## Abstract

Three kinds of “visible” cloud optical thickness *τ*—matching shortwave direct, global, and scattering solar irradiances (*I*_{ds}, *I*_{gs} and *I*_{ss})—are defined, which are marked as *τ*_{d}, *τ*_{g}, and *τ*_{s}, respectively. It is found from radiation calculations that a ratio of *I*_{ss} to *I*_{gs} in the small-*τ* case has a unique characteristic: strong sensitivity to *τ* but weak sensitivity to the cloud scattering phase function. On the basis of this characteristic, a method to retrieve *I*_{ss}-equivalent *τ*_{s} from the ratio is proposed. This method is validated by way of simulation and application tests, in which the Discrete Ordinate Radiative Transfer model (DISORT) is used to calculate irradiances. As shown in simulations with *τ* < 2, there may be unrealistically negative or grossly overestimated *τ* values from *I*_{gs}, owing to the difference between *τ*_{s} and *τ*_{d}, while the new method can lead to a very good agreement of *τ*_{s} retrieval with its input. Furthermore, this method is used to retrieve small *τ* from the pyrheliometer and pyranometer measurements in Lhasa during 2006. It is found that *τ* retrieved from *I*_{gs} was often negative because of cloud inhomogeneity, while the application of the new method resulted in stable yet reasonable *τ*_{s} values. The *I*_{ss} calculations using 1293 sets of *τ*_{s} retrievals fit well into the *I*_{ss} determinations from pyrheliometer and pyranometer measurements with an annual-mean deviation of 0.18%, but the deviation was raised to 46.4% when using *τ*_{g} retrievals.

## 1. Introduction

Clouds are a major radiation modulator in the atmosphere, playing a key role in the energy balance of the Earth–atmosphere system. Clouds are also a most uncertain radiation–climate forcing factor, largely owing to a lack of a long-term systematically compiled statistical database of cloud microphysical parameters based on reliable observations. Among these parameters, the cloud optical thickness *τ* is key to charactering the atmospheric radiation field. Since the 1880s, a densely populated global network of ground-based stations has been developed to measure direct, global, and scattering solar irradiances (*I*_{ds}, *I*_{gs}, and *I*_{ss}, respectively) using relatively inexpensive but reliable pyrheliometers and pyranometers (Roosen et al. 1973). Motivated by the need for *τ* observation, some authors have addressed their efforts on the development of *τ* retrieval methods from broadband pyranometer measurements (Boers 1997; Boers et al. 2000; Qiu 2006) and narrowband and broadband scattering solar irradiance measurements (Barnard and Long 2004; Dong et al. 1997; Min et al. 2003; Leontieva and Stamnes 1996). These methods are based on the assumption of plane-parallel cloud and atmosphere. However, they are generally not applicable to optically thin or broken cloud fields. Boers et al. (2000) estimated errors in *τ* retrieval from pyranometer data, in which the radiative transfer is calculated using the independent pixel approximation. As shown in the study, the negative *τ* retrieval bias increases with reduced cloud cover, and for *τ* < 5, retrievals are mostly invalid. A major factor causing negative *τ* is the cloud inhomogeneity, which will be explained based on physical principles in the text below. Retrieval of *τ* from scattering solar irradiance measurements as reported in the above-cited references, are normally valid for *τ* > 5 in the overcast sky condition. Note that *I*_{ss} is not a monotonic function of *τ*. It increases with increasing *τ* for smaller *τ*, but decreases with increasing *τ* for larger *τ*. Therefore, for a given *I*_{ss} value, two *τ* values are obtained. Thus, the retrieval is not unique. For *τ* > 5, *I*_{ss} monotonously decreases with increasing *τ*, and so a unique *τ* can be retrieved from *I*_{ss}. The broadband *I*_{ss} is usually measured by the pyranometer with a shading band or disc. An inherent problem is the obstructed light correction for the real *I*_{ss} determination, which demands aureole radiance profile. As presented by DeVore et al. (2012), the aureole radiance profile is a dominant variable in the cirrus case, owing to the strong forward-scattering peak for large ice particles, and hence the scattering correction of pyrheliometer measurement varies by a factor of 2, depending upon the pyrheliometer viewing field and the particle effective size. The viewing field of the shadowing band or disc is usually larger than the pyrheliometer field, and so there may be more difficult, yet uncertain, shadowing correction in the thin cloud case, especially for the band case.

