## Abstract

Beer’s attenuation law is the basis for the retrieval of aerosol optical depth (AOD) from sunphotometer data. However, the filter band function causes uncertainty during the retrieval of AOD from sunphotometer data, particularly for channels covering spectral regions of strong gas absorption. In this work, the uncertainty in AOD retrieval due to the filter band function is systematically analyzed by employing fine spectral absorption cross sections obtained from the Molecular Spectroscopy and Chemical Kinetics Group and the line-by-line radiative transfer model (LBLRTM). The uncertainty in AOD retrieval includes the uncertainty due to the wings of the filter band function in the ultraviolet (UV) region and errors in the optical depth calculation for Rayleigh scattering and absorption of O_{3}, NO_{2}, H_{2}O, CH_{4}, and CO_{2}. The results showed that 1) the uncertainty of AOD retrieval by this method, which is called the approximate AOD retrieval method, might become large when the filter band function is not well designed, particularly in the UV region; 2) in the case of a large zenith observation condition, the errors will be nonnegligible if the Rayleigh scattering optical depth is calculated at a central wavelength without including filter band function; 3) the band-weighted absorption coefficients of O_{3} and NO_{2} remain nearly constant when the gas amounts change, except in the case of questionably designed band filters; and 4) these weak-absorption optical depths for H_{2}O, CH_{4}, and CO_{2} cannot be ignored in the 1020- or 1640-nm channels, where an optical depth error of 0.01−0.02 may be introduced.

## 1. Introduction

CIMEL sunphotometers (CIMEL Electronique, Paris), such as the CE318, are widely used by the Aerosol Robotic Network (AERONET) for measuring aerosol optical depth (AOD), aerosol microphysical features, precipitable water, etc. (Holben et al. 1998). Because AOD products derived from sunphotometers are always used to validate estimates from remotely sensed satellite data, it is extremely important to retrieve the AOD accurately from sunphotometer measurements. CIMEL sunphotometers use silicon photodiode detectors to receive direct solar radiation. The typical spectral response range of a detector is 320−1100 nm. Interference filters are placed in front of these detectors and are used to select radiation transmissions in specified spectral channels (Ortiz de Galisteo et al. 2009). An ideal filter should permit radiation only at a single wavelength, blocking all radiation at other wavelengths. However, an actual filter is required to have a spectral response function for a certain spectral channel. The bandwidth of a spectral channel for the CE318 is approximately 2, 10, or 40 nm. A well-designed filter should have 1) high transmittance in a given band (or channel) and 2) negligible out-of-band transmittance. If a filter is not well designed, then it can transmit radiation with wavelengths that significantly differ from the central wavelength of its corresponding spectral channel, particularly for UV channels (Ortiz de Galisteo et al. 2009). These out-of-band transmission function regions are called the “wings” of a filter and can introduce transmittance errors for AOD retrieval when they are not processed properly. Based on the Beer–Lambert–Bouguer law, AOD is obtained by subtracting the Rayleigh scattering and absorption gas optical depths (including that of ozone) from the total optical depths (Holben et al. 1998; Rollin 2000; Hu, 2007). Therefore, the uncertainty in AOD retrieval may be caused by the calibration accuracy of the sunphotometer, the variation in Rayleigh optical depth due to errors in the estimation of ground pressure, and errors in the estimation of gas column amounts. Eck et al. (1999) discussed the above-mentioned error sources in detail and found the following: 1) an uncertainty of ~0.01−0.02 due to the calibration uncertainty of the field sunphotometers, 2) an uncertainty of ~0.0002−0.021 due to extreme variation in atmospheric pressure (~3%), and 3) an uncertainty of ~0.004−0.006 due to a departure of ~50% from total column ozone. A total uncertainty of ~0.010–0.021 is estimated in AOD retrieval for field sunphotometers (which is spectrally dependent, with a higher error in the UV region) under actual atmospheric conditions (Eck et al. 1999). As expected, the uncertainty due to calibration uncertainty dominates the total uncertainty. These conclusions were drawn based on the Beer–Lambert–Bouguer law.

