Three kinds of “visible” cloud optical thickness τ—matching shortwave direct, global, and scattering solar irradiances (Ids, Igs and Iss)—are defined, which are marked as τd, τg, and τs, respectively. It is found from radiation calculations that a ratio of Iss to Igs 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 Iss-equivalent τs 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 Igs, owing to the difference between τs and τd, while the new method can lead to a very good agreement of τs 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 Igs was often negative because of cloud inhomogeneity, while the application of the new method resulted in stable yet reasonable τs values. The Iss calculations using 1293 sets of τs retrievals fit well into the Iss determinations from pyrheliometer and pyranometer measurements with an annual-mean deviation of 0.18%, but the deviation was raised to 46.4% when using τg retrievals.
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 (Ids, Igs, and Iss, 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 Iss 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 Iss value, two τ values are obtained. Thus, the retrieval is not unique. For τ > 5, Iss monotonously decreases with increasing τ, and so a unique τ can be retrieved from Iss. The broadband Iss is usually measured by the pyranometer with a shading band or disc. An inherent problem is the obstructed light correction for the real Iss 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 Reff) (King et al. 1997; Yang et al. 2000). The uncertainty in the SPF (or Reff) 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 ηM of Iss to Igs. On the basis of radiative transfer calculations, we found that using ηM 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, Iss increases with increasing τ, and Igs decreases with increasing τ, and so ηM monotonically increases with τ. Note that Igs equals a sum of horizontal-plane direct solar irradiance Ihds and Iss, and Ihds is not sensitive to SPF variation. Thus, both Iss and Igs have the same variation trend with the SPF (or Reff), implying that their ratio must be nearly independent of the choice of SPF. This unique characteristic of ηM (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 ηM 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.
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 τd, τg, and τs, to match broadband Ids, Igs, and Iss, respectively. The Ids value depends on only τd along the direct solar incident direction, which can be expressed in the plane-parallel atmosphere as follows:
where μ0 is the solar zenith angle cosine, S0(λ) the extraterrestrial solar irradiance at the λ wavelength, λ1 and λ2 are lower and upper spectral limits of the broadband solar spectrum, and τλ,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, λ1 = 0.3 μm and λ2 = 4.0 μm.
The terms Iss and Igs are related to the whole-sky τ distribution. The defined τg indicates Igs-equivalent τ under the assumption of horizontally uniform τ. Using τg, Igs is expressed as
where TTotal is the wavelength dependent total transmittance. Similarly, Igs is given by
where TSca is the scattering (diffuse) transmittance, and τs marks Iss-equivalent τ.
It is found from Figs. 1 and 2 that ηM has some very significant characteristics for small-τs retrieval. The Iss and Igs values, as well as their ratio ηM, are calculated using the Discrete Ordinate Radiative Transfer model (DISORT). Figure 1 shows ηM versus τ (ranging from 0 to 10) at μ0 = 1.0 and 0.5 for the 645-nm ice SPF (marked as MODIS20 in Fig. 2) with Reff = 20 μm, the same as that used by the Moderate Resolution Imaging Spectroradiometer (MODIS) cloud algorithm (King et al. 1997). Figure 2 compares ηM 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 Reff = 10, 20, 50, and 100 μm, respectively, and the plate10, -20, -50, and -100 labels indicate plate SPFs with Reff = 10, 20, 50, and 100 μm, respectively. Other input parameters in Figs. 1 and 2 include 1) column H2O and O3 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 ηM in the thin-τ case are presented as follows:
ηM is a monotonic-increasing function of τ, and it is close to unity (suffering from saturation) as τ > 5 and τ > 2 for μ0 = 1.0 and μ0 = 0.5.
ηM is sensitive to τ (see Fig. 1), because Iss increases, but Igs decreases with increasing τ. For example, ηM increases 308% when τ increases from 0 to 1.0 and μ0 = 1.0, whereas Ids and Igs decrease 172% and 3.9%, respectively. For the same μ0, when τ increases from 0 to 0.02, ηM increases 9.0%, while Ids and Igs decrease 2.0% and 0.1%, respectively. Clearly, ηM is much more sensitive to τ than both Ids and Igs, and thus it is particularly suitable for small-τ retrievals.
