An iterative algorithm is presented in this study for simultaneous determination of both the aerosol optical thickness and the exponent of the Junge power law from the total reflectance data of two satellite-based, near-infrared bands over the ocean. The atmospheric aerosol model is assumed as the Junge power-law size distribution in retrieval of the data. Numerical simulations show that relative errors in retrieval of the aerosol optical thickness and the exponent of the Junge power law are less than 5% when the actual atmospheric aerosol follows the Junge power-law size distribution. For other aerosol size distributions, relative errors of the aerosol optical thickness are less than approximately 10%. The proposed method is applied to a case study of the data of two near-infrared channels of the Sea-Viewing Wide Field-of-View Sensor (SeaWiFS) over the East China Sea area. The results show that reasonable spatial distribution of the exponent of the Junge law and the aerosol optical thickness may be obtained on a pixel-by-pixel basis through use of the proposed retrieval algorithm.
© 2007 Optical Society of America
At the top of the atmosphere, the total outgoing radiance at wavelength λ in the visible and near-infrared regions contains coupling information of both radiance and reflectance between the atmosphere and the ground. If only satellite-derived data are relied on, it is difficult to simultaneously retrieve the two unknown parameters: the atmospheric parameter and the ground optical parameter. It is known that only accumulation and coarse model particles may be detected by using the method of optical remote sensing. With the assumption of spherical aerosol particles, the unknown aerosol optical parameters are size distribution, complex refractive index, and optical thickness. It is necessary for retrieval of aerosol concentration or atmospheric correction of satellite data to assume an aerosol model [1–3], or to select several components externally mixing to form an aerosol-type model by using the optimum inversion method . Such a pre-assumption, which is not based on local practical measurements, will result inevitably in some errors in retrieval of aerosol concentration and atmospheric correction of satellite data.
The Junge power-law model is a simple and popular aerosol model that is usually used to fit a realistic atmospheric aerosol size distribution for most cases. Zhao and Nakajima  and Zhao et al.  suggested that a realistic atmospheric aerosol can be approximated as the Junge power-law size distribution for satellite remote sensing in their algorithm. Zhao and Nakajima  noted that errors in the retrieved aerosol optical thickness are less than 10% with simulated test data by use of the Simplex Method  in the iterative procedure. They made numerical experiments based on wavelength relationships among water-leaving radiances and employed Junge power-law size distribution models with the same complex refractive index for atmospheric correction. Zhao et al.  developed an iterative algorithm for the simultaneous determination of aerosol optical thickness and the exponent of the Junge power law over ocean areas from the upwelling radiances measured in Advanced Very High Resolution Radiometer (AVHRR) visible and near-infrared channels with 1.50-j0.01 of the complex refractive indexes of the aerosols. In the iterative step, the new aerosol size distribution parameters are selected by using the Powell method .
Chomko and Gordon  explored the efficacy of using the Junge power-law size distribution with a variable refractive index. They adopted a nonlinear optimization procedure  for estimating the relevant oceanic and atmospheric optical parameters. Simulated experiments showed that the ocean’s pigment concentration (C) can be retrieved with good accuracy; however, the aerosol optical thickness is retrieved with large errors. They thought that the discrepancy is due to significant differences in the scattering phase functions for bimodal, lognormal, and power-law distributions. The exponent of the Junge power law was determined by comparing the measured values of two near-infrared channels without the wavelength factor in their expressions, which is a possible error source for the retrieval of aerosol optical thickness. Even if the wavelength factor is included, relying on an analytical formula only, our numerical experiments may show that the relative errors in the estimation of size parameter ν are in the 20-40% range and the maximum relative error of the aerosol optical thickness is close to 50%. To improve the accuracy of retrieval, a new iterative algorithm is presented in this study for retrieval of size parameter ν and the aerosol optical thickness. Simulations show that the retrieval accuracy of size parameter ν and the aerosol optical thickness is improved satisfactorily by using the iterative algorithm.
First, a formula is derived in this study for retrieval of the exponent of the Junge power law, and some results are given by using simulated test data. Second, an iterative algorithm is presented and the feasibility of the algorithm is investigated with numerical simulations. Finally, the iterative algorithm is applied to local SeaWiFS imagery.
2. Retrieval scheme and simulated performance
2.1 The equation derived and the numerical simulations
It is assumed in the following retrieval scheme that the signal received by the satellite comes only from atmospheric scattering radiance and that the contributions of ground reflectance can be ignored. It is well-recognized that the assumption can be satisfied for the signals received by satellite at near-infrared bands for open ocean conditions. This generation of ocean-color sensors has two near-infrared channels: the SeaWiFS and the Moderate-Resolution Imaging Spectroradiometer (MODIS).
Ignoring the surface sun glitter and whitecaps in clear water, at the top of the atmosphere the total reflectance at a near-infrared wavelength λ in the single-scattering case can be written as [2,10]
where ρm(θ0,ϕ0,θ,ϕ) is the reflectance by atmospheric molecules (Rayleigh scattering), parameters ωa, τa, and Pa(θ0,ϕ0,θ,ϕ) are the aerosol single-scattering albedo, aerosol optical thickness, and scattering phase function, respectively. Angles θ0 and ϕ0 are the solar zenith and azimuth angles, respectively. Likewise, θ and ϕ are the viewing zenith and azimuth angles.
