## Abstract

There are two important problems in the aerosol optical depth (AOD) retrievals from hourly/daily/monthly accumulated pyrheliometer data, that is, how to select a suitable cosine of the solar zenith angle (*μ*_{0}) and how to eliminate or minimize cloud effects. In this paper, two models of &ldquo=uivalent” *μ*_{0} are developed for two cases of hourly and daily pyrheliometer data. As shown in retrieval simulations, relative standard errors of 0.75-*μ*m-wavelength AOD solutions caused by the model uncertainties are, respectively, 0.77% and 1.93% in the two cases. Then, an approach is proposed to select AOD data from the total optical depths retrieved from hourly/daily pyrheliometer data probably containing cloud contribution. The approach uses additional cloud fraction, surface visibility, and relative humidity data to constrain the AOD upper limit and then to minimize the cloud effect on the monthly/yearly mean AOD estimations. The limit decreases with an increase of cloud fraction. Furthermore, in the case of the AOD retrieval using joint data of total monthly direct solar radiation and sunshine duration, two error factors, that is, the optically thin (solar transparent) cirrus cloud effect and the difference between radiation limits of pyrheliometer and sunshine recorder, are analyzed and an improved inversion method is proposed. The AOD selection approach is tested in two kinds of practical AOD retrieval comparisons. Eleven representative meteorological observatories are selected in the first kind of comparisons of AODs from hourly/daily/monthly pyrheliometer data during 1993–2000. It is found that the AOD selection approach is available to minimize the cloud effect for stable and reasonable monthly/yearly mean AOD estimations. In the case of hourly pyrheliometer data for 1995, the maximum relative deviation between yearly mean AODs, retrieved by using different cloud fraction limits from 0.0 to 60%, is less than 7.9% for all 11 sites. Another kind of comparison between AODs from sun photometer and pyrheliometer data also shows the effectiveness of the AOD selection approach to minimize the cloud effect.

## Introduction

In recent years, some authors have made great efforts to develop broadband extinction methods for retrieving aerosol optical depth (AOD) from pyrheliometer data and their applications (Gueymard 1998; Maxwell et al. 1991; Molineaus et al. 1998; Luo et al. 2001; Qiu 1998, 2001; Qiu and Yang 2000; Zhou et al. 1998), being impelled mainly by two reasons. At first, AOD is an important and convenient parameter for studies of atmospheric pollution, aerosol radiation–climate effect, atmospheric correction in remote sensing from space, and so on. On the other hand, there is a worldwide pyrheliometer network for broadband direct solar radiation observations, and high-quality historic broadband radiation data are available since the 1880s (Roosen et al. 1973; Stothers 1996). Therefore, a quantitative method to retrieve AOD from pyrheliometer data is very useful.

The broadband pyrheliometer method for deriving the Linke turbidity factor was first introduced by Linke (1922) and has been widely used in meteorological observations (WMO 1981). This factor is an appropriate measure to characterize the total extinction by the atmosphere (Kasten 1980). Then, two spectral turbidity definitions, that is, the Ångström (1929, 1964) and Schüpp (1949) turbidity coefficients, are introduced, which are related to the 1- and 0.5-*μ*m AOD, respectively. Both coefficients are usually derived under some assumptions of the relationship between AOD and wavelength. More recently, the broadband pyrheliometer method to derive the broadband or monochromatic AOD was developed. The broadband AOD (BAOD), derived from pyrheliometer data, was first proposed by Unsworth and Monteith (1972). Some recent efforts, contributed by Qiu (1998, 2001), Gueymard (1998), and Molineaux et al. (1998), developed pyrheliometer methods to determine monochromatic AOD. These methods all need to input the cosine of the solar zenith angle (*μ*_{0}). Pyrheliometer data are usually recorded in the format of hourly/daily/monthly accumulated radiation. During the accumulation period, the solar zenith angle is variable, and, thus, it is a problem to select a suitable *μ*_{0} for the AOD retrieval. Because of highly variable and random cloud cover distribution, there may often be a cloud effect on the accumulated direct solar radiation, and, hence, the Mie optical depth (MOD), containing the two components of AOD and cloud optical depth (COD), is often retrieved. Therefore, a more important problem in retrieving the AOD from the accumulated radiation is how to distinguish AOD from COD, or how to eliminate (or minimize) the cloud effect. The problem, however, has not been sufficiently studied yet. This paper is devoted to the two problems through the following three aspects of studies:

models of the “equivalent” cosine of the solar zenith angle for AOD retrievals from hourly/daily/monthly (accumulated) pyrheliometer data and corresponding inversion methods,

the available approach to select the AOD data from the retrieved MOD data, probably containing the two components of AOD and COD, and

comparative tests.

## Equivalent *μ*_{0} models and corresponding inversion methods

In this section, first the broadband extinction method developed by Qiu (2001) for AOD retrieval is introduced, followed by two equivalent *μ*_{0} models and corresponding algorithms for AOD retrievals from hourly/daily/monthly accumulated pyrheliometer data.

Assuming no forward-scattered solar radiation, direct solar radiation (DSR) *S,* detected by a pyrheliometer on the ground, can be expressed as

where *S*_{0,λ} is extraterrestrial solar irradiance at the *λ* wavelength, *μ*_{0} is the cosine of the solar zenith angle (*θ*_{0}), *λ*_{1} and *λ*_{2} are lower and upper spectral limits of pyrheliometer, and *T*_{a} and *T*_{m} are aerosol and molecular (including gas) spectral transmittances, respectively. For the pyrheliometer used in China, *λ*_{1} = 0.3 *μ*m and *λ*_{2} = 4 *μ*m.

In this study, the broadband extinction method developed by Qiu (2001) is used to retrieve AOD from the DSR. Table 1 gives the names and definitions of the main variables used in this text, including the method. The method used is as follows.

First BAOD (*τ*_{BAOD}) is determined as

where *S*_{0} is the solar constant, *R* is the ratio of the extraterrestrial solar radiation in the range of *λ*_{1} ≤ *λ* ≤ *λ*_{2} to the *S*_{0}, *t*_{m} is the broadband molecular (including gas) transmittance in the range, and *m*_{a} is the aerosol optical mass. A parameterization formula, presented by Qiu (2001), is used to calculate the transmittance *t*_{m}. In the formula, *t*_{m} is expressed in terms of four coefficients: the vertical total water vapor amount (*U*), the vertical total ozone amount (*X*), the surface atmospheric pressure (*p*), and the solar zenith angle (*θ*_{0}).