ηM is weakly sensitive to SPF (or Reff) variation (see Fig. 2), because both Iss and Igs have the same variation trend with SPF or Reff. The maximum difference among ηM calculations using nine SPF models (Reff ranging from 10 to 100 μm) is only 2.78% for any τ from 0 to 20. When the column100 with Reff = 100 μm is used instead of the column10 with Reff = 10 μm and τ = 1.0, Iss and Igs increase 4.79% and 4.23%, respectively, while the ηM variation is much smaller, being 0.56%.
The τ retrieval from ηM should characterize Iss data, owing to the very weak sensitivity of Igs to τ variation.
On the basis of the above four characteristics, this paper develops a method to retrieve Iss-equivalent τs. Because of cloud inhomogeneity, τs may be much different from τd, and hence there may be large uncertainty in the τs retrieval if the original Iss-to-Igs ratio is used directly. To avoid the uncertainty, a modified ratio ηM is proposed as follows:
where Ihds = μ0Ids and the modification factor fM is equal to a ratio between two μ0-direction transmittances at τ = τs and τd. Hence, the modified Ihds and Igs are responsible for τs, but not for τd. The final τs value is obtained through an iterative process:
Input a set of Ids, Igs, and Iss data, and retrieve τd from Ids. The input methods for the τ simulation and application tests are presented later.
Set fM = 1.0 (no modification).
Determine ηM according to Eq. (4), using Ids, Iss, and fM.
Retrieve τs using the lookup table (LUT) approach, where the calculated Iss-to-Igs ratio is equal to the value of ηM derived in the last step.
Repeat steps 4 and 5 until the variation of τs 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 Ids and Iss values in step 1 are derived through separate radiation calculations using τd and τs inputs, respectively, and Igs is determined as μ0Ids + Iss. The Iss 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 Ids and Igs inputs, and Iss is determined as
where LC 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; LC is determined through radiative calculations using the τd retrieval in step 1. The correction is closely dependent on Reff, and the effect of Reff uncertainty on τ retrievals is examined in section 4. There are in situ Iss 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 Iss determination using Eq. (6) from pyrheliometer and pyranometer data is treated as the “measured” Iss in this paper. The uncertainties in both pyrheliometer and pyranometer data can result in errors in the Iss 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 H2O and O3 amounts; 2) molecular scattering optical thickness, determined according to the site altitude above sea level (0–4 km); 3) μ0 (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 τd and τs, owing to effects of cloud inhomogeneity. If there is a layer of thin broken cloud in the sky (τs > 0) and there is no cloud around the sun (τd = 0), the true Igs may be larger than the calculated value of Igs 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 (Ihds ≈ 0 at a large τd), and there is no cloud away from the sun (τs ≈ 0), then the true Igs may be very small, and the retrieved τ from the pyranometer data will be unrealistically large. The difference between τs and τd is a major factor causing invalid τg retrieval from Igs. As shown in Fig. 3, two sets of τ distributions are input in simulations: in Fig. 3a, τd = 0.2 and τs changing from 0 to 2.0 and in Fig. 3b, τs = 0.2 and τd 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 τs and τd (i.e., 0.2) are input, both τs and τg retrievals compare well with the input value. When τs input is smaller (or larger) than τd, significantly different τg retrievals are obtained. For instance, when input τs = 0.0 but τd = 0.2, the retrieved τg is as large as 2.14, and when input τs ≥ 0.3 > τd, unreasonably negative τg is obtained. The negative retrieval is determined through interpolating two Igs data points at τ = 0.0 and 0.02. All τs retrievals are exactly equal to their input values ranging from 0 to 1.5. Note that there is a 5.1% underestimation in τs retrieval (1.89) when its input is 2.0. The same conclusions can be drawn from Fig. 3b. When the input τd changes from 0.0 to 2.0, all τs retrievals compare well with their input of 0.2, while negative (or overestimated) τg values are obtained when input τd < τs (or τd > τs). When τd = 2.0 (τs = 0.2) is input, the τg value jumps to 13.39—a gross overestimate. Furthermore, if μ0 = 0.5 and μ0 = 0.3 are taken instead of μ0 = 1.0 in the simulations, the exact τs retrievals are obtained when τ/μ0 < 1.4. Overall, the difference between τs and τd can result in invalid τg retrievals, and satisfactory τs 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 Reff = 10, 20, and 50 μm (King et al. 1997). The SPF of Reff = 20 μm is used to calculate the exact irradiances. There are 40 sets of simulations, covering τ from 0 to 2.0, μ0 = 1.0, and μ0 = 0.5. The maximum deviation of τs retrievals using the SPF of Reff = 10 or 50 μm from those of Reff = 20 μm is 0.0076. The very small deviation is due to the above-analyzed weak sensitivity of ηM to Reff.