For the Junge power-law size distribution, the wavelength dependence of an aerosol optical thickness between 0.4 and 1.1μm is described in the Angstrom formula , which is written as
where the subscripts 1 and 2 denote two near-infrared wavelengths. The Angstrom coefficient α describes the relative spectral course of the extinction coefficient σe in
where β’ is the aerosol extinction coefficient at wavelength 1μm. The formula is valid if the particle size distribution obeys the Junge power-law model. From Eq. (4), the ratio of the single scattering albedo in the two near-infrared bands is given by
where σs is the scattering coefficient. The phase function is expressed as
where φ is the scattering angle and βs(φ) is the scattering function. The ratio of the phase function values in two near-infrared bands is given by
For the Junge power-law size distribution, the aerosol scattering function may be written as 
where C is a constant and η(φ) is a dimensionless function that is used to describe the aerosol scattering direction. 03B7;(φ) is related to the scattering angle, complex refractive index, and size parameter, which may be written as
where x is the size parameter and i is the intensity distribution function. η(φ) is insensitive to wavelength if the scattering angle is more than 4° . The scattering angle is always larger than 4° for the downward observations by satellite sensors. Thus, according to Eqs. (5), (7), and (8), we obtain
So, Eq. (3) can be modified by
According to the Mie scattering theory, the relationship between the exponent of the Junge power-law ν and the Angstrom coefficient α is
Because there was only one near-infrared band for the satellite at that time, further numerical simulation was not done in Ref. . The following simulation results indicate that the errors are large in retrieval of the exponent of Junge power-law v from Eq. (11) only.
A test dataset is simulated by computing ρp(θ0,ϕ0,θ,ϕ) for 2.5, 3.0, 3.5, 4.0, 4.5, and 5.0 of ν with τa(550)=0.3 at a band of 550nm, and the complex refractive index m=1.45+0.0035i. The size range of the particle radius in the calculations is taken as 0.04-10μm. The two near-infrared bands are 0.765 and 0.865μm with a bandwidth of 40nm. Note that the simulated data were given for sensor viewing angles of 15°, 30°, 45°, 60°, and azimuth of 200°, with solar zenith of 30° and azimuth of 180°.
Size distribution parameters for ν are calculated in terms of Eq. (11). The relative errors of ν are listed in Table 1. The τa(550) are calculated iteratively according to the iterative Eq. (16) in the next paragraphs by inputting the ν to the 6S code  for radiative transfer calculation while ignoring the absorption of atmospheric molecules for the ocean’s surface of clear waters. The relative errors in retrievals of aerosol optical thicknesses τa(550) are listed in Table 2.
Tables 1 and 2 show that the relative errors of ν are within 20–40% and the maximum relative error of τa(550) is close to 50%. It is found that computational errors cannot be tolerated in computing the size distribution parameters of ν and τa(550). To see the influence of multiple scattering, we have also carried out simulations with τa(550)=0.1 and 0.05, which are clear atmospheres over the ocean. Unfortunately, the relative errors of ν are still within 20-35% in the case of small aerosol optical thickness, which indicates that multiscattering is not an important factor for the above errors. Simulations suggest that such large errors may not result only from the ignorance of multiple scattering according to Eq. (11). So, it is understood that large errors of τa(550) can be ascribed to the inaccurate ν.
2.2 The new iterative algorithm and the numerical simulations
In order to improve the retrieval accuracy of ν and τa(550), based on the process derived in the above equation, an expression is created:
where αncf is the input value for the 6S mode at the nth iteration. Using the α calculated from Eq. (15), new τa(550) can be obtained with a near-infrared wavelength by using the iterative calculation 
where τna(550) is the input aerosol optical thickness for the 6S mode. This procedure is followed according to Eq. (16) until the convergence criterion of
is satisfied. The oα and the τa(550) in the last step are determined as the actual required values.
Another numerical simulation is carried out by the proposed iterative algorithm with the same preset parameters as above for the mid-latitude winter of the atmospheric model in which multiscattering and molecule absorption are considered. The errors of retrieved values of ν and τa(550) are listed in Tables 3 and 4, respectively. It can be seen from Tables 3 and 4 that the relative errors of the retrieved ν and τa(550) are less than 5%, which indicates that excellent retrievals of both ν and τa(550) can be obtained for the Junge power-law model in the iterative algorithm.