Then, the “effective” wavelength *λ*_{E}, at which AOD is equal to BAOD (*τ*_{BAOD}), using an approximate model from Qiu (2001) is determined as follows:

Here, four coefficients *λ*_{0}, *f*_{size}, *a,* and *b* are expressed in terms of the *τ*_{BAOD} from the last step, the water vapor amount *U,* and the Ångström exponent (*α*). The aerosol optical mass *m*_{a} is determined using a model, presented by Gueymard (1998), given by

Last the *λ*-wavelength AOD is determined. For the Junge-type aerosol, the *λ*-wavelength AOD can be expressed as

where *ν** is the Junge distribution index, *β* is the AOD at *λ* = 1 *μ*m (i.e., Ångström turbidity coefficient), and *α* is the Ångström exponent. Then, using *τ*_{BAOD} and *λ*_{E} from the last two steps, we have

A key problem of the pyrheliometer method for the AOD retrieval is the effect of the aerosol size distribution uncertainty. As shown in an earlier study by Qiu (1998), the 0.75-*μ*m-wavelength AOD retrieval from pyrheliometer data is less sensitive to the uncertainty, and if the Junge distribution with *ν** = 3 is selected in the retrieval, the AOD accuracy better than 5% can generally be expected for usual aerosol size distributions. On the other hand, the typical value of Junge parameter *ν** is usually equal to 3.0 for natural aerosol (Junge 1955). Therefore, *ν** = 3 (*α* = 1) is used in the 0.75-*μ*m AOD retrievals from hourly/daily/monthly pyrheliometer data.

In the above method, two parameterization models of the transmittance *t*_{m} and the effective wavelength *λ*_{E} are used. As shown in numerical simulations presented by Qiu (2001), the deviation of the parameterized *t*_{m} when compared with that from the Moderate Resolution Transmittance (MODTRAN) software (Berk et al. 1996) calculations is within ±0.48% in the range of *θ*_{0} = 0°–85°, and the error of the parameterized *λ*_{E} is within ±0.026 *μ*m for the Junge-type aerosol.

Because the ratio of *β* to *μ*_{0} is larger, the scattering radiation effect on the pyrheliometer measurements is significant. In practical AOD retrievals, a scattering radiation correction factor, named *L*_{C}, is introduced, and *S* in Eq. (2) is replaced by the ratio of *S* to *L*_{C}. The factor *L*_{C} indicates the ratio of the total radiation (DSR plus the scattered solar radiation detected by pyrheliometer) to the DSR. This paper uses a lookup table (LUT) approach to determine this factor. The discrete ordinate method radiative transfer (DISORT) code (Stamnes et al. 1988) is used to calculate the downward sky radiance and then to build the *L*_{C} table. In the table, the *U.S. Standard Atmosphere*, *1976*, Junge-type aerosol, with *ν** varying from 2.0 to 3.6 and *β* from 0.1 to 3.0, the aerosol refractive index of 1.5–0*i,* the pyrheliometer used in China, and the surface albedo of 0.1 are taken.

In the above retrieval method, the instantaneous *S* and *μ*_{0} data are required. Next, the method is developed to retrieve AOD from hourly/daily/monthly accumulated pyrheliometer data.

In the case of hourly/daily/monthly pyrheliometer data, we can obtain a mean value (*S*) of the accumulated radiation, given by

Which *μ*_{0} is suitable for AOD retrieval from the mean DSR is a problem that must be addressed. If the atmosphere is stable during the measurement duration Δ*t,* DSR is a continually varying function of time *t* (or *μ*_{0}), and there must be a time (*t*) within the duration at which the DSR (*S*_{t}) is equal to the mean DSR. If we let *μ*_{0,E} represent the cosine of the solar zenith angle at the time *t,* then we have

This *μ*_{0,E} is regarded as equivalent *μ*_{0}. If *μ*_{0,E} is determined, we can use it and the mean DSR to retrieve AOD according to Eqs. (2)–(6). Next, models of *μ*_{0,E}, corresponding with hourly/daily/monthly DSR data, are developed.

### Case of hourly DSR

Two assumptions are made for the *μ*_{0,E} model development: one is a stable atmosphere with constant gas and aerosol parameters during the DSR-accumulated period; the other is a Junge-type aerosol.

Using the two assumptions and many DSR calculations using MODTRAN, a model of *μ*_{0,E} is empirically derived as

where *μ*_{00} is the cosine of the solar zenith angle at the beginning time, and *μ*_{60} is the cosine at the 60th minute. Taking the *μ*_{0} and mean DSR (*S*) as *μ*_{0} and *S,* the AOD can be easily determined according to Eqs. (2)–(6).

This equivalent *μ*_{0} model is tested in the AOD retrieval simulations in which values of *α* and *β,* varying from 0 to 2.0 and 0.05 to 2.0, respectively, are selected as follows:

In these simulations, the maximum *θ*_{0} is 80°. The minimum *θ*_{0} in a day depends on latitude and date. Numerical simulations use six latitude values of 0°, 10°, 20°, 30°, 40°, and 50°N, and the 15th day of every month from January to June. The “true” hourly exposed DSRs are calculated using MODTRAN. There are a total of 28 600 sets of simulations using the *U.S. Standard Atmosphere, 1976.* As shown in the simulations, the relative standard error among 28 600 sets of 0.75-*μ*m-wavelength AOD retrievals, caused by the uncertainty in the equivalent *μ*_{0} model, is 0.77%, and the maximum error is 7.82%.

### Case of daily DSR

For the optically stable and horizontally homogeneous atmosphere, DSR decreases with an increase of the solar zenith angle. A pyrheliometer can detect DSR when it is larger than the detectable radiation limit (*S*_{Min}) of the pyrheliometer. So in the case of daily DSR, the pyrheliometer measurement duration (Δ*t*) in Eq. (7) needs to be defined. Let *μ*_{Max} stand for the maximum value of *μ*_{0} in a day and *μ*_{Min} indicate the minimum *μ*_{0}, corresponding with the *S*_{Min}. And let *t*_{am} (*t*_{pm}) represent the time when *S* = *S*_{Min} in the morning (afternoon). Now, the difference between *t*_{pm} and *t*_{am} is defined as the measurement duration, and this paper uses a model on the position of the sun presented by Spincer (1971) to calculate *t*_{am} and *t*_{pm} from the *μ*_{Min}, local time, longitude, and latitude.

According to daily DSR data from MODTRAN calculations, it is found that *μ*_{0,E} mainly depends on three parameters: *μ*_{Max}, *β,* and *U* (vertical total water vapor amount), especially the first two. Based on the property, a model of *μ*_{0,E} is proposed as follows:

Here, the coefficient *f*_{1} is introduced to characterize the dependence of *μ*_{0,E} with *μ*_{Max} and *β,* and the coefficient *f*_{2} is used to indicate the relationship between *μ*_{0,E} and *U.* Using the least squares technique and the *U.S. Standard Atmosphere, 1976, **f*_{1} is expressed in terms of *μ*_{max} and *β.* Using five atmospheric models in MODTRAN (i.e. tropical, midlatitude summer, midlatitude winter, subarctic winter, and *U.S. Standard Atmosphere, 1976*), *f*_{2} is empirically determined, being only relative to *U.* Last, *f*_{1} and *f*_{2} are expressed as

According to Eq. (2), the aerosol optical mass *m*_{a}(*μ*_{Min}), corresponding to the *μ*_{Min}, meets the following equation:

where *L*_{C} is the scattering radiation correction factor when *μ*_{0} = *μ*_{Min}. Combining Eq. (15) with Eq. (4), the *μ*_{Min} used in the above *μ*_{0,E} model can be determined if *β, **α, **τ*_{BAOD}, and *t*_{m} are known.

