## 1. Introduction

Clouds play an important role in the climatic system because of their effects on the earth’s energy balance and on the hydrological cycle. The Fourth Assessment Report (AR4) of the Intergovernmental Panel on Climate Change (IPCC) indicates (Randall et al. 2007 and references therein) the following: “Recent studies reaffirm that the spread of climate sensitivity estimates among [climate] models arises primarily from inter-model differences in cloud feedbacks.... The relatively poor simulation of these [low and mid-level] clouds is a reason for some concern [when modeling the climatic system]. The response to global warming of deep convective clouds is also a substantial source of uncertainty in projections since current models predict different responses of these clouds. Observationally based evaluation of cloud feedbacks indicates that … it is not yet possible to determine which estimates of the climate change cloud feedbacks are the most reliable.” Consequently, our knowledge of the physical and, in particular, of optical characteristics of clouds is essential in describing and predicting potential cloud feedbacks that may affect the climate.

The transmission, reflection, and absorption by clouds depend on the solar zenith angle, surface albedo, and cloud physical properties themselves. Among the cloud properties, the cloud optical depth (hereinafter COD or *τ _{c}*) and effective radius

*r*of the droplet size distribution are the most relevant. Experimental determination of these properties is usually based either on measurements (from satellites or airplanes) of reflected radiation within the visible or the near-infrared bands, or on ground measurements of radiation transmitted through the clouds. The scope of these methods is shown in a review by Clothiaux et al. (2005).

_{e}Optically thick water clouds allow the presence of the diffuse component of shortwave radiation but not the direct component. The cloud optical depths of such clouds can be evaluated from measurements of atmospheric transmittance, which is defined as the ratio between the horizontal irradiance at ground level and that at the top of the atmosphere. Several spectral bands can be used for such a goal. Generally speaking, most methods require an assumption about the droplet effective radius, but this value can be simultaneously estimated when a concurrent measurement of the liquid water path (LWP) is available. This variable can be continuously measured at a specific site by using a microwave radiometer (MWR).

Thus, Leontieva and Stamnes (1994), Dong et al. (1997), Barnard and Long (2004), and Qiu (2006) have suggested several methods of determining COD from measurements of broadband shortwave irradiances taken by pyranometers. More recently, Barnard et al. (2008) have applied to thin, ice clouds the algorithm proposed by Barnard and Long (2004) for thick clouds. All these methods assume plane-parallel conditions, so they are suitable for stratiform clouds. Applying these methods to inhomogeneous cloud fields may result in an “effective” COD (Leontieva and Stamnes 1994). Other authors have suggested methods for COD determination that are based on measurements of monochromatic (or narrowband) atmospheric transmissions. Specifically, Leontieva and Stamnes (1996) introduced a technique for COD determination based on multichannel radiometer measurements, and applied it to Multifilter Rotating Shadowband Radiometer (MFRSR) data. This instrument was initially designed to obtain routine measurements of the aerosol optical depth (Harrison et al. 1994; Harrison and Michalsky 1994).

To obtain cloud properties, the Min and Harrison algorithm (Min and Harrison 1996a; Min and Harrison 1996b, hereinafter MH96), uses an iterative process based on an inversion of the adjoint treatment of the radiative transfer. This treatment allows us to obtain the atmospheric transmittance for any solar angle in a single computation. The MH96 method has been applied to measurements of transmittance at 415 nm, obtained at several sites, and specifically to measurements taken by the MFRSR instrument at the Atmospheric Radiation Measurement Program (ARM) Southern Great Plains (SGP) site (Min and Harrison 1996b). Results have been validated against vertical profiles obtained in situ by aircraft measurements during the second ARM Enhanced Shortwave Experiment (ARESE II) field campaign (Min et al. 2003). Selection of the 415-nm band (the first channel of the MFRSR) is justified by two reasons mainly: first, this band is free of absorption by ozone and by permanent atmospheric gases, and, second, the surface albedo at this wavelength is low and stable for natural surfaces (about 0.04, although it can reach values above 0.15 for white sand), except in the case of snow-covered surfaces. For the SGP site, a value of 0.036 was used in previous works (Min and Harrison 1996b; Kim et al. 2003). On the other hand, an absolute calibration of the MFRSR is not necessary to obtain the transmittance; in fact, the ratio between the ground-level-measured signal and the extrapolation for the top of the atmosphere (Langley calibration) is the only requirement.

The relationship between COD and the atmospheric transmittance shows a remarkable linearity when absorption is absent. Parameters of the linear relation mainly depend on solar height and surface albedo and, less importantly, on the droplet effective radius (Kim et al. 2003). This latter fact is a problem when trying to estimate the effective radius from transmittance measurements, but it is a great advantage when the goal is obtaining COD, since it means that the exact value of the effective radius does not necessarily have to be known.