However, we must note that the Beer–Lambert–Bouguer law holds strictly for monochromatic radiation and remains approximate for a narrow bandpass, such as the channels of sunphotometers (Shi 2007). To obtain AOD accurately from sunphotometer measurements, filter band functions and detector functions must be known (Ortiz de Galisteo et al. 2009). The direct solar irradiance recorded by a sunphotometer is the integral of the irradiance that reaches the collimator, depending on the filter band function and the detector function. Therefore, these two types of response functions need to be considered when the Beer–Lambert–Bouguer law is used to retrieve AOD.

Additional errors in AOD retrieval are caused when these issues are not resolved, and this problem has attracted the attention of researchers. The AOD in UV channels (340 and 380 nm) is underestimated when only the contribution of the filter band function is considered and the contribution of the detector function is not taken in account (Ortiz de Galisteo et al. 2009). AERONET direct-sun algorithms include the contribution of the filter band function to the total optical depths but provide constant values for the weak-absorption optical depths of CO_{2} and CH_{4} (http://aeronet.gsfc.nasa.gov/version2_table.pdf). Although the filter band function has been considered for retrieving AOD from sunphotometer data in previous reports, the filter band function was always used to separately obtain the band-weighted optical depths of the Rayleigh extinction, ozone, water vapor, and other gases (Smirnov et al. 2004; Ortiz de Galisteo et al. 2009). We have previously referred to this widely used AOD retrieval algorithm as the approximate AOD retrieval method. This method can produce errors because the Beer–Lambert–Bouguer law cannot be applied directly to a spectral band.

Therefore, the overall uncertainty of AOD retrieval can be divided into the following two categories: 1) the uncertainty due to various factors in the AOD retrieval process, such as the calibration uncertainty of the instrument, the Rayleigh optical depth, and the gas column amounts; and 2) the uncertainty due to an improper treatment of the filter band function and the detector function. It is important to consider these two types of uncertainties to reduce the overall uncertainty in AOD retrieval, even if the first type of uncertainty is dominant. The first issue was discussed in detail by Eck et al. (1999); therefore, it is not discussed here. However, the second issue has not been thoroughly analyzed. Because of the lack of detailed response data for the detector, we mainly analyzed the effects of the filter band function on AOD retrieval from UV channels to near-infrared (NIR) channels. In the following sections, the uncertainty of AOD retrieval particularly refers to the uncertainty due to the filter band function, excluding the uncertainty associated with the calibration of the sunphotometer and other atmospheric parameter estimation uncertainties. The following four topics related to filter band functions are discussed in the present work:

The uncertainty of AOD retrieval from sunphotometer measurements by the approximate AOD retrieval method. The measured transmittance by the sunphotometer is the band-weighted transmittance, which is calculated by integrating the product of the spectral transmittance of each individual component and the filter band function. Strictly speaking, the AOD of a particular band is the logarithm of the aerosol extinction band transmittance, which is derived from the measured transmission divided by the band-weighted transmittance of other gas components. However, the band-weighted transmittance of these other components, such as Rayleigh and ozone, can be calculated either by the band-weighted value of the products of each individual component’s spectral transmittance or by the products of the band-weighed transmittance of each individual component. The former process is rigorous, but the calculation process is considerably complicated; therefore, the latter is always adopted because it is easy to calculate the band-weighted transmittance of the individual component, which has been mentioned as the approximate AOD retrieval method and has consistently been used in the past. We have mentioned the uncertainty in AOD retrieval by the approximate AOD retrieval method, which excludes the uncertainty due to calibration uncertainty and atmospheric parameter estimation uncertainties, as discussed by Eck et al. (1999). Therefore, in the current study, we clarify the concept of uncertainty in AOD retrieval when the approximate AOD retrieval method is used. This study only considers the contribution to uncertainty of using the approximate method instead of the more rigorous method.

The possibility of ignoring the filter band function when the Rayleigh optical depth is calculated. It is necessary to analyze the influence of the filter band function on the band-weighted Rayleigh optical depth under different zenith angles.

The changes in the band-weighted coefficients of the absorption gases (O

_{3}and NO_{2}) as path gas concentrations vary with the zenith angle.The possibility of ignoring or treating as constants the weak-absorption optical depths, such as H

_{2}O absorption in the 1020- and 1640-nm bands and CH_{4}and CO_{2}absorption in the 1640-nm band. These problems are studied using numerical simulation calculations and three commercial CE318s, which were purchased by the Institute of Remote Sensing Applications, Chinese Academy of Sciences (CAS), and the Center of Earth Observation and Digital Earth, CAS, and are identified by the following CIMEL numbers: #469, #478, and #681. Along with the sunphotometers, the filter band functions are provided by CIMEL Electronique (http://www.cimel.fr/index_us.html). These three CE318s are employed in many of the in situ experiments described here.

