functions, and Plume Locationb.
103 Δρatm Fixed Parameters Sensor Altitude
Standard Atmospheric Model
Plume Phase Function
Plume Location
103 ΔT atm
103 ΔSatm
103 Δρsensor
Bands
Λ1
Λ2
Λ3
Λ1
Λ2
Λ3
Λ1
Λ2
Λ3
Λ1
Λ2
Λ3
2 km 5 km 10 km 99 km 20 km Tropical Mid-L sum Mid-L win Sub-A sum Sub-A win U.S. 1976 Maritime Rural Urban 0–0:5 km 0–1 km 0–2 km 0–3 km 0–4 km 0–6 km 0:5–1 km 2–3 km 3–5 km 6–8 km
1.8 1.5 1.4 1.5 1.5 1.6 1.6 1.6 1.6 1.6 1.5 9.4 3.9 1.5 1.5 2.0 5.0 6.7 12 17 3.8 22 36 51
1.4 1.3 1.3 1.4 1.3 2.1 1.9 1.6 1.7 1.6 1.3 8.5 4.8 1.3 1.3 1.4 1.7 2.5 3.6 4.2 1.3 11 14 17
1.2 1.2 1.2 1.2 1.2 1.8 1.6 1.3 1.4 1.4 1.2 2.7 1.6 1.2 1.2 1.2 1.2 1.6 1.6 2.1 1.3 5.8 6.7 6.6
5.2 3.1 2.3 2.4 2.4 2.3 2.3 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.5 2.6 3.7 6.2 7.3 2.4 5.0 5.4 7.3
2.9 2.3 2.0 1.9 1.9 2.1 2.0 1.9 1.9 2.0 1.9 1.9 1.9 1.9 1.9 1.9 2.3 3.4 7.5 8.1 2.0 4.7 4.9 5.6
1.9 1.8 1.8 1.7 1.7 2.3 2.0 1.9 1.8 2.3 1.7 1.7 1.7 1.7 1.7 1.7 2.2 2.2 4.3 4.8 2.1 2.3 2.3 2.3
0.9 0.9 0.9 0.9 0.9 1.0 1.0 1.0 0.9 0.9 0.9 0.9 0.9 0.9 0.9 1.2 2.9 4.7 5.8 6.4 5.3 22 32 51
0.9 0.9 0.9 0.9 0.9 1.1 1.0 0.9 1.0 1.0 0.9 0.9 0.9 0.9 0.9 0.9 1.3 1.7 1.9 2.1 1.5 5.9 8.1 13
1.7 1.7 1.7 1.7 1.7 1.7 1.6 1.8 1.7 1.9 1.7 1.7 1.7 1.7 1.7 1.7 1.8 2.0 2.2 2.4 1.7 4.1 4.1 4.7
2.1 1.6 1.2 1.6 1.4 1.5 1.5 1.6 1.5 1.6 1.4 9.4 3.6 1.4 1.4 2.1 4.8 5.9 10 15 3.7 19 34 47
1.6 1.5 1.3 1.5 1.3 2.1 1.9 1.5 1.7 1.5 1.3 8.6 4.8 1.3 1.3 1.3 1.5 2.0 2.2 3.0 1.3 12 14 17
1.4 1.4 1.3 1.3 1.3 2.1 1.7 1.4 1.5 1.7 1.3 2.7 1.8 1.3 1.3 1.3 1.4 1.6 1.6 1.8 1.3 8.3 6.3 6.3
a Λ1 ¼ ½0:4; 0:5 μm, Λ2 ¼ ½0:5; 0:8 μm, Λ3 ¼ ½0:8; 2:5 μm.bThe standard deviations associated with the mean errors approximately equal them.
carbonaceous matter [23]), the model is well adapted to our applications. We consider now the effect of the plume location (see Table 4). When the plume dilates at the altitudes 0–1 km, 0–2 km, and 0–3 km, the modeling errors remain quite low: the mean error is below 0.007 for ρatm , Satm , and ρsensor outside strong gaseous bands and below 0.004 for T atm. In the worst cases, Δρatm and Δρsensor can reach values of 0.02 but they correspond to relative errors below 5%. For greater plume vertical extensions (0–4 km, 0–6 km), errors become unacceptable (Δρsensor exceeds 0.01 on average for short wavelengths and can reach values above 0.03). When the plume bottom does not lie on the ground, the modeling errors are higher for the shorter wavelengths. Actually, the Rayleigh scattering due to the atmospheric layer below the plume may be masked by the plume, especially for high optical depths. As APOM does not take into account this phenomenon, it tends to overestimate ρatm . Modeling errors remain low for 0:5–1 km but are quite high for 2–3 km, 3–5 km, and 6–8 km (Δρsensor exceeds 0.02 on average for short wavelengths). For plume locations that cannot be accurately modeled, the analytical formulation of APOM may be valid, but the numerical coefficients should be then recalculated. It has been proved that APOM could be used in the presence of natural aerosols, whatever the sensor altitude, the standard atmosphere and for plumes that extend from the ground (or close to the ground)
to a height below 3 km. Then, for a particular scene containing a plume, an APOM user should compute atm atm ρatm in the exact conditions of the 0 , T 0 , and S0 scene without a plume, and afterward introduce them and the desired optical properties as inputs in the model to obtain quasi-instantaneously ρatm, T atm , Satm , and ρsensor . F.