In the present work, the previously mentioned linear relationship is used to develop a simple method to obtain COD for liquid water clouds. When measurements of liquid water path are available, the cloud droplet effective radius can be estimated as well. Our proposed method is very easy to implement for this, as well as for extensive assessment of cloudiness for climatic studies. On the other hand, it adds the possibility of introducing the aerosol optical depth (the actual measured value or, in other words, the climatic value) to improve cloud characterization. Section 2 presents and justifies the method (the most technical questions are treated in the appendix), and section 3 is devoted to explaining and discussing the results of its application to a dataset for which the MH96 algorithm (adopted here as a reference method) has also been applied. Special attention is devoted to the effects of introducing aerosol information on the cloud properties retrieval. Section 4 summarizes our conclusions for this study.

## 2. Method

*τ*and the reciprocal of its transmittance

*T*can be justified by solving the radiative transfer equation by simple resolution methods. Thus, two-stream methods for diffuse radiation in a homogeneous environment are usually presented as a system of two differential equations, as shown by Lenoble (1992) and Liou (2002). In particular, when absorption is absent, and for isotropic incident radiation, the relationship may be written aswhere

*g*is the asymmetry parameter of the medium and

*a*is the albedo of the bottom boundary (considered Lambertian).

*τ*> 10, and where

*μ*

_{0}is the cosine of the solar zenith angle (SZA). In the case of clouds in the atmosphere,

*g*is related to the droplet size distribution, which can be parameterized through the droplet effective radius

*r*(i.e., the ratio between the third and the second moments of the size distribution).

_{e}The linearity between optical depth and the reciprocal of transmittance is rapidly broken in absorbing mediums (even if absorption is low) for large enough values of *τ*. This is caused by the effects of the increasing number of interaction events leading to a higher probability of absorption. Fortunately, as mentioned above, the 415-nm band is almost free of absorption in the atmosphere, and irradiance in this band is routinely measured by the MFRSR. On the other hand, it is known (Min and Harrison 1996b; Kim et al. 2003) that the surface albedo is low and remarkably stable for this wavelength. Moreover, for this wavelength, both the asymmetry parameter and the single-scattering albedo (which is very close to unity) show little dependency on droplet size. This means that the relationship between the COD and *T* is hardly sensible to *r _{e}*, which all together makes it suitable for the estimation of COD even when

*r*is unknown.

_{e}Several simulations by means of a rigorous radiative transfer model were performed in order to determine more precisely the effect of cloud characteristics, its geometry, and the surface albedo on the linearity of the COD–transmittance relationship at 415 nm. Results were also used to derive a parameterization of the relationship on the most relevant variables. Specifically, we used version 2.4 of the Santa Barbara DISORT Atmospheric Radiative Transfer (SBDART; Ricchiazzi et al. 1998) multilayer and multispectral model. This code implements the Discrete Ordinates Radiative Transfer Program for a Multi-Layered Plane-Parallel Medium (DISORT) method, and was used with eight streams in the current study. The atmosphere was described using 65 layers, with a maximum resolution of 100 m in the lowest levels of the troposphere (between the surface and 4.5-km altitude). The solar extraterrestrial spectrum was taken from version 7 of the Low Resolution Transmission model (LOWTRAN 7) with a spectral resolution within the analyzed band of 20 cm^{−1}. The atmospheric vertical profile was chosen from the *U.S. Standard Atmosphere, 1962*. For describing cloud scattering, the Henyey–Greensteen phase function was adopted, after Ricchiazzi et al. (1998). Finally, to describe tropospheric aerosol, a rural model was considered.

*T*is remarkable. Linear regression between these variables, in the range of COD values from 10 to 150, gives high correlation coefficients (

*R*

^{2}being always higher than 0.9999). The slope of the relationship between COD and 1/

*T*has a linear dependency on the cosine of SZA, while the intercept is almost independent. The effects of surface albedo match the pattern of behavior that is derived from Eqs. (1) and (2). The dependence on the effective radius—which is implicitly included in the asymmetry parameter in Eq. (2)]—is adequately described by a weak rational function. All these considerations together result in the following expression:with parameters

*P*

_{1}–

*P*

_{3}to be adjusted to minimize the squared residuals against the SBDART results. It turns out that these parameters depend linearly on the aerosol optical depth (AOD):

This linear dependency has been obtained from simulations performed for values of AOD ranging from 0 to 0.2 (see the appendix). This interval might be considered to be a conservative range of validity of Eqs. (3) and (4), but in fact the linearity is very high, so one could expect that Eqs. (3) and (4) may be valid for values of AOD well above 0.2.