## 2. Methodology

A sunphotometer measures the direct irradiance transmitted through the atmosphere, attenuated by aerosol scattering, aerosol absorption, Rayleigh scattering, ozone absorption, water vapor absorption, and the absorption of other gases (Iqbal 1983):

where *E*_{0} is the extraterrestrial irradiance at the mean earth–sun distance; *R* is the earth–sun distance in astronomical units (AU) at the time of the observation; *T* is the transmittance; the subscripts *a*, *r*, *o*, *g*, and *w* represent the scattering or absorption optical depths of aerosol, Rayleigh extinction, ozone, other gases, and water vapor, respectively; and *λ* represents the corresponding value at a particular wavelength. With the exception of water vapor transmittance, the transmittances (e.g., *T _{a}*,

*T*,

_{r}*T*,

_{o}*T*) in the bands of the sunphotometer can be calculated as the product of the relative optical air mass

_{g}*m*and the corresponding optical depths, according to the Beer–Lambert–Bouguer law [Eq. (2)]:

where *τ* is the optical depth of a certain air component; the subscript *x* denotes aerosol, Rayleigh extinction, ozone, and other absorption gases (CO_{2}, CH_{4}, or NO_{2}); *a* and *b* are coefficients of water vapor transmittance (Reagan et al. 1987); *w* is the vertical columnar water vapor; and *m _{w}* is the relative optical mass of water vapor. We note that water vapor transmittance follows Eq. (3) at a wavelength of 940 nm but follows Eq. (4) at the weak-absorption bands around 1020 or 1640 nm (Reagan et al. 1987; Smirnov et al. 2004). The relative optical mass of dry air, water vapor, and ozone can be calculated as follows (Kasten 1965; Robinson 1966):

where *θ _{z}* is the solar zenith angle,

*r*is the mean earth radius (6370 km), and

_{e}*z*

_{3}is the height of ozone layer concentration (km).

Equation (5) is applicable at a standard pressure of 1013.25 hPa at sea level; for other pressures, *m _{r}* needs to be modified by the factor

*P/P*

_{0}:

Because we do not have more information about the complicated properties of aerosols, such as their size, distribution, composition, and optical properties, the relative aerosol optical mass *m _{a}* is replaced by

*m*[Eq. (5)]. Furthermore, the uncertainty involved in the accurate determination of the concentrations of H

_{g}_{2}O, NO

_{2}, CO

_{2}, and CH

_{4}is greater than the uncertainty in the determination of their relative optical masses. Therefore, Eq. (5) is adopted for calculating the relative optical mass of these gas components and does not need to be modified according to the actual pressure (Iqbal 1983). The relative optical mass

*m*is replaced by

_{g}*m*for clarity. The AOD

*τ*is calculated after subtracting the optical depths of other components (Rayleigh extinction

_{a}*τ*, ozone

_{r}*τ*, water vapor

_{o}*τ*, and other gases

_{w}*τ*) from the total optical depth

_{g}*τ*. The AOD at a given wavelength

*λ*is given by Ortiz de Galisteo et al. (2009):

In fact, the filter band function contributes to the optical depth of each component, such as *τ*, *τ _{r}*,

*τ*,

_{o}*τ*, and

_{w}*τ*.

_{g}The three CE318s identified as #469, #478, and #681 are used in the case study to analyze the influence of filter band function on the accuracy of AOD retrieval. The center wavelength (CW), bandwidth (BW), and absorption gases of each CE318 are listed in Table 1.

