**2. Methods**

**1. Introduction**

110 Aerosols - Science and Case Studies

Aerosols, suspended particulate matter in air, act as a crucial factor in global climatic fluctu‐ ations [1]. Aerosols can affect the climate through absorption and scattering of solar radiation [2] and therefore perturb the radiation budget and contribute to radiative forcing [3]. Aerosols may change the size and density of cloud droplets, thus modify the cloud albedo, the cloud lifetime and the precipitation [4]. Aerosols also influence air quality and therefore affect human health [5]. Current uncertainties of aerosols in the Earth radiation budget limit our under‐ standing of the climate system and the potential for global climate change [1]. Satellite observations are needed to understand the distribution and impact of aerosols on regional and global scales [6]. Satellites can monitor some aerosol optical properties, e.g. aerosol optical thickness (AOT) and Angstrom exponent, the key factors for climate change research [7]. In fact, these properties can be retrieved during the atmospheric correction of satellite images.

Ever since Gordon [8] designed an atmospheric correction approach based on the black ocean assumption (BOA) at two near‐infrared (NIR) bands, this approach has been widely applied to process satellite ocean colour remote sensing data. The performance of the approach was improved significantly [9, 10]. However, this approach still faces problems in Case 2 waters [11]. Some other algorithms of atmospheric corrections have been developed especially for the coastal waters. These include the use of the assumption of spatial homogeneity of the NIR band ratio [12], the spectral shape matching methods [13], an iterative fitting algorithm with the bio‐ optical models [14], the BOA method using the short wave infrared (SWIR) bands over turbid

The atmospheric correction over land meets more complicated situations. Similar to the BOA approach for the ocean colour remote sensing, the dark target (DT) approach has been widely used to estimate the optical properties of aerosols over land [17]. Other approaches have been developed using different methods, for example, the invariant object approach by Hall et al. [18], the histogram matching by Richter [19] and the radiative transfer model by Gao et al. [20]. Traditionally, different approaches of the atmospheric correction are necessary for land and ocean to optimize each case. Recently, Mao et al. [21, 22] developed an approach to estimate the aerosol scattering reflectance over turbid waters based on a look‐up table (LUT) of in situ measurements. Following this approach, a unified atmospheric correction (UAC) approach is

Over the last several decades, satellite remote sensing has provided an increasingly detailed view of aerosols and clouds [24] but limited with column‐averaged aerosol properties. Aerosols in the lowest part of the atmosphere are likely to be removed quickly by the rain, those in higher altitudes are much more likely to travel long distances and affect air quality in distant regions. The Cloud‐Aerosol Lidar and infrared pathfinder satellite observation (CALIPSO) satellite provides new capabilities to distinguish aerosol optically thin boundary layer from cloud by considering the vertical thickness and location of the layers as well as from the spectral behaviour of the lidar backscatter [25], useful in studying the interactions between

waters [15] or the algorithm using the ultraviolet bands [16].

aerosols and clouds with their roles in the climate system.

developed for both land and ocean [23].

#### **2.1. Retrieving the aerosols based on the BOA method**

In the atmospheric correction procedure, the aerosol scattering reflectance needs to be estimated from remote sensing data, relying on the condition that it can be clearly separated from the total satellite-measured reflectance. This condition can be met using the BOA method over clear oceanic waters when the water-leaving reflectance in the NIR bands is negligible. The satellite-received reflectance at the TOA was defined by:

$$\rho\_{\rm r}(\mathcal{\lambda}) = \pi L\_{\rm r}(\mathcal{\lambda}) / \left( F\_0(\mathcal{\lambda}) \cos \theta\_0 \right) \tag{1}$$

where *Lt* (*λ*) is the satellite-measured radiance, *F*0(*λ*) is the extra-terrestrial solar irradiance, and *θ*0 is the solar-zenith angle. Wang [26] partitioned the term *ρt* (*λ*) into components corresponding to distinct physical processes by:

$$
\rho\_r(\boldsymbol{\lambda}) = \rho\_r(\boldsymbol{\lambda}) + \rho\_\boldsymbol{\lambda}(\boldsymbol{\lambda}) + t(\boldsymbol{\lambda})\rho\_\boldsymbol{w}(\boldsymbol{\lambda}) + T(\boldsymbol{\lambda})\rho\_\boldsymbol{g}(\boldsymbol{\lambda}) + t(\boldsymbol{\lambda})\rho\_\boldsymbol{w}(\boldsymbol{\lambda})\tag{2}
$$

where *ρr*(*λ*) is the Rayleigh scattering reflectance due to the air molecules, *ρA*(*λ*) is the aerosol scattering reflectance including the Rayleigh-aerosol interactions, *ρwc*(*λ*) is the reflectance of the ocean whitecaps, *ρg*(*λ*) is the reflectance of Sun glitter off the sea surface and *ρw*(*λ*) is the water-leaving reflectance. The terms *t*(*λ*) and *T*(*λ*) are the diffuse and direct transmittances of the atmosphere, respectively.