Error Budget
Modeling errors in ρatm , T atm , and Satm do not have the same effect on ρsensor . Then we propose to assess their respective effects by differentiating Eq. (2): dρsensor ¼ dρatm þ ðρUÞdT atm þ ðT atm ρ2 U 2 ÞdSatm þ ðT atm U 2 Þdρ;
ð15Þ
where U ¼ ð1 − Satm ρÞ−1 . Considering the previous ground reflectance of 0.3 and the mean value 0.15 over all simulations for Satm, Eq. (15) becomes dρsensor ¼ dρatm þ 0:3dT atm þ 0:1T atm dSatm þ 1:1T atm dρ:
ð16Þ
By considering the mean values of Δρatm, ΔT atm and ΔSatm outside strong gaseous absorption bands (see Table 2) and Eq. (16), it clearly appears that ρatm has the dominating effect and Satm has a second-order effect (0 ≤ T atm ≤ 0:8 in plumes). However, inside 10 April 2008 / Vol. 47, No. 11 / APPLIED OPTICS
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Fig. 9. Study of standard atmospheric model (see Fig. 8 for details). Errors are amplified inside strong gaseous absorption bands in comparison to 1976 U.S. standard model. This is due to different gases concentrations from one standard model to another.
strong gaseous absorption bands, ΔSatm can reach high values as 0.04 [see Fig. 9(c)]. Fortunately, such cases correspond to low transmittances (T atm ≤ 0:3), and then the effect on Δρsensor is insignificant (below 0.001). Adding the absolute values of the weighted errors from each atmospheric term do not illustrate the global error on ρsensor . Indeed, comparisons between Figs. 8(a) and 8(d) show that the spectral errors in ρatm and ρsensor have about the same distributions (for all values and on average). It clearly means that errors in ρatm , T atm , and Satm compensate; otherwise, Δρsensor would have at least on average higher values than Δρatm . A parenthesis is made about the knowledge on the ground reflectance ρ. In the purpose of retrieving the aerosol optical properties over a nondark ground (for example, by using lookup tables), the ground reflectance under the plume is assumed with a given uncertainty Δρ. Equation (15) allows us to assess the effect of this uncertainty on ρsensor . By assuming a mean value of 0.15 for Satm, the begotten uncertainty on ρsensor is 0:88 Δρ for a low τ around 0.3 (T atm ¼ 0:8), of 0:44 Δρ for a high τ around 1.0 (T atm ¼ 0:4), and of 0:11 Δρ for a very high τ around 2 (T atm ¼ 0:1). Δρ can reach values higher than 0.1, which yields huge errors up to 0.09 in ρsensor (for low optical depths). 1864
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Then, to minimize the ground effect, one should first retrieve the optical properties from the plume parts for which the ground located below is known quite accurately (e.g. dark surfaces) or from the denser parts of the plume that correspond to mostly masked grounds. The retrieved optical properties could then be used to invert by extrapolation the other parts of the plume. The assumption of a Lambertian ground reflectance used by APOM is now discussed. As nadir viewing is considered, the bidirectional character of the surface does not intervene much (except for a Sun close to nadir). Moreover, the ground effect decreases as the plume optical depth increases [and then the atmospheric transmittance decreases, see Eq. (16))]. However, further work is required to assess accurately this point. G.