As described in section 1, the uncertainty in AOD retrieval by the approximate AOD retrieval method versus the more rigorous method needs to be discussed first. The following two processes are used to calculate AOD: 1) The transmittance of spectral gases and the Rayleigh scattering transmittance are considered together to calculate the weighted transmittance by integrating the filter band function, as shown in Eq. (10). This process results in precise AOD retrieval, but it is extremely complicated because the radiative transfer model is required. 2) The weighted transmittance of gases and Rayleigh scattering are calculated separately, as shown in Eq. (11). The approximate AOD retrieval method has consistently been used in the past because it is easier to compute the band-weighted transmittance of the individual components. However, the total transmittance on the left side of Eq. (11) can only be approximated by the multiplication of band-weighted transmittances from each individual component:

The uncertainty in the approximate AOD retrieval method can be represented by

We use the line-by-line radiative transfer model (LBLRTM) (http://rtweb.aer.com/lblrtm_frame.html) to calculate *T*_{tot}, *T _{r}*,

*T*,

_{o}*T*,

_{g}*T*, and Δ

_{w}*τ*in Eqs. (10) and (11).

The second topic relates to the influence of the filter band function on the Rayleigh optical depth calculation, for which the equation suggested by Bodhaine et al. is relatively more accurate (Bodhaine et al. 1999) at a given wavelength. However, we mainly analyze the influence of the filter band function on the Rayleigh scattering optical depth by comparing the Rayleigh optical depth at the central wavelength with that of the filter band function under consideration. Consequently, similar conclusions will be drawn from analyzing the influence of the filter band function on the Rayleigh optical depth calculation, regardless of whether more or less accurate models are employed for the Rayleigh optical depth calculation at a given wavelength. Therefore, we employ a simplified equation to calculate the Rayleigh scattering optical depth (Hansen and Travis 1974):

where *P* is the actual pressure (hPa) and *A* is the elevation (km).

The difference between *τ _{r}* at a central wavelength and that of the filter band function under consideration is

The optical depth of O_{3} or NO_{2} in Eq. (9) is estimated from the vertical gas columnar concentration *U _{g}* and the spectral absorption coefficient

*k*(

_{g}*λ*), for a given wavelength, according to

For a channel of the sunphotometer, the effective absorption coefficient is calculated by a convolution of the filter band function *f*(*λ*), the gas spectral absorption coefficient *k _{g}*(

*λ*), the vertical gas columnar concentration

*U*, and the relative optical depth according to

_{g}We use the fine (~0.2 nm) spectral absorption cross sections from the Scanning Imaging Absorption Spectrometer for Atmospheric Cartography (SCIAMACHY) satellite spectrometer measurements prior to its launch on board the European Space Agency’s (ESA) *Environmental Satellite* (*Envisat-1*) in 2001 to calculate band-weighted optical depths (Burrows et al. 1998, 1999) and to analyze whether the band-weighted coefficients of these absorption gases change with the path gas concentrations (*U _{g} m*).

Because H_{2}O, CO_{2}, and CH_{4} have complicated absorption lines in the 1020- and 1640-nm spectral channels, we employ LBLRTM to calculate transmittances and to obtain optical depths.

## 3. Results and discussion

### a. Uncertainty in AOD retrieval

LBLRTM was used to perform numerical transmittance simulation calculations in order to evaluate the uncertainty in AOD retrieval by the approximate AOD retrieval method [Eq. (11)]. We applied the following standard atmosphere conditions in the simulation calculations using the LBLRTM (Clough et al. 2005): 1) the atmospheric model was the *U.S. Standard Atmosphere, 1976*; 2) the aerosol type was rural, and the visibility was 23 km; 3) the height of the sensor was 100 km, and the ground altitude was 0 km; 4) the view zenith angle was 0 (i.e., the view direction was vertical from the sensor to the ground); and 5) the values of the other parameters, such as the profiles of H_{2}O, CO_{2}, and O_{3}, were the defaults in LBLRTM. The AOD retrieval uncertainty Δ*τ* by the approximate AOD retrieval method was calculated for three CE318s [Eq. (12)]. The results are listed in Table 2.