Evidently, *μ*_{0,E}, *L*_{c}, and *μ*_{Min} are all relative to *β.* Thus, an iterative algorithm for the AOD retrieval from the daily pyrheliometer data is proposed as follows:

set

*β*= 0,*μ*_{Min}= 0.087 (*θ*_{0}= 85°), and*μ*_{0,E}=*μ*_{Max}/2;calculate the measurement duration (Δ

*t*) and then mean DSR, responsible for the*μ*_{Min};calculate the transmittance

*t*_{m}and the scattering radiation correction factor*L*_{c}when*μ*_{0}=*μ*_{0,E}and*μ*_{0}=*μ*_{Min};letting

*μ*_{0}=*μ*_{0,E}and*S*=*S*, and using the*t*_{m}and*L*_{c}from the last step, determine*τ*_{BAOD},*λ*_{E}, and*β*according to Eqs. (2)–(6);using

*β,*the detectable radiation limit (*S*_{Min}) of the pyrheliometer, and the*L*_{c}(when*μ*_{0}=*μ*_{Min}) from the step 3, and combining Eq. (4) with Eq. (15), determine*μ*_{Min};using the

*μ*_{Min}and*β,*determine*μ*_{0,E}according to Eqs. (12)–(14); andrepeat steps 2–6 until a stable solution is reached.

In the above algorithm, *U, **X,* surface atmospheric pressure, and *α* are needed as input. The first three parameters are for calculating the transmittance *t _{m}*.

Some AOD-retrieval simulations using this equivalent (or daily mean) *μ*_{0} model are made for accuracy checks. There are a total of 4356 sets of simulations, which use 11 values of *α*(0–2.0) and 13 values of *β*(0.05–2.0), as shown in Eq. (11), five MODTRAN atmospheres (tropical, midlatitude summer, midlatitude winter, subarctic winter, and *U.S. Standard Atmosphere, 1976*), six latitude values of 0°–50°N, and the 15th day of every month from January to June. In these simulations, there is a limit in *μ*_{Max}, being larger than 0.2588 (*θ* ≤ 70°), and *S*_{Min} = 1.0 W m^{−2} is taken. In addition, there is not any error in all the input parameters except for an uncertainty in the equivalent *μ*_{0} determination. As shown in Table 2, the relative standard error and maximum error among the total 4356 sets of 0.75-*μ*m-wavelength AOD retrievals are, respectively, 1.93% and 8.96% when the equivalent *μ*_{0} model is used. When the daily mean *μ*_{0} model is used, the standard and maximum errors are up to 23.7% and 43.9%, respectively.

### Case of monthly DSR

The monthly pyrheliometer data always contain a cloud contribution. A main difficulty of using the data for the AOD retrieval is how to eliminate the cloud effect. Luo et al. (2001) proposed a method in which the monthly mean DSR is expressed as

and the daily mean *μ*_{0} at the 15th day in a month, being equal to *μ*_{Max}/2, is treated as monthly mean value of *μ*_{0}.

In 1981, WMO (1983) defined 120 W m^{−2} as the detectable radiation limit of a sunshine recorder, but the radiation limit detected by a pyrheliometer is much smaller, usually being 1 W m^{−2}. This difference can result in an overestimated *S* and then an underestimated AOD solution in Luo et al.'s method.

On the other hand, optically thin cirrus clouds can be solar transparent. A larger optical depth has a smaller contribution to the accumulated DSR for the same sunshine duration. Thus, the thin cirrus cloud can result in an underestimated *S* and, then, an overestimated AOD retrieval in Luo et al.'s method.

Clearly, the above-mentioned two factors can cause very different errors in Luo et al.'s AOD solution. As a result, both underestimated and overestimated solutions may be obtained. In addition, as pointed out above, if *μ*_{0} = *μ*_{Max}/2 is taken for the AOD retrieval from daily DSR, a large AOD error may be obtained. For these reasons, a modified method is proposed. First, the daily mean *μ*_{0} (= *μ*_{Max}/2) is replaced by the present equivalent *μ*_{0} for the daily DSR. Second, using two modification coefficients (marked as *f*_{1} and *f*_{2}), monthly mean DSR is expressed as

In Eq. (18), *h*_{sun} and *h*_{pyr} are monthly mean values of daily sunshine duration and pyrheliometer measurement duration, calculated assuming a cloudless but aerosol-loading atmosphere, in which AOD is equal to the monthly mean AOD. In the calculations of *h*_{sun} and *h*_{pyr}, this paper takes the 15th day in a month for the *μ*_{0} determination. If detectable radiation limits of sunshine recorder and pyrheliometer are the same, *h*_{sun} = *h*_{pyr}, and, thus, *f*_{1} = 1.0. Because of the smaller radiation limit of pyrheliometer, *h*_{sun} < *h*_{pyr}, and, hence, *f*_{1} < 0, resulting in a decrease of *S* and a corresponding increase of the AOD retrieval.

In Eq. (19), the parameter *T*_{C}, named the cloud transparent factor, is introduced to indicate the transparent characteristic of the thin cirrus cloud. Before presenting the definition of *T*_{C}, a relative parameter, that is, the total cloudless days (*N*_{Clear}) in a month, is defined as

In Eq. (20), *C*_{Fm} is the monthly mean cloud fraction, varying between 0 and 1.

If there is is not any solar-transparent cirrus cloud, the ratio of the total monthly sunshine duration (*t*_{sun,m}) to *N*_{Clear} leads to the mean daily sunshine duration in the cloudless *N*_{Clear} days, being equal to *h*_{sun} in the perfect case of no error in the *t*_{sun,m} and *N*_{Clear} determinations. Based on this analysis, this paper defines the cloud transparent factor *T*_{C} as

In the perfect case of no error in the *T*_{C} determination, if there is no transparent cloud, *T*_{C} = 0, then *f*_{2} = 1.0, implying no transparent cloud correction. In the presence of the transparent cloud, theoretically we have *T*_{C} > 0 and then *f*_{2} > 0, resulting in an increase of *S* and a corresponding decrease of the AOD retrieval. In fact, because of some error factors, including cloud cover and sunshine duration uncertainties, the above analysis may not be strictly suitable.

As shown in the latter section, the two proposed modifications to *S* can result in more reasonable AOD retrievals.