$$\rho\_A(\mathcal{\lambda}) = \rho\_i(\mathcal{\lambda}) \cdot \rho\_r(\mathcal{\lambda}) - T(\mathcal{\lambda})\rho\_\mathcal{g}(\mathcal{\lambda}) - t(\mathcal{\lambda})\rho\_{\text{uc}}(\mathcal{\lambda}) - t(\mathcal{\lambda})\rho\_w(\mathcal{\lambda}) \tag{3}$$

When the water-leaving reflectance in the two NIR bands is assumed to be zero, the aerosol scattering reflectance can be obtained and used to compute the aerosol single scattering reflectance from:

$$\rho\_{\rm AS}(\lambda) = \frac{-b + \sqrt{b^2 - 4c(a - \rho\_\lambda(\lambda))}}{2c} \tag{4}$$

The epsilon spectrum of the aerosol single scattering reflectance is then defined as follows:

$$\varepsilon\left(\hat{\lambda}\_{\text{l}},\hat{\lambda}\_{\text{0}}\right) = \frac{\rho\_{\text{AS}}\left(\hat{\lambda}\_{\text{l}}\right)}{\rho\_{\text{AS}}\left(\hat{\lambda}\_{\text{0}}\right)}\tag{5}$$

Angstrom exponent *n* is computed from:

$$m = \frac{\lambda\_{\text{l}}}{\lambda\_{\text{p}}} \cdot \varepsilon \left(\mathcal{A}\_{\text{l}}, \mathcal{A}\_{\text{0}}\right) \tag{6}$$

According to Ref. [27], the aerosol optical thickness (AOT) can be calculated from the aerosol single scattering reflectance as follows:

$$\pi\_a(\lambda) = \begin{aligned} \rho\_{A^S}(\lambda) & \Bigg\prime \alpha\_a(\lambda) P\_a(\lambda) \end{aligned} \tag{7}$$

where *ωa*(*λ*) is the aerosol single scattering albedo and *Pa*(*λ*) is the aerosol scattering phase function.

#### **2.2. Retrieving the aerosols based on the UAC model**

The assumption of the BOA method becomes invalid over turbid waters and lands, leading to the failure of the standard atmospheric correction. A new approach needs to be developed to retrieve the aerosols from the satellite data. We define *ρAW*(*λ*) as a term of the aerosol-water reflectance, which includes aerosol scattering reflectance and the ground reflectance at the top of the atmosphere (TOA), derived as follows.

$$
\rho\_{\cdot \cdot W}(\mathcal{\lambda}) = \rho\_{\cdot}(\mathcal{\lambda}) \cdot \rho\_{\cdot}(\mathcal{\lambda}) - T(\mathcal{\lambda})\rho\_{\cdot \cdot \mathcal{\mathcal{E}}}(\mathcal{\lambda}) - t(\mathcal{\lambda})\rho\_{\cdot \cdot \mathcal{C}}(\mathcal{\lambda}) \tag{8}
$$

Then, aerosol scattering reflectance *ρA*(*λ*) can be obtained using:

$$
\rho\_A(\mathcal{Z}) = \rho\_{A\mathcal{W}}(\mathcal{Z}) - t(\mathcal{Z}) \cdot \rho\_w(\mathcal{Z}) \tag{9}
$$

It is derived from the normalized water-leaving reflectance, which is selected from a lookup table of the in situ measurements using the UAC method.

The epsilon spectrum is used to match the two closest aerosol models and obtain the corrected epsilon values *εc* (*λi* , *λ*0) following the approach of Mao et al. (2013). The Angstrom exponent *η*(*λi* ) is obtained from:

$$\eta\left(\mathcal{J}\_{\boldsymbol{\gamma}}\right) = \ln(\varepsilon^{\boldsymbol{\varepsilon}}\left(\mathcal{J}\_{\boldsymbol{\gamma}}, \mathcal{J}\_{\boldsymbol{0}}\right)) \cdot \mathcal{J}\_{\boldsymbol{0}} \;/\; \mathcal{J}\_{\boldsymbol{\gamma}} \tag{10}$$

We defined *λM* as a new reference wavelength obtained from the mean value of the band wavelengths.

$$\mathcal{A}\_{\mathcal{M}} = \sum \mathcal{A}\_{i} \,/\,\,\text{m} \tag{11}$$

The corrected epsilon spectrum is adjusted by a new reference wavelength, defined as follows:

$$\mathcal{L}\_M\left(\mathcal{A}\_l, \mathcal{A}\_M\right) = \left(\frac{\mathcal{A}\_M}{\mathcal{A}\_l}\right)^{\eta(\mathcal{A}\_l)}\tag{12}$$