Except for the 340-nm channel of #469, the absolute values of Δ*τ* were less than 0.001, which was evidently lower than the AOD measuring error obtained by CE318 (~0.01−0.02) (spectrally dependent, with a higher error in the UV region, e.g., ~0.02 for the 340-nm channel) (Eck et al. 1999), and can be ignored. Therefore, the approximate AOD retrieval method could unequivocally be adopted to retrieve the AOD accurately in these channels, except for the 340-nm channel of #469, in which Δ*τ* was approximately 0.007, which was about one-third of the AOD measuring error obtained by the CE318 in the UV spectrum (~0.02). This exception arose because the filter band function of the 340-nm channel of #469 had a relatively wider cutting spectral band; therefore, significant transmittances occurred because of the strong absorption of O_{3} in the Hartley bands (200−300 nm). The error in the approximate AOD retrieval method due to the questionable filter band design could be considered a systematic error. A comparison of transmittances in the 340-nm channels of #469, #478, and #681 is shown in Fig. 1. Compared with other sunphotometers, the spectral response range was limited for #681 at the 340-nm channel. The response values were given by CIMEL Electronique for #681 in the spectrum from 335 to 343 nm, which excluded values less than 0.03%. The limited spectral range used for 681 in this work may have occurred if the lower values had been eliminated by the manufacturer. The absorption cross section of O_{3} in the spectral region of 290−390 nm is shown in Fig. 2.

It was evident that the band filters of #478 and #681 for the 340-nm channel were well designed. The transmittances were almost greater than 0.01% at wavelengths of 335−343 nm and less than 0.01% outside this spectral region. However, the questionable CE318 for the 340-nm channel of #469 had significant transmittance values (more than 0.1% at most wavelengths) outside the 335−343-nm region. In addition, the ozone absorption cross sections increased almost exponentially with decreasing wavelength. Therefore, the weak transmittance of ozone in spectral regions below 330 nm would result in a highly uncertain AOD retrieval if the ozone band-weighted transmittance were calculated using Eq. (11). Therefore, the uncertainty in AOD retrieval by the approximate AOD retrieval method, compared with the AOD retrieval using the more rigorous method, could be ignored except for the poorly designed filter band functions, such as the 340-nm channel of #469. In such a case, the approximate AOD retrieval method would underestimate the AOD value by 0.007. This magnitude of error would greatly affect the AOD measurement in polar regions or in high-altitude stations (http://aeronet.gsfc.nasa.gov/), where the AOD can be as low as 0.05 in the 340-nm channel (South_Pole_Obs_NOAA of AERONET station in 2011). The Mauna Loa Observatory (MLO)–calibrated reference instruments have a very small estimated uncertainty of ~0.002–0.009 (Eck et al. 1999). Compared with that small uncertainty, the calculated Δτ using the approximate AOD retrieval method from Table 2 would be taken into account in more channels (e.g., the 340-nm channel and the 1020-nm channels of #469 and #478, which have an uncertainty of 0.0005).

Only the filter response data of more than 0.001% are plotted in Fig. 1. Values less than 0.001% may be caused by the limited measurement sensitivity for transmittances. Many applications require the blocking of transmittance for a UV filter less than 0.001% (Schott 2008; Anderson and Archard 2011). The measurement accuracy of the spectral transmittance can reach as high as 10^{−6}–10^{−8} in spectral ranges from 200 to 400 nm (Xue et al. 2007). Therefore, irregularities may have occurred in the band filters of #469 for the 340-nm channel due to the problematic design. Similar problems in the UV filter have been identified by CIMEL Electronique (Buis 2007).

Although the out-of-band response of the 675-nm channel of #681 was less than that of the 340-nm channel of #469 (Fig. 3), the response could explain the higher AOD retrieval uncertainty in the 675-nm channel of #681 by the approximate AOD retrieval method (Δτ = −0.001, Table 2). The transmittance value of the filter band function of #681 for the 675-nm channel was more than 0.01% at most wavelengths around 720 nm. In this spectral region, known as an overtone and combination band, the water vapor absorption was still significant (Liou 2002). The water absorption affected the other sunphotometers less because they had lower out-of-band responses.

In the above-mentioned simulation, the zenith angle equaled zero (i.e., *m* = 1). To further demonstrate that the uncertainty of AOD retrieval can be ignored under most conditions, the uncertainty Δ*τ*, which related to the filter band function, was also calculated under the condition of the relative optical mass *m* varying from 1 to 5. The result from the 340-nm channel of #469 was considered typical and is listed in Table 3.

For #469, Δ*τ* (−0.006 or −0.007) in the 340-nm channel was almost constant when the zenith angle changed from 0° to 78°. In other words, despite the variation in *m*, the AOD retrieval uncertainty of the approximate AOD retrieval method remained constant. Therefore, we concluded that the approximate AOD retrieval method could also be applied to all of the channels to retrieve AOD from the sunphotometer data, even for poorly designed channels such as the 340-nm channel of #469. In the case of a poorly designed channel, we note that a correction constant, which can be precomputed from the radiative transfer models, needs to be applied to the retrieved AOD when the approximate AOD retrieval method is used.