In Eq. (19), the coefficient *B*_{C} = 0.45 is empirically found through comparative tests presented in the latter section. The rest of the presentation is how to determine *h*_{sun} and *h*_{pyr} and retrieve AOD. An iterative algorithm for this purpose is proposed as follows:

set

*β*= 0,*μ*_{Min}= 0.087 (*θ*_{0}= 85°),*μ*_{0,E}=*μ*_{Max}/2,*f*_{1}= 1.0, and*f*_{2}= 1.0;calculate

*C*_{Fm}and*N*_{Clear}from cloud cover records;calculate the measurement duration and the mean DSR, being responsible for the

*μ*_{Min}and the 15th day in a month;calculate the transmittance

*t*_{m}and the scattering radiation correction factor*L*_{c}when*μ*_{0}=*μ*_{0,E}and*μ*_{0}=*μ*_{Min};letting

*μ*_{0}=*μ*_{0,E}and*S*=*S*, and using the*t*_{m}and*L*_{c}from the last step, determine*τ*_{BAOD},*λ*_{E}, and*β*according to Eqs. (2)–(6);using the

*β,*radiation limit*S*_{Min}, and*L*_{c}(when*μ*_{0}=*μ*_{Min}) from step 5, and combining Eq. (4) with Eq. (15), determine*μ*_{Min}for the 15th day in a month;using the

*μ*_{Min}and*β,*determine*μ*_{0,E}according to Eqs. (12)–(14);using

*t*_{m}and*τ*_{BAOD}, calculate*h*_{sun}(sunshine duration) and*h*_{pyr}for given radiation limits of the sunshine recorder and pyrheliometer according to Eq. (4) and Eq. (15);calculate

*f*_{1}and*f*_{2}according to Eqs. (18)–(19) and Eq. (21); andrepeat steps 4–10 until a stable solution is reached.

Later on, the above-modified method is called Qiu's method to distinguish it from Luo et al.'s method, and AODs by the two methods are marked as *τ*_{a,mQ} and *τ*_{a,mL}, respectively.

## AOD selection approach

In the last section, the method to determine the monthly mean AOD from monthly DSR data was introduced. In this section, an approach is developed for representative and reasonable estimation of the monthly/yearly mean AOD from hourly/daily DSR data. The development is based on the following facts:

If there are no clouds between the sun and the pyrheliometer, there is no effect of clouds on DSR measurement. Therefore, some pyrheliometer data detected on a cloudy day can be used in AOD retrieval if there is an available approach to select only the portions in which the direct solar beam was cloud free.

There are several times of cloud cover (or fraction) over most meteorological observatories in the world. Even if there are no clouds during recording time, there may be cloud effects on hourly or daily pyrheliometer measurements owing to highly variable and random cloud cover distribution.

There may not be one or more days without any clouds during the daytime in a month over some regions. If all pyrheliometer data detected in cloudy days are not used, representative AOD estimation may not be obtained.

Owing to the above facts, MOD, containing the two components of AOD and COD, is often retrieved from hourly/daily pyrheliometer data; a problem is how to select the AOD from MODs that probably contain a cloud contribution. Based on the following theoretical analysis and comparative tests presented in section 4, an AOD selection approach is proposed in which two key points are (a) constraints to the AOD upper limit using additional visibility, humidity, and cloud fraction data and (b) empirical coefficients used in the constraints and determined through comparative tests.

Assume that aerosol extinction coefficient (*σ*_{a}) decreases with height (*z*) in an exponent law, such as

where *σ*_{a}(0, *λ*) is the *λ*-wavelength aerosol extinction coefficient at the surface level and *z*_{a} is a constant. Under this assumption and according to definition of visibility (*V*), the 0.75-*μ*m-wavelength AOD (marked as *τ*_{a,vis}) can be expressed as

In Eq. (23), *α* is the Ångström exponent.

Here, *z*_{a} indicates the decreasing rate of aerosol extinction coefficient, being regarded as the “standard” height of the aerosol layer, at which ln[*σ*_{a}(0, *λ*)/*σ*_{a}(*z*_{a}, *λ*)] = 1. The smaller *z*_{a} is, the larger the decreasing rate, and, thus, the smaller *τ*_{a,vis} is for the same visibility, implying a larger contribution to *τ*_{a,vis} from the lower atmosphere. If MOD (*τ*_{Mie}) retrieved from DSR includes the contribution from a cloud layer in the sky, it can result in a MOD retrieval larger than *τ*_{a,vis}. In other words, the larger ratio of *τ*_{Mie} to *τ*_{a,vis} implies the larger possibility of the cloud contribution to *τ*_{Mie}. Therefore, this paper constrains the ratio to less than a certain value, named the constraint coefficient (*f*_{c}), to minimize the cloud effect. For the AOD (*τ*_{a}) retrieval from hourly/daily DSR data, the following two constraints are used for the AOD selection from MOD,

In the above expressions, *C*_{F}, *V,* and *H _{u}* are hourly (daily) mean cloud fraction, surface visibility, and relative humidity for the AOD retrievals from hourly (daily) DSR data. The monthly mean cloud fraction is

*C*

_{Fm}, and

*C*

_{up}is the upper limit of cloud fraction taken for the AOD selection. Later, the AOD, derived using this AOD selection approach, is marked as

*τ*

_{a,hv}and

*τ*

_{a,dv}for two cases of hourly/daily DSR data, respectively.

The factor *d* in Eqs. (25)–(26), controlling the variation range of the constraint coefficient *f*_{c}, is equal to 0.8 and 0.4 for two cases of hourly and daily DSR data, respectively. The true aerosol extinction coefficient profile is often extremely variable (Spinhirne et al. 1980; Qiu et al. 1988); thus, there is a large variation range of the standard height *z*_{a}. Based on this property and the below-presented comparative tests, *d* = 0.8 is empirically selected for the hourly DSR case. In the *d* selection, the constraint coefficient *f*_{c} varies between 1 and 1.8. This implies that the AOD upper limit, decreasing with an increase of the cloud fraction, can be 1–1.8 times larger than *τ*_{a,vis}. If the retrieved MOD is beyond the limit, it is not used in the AOD estimation. In the daily DSR case, *d* = 0.4 is confirmed through many comparisons between *τ*_{a,dv} and *τ*_{a,hv}.

As shown in Eq. 27, the upper cloud fraction constraint of *C*_{up} = 0.8 *C*_{Fm} is proposed and used in later practical retrievals. If there are almost clear (cloudless) days in a month, *C*_{Fm} ≅ 0, and then *C*_{up} ≅ 0, implying that the AOD data are from days that are almost clear. In the case of *C*_{Fm} ≈ 100%, *C*_{up} ≈ 80%. This implies that the MOD, retrieved for the case of cloud fraction within 80%, may be used in the AOD estimation. It is emphasized that the DSR data from the cloudy days can be selected in the AOD retrieval, but they can be not used in the *z*_{a} determination. Only the DSR data from the very clear days with *C*_{F} = 0 are selected for the *z*_{a} determination in this paper, which is important to minimize the cloud effect on the monthly/yearly mean AOD estimation.