The aerosol single scattering reflectance is used to obtain the mean value, defined as follows:

$$\mathcal{D}\_{\rm AS}(\mathcal{A}\_{\rm M}) = \sum \mathcal{D}\_{\rm AS}(\mathcal{A}\_{\rm l}) / n \tag{13}$$

Then, a new aerosol single scattering reflectance is obtained from:

( ) <sup>0</sup>

According to Ref. [27], the aerosol optical thickness (AOT) can be calculated from the aerosol

wl

where *ωa*(*λ*) is the aerosol single scattering albedo and *Pa*(*λ*) is the aerosol scattering phase

The assumption of the BOA method becomes invalid over turbid waters and lands, leading to the failure of the standard atmospheric correction. A new approach needs to be developed to retrieve the aerosols from the satellite data. We define *ρAW*(*λ*) as a term of the aerosol-water reflectance, which includes aerosol scattering reflectance and the ground reflectance at the top

( ) ( )- ( ) ( ) ( ) ( ) ( )

() () () () *<sup>A</sup> AW <sup>w</sup>*

 l rl

It is derived from the normalized water-leaving reflectance, which is selected from a lookup

The epsilon spectrum is used to match the two closest aerosol models and obtain the corrected

We defined *λM* as a new reference wavelength obtained from the mean value of the band

 l l

( ) 0 0 ln( ( , )) / *<sup>c</sup>*

/ *M i*

 l

l

 e ll

 lr l  lr l

, *λ*0) following the approach of Mao et al. (2013). The Angstrom exponent

*AW* = -- *t r T t <sup>g</sup> wc* (8)

= -× *t* (9)

*ii i* = × (10)

<sup>=</sup> å *<sup>n</sup>* (11)

 l

( ) ( ) () () *AS <sup>a</sup> a a <sup>P</sup>* r l

t l

Then, aerosol scattering reflectance *ρA*(*λ*) can be obtained using:

rl

hl

table of the in situ measurements using the UAC method.

 r l

**2.2. Retrieving the aerosols based on the UAC model**

of the atmosphere (TOA), derived as follows.

r l r l r l

= × (6)

= (7)

ell

0 , *<sup>i</sup> <sup>i</sup> n* l

l

single scattering reflectance as follows:

112 Aerosols - Science and Case Studies

function.

epsilon values *εc*

wavelengths.

) is obtained from:

*η*(*λi*

(*λi*

$$
\boldsymbol{\rho}^{\boldsymbol{\pi}}\_{\ \boldsymbol{\Lambda} \\$} (\boldsymbol{\lambda}\_{\!\!\!}) = \boldsymbol{\varepsilon}\_{\ \!\!\!} \left(\boldsymbol{\lambda}\_{\!\!\!\!}, \boldsymbol{\lambda}\_{\ \!\!\!\!}\right) \cdot \boldsymbol{\rho}\_{\ \!\!\!\!\!} (\boldsymbol{\lambda}\_{\ \!\!\!\!}) \tag{14}$$

This approach relies on the assumption that the aerosol scattering reflectance follows the Angstrom law instead of the BOA. This new assumption eliminates the effects of non-zero water-leaving reflectance in the atmospheric correction procedure. Due to the amplifying effects of the epsilon spectrum in estimating the aerosol scattering reflectance, the waterleaving reflectance may be estimated with large errors using the BOA method. One advantage of the UAC model is that the epsilon spectrum can be obtained to find the aerosol model, instead one value in the NIR band. The other advantage is that the difference of the ground reflectance among different objects is represented by different spectra in the LUT. Therefore, the LUT of the ground reflectance helps the UAC model to eliminate the different effects of the atmospheric correction due to the large difference of the reflectance between land and ocean.

#### **2.3. Comparison between the UAC model and the BOA method**

It is well known that the processing of satellite data using the standard atmospheric correction usually fails over turbid coastal regions. One of the main reasons is the difficulty in accurately determining the epsilon values. Epsilon is used to determine the magnitude of the aerosol scattering reflectance and deficiencies in its determination degrades the accuracy of the atmospheric correction of satellite remote sensing data. Normally, epsilon is estimated from aerosol scattering reflectance under the assumption of zero water-leaving radiance in the two NIR bands. However, the water-leaving radiance values in the two NIR bands are usually much higher than zero in the coastal waters [28], causing errors in the epsilon estimation.

This overestimation is caused by that the water-leaving reflectance is wrongly attributed as part of the aerosol scattering reflectance to overestimate the epsilon value. A small bias of the epsilon may easily lead to a relatively large error in the water-leaving reflectance due to the amplifying effects of the extrapolation of epsilon. The aerosol scattering reflectance obtained by the UAC model is different from that by the BOA method, especially in the coast regions. A simple comparison is made based on three selected typical locations (marked by the green crosses in **Figure 3**), and the results are shown in **Figure 1**.