For the column densities of O_{3} and NO_{2} in the LBLRTM, we used the default values of 344 DU and 5.538 × 10^{15} molec cm^{−2}, respectively (http://rtweb.aer.com/lblrtm_frame.html). These two main components may cause uncertainty in the approximate AOD retrieval method due to their strong absorption in the 340-nm channel. Under actual atmospheric conditions, the column density of O_{3} from the equatorial region to the polar region may change from 250 to 500 DU in different seasons (Scannell et al. 2011). Furthermore, the mass of NO_{2} may increase from 10^{15} molec cm^{−2} under clean atmosphere conditions to 20 × 10^{15} molec cm^{−2} under polluted conditions (Castellanos and Boersma 2012). Therefore, the following extreme conditions were also used to further evaluate the uncertainty of AOD using the approximate AOD retrieval method: 1) the column density of O_{3} changed from 250 to 500 DU, while other conditions remained the same, as the above; and 2) the column density of NO_{2} changed from 10^{15} molec cm^{−2} to 20 × 10^{15} molec cm^{−2}, while other conditions remained the same, as described above. The values of Δ*τ* that were calculated for the 340-nm channel of #469 varied from −0.0061 to −0.0071 and from −0.0066 to −0.0065 under conditions 1 and 2, respectively. The calculated Δ*τ* changed only slightly when the column density of O_{3} or NO_{2} varied widely. Therefore, we could apply the approximate AOD retrieval method to the measurement data from the sunphotometers under all atmospheric conditions.

The detector spectral response functions were not considered in the above-mentioned discussion, owing to the lack of detailed response data from the detector. However, we used the detector spectral response function (Ortiz de Galisteo et al. 2009) to estimate the AOD retrieval uncertainty of the approximate AOD retrieval method in the 340-nm channel. The detector response value changed linearly from 300 to 400 nm and equaled 0.1 and 0.18 at the two endpoints. Therefore, the detector response function could be approximately represented by

A similar calculation was performed to obtain the uncertainty of AOD retrieval in the 340-nm channel. In Eqs. (10) and (11), *f*(*λ*) was replaced by the product of the filter band function and the detector response function [*f*(*λ*) *D*(*λ*)]. For the 340-nm channel of #469, Δ*τ* was −0.003, which was less than the uncertainty when only the filter band function was considered and consistent with Ortiz de Galisteo et al. (2009). Because of the lack of information about the detector response functions of the sunphotometer, we did not consider the detector response functions when calculating the AOD uncertainty in the other channels. However, it would not be a technical issue if the detector response function was known. We suggest that the sunphotometer should be examined periodically to measure its entire response function, including the filter and the detector, which is expected to change over time.

### b. Influence of filter band functions on Rayleigh optical depth calculation

We compared the weighted Rayleigh optical depths by integrating the filter band function and the Rayleigh optical depth only for the center wavelength. Hansen’s empirical equations [Eqs. (13) and (14)] were used in the calculation. When *m* = 1, the difference between these two Rayleigh optical depths reached a magnitude of ~10^{−3} in some channels (Table 4), such as the first three channels of #681 (0.002 87, 0.001 33, and 0.001 22) and the first channel of 478 (0.001 65). We also calculated the Rayleigh optical depth difference at different zenith angles; the difference increased linearly as the relative mass (*m*) increased. Figure 4 shows the difference for #469 in the 340-nm channel under different zenith angles. Thus, the filter band function should be considered when calculating the Rayleigh optical depth, particularly under large-zenith-angle observation conditions.

### c. O_{3} and NO_{2} band-weighted absorption coefficients

As expected, it was complicated to calculate the transmittances of O_{3} and NO_{2} using the filter band functions in Eq. (11). As such, the effective absorption coefficient in Eq. (16) was always adopted to calculate transmittances. A mostly constant was expected for a given CE318 channel and a given gas. Therefore, we needed to determine whether in Eq. (16) varied significantly when *U _{g}* or

*m*changed.