On a foggy day there is usually low visibility and high humidity in the surface layer. A dust storm event can also result in a very low visibility, but the humidity is usually low. For the low-visibility case, the *τ*_{a,vis}, used in the first constraint, can be very large, resulting in a failure to eliminate a fog effect on the AOD estimation. Thus, the second constraint using the humidity is introduced to minimize the fog effect on AOD estimation.

The first constraint is usually much more important than the second one.

## Comparative tests

In this section, two kinds of comparisons for validation tests are analyzed. First, comparisons among AODs retrieved from hourly/daily/monthly accumulated pyrheliometer data and by the above AOD selection approach are made. Second, comparisons between AODs from pyrheliometer and sun photometer data are analyzed.

The sun photometer has a 1° field of view and eight interference filters with central wavelengths at 0.4, 0.44, 0.52, 0.612, 0.67, 0.78, 0.88, and 1.03 *μ*m. Controlled by a microcomputer, it can work with functions that actively track the sun's center, transforming filters, sampling, and storing the data on time. The sun photometer is calibrated by the Langley method at the Astronomy Observatory (on top of a hill) of the Chinese Academy of Sciences, and a calibration accuracy better than 0.0069 is estimated for the 0.78-*μ*m wavelength (Qiu 1998).

The pyrheliometer has a spectral response of 0.3–4 *μ*m, a field of view angle of 6°42′, and detectable limit of 1 W m^{−2}. It is calibrated using a first-class normal incidence pyrheliometer (NIP) once every 2 yr. The calibration error is within ±1%.

Next, these two kinds of comparisons are analyzed.

### Comparisons among AODs from hourly/daily/monthly pyrheliometer data

This kind of comparison selects 11 representative meteorological observatories. As shown in Fig. 1, these observatories cover most areas of China. As shown in Table 3, over the 11 observatories the sea level varies from 4 (Shanghai) to 2808 m (Geermu); yearly mean cloud fraction, sunshine duration, and AOD in 1995 vary from 41.4% (Beijing) to 83.6% (Chengdu); 2.81 h (Chengdu) to 8.63 h (Geermu); and 0.222 (Geermu) to 0.737 (Chengdu), respectively. The cloud fraction is determined from the cloud cover data, from the visual observation by meteorological personnel, and recorded in terms of 11 levels from 0 to 10 (100% cloud fraction) with a step of unity. Furthermore, Fig. 2 shows monthly mean cloud fraction and sunshine duration over Beijing and Chengdu in 1995. As shown in Fig. 2, in Beijing the monthly mean cloud fraction in February is 22%, being the lowest, and the sunshine duration in May is 11.2 h, being the longest. In Chengdu, the cloud fraction is higher than 60% for all months, with its highest value of 95% in January, and the sunshine duration in January is only 0.2 h, being the shortest. The 11 sites are representative, covering wide aerosol and cloud conditions. Table 3 also shows the yearly mean aerosol standard heights (*Z*_{a}) and its standard deviation over the 11 sites in 1995. In this paper, only AOD data of *τ*_{a} ≤ 1.0 from the cloudless days are used to determine the height according to Eq. (23); yearly mean *Z*_{a} is used in the AOD selection approach. The height is larger than 2 km for all 11 sites except Shengyang, which is larger than the height (1.94 km) from MODTRAN's rural aerosol model, with the 23-km visibility. Over Urumqi, the height is up to 3.72 km, implying many aerosol particles in the upper atmosphere. Over Shengyang, the height is 1.47 km, implying serious aerosol pollution in the lower atmosphere. The *z*_{a} standard deviation changes between 0.69 and 1.56 km.

The present AOD retrievals from pyrheliometer data require an input of hourly/daily/monthly mean cloud cover, visibility, and relative humidity. Cloud cover, visibility, and relative humidity at meteorological observatories in China are recorded at 0200, 0800, 1400, and 2000 LT each day. The daily mean cloud cover computes the mean value of the three records at 0800, 1400, and 2000 LT, and the hourly mean cover is obtained from the three records through linear interpolation. The monthly mean cloud cover takes the mean value of all daily covers in a month. The same procedure yields the hourly/daily/monthly mean visibility and humidity.

In addition, column precipitable water vapor and ozone amounts are needed to calculate molecular (including gas) broadband transmittance. In this paper, the total ozone amounts in Beijing and Kunming are determined from monthly mean Dobson observation data. The ozone amounts at the rest of the sites use monthly mean satellite-based Total Ozone Mapping Spectrometer (TOMS) data (McPeters et al. 1996). The total vertical water vapor amount (*U*) is determined from local surface water vapor pressure (*e*), using the following empirical expression (Yang and Qiu 2002):

This expression selects site-dependent empirical coefficients (*a*_{0}, *a*_{1}, and *a*_{2}), and its standard error is in the range from 0.225 (Urumqi) to 0.65 cm (Shanghai). The AOD retrieval error, caused by a ±0.5-cm uncertainty in *U,* is within ±0.023 as *U* ≥ 1 cm (Qiu 1998).

The Ångström exponent (*α*) is also needed. The typical value of the Junge parameter *ν** is usually equal to 3.0 (*α* = 1.0) for natural aerosols (Junge 1955). Owing to a lack of the true *α* data, and considering the weaker effect of an uncertainty in *α* on the 0.75-*μ*m AOD retrieval from pyrheliometer data (Qiu 1998), *α* = 1.0 is selected for all 11 sites.

Because the pyrheliometer data from the cloudy days are used in the AOD retrieval, stability of the AOD selection approach under different cloud fraction limits would be very important. Next, the stability of monthly/yearly mean AODs, retrieved from the hourly/daily DSR and by the AOD selection approach, is analyzed according to Figs. 3–5 and Tables 4–5.

Figure 3 shows the number of yearly total days, having the AOD data retrieved from hourly/daily DSR data and by the AOD selection approach, for six cases of cloud fraction upper limits. In Fig. 3, and later in Figs. 4–5, the digits and 0.8 *C*_{m} (0.8 times of monthly mean cloud fraction *C*_{m}) in the legend indicate the cloud fraction limits (*C*_{up}). For the hourly DSR case (see Fig. 3a), if the AOD only from cloudless days is used, there are more than 110 days of AOD data for six sites in northern China [from the first station (Harbin) to the sixth (Zhengzhou) in Fig. 3a]; but only 25 days in Chengdu. If *C*_{up} = 0.8*C*_{m} is taken, it is more than 132 days for all 11 sites except for Guangzhou and Chengdu, where it is 83 and 51 days, respectively. For the daily DSR case (Fig. 3b), the day number is smaller. As *C*_{up} = 0, it is less than 58 days for all 11 sites, especially only 2 days over Chengdu. As *C*_{up} = 0.8*C*_{m}, the day number gets much larger; over Chengdu, it is up to 21 days. Therefore, an available approach using a large value of *C*_{up}, but eliminating cloud effect, is very significant for such sites as Chengdu to yield more representative AOD estimation.