4. Application tests
In the application tests, we compare τg and τs retrievals from hourly-mean Ids and Igs 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 Ids and Igs 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 Ids data is corrected by using the τd retrieval. Except for cloud SPF, other parameters (including aerosol, H2O, 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 τi and τw, 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 Igs measurements for the case of μ0 > 0.3 and Igs > 10 W m−2 in Lhasa during the year 2006. The selection criteria of τs retrievals are 1) τs/μ0 < 1.4, 2) Iss/Igs < 0.9, and 3) μ0 > 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 τs retrievals, spanning 305 days. Unreasonably negative τ retrievals are sometimes obtained, especially for τg retrievals. The proportion of negative τg 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 τs 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)—τs > τd and the τg retrievals are negative. At the other five times, τs < τd, and overestimated τg retrievals are obtained. For example, at 1530 LT, τs (0.237) is 0.376 smaller than τd (0.613), and the corresponding τg reaches 4.403. The Igs value (508 W m−2) calculated using τg is exactly equal to its measurement (508 W m−2), but the corresponding Ids and Iss 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 Iss calculation by using the τd retrieval, but Iss from τs has only a −0.96% (−2.63 W m−2) deviation from its measurement.
In the next comparison, we selected the cases where τg, τs, and τd retrievals are all positive. There are a total of 2000 positive τs retrievals, but there are only 1293 sets of τ comparisons, mostly owing to negative τg retrievals. Figure 5 compares monthly- and annual-mean τ retrievals. All monthly- and annual-mean τg values are considerably larger than τd and τs. Annual-mean τg, τd, and τs values are 2.08, 0.543, and 0.348, respectively. Figure 6 compares uncertainties in monthly- and annual-mean Igs, Iss, and Ids values calculated using these 1293 sets of τ retrievals. When the τg retrieval is used, monthly- and annual-mean Igs calculations (Fig. 6a) agree well with their measurements (0.0% deviations), but there are very large deviations in both Ids and Iss 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 τg retrievals are overestimated (see Fig. 5). The τd case is quite different: Ids calculations from τd and their measurements match very well, but both Igs and Iss calculations are obviously overestimated. Only τs retrievals (Fig. 6b) by the present method achieves a very good match between Iss calculations and measured data (annual-mean deviation of −0.18%), but the τs retrieval may not be suitable for the Igs or Ids data. Owing to cloud nonuniformity, τs and τd may be much different. If τd > τs, an unreasonably large τg retrieved from Igs can be obtained, which matches Igs measurements but do not match Ids and Iss measurements. The Igs calculation using τd + τs in Fig. 6a stands for a sum between Ihds and Iss determined from τd and τs retrievals, respectively. Please note that the Igs calculation is in excellent agreement with its measurement, having only a 0.08% annual-mean deviation. In summary, τs + τd retrievals can result in exact Ids, Igs, and Iss calculations in the case of thin cloud (including broken cloud).
The following analysis is for studying the effect of Reff uncertainty on τd and τs 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 τd is, the larger the scattering light correction of pyrheliometer data must be, and thus the larger the retrieved τd deviation is, caused by the Reff variation. In the case of τ < 0.5, the maximum deviation of τd retrievals by the four column SPFs (Reff changing from 10 to 100 μm) with those by MODIS20 (Reff = 20 μm) 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 τs retrievals by the new method have almost the same uncertainties in both magnitude and sign as the τd retrievals. The τd uncertainty is transmitted into τs through fM in Eq. (5). The acceptable τd and τs retrievals, using different SPF with Reff from 10 to 100 μm, 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.
The ratio of Iss to Igs 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 Igs is a sum of Ihds and Iss, which are closely correlated with τd and τs, respectively. Owing to cloud inhomogeneity, τd and τs may be much different. The difference in the thin cloud case is the essential cause of unreasonably negative or grossly overestimated τ retrievals from Igs data. As shown in retrieval simulations and applications, the new method can overcome this problem, yielding stable and reasonable τs 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 Iss-to-Igs ratio is treated as the Iss-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.
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).