For Eq. (16), we considered *U _{g}* and

*m*together to evaluate whether would remain constant as the total path mass gas changed . The fine spectral absorption cross sections were from the SCIAMACHY results of the Molecular Spectroscopy and Chemical Kinetics Group. Figure 5 shows the absorption cross sections of O

_{3}and NO

_{2}at 273 K (Burrows et al. 1998, 1999). The coefficient was calculated for O

_{3}and NO

_{2}at 203, 223, 243, 273, and 293 K (http://www.iup.physik.uni-bremen.de/gruppen/molspec/databases/index.html). The path gas masses of O

_{3}and NO

_{2}fluctuated between 200 and 1100 DU and from 10

^{15}to 28 × 10

^{15}molec cm

^{−2}, respectively, which were very large ranges (Table 5).

The mean value and the relative standard deviation (RSD) of are shown in Figs. 6 and 7 for the channels of three different CE318s at different temperatures, where the mean value is the average for different gas concentrations and RSD is the ratio of the standard deviation to the mean value. Note that the channels with values of less than 0.001 are omitted in Figs. 6 and 7.

The following conclusions can be drawn based on the above-mentioned results:

The coefficient remained nearly constant for a certain temperature and channel, with a maximum difference of 1% except in the 340-nm channel of #469. The absorption of O

_{3}greatly increased below the 340-nm spectral region, where substantial transmittance occurred. This increased absorption resulted in of O_{3}that varied considerably when the path gas mass changed. This was not the case for NO_{2}, where the absorption did not change to the same extent as O_{3}, and of NO_{2}did not vary with the path gas concentration, even for the 340-nm channel of #469. The RSD of (less than 1%) contributed negligibly to the AOD uncertainty (less than 10^{−4}), except in the 340-nm channel.The coefficient differed at different temperatures; hence, it should be integrated across the actual gas profile. We simplified the process and calculated at the temperature closest to the average temperature of each gas. For example, 223 K was used for O

_{3}(Liou 2002). Table 6 lists the values of O_{3}calculated by the methods developed by Hu (2007), using moderate resolution atmospheric transmission version 4 (MODTRAN4), and the method used in this paper for #469. These two results were similar, showing that our calculation for O_{3}using SCIAMACHY data at 223 K was reasonable (note that the in the 340-nm channel could not be used because it varied with path gas mass, as previously mentioned).The coefficient was different for different sunphotometers. Therefore, needed to be calculated according to the actual filter band functions of different sunphotometers. Under actual atmospheric conditions, the optical depths of O

_{3}or NO_{2}in some channels can be very low, and the absorption effects can be ignored. We adopted the following calculation conditions to analyze which channels were actually responsible for O_{3}or NO_{2}absorption: 300 DU was the average column density of O_{3}(Scannell et al. 2011), 1 DU (26.9 × 10^{15}molec cm^{−2}) represented the NO_{2}concentration under highly polluted conditions (Castellanos and Boersma 2012), and an optical depth of less than 10^{−4}was ignored. Therefore, the absorption channels for O_{3}were 340, 440, 500, and 675 nm, and those for NO_{2}were 380, 440, and 500 nm, which is consistent with Table 1.

### d. H_{2}O, CO_{2}, and CH_{4} absorption optical depth

According to Table 1, H_{2}O had weak absorption at 1020 and 1640 (or 1610 nm), and CO_{2} and CH_{4} had weak absorption at 1640 (or 1610 nm). Because the concentration of CO_{2} reached 379 ppm in the Intergovernmental Panel on Climate Change (IPCC) report (Pachauri and Reisinger 2007), we used 380 ppm for CO_{2} in the present study. These three gases had complicated absorption lines in these channels; hence, the LBLRTM was used to complete the optical depths for these two gases (Smirnov et al. 2004). The optical depths of CO_{2} and CH_{4} could be calculated directly using the LBLRTM because the concentrations of these two gases are relatively stable. The absorption optical depths of water vapor in the 1020- and 1640-nm channels were linear fitted by water vapor volume according to the LBLRTM calculation (Smirnov et al. 2004):

where PW is the vertical columnar water vapor and *a* and *b* are the fitting coefficients.

The following conditions were established for the calculation: the zenith angle was zero (*m* = 1), the vertical columnar water vapor changed from 0.5 to 8 g cm^{−2}, and the atmosphere models used were the *U.S. Standard Atmosphere, 1976* (U.S.), midlatitude summer (MS), and midlatitude winter (MW).