The solar radiative parameters for thin cirrus strongly depend upon *τ* and scattering phase function (SPF), while the SPF is closely correlated with particle effective radius (marked as *R*_{eff}) (King et al. 1997; Yang et al. 2000). The uncertainty in the SPF (or *R*_{eff}) is a major weakness in the small-*τ* retrieval using solar radiation measurements. Baum et al. (2005) examined bulk single-scattering properties of ice clouds—including single-scattering albedo, asymmetry factor, and phase function—for a set of 1117 particle size distributions. The examination results indicate that there are substantial differences in the bulk-scattering properties of ice clouds formed in areas of deep convection and those that exist in areas of much lower updraft velocities. Comstock et al. (2007) reported on recent progress in the remote retrieval of microphysical properties for optically thin cirrus. They found that characterizing ice crystal shape and particle size distribution for optically thin cirrus is still challenging. They compared cloud visible optical thickness retrieved by 14 different combinations of instruments and algorithms and found that the *τ* measurements varied by up to 1.5 orders of magnitude for thin (*τ* < 0.3) and by nearly an order of magnitude for thicker (0.3 < *τ* < 5.0) cirrus. Therefore, a reliable method for the small-*τ* retrieval should have weak sensitivity to cloud SPF.

Taking into account the aforementioned problems in the small-*τ* retrieval from ground-based solar irradiance measurements, we propose a new method to retrieve *τ* using a ratio *η*_{M} of *I*_{ss} to *I*_{gs}. On the basis of radiative transfer calculations, we found that using *η*_{M} has some very significant advantages for the small-*τ* retrieval. First, the ratio is a monotonically increasing function of *τ*, and so a unique, yet stable, *τ* retrieval can be achieved. In the small-*τ* case, *I*_{ss} increases with increasing *τ*, and *I*_{gs} decreases with increasing *τ*, and so *η*_{M} monotonically increases with *τ*. Note that *I*_{gs} equals a sum of horizontal-plane direct solar irradiance *I*_{hds} and *I*_{ss}, and *I*_{hds} is not sensitive to SPF variation. Thus, both *I*_{ss} and *I*_{gs} have the same variation trend with the SPF (or *R*_{eff}), implying that their ratio must be nearly independent of the choice of SPF. This unique characteristic of *η*_{M} (its strong sensitivity to *τ* and weak sensitivity to SPF) is the principal physical basis of the new method. Section 2 presents this characteristic of *η*_{M} according to radiation calculations and the retrieval method. The method is validated by way of simulation and application tests. Sections 3 and 4 report results of the two groups of tests, including analysis of the effect of SPF uncertainty on *τ* retrievals.

## 2. Methodology

In the shortwave solar spectrum, *τ* is not sensitive to wavelength and one can speak of the “visible” optical thickness (DeVore et al. 2012; Comstock et al. 2007). This paper defines three visible cloud optical thicknesses *τ*_{d}, *τ*_{g}, and *τ*_{s}, to match broadband *I*_{ds}, *I*_{gs}, and *I*_{ss}, respectively. The *I*_{ds} value depends on only *τ*_{d} along the direct solar incident direction, which can be expressed in the plane-parallel atmosphere as follows:

where *μ*_{0} is the solar zenith angle cosine, *S*_{0}(*λ*) the extraterrestrial solar irradiance at the *λ* wavelength, *λ*_{1} and *λ*_{2} are lower and upper spectral limits of the broadband solar spectrum, and *τ*_{λ,Clear} is the wavelength total optical thickness of all atmospheric components except for cloud. Optical thicknesses of these components are determined using the same approaches with those as in Qiu (2006). In this paper, *λ*_{1} = 0.3 *μ*m and *λ*_{2} = 4.0 *μ*m.

The terms *I*_{ss} and *I*_{gs} are related to the whole-sky *τ* distribution. The defined *τ*_{g} indicates *I*_{gs}-equivalent *τ* under the assumption of horizontally uniform *τ*. Using *τ*_{g}, *I*_{gs} is expressed as

where *T*_{Total} is the wavelength dependent total transmittance. Similarly, *I*_{gs} is given by

where *T*_{Sca} is the scattering (diffuse) transmittance, and *τ*_{s} marks *I*_{ss}-equivalent *τ*.