2.3 Applications to other aerosol size distributions
Atmospheric aerosol has large spatial-temporal variations. It is possible that actual atmospheric aerosol may not obey Junge power-law size distribution. The iterative algorithm is derived from the Junge power-law aerosol model. It is a key point, obviously, in its application whether the iterative scheme is still valid for another aerosol model. Many in-situ observations have shown that lognormal distributions are more appropriate to describe realistic aerosol size distribution [16,17]. Generally, any aerosol size distribution may be described by the combination of three lognormal distribution functions. As in size distribution, lognormal distributions are applied to each component i:
where r modN,i is the mode radius, σi measures the width of distribution, and Ni is the total particle number density of the component i in particles per cubic centimeter. The microphysical characteristics of the four components for the standard 6S aerosol types  are listed in Table 5. From the four basic components, the three aerosol types (maritime, continental, and urban models) are selected by mixing them with the following volume percentages in Table 6. The values of the brackets in Table 6 are the percentage density of particles. The complex refractive index of aerosol is assumed to be uniform as 1.45+0.0035i at the two near-infrared bands (0.765 and 0.865μm).
The retrieved values and errors of both ν and τa(550) can be obtained by using the proposed iterative algorithm with the microphysical characteristics in Table 5 and the volume percentages of the two basic components with maritime in Table 6, which are tabulated in Table 7. The true τa(550) is 0.3. The sun-viewing geometries are uniform in the above tables. For the four cases of sun-viewing geometry, the difference of size distribution of parameter ν is little. These values of ν can be evaluated as equivalent exponents of the Junge power law for lognormal distributions. It can be found in Table 7 that the retrieved values of τa(550) are in relative error by <5%.
Assuming that the three components of the lognormal distribution are externally mixed to form aerosol types with the volume percentages of the three basic components with the continental model in Table 6, the examples of performance of the iterative algorithm provide that ν is around 4.7 and the relative error of the τa(550) is around 7%, which is shown in Table 8.
In the same way, we have applied the iterative algorithm for the urban aerosol type. The results are shown in Table 9.
It can be seen from Tables 7–9 that the equivalent exponents of the Junge power law are about 4.2, 4.7, and 5.0, with the complex refractive index of 1.45+0.0035i for the maritime, continental, and urban aerosol models, respectively. The difference among the values of retrieved size distribution parameter ν is small with various sun-viewing geometries for the same aerosol model. The relative errors in retrieved values of τa(550) should be <10%. It can be summarized that it is responsible to approximate the actual atmospheric aerosol as the Junge power-law size distribution in the course of retrieval of aerosol concentrations in this iterative algorithm.
The new algorithm is a physical iterative method, which is different from others and has better accuracy in simultaneous determination of aerosol optical thickness and the Junge power-law size distribution parameter that is relied on in satellite-derived data only.
3. Example of application to SeaWiFS data
By using the iterative algorithm, the values of ν and τa(550) were retrieved from SeaWiFS measurements on January 1, 2001, over the Taiwan Strait region with an aerosol complex refractive index of 1.45+0.0035i . The spatial varieties of ν and τa(550) are shown in Figs. 1 and 2. The black color represents clouds or land, or Case 2 waters in Figs. 1 and 2, which is distinguished from Case 1 waters in terms of the criterion of α > 4 or α≤ 0 during the iterative retrieval process. It can be seen from Fig. 1 that the values of ν can be divided into three groups: 3.0-3.7 near the cloud region; 4.5-5.0 on the coast region, which is close to the one in the continental or urban aerosol models discussed above; and 3.7-4.5 on most open oceans in the Taiwan Strait. In the red areas in Fig. 1, the values of ν between 5.25 and 6.0 are likely to be the influence of the uncertain aerosol model. Although the retrieved equivalent Junge index in the red areas is a little bit large in those cases, the corresponding optical depth that is retrieved is still reasonable (see Fig. 2). It can be seen from Fig. 2 that most of τa(550) values vary within the range of 0.20 to 0.28. Though νis also in the 4.5-5.0 range on the east side of Taiwan, the τa(550) is less than 0.15, which indicates clean air in that region. On the east side of Taiwan and most regions in the center of the Taiwan Strait, the aerosols possibly might obey the Junge power-law size distribution.
By using the Junge power-law size distribution to approximate the actual atmospheric aerosol model, the proposed iterative algorithm avoids the uncertainty of the aerosol model in the retrieval of aerosol concentrations from satellite measurements. The exponent of the Junge power law can be retrieved for each pixel over an entire image, which also avoids the ordinary assumption of uniform aerosol size distribution over an entire image or just a part of an image. The iterative method has performed radiative transfer simulations of the retrieval process. The simulated experiment indicates that the relative error of the retrieved ν and τa(550) is less than 5% if the actual atmospheric aerosol follows the Junge power-law size distribution. If actual atmospheric aerosol is the lognormal distribution, the equivalent exponent of the Junge power law can be obtained by the iterative method, and the relative error of retrieved τa(550) is approximately <10%. The method has been also applied to SeaWiFS local data. It has been demonstrated that the values of retrieved ν and τa(550) are reasonable, and that ν has a large spatial variation over a local-area image. It can be concluded that the assumption of uniform aerosol size distribution parameters over an entire image or a segmented image undoubtedly will lead to retrieval errors of aerosol concentrations.
A fixed aerosol complex-refractive index is assumed for the Junge power-law model in the current algorithm. The assumption would produce additional errors, more or less. The influence of an aerosol complex-refractive index in the retrieval of aerosol concentration is worthy of further investigation.
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