Throughout the calculation, coefficients *a* and *b* were nearly the same under all three atmosphere models. The coefficients calculated under the U.S. model are shown in Table 7. Figure 8 shows the water vapor absorption optical depths for #681 changing with water vapor density under the U.S. model.

The optical depths of CO_{2} and CH_{4} in the 1640-nm channel were calculated for three CE318s using the LBLRTM under the three atmosphere models. To analyze the influence of elevation on the absorption optical depth, different elevation settings were used in the calculation. The results are listed in Table 8.

Table 8 shows that 1) the differences between optical depths due to different atmospheres were extremely small (the maximum value was ~10^{−4}) and 2) the optical depths did not vary widely with elevation; for example, a 1000-m elevation difference resulted in an ~10^{−3} error in optical depths. However, different instruments with different filter band functions varied widely in optical depths for CO_{2} and CH_{4}. For example, the absorption optical depth of #469 for CO_{2} was 0.018, and that of #681 was only 0.005. Therefore, the optical depths of CO_{2} and CH_{4} need to be calculated according to the filter band functions of the instrument.

These results demonstrate that the optical depths of H_{2}O in the last two channels of the CE318 and those of CO_{2} and CH_{4} in the last channel cannot be ignored, and they may be as large as the AOD measuring error of CE318 (~0.01−0.02). The most important point is that these optical depths need to be recalculated under actual conditions.

## 4. Conclusions

In this work, the uncertainty in AOD retrieval due to the filter band function was analyzed systemically by employing fine spectral absorption cross sections and the LBLRTM. The following conclusions can be drawn:

The Beer–Lambert–Bouguer law applies to AOD retrieval from sunphotometer data; that is, the approximate AOD retrieval method is valid for a known bandpass with a 10-nm spectral resolution. In other words, excluding the calibration uncertainty of the instrument or other uncertainties in atmospheric parameter estimation, the approximate AOD retrieval method can be applied in place of the rigorous method to retrieve AOD from sunphotometer data with an uncertainty of less than 10

^{−3}. Therefore, the Rayleigh scattering optical depth and the absorption gas optical depths or transmittances can be calculated individually by the integrating of the filter band functions.The filter band function is critically important in retrieving the AOD, particularly in the UV spectral region, such as at 340 nm. For example, the approximate AOD retrieval method would underestimate AOD by 0.007 for the 340-nm channel of #469. This underestimation occurs because the filter band function of #469 CE318 in the 340-nm channel has wide wings around the center wavelength. With wings shorter than 340 nm, the strong ozone absorption effect causes large uncertainty in AOD retrieval (Figs. 1 and 2). However, the uncertainty is nearly constant for a specific filter band function. In that case, we recommend the precalculation of the systemic AOD retrieval uncertainty based on the radiative transfer model.

Under large zenith observation conditions, nonnegligible errors may be introduced if the Rayleigh scattering optical depth is calculated directly at the central wavelength without including filter band functions. Errors of approximately 0.01 will appear in the 340-nm channel when the zenith angle exceeds 75°. Therefore, we suggest that filter band functions should be considered when calculating the Rayleigh scattering optical depth.

The calculations using different gas concentrations demonstrate that the effective absorption coefficients of O

_{3}and NO_{2}remain nearly constant when gas concentrations increase, except for #469 CE318 in the 340-nm channel. However, the gas absorption coefficients vary among different CE318s; hence, the band-weighted absorption coefficients need to be recalculated when a new CE318 is used in Eq. (16).The optical depths of H

_{2}O in the last two channels of the CE318 and those of CO_{2}and CH_{4}in the last channel cannot be ignored. These optical depths need to be recalculated according to actual conditions.

The response of the filter or the detector changes over time; thus, we suggest periodic measurement of the response of the sunphotometer. In addition, the uncertainty in AOD retrieval using the method described in the present paper needs to be reevaluated when the detector response functions are known; this will be a topic for our future research.

## Acknowledgments

This work has been supported by the National Natural Science Foundation of China (Grant 41001206), the National Natural Science Foundation of China (Grant 41271370), and the National Basic Research Program (973 Program) (Grant 2009CB723902).

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_{2}in the 231-794 nm range

_{3}in the 231–794 nm range

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