It is found from Figs. 1 and 2 that *η*_{M} has some very significant characteristics for small-*τ*_{s} retrieval. The *I*_{ss} and *I*_{gs} values, as well as their ratio *η*_{M}, are calculated using the Discrete Ordinate Radiative Transfer model (DISORT). Figure 1 shows *η*_{M} versus *τ* (ranging from 0 to 10) at *μ*_{0} = 1.0 and 0.5 for the 645-nm ice SPF (marked as MODIS20 in Fig. 2) with *R*_{eff} = 20 *μ*m, the same as that used by the Moderate Resolution Imaging Spectroradiometer (MODIS) cloud algorithm (King et al. 1997). Figure 2 compares *η*_{M} calculations using nine sets of 645-nm ice SPFs, including the MODIS20; four hollow column SPFs; and four plate SPFs (Yang et al. 2000). The column10, -20, -50, and -100 labels denote column SPFs with *R*_{eff} = 10, 20, 50, and 100 *μ*m, respectively, and the plate10, -20, -50, and -100 labels indicate plate SPFs with *R*_{eff} = 10, 20, 50, and 100 *μ*m, respectively. Other input parameters in Figs. 1 and 2 include 1) column H_{2}O and O_{3} amounts of 1.0 and 0.35 cm, respectively; 2) aerosol optical thickness (AOT) of 0.15 at 750-nm wavelength, and Ångström exponent of 1.0; and 3) broadband surface albedo of 0.15. According to Figs. 1 and 2, four significant characteristics of *η*_{M} in the thin-*τ* case are presented as follows:

*η*_{M}is a monotonic-increasing function of*τ*, and it is close to unity (suffering from saturation) as*τ*> 5 and*τ*> 2 for*μ*_{0}= 1.0 and*μ*_{0}= 0.5.*η*_{M}is sensitive to*τ*(see Fig. 1), because*I*_{ss}increases, but*I*_{gs}decreases with increasing*τ.*For example,*η*_{M}increases 308% when*τ*increases from 0 to 1.0 and*μ*_{0}= 1.0, whereas*I*_{ds}and*I*_{gs}decrease 172% and 3.9%, respectively. For the same*μ*_{0}, when*τ*increases from 0 to 0.02,*η*_{M}increases 9.0%, while*I*_{ds}and*I*_{gs}decrease 2.0% and 0.1%, respectively. Clearly,*η*_{M}is much more sensitive to*τ*than both*I*_{ds}and*I*_{gs}, and thus it is particularly suitable for small-*τ*retrievals.*η*_{M}is weakly sensitive to SPF (or*R*_{eff}) variation (see Fig. 2), because both*I*_{ss}and*I*_{gs}have the same variation trend with SPF or*R*_{eff}. The maximum difference among*η*_{M}calculations using nine SPF models (*R*_{eff}ranging from 10 to 100*μ*m) is only 2.78% for any*τ*from 0 to 20. When the column100 with*R*_{eff}= 100*μ*m is used instead of the column10 with*R*_{eff}= 10*μ*m and*τ*= 1.0,*I*_{ss}and*I*_{gs}increase 4.79% and 4.23%, respectively, while the*η*_{M}variation is much smaller, being 0.56%.The

*τ*retrieval from*η*_{M}should characterize*I*_{ss}data, owing to the very weak sensitivity of*I*_{gs}to*τ*variation.

On the basis of the above four characteristics, this paper develops a method to retrieve *I*_{ss}-equivalent *τ*_{s}. Because of cloud inhomogeneity, *τ*_{s} may be much different from *τ*_{d,} and hence there may be large uncertainty in the *τ*_{s} retrieval if the original *I*_{ss}-to-*I*_{gs} ratio is used directly. To avoid the uncertainty, a modified ratio *η*_{M} is proposed as follows:

where *I*_{hds} = *μ*_{0}*I*_{ds} and the modification factor *f*_{M} is equal to a ratio between two *μ*_{0}-direction transmittances at *τ* = *τ*_{s} and *τ*_{d}. Hence, the modified *I*_{hds} and *I*_{gs} are responsible for *τ*_{s}, but not for *τ*_{d}. The final *τ*_{s} value is obtained through an iterative process:

Input a set of

*I*_{ds},*I*_{gs}, and*I*_{ss}data, and retrieve*τ*_{d}from*I*_{ds}. The input methods for the*τ*simulation and application tests are presented later.Set

*f*_{M}= 1.0 (no modification).Determine

*η*_{M}according to Eq. (4), using*I*_{ds},*I*_{ss}, and*f*_{M}*.*Retrieve

*τ*_{s}using the lookup table (LUT) approach, where the calculated*I*_{ss}-to-*I*_{gs}ratio is equal to the value of*η*_{M}derived in the last step.Recalculate

*f*_{M}[Eq. (5)] and then*η*_{M}[Eq. (4)] using*τ*_{s}and*τ*_{d}*.*Repeat steps 4 and 5 until the variation of

*τ*_{s}is smaller than a convergence criterion.

The maximum and mean iterative numbers for simulation test results to be presented in the next section are 54 and 7.01 for the convergence criterion of 0.001. If the convergence criterion is changed to 0.005, then the mean iterative number is 4.08.

In simulation tests, exact *I*_{ds} and *I*_{ss} values in step 1 are derived through separate radiation calculations using *τ*_{d} and *τ*_{s} inputs, respectively, and *I*_{gs} is determined as *μ*_{0}*I*_{ds} + *I*_{ss}. The *I*_{ss} calculation uses the DISORT algorithm (Stamnes et al. 1988) based on the assumption of horizontally uniform cloud optical properties. In application tests, pyrheliometer and pyranometer measurements at the Lhasa site are taken as the *I*_{ds} and *I*_{gs} inputs, and *I*_{ss} is determined as

where *L*_{C} is a scattering light correction factor of the pyrheliometer data, equaling 1 plus a ratio of the scattering light flux into pyrheliometer field to the direct solar irradiance; *L*_{C} is determined through radiative calculations using the *τ*_{d} retrieval in step 1. The correction is closely dependent on *R*_{eff}, and the effect of *R*_{eff} uncertainty on *τ* retrievals is examined in section 4. There are in situ *I*_{ss} measurements at the Lhasa site using the China-made DFY-3 pyranometer with a shadow band. The shadowing correction factor is determined using an empirical model (China Meteorological Administration 1993). The band has about an 18.6° shadowing field in the width. There may be a very large uncertainty in the shadowing correction on a cloudy day, and so the *I*_{ss} determination using Eq. (6) from pyrheliometer and pyranometer data is treated as the “measured” *I*_{ss} in this paper. The uncertainties in both pyrheliometer and pyranometer data can result in errors in the *I*_{ss} determination. This error will be a continual problem in the future.

The LUT approach was presented by Nakajima and Nakajima (1995) in cloud microphysical property retrievals from National Oceanic and Atmospheric Administration (NOAA) Advanced Very High Resolution Radiometer (AVHRR) measurements, and it was used by Qiu (2006) in *τ* retrievals from pyranometer data. Here we use the same approach for all *τ* retrievals, and the irradiance (flux) and aureole radiance data in the LUT are calculated by DISORT using 8-stream and 60-stream models, respectively. The radiance data are used to yield a scattering light correction of pyrheliometer data. An eight-layered atmosphere is assumed, in which the first layer is the altitude range from 10 to 30 km, the second from 6 to 10 km, and the other six layers located from 0 to 6 km with a 1-km step. The main input parameters in the LUT are 1) column H_{2}O and O_{3} amounts; 2) molecular scattering optical thickness, determined according to the site altitude above sea level (0–4 km); 3) *μ*_{0} (0.2–1.0); 4) 750-nm AOT (0.02–1.0), and Ångström exponent of 1.0; 5) broadband surface albedo (0.05–0.6); 6) *τ* (0–200); and 7) MODIS ice and water cloud SPF. The LUT file size is about 38 Mbit.