Figure 4 shows monthly mean AODs, retrieved from hourly pyrheliometer data in 1995 and, by the AOD selection approach, for the three sites of Beijing, Geermu, and Chengdu. As shown in Figs. 4a and 4b, over Beijing and Geermu the AOD estimations are stable for all months, being insensitive to the variation of the cloud fraction constraint *C*_{up}. The maximum difference among AODs from the six selections of *C*_{up} = 0, 10%, 20%, 40%, 60%, and 0.8C_{m} is 12.9% for Geermu case, and 14.2% for Beijing. It is noted that the difference can be large for some months over the Chengdu site (Fig. 4c). Here, there are no AOD data from the cloudless days (*C*_{up} = 0) in the 2 months of February and September.

As shown in Fig. 5, the yearly mean AOD estimations from hourly DSR data in 1995 are very stable. The difference among AODs from the different cloud fraction constraints of *C*_{up} = 0, 10%, 20%, 40%, 60%, and 0.8*C*_{m} is less than 7.9% for all 11 sites, including Chengdu. For the 11-site mean yearly mean AOD, the difference is only 0.42%. Over such sites as Beijing and Kunming, the AOD has a weakly increasing trend when *C*_{up} increases, and over Guangzhou it has a weakly decreasing trend. The increasing (decreasing) trend can be explained if the mean AOD in the cloudy days is larger (smaller) than that in the clear (no cloud) days. Over Chengdu, a little of the AOD data in some months can result in an unstable monthly mean AOD estimation (see Fig. 4c), but the yearly mean AOD estimations are also stable (Fig. 5). Therefore, the present AOD selection approach is particularly available for yearly mean AOD estimations.

Table 4 compares the 11-site mean yearly mean AODs by taking *C*_{up} = 0 with those by *C*_{up} = 0.8*C*_{m} in the two cases of hourly/daily DSR data. For the hourly DSR case, the maximum difference between AODs by *C*_{up} = 0 and *C*_{up} = 0.8*C*_{m} is 2.98% for every year during 1993–2000, showing a very good agreement. For the daily DSR case, there is not any AOD data from *C*_{up} = 0 over Chengdu in 2000, thus, there is a blank in this year. From 1993 to 1999, the maximum difference between AODs by *C*_{up} = 0 and *C*_{up} = 0.8*C*_{m} is 9.03%. It is interesting that as *C*_{up} = 0.8*C*_{m}, there is a very good agreement between both AODs from hourly and daily DSR data, the maximum deviation being only 2.5%. It is worth pointing out that the above-mentioned selection of *d* = 0.4 [in Eq. (26)] for the daily DSR case is based on this agreement.

Table 5 compares yearly mean AODs derived using the AOD-selection approach from hourly DSR data over six sites in 2000 with those without the selection. As shown in Table 5, the AOD retrievals by the selection approach weakly depend on the cloud fraction limit (*C*_{up}). The maximum difference among AODs from eight choices of *C*_{up} (0–60%, and 0.8*C*_{m}) is less than 9.8% for every site among Beijing, Geermu, Kunming, Guangzhou, Shanghai, and Chengdu; especially for the first four sites, the difference is less than 2.7%. However, the AODs (second value of each site, Table 5) without the selection can be greatly overestimated, especially for a large *C*_{up}. It is noted that as *C*_{up} = 0, the AODs without the selection can also be overestimated. This may be due to the fact that the hourly mean cloud fraction, used in the AOD-selection approach, is determined from observations of 0800, 1400, and 2000 LT, using linear interpolation.

Clearly, the present AOD-selection approach is very significant for minimizing the cloud effect on monthly/yearly mean AOD estimations from hourly pyrheliometer data. In the approach, the AOD upper-limit constraint from visibility and data, defined by Eq. (24), is particularly important to guarantee stability. Based on stability and reasonability of the *τ*_{a,hv} estimation, it is regarded as a standard in the next analysis to test the other three sets of AODs (*τ*_{a,dv}, *τ*_{a,mQ}, and *τ*_{a,mL}), as shown in Tables 5–6 and Fig. 6. In Fig. 6 and Tables 5–6, *C*_{up} = 0.8*C*_{m} is used for the *τ*_{a,hv} and *τ*_{a,dv} estimations.

Figure 6 compares four sets of yearly mean AODs in 1995, that is, *τ*_{a,hv} and *τ*_{a,dv} retrieved from hourly/daily DSR data, *τ*_{a,mI} and *τ*_{a,mL} from monthly DSR data, and by Qiu's and Luo's methods, respectively. There is a good coincidence between *τ*_{a,hv} and *τ*_{a,dv} for all 11 sites. Over Chengdu, there is the maximum absolute deviation between them, equal to 0.072 (14.3% relative standard error). As far as the 11-site mean yearly mean AOD is concerned, the deviation is 0.002 (0.4%). Again, see Fig. 6, there is the better agreement between *τ*_{a,mQ} and *τ*_{a,hv} than with *τ*_{a,mL} for all 11 sites. Over the Geermu and Kunming sites, *τ*_{a,mL} > *τ*_{a,hv}, and over the remaining nine sites, *τ*_{a,mQ} < *τ*_{a,hv}. The maximum deviation is up to 39.6%. The larger the AOD, the smaller the DSR. In the larger AOD case over such sites as Chengdu, the total daily DSR is smaller and, hence, contains a relatively larger contribution from radiation less than 120 W m^{−2}, resulting in an underestimated AOD retrieval by Luo's method. Over Geermu, the AOD is smaller; correspondingly, the AOD retrieval by Luo et al.'s method is overestimated.

Table 6 compares four sets of 8-yr (1993–2000) mean yearly mean AODs (*τ*_{a,hv}, *τ*_{a,dv}, *τ*_{a,mQ}, and *τ*_{a,mL}) over the 11 sites. The deviation between *τ*_{a,dv} and *τ*_{a,hv} is in the range from 0.003 (0.7% relative deviation) over Harbin to 0.056 (9.3%) over Zhengzhou. Over the five sites of Beijing, Shenyang, Zhengzhou, Shanghai, and Wuhan, *τ*_{a,dv} < *τ*_{a,hv}, and over the remaining six sites, *τ*_{a,dv} > *τ*_{a,hv}. The difference between the 11-site mean 8-yr mean yearly mean *τ*_{a,dv} and *τ*_{a,hv}, is only 0.008. The deviation between *τ*_{a,mQ} and *τ*_{a,hv} ranges from 0.008 (3.6%) over Geermu to 0.054 (16.4%) over Kunming, and the difference between the 11-site *τ*_{a,dv} and *τ*_{a,hv} is 0.013, also showing good agreement. However, the deviation of *τ*_{a,mL} with *τ*_{a,hv} is large, the maximum deviation being 0.14 for the Zhengzhou site.