**Figure 1.** Comparison between the aerosol scattering reflectance using the UAC and the BOA method.


**Table 1.** The difference between the aerosol reflectance obtained from the BOA method and that from the UAC model over the different epsilon ranges.

The aerosol scattering reflectance calculated from the two methods are similar to each other in oceanic waters (e.g. Location 3). The differences of the two reflectance spectra become larger near the coastal regions, with a mean relative error over 50% (e.g. Location 2). The difference of the two aerosol scattering reflectance values is very small in Band 8 and large in Band 1. This difference is attributed to the different epsilon values of the two methods, with 1.17 using the BOA method and 1.03 using the UAC model, respectively. A larger epsilon value will obviously overestimate the aerosol scattering reflectance, but this error is difficult to detect until a negative value of the water-leaving reflectance is produced. The BOA reflectance in Band 1 is 5.4 times larger than the UAC reflectance in highly turbid waters (e.g. Location 1). The value in Band 8 is also overestimated using the BOA method.

A simple comparison is made based on three selected typical locations (marked by the green

**Figure 1.** Comparison between the aerosol scattering reflectance using the UAC and the BOA method.

**Epsilon Band 1 Band 2 Band 3 Band 4 Band 5 Band 6 Band 7 Band 8** 1–1.04 8.94 8.44 7.72 7.42 6.74 5.06 4.26 4.51 1.04–1.07 11.77 11.21 10.41 10.09 9.37 7.80 7.03 6.86 1.07–1.1 74.38 70.16 64.40 62.17 57.49 47.43 41.04 37.29 1.1–1.2 206.63 186.90 161.39 151.86 132.73 94.93 72.27 55.23 1.2–1.3 453.50 394.95 323.45 297.96 248.86 159.89 111.68 75.14 >1.3 961.04 796.23 612.95 552.07 441.02 259.90 172.08 110.42

**Table 1.** The difference between the aerosol reflectance obtained from the BOA method and that from the UAC model

The aerosol scattering reflectance calculated from the two methods are similar to each other in oceanic waters (e.g. Location 3). The differences of the two reflectance spectra become larger near the coastal regions, with a mean relative error over 50% (e.g. Location 2). The difference of the two aerosol scattering reflectance values is very small in Band 8 and large in Band 1. This difference is attributed to the different epsilon values of the two methods, with 1.17 using the BOA method and 1.03 using the UAC model, respectively. A larger epsilon value will obviously overestimate the aerosol scattering reflectance, but this error is difficult to detect until a negative value of the water-leaving reflectance is produced. The BOA reflectance in Band 1 is

over the different epsilon ranges.

crosses in **Figure 3**), and the results are shown in **Figure 1**.

114 Aerosols - Science and Case Studies

To evaluate the difference between the aerosol scattering reflectance using the two methods, the relative errors of the reflectance are computed according to the epsilon ranges (**Table 1**), in which the values using the UAC model are taken as the truth data.

From **Table 1**, the relative differences of the aerosol reflectance from the BOA method and the UAC model vary largely over the different epsilon ranges. The differences in all bands are relatively small for the epsilon value range of 1–1.07. The differences become higher than 50% for the range of 1.07–1.1. The differences become too much large when epsilon values are higher than 1.1, with a mean difference of 488% for the range of >1.3. Therefore, the BOA method is valid only with the epsilon value of less than 1.07.

A small error of the reflectance in NIR bands will cause somehow relatively larger error of the aerosol scattering reflectance using the BOA method. For example, two aerosol-water reflectance spectra are selected from two neighbouring pixels and shown in **Figure 2**, together with two aerosol scattering reflectance spectra obtained by the BOA method.

**Figure 2.** Comparison between the reflectance spectra at two neighbouring pixels. The red lines represent the aerosolwater reflectance from satellite data and the green lines are the aerosol scattering reflectance using the BOA method.

The magnitudes of the two aerosol-water reflectance are close to each other. But the differences of the reflectance in NIR bands lead to obtain two different epsilon values of 1.031 and 1.036 using the BOA method. Due to the amplifying effects, the difference in the two aerosol scattering reflectance is 2.26 × 10−4 sr−1 in Band 1, about 10-folds higher than that in Band 8. This difference tends to spread to the water-leaving reflectance in the atmospheric correction procedure with 2.1 × 10−4 sr−1 in Band 1, about threefolds of the difference between the satellite reflectance. The difference of the two aerosol scattering reflectance in Band 2 is up to 50-folds higher than the satellite TOA values. Therefore, magnitudes of the aerosol scattering reflec‐ tance in the visible bands are very sensitive to the bias of the reflectance in the NIR bands using the BOA method. This problem can be significantly reduced by the UAC model.