## 3. Retrieval simulation tests

There may be a large difference between *τ*_{d} and *τ*_{s}, owing to effects of cloud inhomogeneity. If there is a layer of thin broken cloud in the sky (*τ*_{s} > 0) and there is no cloud around the sun (*τ*_{d} = 0), the true *I*_{gs} may be larger than the calculated value of *I*_{gs} for *τ* = 0, and a negative *τ* may be retrieved from the pyranometer data. On the other hand, if a thick patch of cloud shields the sun (*I*_{hds} ≈ 0 at a large *τ*_{d}), and there is no cloud away from the sun (*τ*_{s} ≈ 0), then the true *I*_{gs} may be very small, and the retrieved *τ* from the pyranometer data will be unrealistically large. The difference between *τ*_{s} and *τ*_{d} is a major factor causing invalid *τ*_{g} retrieval from *I*_{gs}. As shown in Fig. 3, two sets of *τ* distributions are input in simulations: in Fig. 3a, *τ*_{d} = 0.2 and *τ*_{s} changing from 0 to 2.0 and in Fig. 3b, *τ*_{s} = 0.2 and *τ*_{d} from 0 to 2.0. Other input parameters except for *τ* are the same as those given in Fig. 1.

As shown in Fig. 3a, when the same *τ*_{s} and *τ*_{d} (i.e., 0.2) are input, both *τ*_{s} and *τ*_{g} retrievals compare well with the input value. When *τ*_{s} input is smaller (or larger) than *τ*_{d}, significantly different *τ*_{g} retrievals are obtained. For instance, when input *τ*_{s} = 0.0 but *τ*_{d} = 0.2, the retrieved *τ*_{g} is as large as 2.14, and when input *τ*_{s} ≥ 0.3 > *τ*_{d}, unreasonably negative *τ*_{g} is obtained. The negative retrieval is determined through interpolating two *I*_{gs} data points at *τ* = 0.0 and 0.02. All *τ*_{s} retrievals are exactly equal to their input values ranging from 0 to 1.5. Note that there is a 5.1% underestimation in *τ*_{s} retrieval (1.89) when its input is 2.0. The same conclusions can be drawn from Fig. 3b. When the input *τ*_{d} changes from 0.0 to 2.0, all *τ*_{s} retrievals compare well with their input of 0.2, while negative (or overestimated) *τ*_{g} values are obtained when input *τ*_{d} < *τ*_{s} (or *τ*_{d} > *τ*_{s}). When *τ*_{d} = 2.0 (*τ*_{s} = 0.2) is input, the *τ*_{g} value jumps to 13.39—a gross overestimate. Furthermore, if *μ*_{0} = 0.5 and *μ*_{0} = 0.3 are taken instead of *μ*_{0} = 1.0 in the simulations, the exact *τ*_{s} retrievals are obtained when *τ*/*μ*_{0} < 1.4. Overall, the difference between *τ*_{s} and *τ*_{d} can result in invalid *τ*_{g} retrievals, and satisfactory *τ*_{s} retrievals are obtained with our method in the smaller-*τ* case.

Furthermore, the effect of ice SPF on the new *τ* retrieval is tested through simulations using three sets of MODIS ice SPFs with *R*_{eff} = 10, 20, and 50 *μ*m (King et al. 1997). The SPF of *R*_{eff} = 20 *μ*m is used to calculate the exact irradiances. There are 40 sets of simulations, covering *τ* from 0 to 2.0, *μ*_{0} = 1.0, and *μ*_{0} = 0.5. The maximum deviation of *τ*_{s} retrievals using the SPF of *R*_{eff} = 10 or 50 *μ*m from those of *R*_{eff} = 20 *μ*m is 0.0076. The very small deviation is due to the above-analyzed weak sensitivity of *η*_{M} to *R*_{eff}.

## 4. Application tests

In the application tests, we compare *τ*_{g} and *τ*_{s} retrievals from hourly-mean *I*_{ds} and *I*_{gs} records at a Chinese meteorological observatory (29°40′N, 91°8′E) in Lhasa during 2006. There are often thin cirrus clouds over the observatory located on the Tibetan plateau. The *I*_{ds} and *I*_{gs} values are measured by the DFY-3 pyrheliometer and pyranometer, made by the Changchun Meteorological Instrument Company, China. They are calibrated using international standard instruments (PMO-6 absolute radiometer) once every 2 years (China Meteorological Administration 1993). The pyrheliometer field of view is 6°42′, and the effect of scattering light on *I*_{ds} data is corrected by using the *τ*_{d} retrieval. Except for cloud SPF, other parameters (including aerosol, H_{2}O, molecular scattering, ozone, surface albedo, etc.) are input using the same approaches as those in Qiu (2006). AOT and aerosol scaling height (ASH) on the cloud-free day are retrieved from the pyrheliometer data using a broadband extinction method (Qiu 1998; 2003), and AOT on the cloudy day is determined using the monthly-mean ASH and surface visibility record (Qiu 2006). The 645-nm ice and water cloud SPF in the MODIS cloud algorithm (King et al. 1997) is used, and ice- and water-cloud layers are located at 4–5 km and 1–2 km (over the ground), respectively. The *τ* retrievals, using ice- and water-cloud SPF, are marked *τ*_{i} and *τ*_{w}, respectively. Low- and total-cloud fractions are observed by meteorological personnel at the Chinese meteorological observatories. Using the observation records, the final *τ* is determined as