Until now, the Ångström exponent of *α* = 1 is used in all retrievals. Its uncertainty can cause an AOD error, owing to the fact that the scattering radiation correction and the AOD retrieval from pyrheliometer data are relative to *α.* Next, the error is analyzed and shown in Table 7, which gives the yearly mean AODs over Geermu and Chengdu in 2000, derived from hourly DSR data and using *α* = 0, 0.5, 1.0, and 1.5, respectively. The smaller the *α,* the stronger the near-forward-scattering radiation effect on the pyrheliometer measurement, especially when the AOD is large. Correspondingly, a larger *α* can result in a smaller AOD retrieval. It is estimated from Table 7 that the error in the yearly mean AOD is within ±0.015 for a ±0.5 uncertainty of *α* in the Chengdu case with the large AOD. In the Geermu case, with the smaller AOD, the error is smaller.

The aerosol standard height *z*_{a} is an important factor, introduced in the AOD selection approach. Some comparative retrievals from hourly DSR data in 1995, using *z*_{a} = 2 km and the *z*_{a} values listed in Table 3, are made to check the AOD error, caused by an uncertainty of *z*_{a}. An overestimated *z*_{a} can result in an overestimated AOD retrieval. For the nine sites, except Wuhan and Urumqi, the deviation of the *z*_{a} in Table 3 to 2 km is within ±0.5 km, and the yearly mean AOD difference, caused by the deviation, is within ±0.02. Over Urumqi, the *z*_{a} deviation is 1.72 km, and the corresponding AOD difference is up to 0.048. It is expected that the AOD error, caused by a 0.5-km uncertainty of *z*_{a}, is within 0.02.

### Comparisons of AODs retrieved from pyrheliometer and sun photometer data

The AODs from sun photometer and pyrheliometer data are comparatively analyzed according Fig. 7 and Table 8. Figure 7 compares 10 days of the 0.75-*μ*m-wavelength AODs during March–April of 1995, retrieved from hourly pyrheliometer data and using *α* = 1, with those from sun photometer measurements. Table 8 gives the cloud fraction and visibility data at 0800, 1400, and 2000 LT, the Ångström exponent (*α**) from sun photometer measurements, and six kinds of the daily mean AODs in the 10 days. The index *α** is derived from the eight-wavelength AODs, measured by the sun photometer. The 0.75-*μ*m-wavelength AOD from the sun photometer, marked as *τ*_{a,sun} is derived by linear interpolation from AODs at *λ* = 0.67 and 0.78 *μ*m. The *τ*_{a,hv} and *τ*_{a,dv} in Table 8 are derived from hourly/daily pyrheliometer data and by using an AOD selection approach; *τ*_{a,h} and *τ*_{a,d} are from the same pyrheliometer data but do not use the selection. There are two selections of *α* = 1 and *α* = *α** for the *τ*_{a,hv} retrieval, and only *α* = 1 is taken for other AOD retrievals from pyrheliometer data. In the *τ*_{a,dv} column of Table 8, “Yes” is given because *τ*_{a,dv} = *τ*_{a,d}, and “No” is given because *τ*_{a,dv} fails to meet the criterion of the AOD selection approach. For the 3 days of 26 March, 6 April, and 14 April, sun photometer measurement stopped in the afternoon owing to a cloud or strong dust storm event. Thus, there are two values in the *τ*_{a,hv} column of Table 8 for the 3 days. The first value is the mean AOD, corresponding with the sun photometer measurement period, and the second corresponds to the longer pyrheliometer measurement period. Two values of the mean pyrheliometer AODs in Figs. 7d, 7f, and 7h have the same meaning.

There is usually a very strong daily variation of AODs and a good agreement between AODs from the sun photometer and pyrheliometer. As shown in Fig. 7a, from 0820 to 1015 LT on 21 March the sun photometer's AOD increases from 0.22 to 0.58, then from 1030 to 1230 LT it has a decreasing trend, and then from 1230 to 1530 LT it again increases from 0.34 to 0.64. The pyrheliometer's AOD has a similar but smooth daily variation trend. The difference between the two daily mean AODs is only 1.3%. For the case of 4 April (Fig. 7f), AODs vary between 0.96 and 2.04, and there is very good agreement between the sun photometer and pyrheliometer AODs during 0730 and 1630 LT. The difference between the daily mean AODs is 3.7%. In some days, for example, 25 March (Fig. 7c) and 26 March (Fig. 7d), there is a larger and systematic deviation between both AODs, but they also have a similar daily varying trend, and the maximum difference of daily mean AODs is less than 18.3%.

In four such days, 21 and 26 March and 11 and 14 April,

*τ*_{a,h}and*τ*_{a,hv}are considerably different, usually corresponding with the cloud-day case. In the morning of 14 April there are no clouds, but in the afternoon the cloud fraction ranging from 90% to 100% was recorded and the sun photometer measurement stopped. The AOD from pyrheliometer data strongly increases with the maximum value of 3.31 (Fig. 7i). The AOD data in the afternoon containing the cloud contribution fails to meet the criterion of the AOD selection approach; thus, it is not used in AOD (*τ*_{a,hv}) estimation. As a result,*τ*_{a,hv}(1.16) is close to*τ*_{a,sun}(1.269), measured in the cloudless morning.There are a total of 2 days (26 March and 14 April) having “No” in the

*τ*_{a,dv}column. On these days*τ*_{a,d}fails to meet the criterion of the AOD selection approach, owing to a higher cloud fraction. As shown in Fig. 7i, there is a strong effect of clouds on AOD retrievals during the afternoon of 14 April, resulting in a large value of*τ*_{a,d}= 1.528. If the*τ*_{a,d}is used, an overestimated AOD must be obtained.On 4 April there are no clouds during the entire daytime, and very serious aerosol pollution results in a very low surface visibility of less than 4 km (see Table 8), and, thus, has very large AODs with the maximum value of 2.04 (see Fig. 7f). Because all AOD data from hourly pyrheliometer data meet the criterion of the AOD-selection approach, we have

*τ*_{a,hv}=*τ*_{a,h}(1.278).On 6 April, a very strong dust storm occurred over the Beijing area causing a very quick decrease of surface visibility (from 30 km at 0800 LT to 1 km at 1400 LT) and a very strong increase of AODs, varying from 0.22 at 0800 LT to 3.24 at 1400 LT (Fig. 7g). During the day, no cloud cover was reported; thus, the AOD value of

*τ*_{a,h}= 1.592 is selected as*τ*_{a,hv}. The first digit (0.483) in the*τ*_{a,h}column is the mean AOD in the sun photometer measurement period (0700–1100 LT), being close to*τ*_{a,sun}(0.524). It is noted that*τ*_{a,d}and*τ*_{a,dv}are the same (0.929), but evidently smaller than*τ*_{a,hv}(1.592). Because of a very strong increasing trend of AODs, dominant contributions to daily DSR data are from that in the morning, and so the*τ*_{a,d}from the data is responsible mainly for the smaller AOD in the morning. Evidently, the AOD from hourly exposed DSR is usually more reliable than that from daily DSR.Values in the last two rows of Table 8 are total 10- and 9-day (except 6 April) mean values of daily AODs. Because of a strong dust storm event, sun photometer measurements stopped in the afternoon of 6 April, and, thus, a smaller