Using Aerosol Plume Optical Model for Inversion
In this section, investigations are done to characterize the optical properties of plumes from hyperspectral images with APOM. At first, APOM requires the plume spectral optical properties as inputs. As aerosols exhibit slow spectral variations (see Fig. 5), Alakian et al. [35] showed that it is possible to model accurately the spectral dependency of the plume
optical properties in the spectral range ½0:4; 1:1 μm, for which the plume effect is maximum, as τðλÞ ¼ a0 þ a1 ln λ þ a2 ln2 λ; ω0 ðλÞ ¼ b0 þ b1 ln λ þ b2 ln2 λ;
gðλÞ ¼ c0 þ c1 λ;
where a0 , a1 , a2 , b0 , b1 , b2 , c0 , and c1 are eight modeling coefficients. These equations generalize the Angström law τ ¼ βλ−α [36] and the work from King and Byrne [37]. Outside gaseous absorption bands, the model APOM and the modeling of the spectral optical properties enable us to characterize the spectral at-sensor reflectance ρsensor ðλÞ from the plume coefficients a0, a1 , a2 , b0 , b1 , b2 , c0 , and c1 (unknown), the cosine of the solar zenith angle μs (known), the sensor spectral bands λ (known), the ground reflectance ρðλÞ (unknown) and the standard atmospheric terms ρatm 0 ðλÞ, atm T atm ðλÞ, S ðλÞ (unknown). 0 0 The standard atmospheric terms can be estimated from pixels that are not contaminated by the plume. The ground reflectance, if unknown, can be estimated using the shortwave infrared bands [38]. Then, the eight unknown aerosol parameters need to be retrieved. By using hyperspectral sensors that provide tens or hundreds of bands, this can be done by minimizing the least squares between observed and modeled ρsensor . The main advantage of such an inverse method is that it does not assume any a priori knowledge on the type of particles (on the contrary to lookup tables). Then, it could be used to invert plumes for which emitted particles are completely unknown (e.g., industrial plumes). The development of this method is still ongoing.
4. Conclusion and Discussion
The semianalytical forward model APOM for the radiative effects of aerosol plumes in the spectral range ½0:4; 2:5 μm has been presented in the case of nadir viewing. It is devoted to the analysis of plumes arising from single strong emission events (high optical depths) such as fires or industrial discharges. The scene is represented by an atmospheric layer (molecules and natural aerosols) located above the plume layer. The at-sensor reflectance is completely defined by the atmospheric upwelling reflectance, the total atmospheric transmittance, the atmospheric spherical albedo, and the ground reflectance. Each atmospheric term is modeled as a function of the solar zenith angle, the aerosol optical depth, the singlescattering albedo, the asymmetry parameter, and the wavelength. The mathematical formula as well as the numerical coefficients are derived from MODTRAN4 radiative transfer simulations. The DISORT option is used with 16 fluxes to provide a sufficiently accurate calculation of multiple scattering effects, which are important for dense smokes. A large set of simulations is performed in the case of biomass burning and industrial plumes in order to
validate the model. APOM proves to be accurate for solar zenith angles between 0° and 60°. The modeling errors in ρsensor are on average below 0.002. The highest errors can reach values of 0.01 that correspond to low relative errors (below 3%). The addition of natural aerosols in the atmosphere does not affect APOM accuracy except for some particular cases (nonabsorbing plume particles and absorbing natural particles) that can be processed by directly considering the optical properties of the external mixture between plume aerosols and natural aerosols (rather then considering them separately). APOM preserves its accuracy whatever the sensor altitude, the standard atmospheric model and for plumes whose bottom zmin is close to the ground (≤0:5 km) and top zmax is below 3 km. For other plume locations, the analytical formulation of APOM could still be used, but the associated numerical coefficients should be recalculated. APOM also proves to be accurate for particles for which the phase function is defined from the urban and the rural types in MODTRAN4. An airborne experiment using a hyperspectral sensor is planned to validate APOM on real data. The model is modular and may be improved by including the sensor direction (zenith and azimuth angles) as a parameter. It would be performed in the same manner as for the solar zenith angle by replacing in Eq. (10) bijk by bijk cos−1 (viewing zenith angle). Further work is required to achieve this extension (maybe a dependency on azimuth should also be added). Then, APOM could be used for multiangle imaging sensors. This model may be used in forward modeling and retrieval of optical properties of aerosols. In forward modeling, it allows quick simulations of hyperspectral (and multispectral) images of plumes. To give an idea about the gain in time, the simulation of one pixel with MODTRAN4 at AVIRIS spectral resolution with given optical properties requires about one hour with current computers (e.g., AMD Opteron 2:40 GHz). Once model coefficients have been computed, APOM allows quasi-instantaneous simulations. This model can also help in designing a sensor (choice of the wavelengths and viewing angles) to optimally estimate the aerosol properties. It also allows to assess the sensitivity of the sensor signal to the optical properties. For retrieval, the generation of huge lookup tables is now possible very quickly with a high accuracy (DISORT 16 fluxes accuracy). It is also possible to develop more accurate methods of retrieval by adapting the existing methods in signal processing that use analytical models (gradient descents). If the standard atmospheric conditions are known or can be estimated (for example, outside the plume), the retrieval process to characterize the plume optical properties only depends on the plume itself. A method that models the plume optical properties with a few coefficients and combine them with APOM is currently in development. 10 April 2008 / Vol. 47, No. 11 / APPLIED OPTICS
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