where *x* is the ratio of low cloud cover to total cloud cover.

There are in total 2961 hourly-mean *I*_{gs} measurements for the case of *μ*_{0} > 0.3 and *I*_{gs} > 10 W m^{−2} in Lhasa during the year 2006. The selection criteria of *τ*_{s} retrievals are 1) *τ*_{s}/*μ*_{0} < 1.4, 2) *I*_{ss}/*I*_{gs} < 0.9, and 3) *μ*_{0} > 0.3. The former two criteria are chosen according to the results of the last section’s simulations, guaranteeing a *τ* retrieval below 2. Under these selections, there is a total of 2000 sets of positive *τ*_{s} retrievals, spanning 305 days. Unreasonably negative *τ* retrievals are sometimes obtained, especially for *τ*_{g} retrievals. The proportion of negative *τ*_{g} retrievals is 39.6% when cloud fraction ranges from 0% to 50%, and 18.4% in the range from 50% to 100%. The negative retrievals increase with decreasing cloud cover, which is consistent with the study of Boers et al. (2000). As far as the *τ*_{s} retrieval is concerned, the proportions are much lower: 0.46% and 0.56% in the same two cloud-fraction ranges. Figure 4 compares nine sets of *τ* retrievals on 1 August. Interestingly, there are characteristics very similar to those presented in the last section. At four times of the day—0830, 1030, 1130, and 1230 local time (LT)—*τ*_{s} > *τ*_{d} and the *τ*_{g} retrievals are negative. At the other five times, *τ*_{s} < *τ*_{d}, and overestimated *τ*_{g} retrievals are obtained. For example, at 1530 LT, *τ*_{s} (0.237) is 0.376 smaller than *τ*_{d} (0.613), and the corresponding *τ*_{g} reaches 4.403. The *I*_{gs} value (508 W m^{−2}) calculated using *τ*_{g} is exactly equal to its measurement (508 W m^{−2}), but the corresponding *I*_{ds} and *I*_{ss} calculations (1.97 and 508 W m^{−2}) do not match their measurements (539 and 274 W m^{−2}). At 1530 LT, there is also a 50.29% (137.9 W m^{−2}) overestimated *I*_{ss} calculation by using the *τ*_{d} retrieval, but *I*_{ss} from *τ*_{s} has only a −0.96% (−2.63 W m^{−2}) deviation from its measurement.

In the next comparison, we selected the cases where *τ*_{g}, *τ*_{s}, and *τ*_{d} retrievals are all positive. There are a total of 2000 positive *τ*_{s} retrievals, but there are only 1293 sets of *τ* comparisons, mostly owing to negative *τ*_{g} retrievals. Figure 5 compares monthly- and annual-mean *τ* retrievals. All monthly- and annual-mean *τ*_{g} values are considerably larger than *τ*_{d} and *τ*_{s}. Annual-mean *τ*_{g}, *τ*_{d}, and *τ*_{s} values are 2.08, 0.543, and 0.348, respectively. Figure 6 compares uncertainties in monthly- and annual-mean *I*_{gs}, *I*_{ss}, and *I*_{ds} values calculated using these 1293 sets of *τ* retrievals. When the *τ*_{g} retrieval is used, monthly- and annual-mean *I*_{gs} calculations (Fig. 6a) agree well with their measurements (0.0% deviations), but there are very large deviations in both *I*_{ds} and *I*_{ss} calculations (Figs. 6b and 6c), −39.8% (−268 W m^{−2}) annual-mean underestimation, and 46.4% (140 W m^{−2}) annual-mean overestimation, respectively. This implies that *τ*_{g} retrievals are overestimated (see Fig. 5). The *τ*_{d} case is quite different: *I*_{ds} calculations from *τ*_{d} and their measurements match very well, but both *I*_{gs} and *I*_{ss} calculations are obviously overestimated. Only *τ*_{s} retrievals (Fig. 6b) by the present method achieves a very good match between *I*_{ss} calculations and measured data (annual-mean deviation of −0.18%), but the *τ*_{s} retrieval may not be suitable for the *I*_{gs} or *I*_{ds} data. Owing to cloud nonuniformity, *τ*_{s} and *τ*_{d} may be much different. If *τ*_{d} > *τ*_{s}, an unreasonably large *τ*_{g} retrieved from *I*_{gs} can be obtained, which matches *I*_{gs} measurements but do not match *I*_{ds} and *I*_{ss} measurements. The *I*_{gs} calculation using *τ*_{d} + *τ*_{s} in Fig. 6a stands for a sum between *I*_{hds} and *I*_{ss} determined from *τ*_{d} and *τ*_{s} retrievals, respectively. Please note that the *I*_{gs} calculation is in excellent agreement with its measurement, having only a 0.08% annual-mean deviation. In summary, *τ*_{s} + *τ*_{d} retrievals can result in exact *I*_{ds}, *I*_{gs}, and *I*_{ss} calculations in the case of thin cloud (including broken cloud).