*τ*_{a,sun}, only indicating the AOD in the morning, is obtained. For the 9-day mean AOD, differences of*τ*_{a,hv}and*τ*_{a,dv}from hourly/daily pyrheliometer data with*τ*_{a,sun}from the sun photometer are 1.8% and 5.4%, respectively, but*τ*_{a,h}and*τ*_{a,d}from the same pyrheliometer data are evidently larger than*τ*_{a,sun}, owing to the cloud effect. Therefore, the AOD selection approaches are very significant in minimizing the cloud effect and then yielding reasonable AOD estimation.The Ångström exponent (

*α**) from sun photometer measurements changes between 0.18 and 0.94, with a mean of 0.49. Using a larger wavelength index can underestimate the scattering radiation effect on pyrheliometer measurements and, thus, result in an underestimated AOD retrieval. As a result, the*τ*_{a,h}by*α*=*α** is larger than the*τ*_{a,h}by*α*= 1.0 (see Table 8). For days such as 26 March (Fig. 7d), the*τ*_{a,h}by*α*= 1 is systematically larger than*τ*_{a,sun}from the sun photometer. If*α*=*α** is used, a little larger*τ*_{a,h}can be obtained, which cannot explain the deviation. The deviation may be caused mainly by an error in sun photometer or pyrheliometer calibration.

## Conclusions and discussion

When hourly/daily/monthly accumulated direct solar radiation data are used in monthly/yearly mean aerosol optical depth estimations, an approach to eliminate cloud effect is important. In this paper, an approach is proposed to select the AODs from the retrieved Mie optical depth data probably containing cloud contribution. Physically, the approach is based upon two aspects. At first, surface visibility is a convenient indicator of aerosol pollution in the surface layer, and aerosol particles in the lower atmosphere layer usually contribute to a dominant part of the total column AOD; thus, there is often good correlation between AOD and visibility (Qiu et al. 1997). The second physical basis is that the aerosol extinction coefficient in the troposphere usually has a decreasing trend with the height. Based on this property, this paper defines the aerosol standard height (*z*_{a}) to characterize the decreasing rate. In the case of the exponent-type extinction coefficient profile, the AOD (*τ*_{a}) is equal to the height multiplied by the surface extinction coefficient (*σ*_{0}), that is, *τ*_{a} = *z*_{a}*σ*_{0}. Using this relationship and the surface visibility for the *σ*_{0} determination, a constraint criterion [Eq. (24)] is proposed to restrict the AOD upper limit and then minimize the cloud effect. It is emphasized that the AOD limit decreases with an increase of the cloud fraction. The MOD beyond the limit is not selected for the AOD estimation. Another key point is that the *z*_{a} is determined using the AOD and visibility data in the case of very clear (zero cloud fraction) days. The true aerosol extinction coefficient profile in the troposphere is usually not an exponent-type profile, but the statistically mean aerosol profile in the troposphere can be fitted by an exponent function. For example, if the MODTRAN rural aerosol extinction profile with the 23-km visibility is fitted in terms of the exponent function, the absolute (relative) standard error of the fitted extinction coefficients in the altitude range of 0–10 km is 0.0022 km^{−1} (19.2%). In addition, the relationship between AOD and visibility may be very variable, but the monthly/yearly mean visibility can better characterize the mean AOD (Qiu et al. 1997). Therefore, the AOD-selection approach is available for the monthly/yearly mean AOD estimations from hourly/daily accumulated DSR data.

As shown in two kinds of comparative tests, the AOD-selection approach can minimize cloud effects to yield stable and reasonable AOD estimations. In the case of hourly pyrheliometer data, the maximum relative deviation between yearly mean AODs, retrieved using different cloud fraction limits from 0.0 to 60%, is less than 7.9% for all 11 sites in 1995. As shown in comparisons of AODs from pyrheliometer and sun photometer data, if the selection approach is not used, the daily mean AOD retrieved from pyrheliometer data may be overestimated for the cloud day, but the AODs from same pyrheliometer data using the selection approach have good agreement with those from sun photometer data.

Because the ratio of the total monthly DSR to the total monthly sunshine duration is treated as monthly mean DSR in the AOD retrievals, like Luo et al.'s method, there are some error sources. One is the different radiation limits of pyrheliometer and sunshine recorders. Another is the effect of solar-transparent thin cirrus clouds. For these reasons, this paper proposed some modifications to Luo et al.'s method. Comparative tests show that the modified method can result in more reasonable AOD estimations.

During hour/daily period, *μ*_{0} is highly variable, especially in the day case. So, it is important to select a suitable *μ*_{0} for AOD retrieval from hourly/daily/monthly accumulated DSR data. Two models of &ldquo=uivalent” *μ*_{0} are developed. In the case of daily pyrheliometer data, typical AOD error caused by the model uncertainty is 1.93%, showing a very good accuracy, and much worse accuracy may be obtained if a daily mean *μ*_{0} is used instead of the equivalent *μ*_{0}.

The AOD is often strongly variable during the daily period. The smaller the AOD, the larger the DSR. The daily variation of AOD may result in an underestimated AOD solution, owing to larger contributions to the hourly/daily accumulated DSR from the atmosphere with a smaller AOD during same accumulated period. Therefore, the AOD retrieved from hourly (in a shorter period) accumulated DSR is usually more reliable than the AOD from daily or monthly DSR. So, the monthly/yearly mean AOD retrievals from hourly DSR data are regarded as a “standard” to test other AODs from daily/monthly DSR data. The factors *B*_{C} (= 0.45) in Eq. (19) and *d* (= 0.4 for the daily DSR case) in Eq. (26) are confirmed through the comparative tests. If the variable values of *B*_{C} and *d* are selected for different sites, more reasonable AOD retrievals would be obtained.

Because of a larger view field of the pyrheliometer, the scattering radiation effect on pyrheliometer measurements may be significant, close to the Ångström exponent (*α*). The smaller *α* is, the stronger the near-forward-scattering radiation effect on the pyrheliometer measurement, especially when the AOD is large. Correspondingly, a larger *α* can result in a smaller AOD retrieval. It is estimated that the error of the yearly mean AOD determination is within ±0.015 for a ±0.5 uncertainty of *α.* In addition, when there is a dust layer with a larger optical depth in the sky, but the surface visibility is higher, the dust optical depth may be treated as the cloud optical depth not being used in the AOD estimation.

## Acknowledgments

This research was supported by the National Development Project of Fundamental Research (G1999045700) and the National Natural Science Foundation of China (Grant No. 40175009).

## REFERENCES

## Footnotes

*Corresponding author address:* Jinhuan Qiu, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China. jhqiu@mail.iap.ac.cn