The following analysis is for studying the effect of *R*_{eff} uncertainty on *τ*_{d} and *τ*_{s} retrievals using radiation measurements on 1 August and five sets of SPF, which are MODIS20 and column10, -20, -50, and -100. There are six sets of *τ* retrievals below 0.5 and three above 0.5. The larger the *τ*_{d} is, the larger the scattering light correction of pyrheliometer data must be, and thus the larger the retrieved *τ*_{d} deviation is, caused by the *R*_{eff} variation. In the case of *τ* < 0.5, the maximum deviation of *τ*_{d} retrievals by the four column SPFs (*R*_{eff} changing from 10 to 100 *μ*m) with those by MODIS20 (*R*_{eff} = 20 *μ*m) is 0.039, and the mean deviation is 0.010. In the case of *τ* > 0.5, the maximum and mean deviations are 0.273 (31.2%) and 0.092 (8.03%). The *τ*_{s} retrievals by the new method have almost the same uncertainties in both magnitude and sign as the *τ*_{d} retrievals. The *τ*_{d} uncertainty is transmitted into *τ*_{s} through *f*_{M} in Eq. (5). The acceptable *τ*_{d} and *τ*_{s} retrievals, using different SPF with *R*_{eff} from 10 to 100 *μ*m, are obtained, especially for *τ* < 0.5. The above analysis is available for *τ* < 1.0. The case of *τ* > 1.0 remains a problem for future study.

## 5. Summary

The ratio of *I*_{ss} to *I*_{gs} is a monotonic function of *τ*, and it is strongly sensitive to *τ* variation but weakly sensitive to SPF in the small-*τ* case. Based on these properties, a new method is proposed to retrieve small *τ* values (<2) by using this ratio. The term *I*_{gs} is a sum of *I*_{hds} and *I*_{ss}, which are closely correlated with *τ*_{d} and *τ*_{s}, respectively. Owing to cloud inhomogeneity, *τ*_{d} and *τ*_{s} may be much different. The difference in the thin cloud case is the essential cause of unreasonably negative or grossly overestimated *τ* retrievals from *I*_{gs} data. As shown in retrieval simulations and applications, the new method can overcome this problem, yielding stable and reasonable *τ*_{s} retrievals. There is a potential application of the new method to thin cirrus *τ* retrievals from worldwide long-term pyrheliometer and pyranometer measurements.

The *τ* retrieval from the *I*_{ss}-to-*I*_{gs} ratio is treated as the *I*_{ss}-equivalent *τ* value. The physical meaning of the equivalency will be further studied through a large amount of 3D radiation calculations. Another problem left for future study is the uncertainty of the *τ* retrievals, caused by noise in the radiation measurements.

## Acknowledgments

This research is supported by National Natural Science Foundation of China (Grant 41175029), the National Basic Research Program (Grant 2011CB403401), and the “Strategic Priority Research Program” of the Chinese Academy of Sciences (XDA05100300).

## REFERENCES

*Meteorological Radiation Observation Method.*Meteorological Press, 165 pp.