More details on the derivation of Eqs. (3) and (4) are given in the appendix. The agreement between the parameterized expression [Eq. (3)] and the exact SBDART simulations is very good in the range of CODs between 10 and 100: the mean deviation (in units of optical depth) is 0.01 and the standard deviation of the differences is 0.24 for the simulations performed within this range. For COD = 10, differences are between −0.26 (−2.6%) and 0.42 (4.2%), whereas for COD = 100, the differences are between −0.74 (−0.74%) and (0.82) 0.82%. For cloud optical depths lower than 10, Eq. (3) is still well centered on the SBDART results (mean deviation is only 0.10), but the relative deviation can reach very high values. Thus, Eq. (3) should not be used for CODs lower than 10. For COD > 100, Eq. (3) tends to increasingly overestimate the SBDART simulations as COD increases; the relative differences are in the range from −0.31% to 3.80% for COD = 300. The comparison of Eq. (3) against a reference method will be carried out below for the range of cloud optical depths between 10 and 100.

*r*is not known, so Eq. (3) allows obtaining COD by introducing a fixed value of

_{e}*r*on the basis of cloud type. A typical value for stratiform clouds is

_{e}*r*= 8

_{e}*μ*m. There exists, however, a relation between COD and LWP, assuming a size distribution

*n*(

*r*) of droplets with radius

*r*and with an effective radius

*r*:where

_{e}*ρ*is the water density and

_{w}*Q*(

_{λ}*r*) is defined here as the extinction efficiency weighted by the droplet size distribution, which isIn the above expression,

_{e}*q*(

_{λ}*r*) is the extinction efficiency for wavelength

*λ*and for droplets with radius

*r*. The weighted extinction efficiency

*Q*(

_{λ}*r*) can be approximated, for

_{e}*λ*= 415 nm, and for a modified gamma size distribution with width

*p*= 7 (Ricchiazzi et al. 1998), by

When concurrent measurements of atmospheric transmittance and liquid water path are available, a simple iterative process may be applied to estimate COD and *r _{e}* simultaneously. First, Eq. (3), along with Eq. (4), are used by introducing the measured transmittance, and assuming

*r*= 8

_{e}*μ*m to obtain a first estimation of COD. Then,

*Q*

_{415nm}is estimated from Eq. (7), assuming again

*r*= 8

_{e}*μ*m. Subsequently, and by introducing the measured value of LWP, Eq. (5) is used to obtain a better estimation of the effective radius, which can be introduced again in Eqs. (3) and (7) to improve the estimations of COD and

*Q*

_{415nm}. This procedure results, after a few iterations (e.g., three or four iterations that can be easily implemented in a spreadsheet), in stable values of COD and

*r*that are considered the results of this method. This iterative use of Eqs. (3)–(5) and (7) constitutes the basis of our inversion method.

_{e}Here, we must remember that Eqs. (1) and (2) are obtained from very strong simplifications of the actual radiative transfer processes in the atmosphere: the two-stream method and the delta-Eddington approximation for a homogeneous nonabsorbing medium, respectively. On the other hand, Eqs. (3) and (4) have been obtained by fitting to SBDART results corresponding to simulating single-layer, liquid clouds. SBDART treats the atmosphere as a plane-parallel system, so vertical inhomogeneities of the bare atmosphere and of the aerosol distribution can be taken into account. On the other hand, the probable vertical inhomogeneities in clouds and other possible 3D effects cannot be included. For conditions with vertical inhomogeneities, such as a liquid water cloud with an effective radius increasing from bottom to top (which is actually a typical case), the presence of an ice layer above the liquid water cloud, or when horizontal inhomogeneities generate important 3D effects, it is expected that Eqs. (1)–(3) will lead to nonnegligible errors. This also applies to the MH96 algorithm, and to any other method that assumes the cloud to be described as a single homogeneous layer containing only water droplets.

To be more specific, it is necessary to note that three-dimensional effects can be very important. As an example, it is known that reflections from scattered convective clouds that do not intercept the direct solar beam can give total radiative levels at the surface higher than that corresponding to a cloudless sky. This is the so-called cloud enhancement effect. Assuming plane-parallel conditions in these situations would lead to a calculated cloud optical depth that is much lower than the results obtained if the cloud intercepts the solar beam. In a more extreme case, if radiative levels at the surface were higher than the incident radiation at the top of the atmosphere, a physically nonsensical negative COD could be obtained. Three-dimensional radiative transfer should be treated with methods such as the Monte Carlo approach (see Marshak and Davis 2005). However, the restrictions (minimum cloud fraction of 99%) applied to the conditions analyzed in this study should minimize the probability of having important 3D effects.

Therefore, it is clear that our proposed method, and the MH96 method, have not been designed to treat these kinds of skies. However, we consider that these methods can still be useful in describing cloud conditions and their changes if one keeps in mind that the retrieved optical properties are “effective,” in the same manner as Leontieva and Stamnes (1994) state for horizontally inhomogeneous clouds, “cloud optical thickness is considered in the sense of the optical depth *τ* of some ‘effective’, plane-parallel, homogeneous cloud layer, resulting in the same radiation field at the surface as real cloudiness.” In fact, this is often the only choice, since no information about the spatial structure of the cloud cover is usually available. Here, to give a less misleading definition of what an effective optical depth should be, “the same radiation field” should be interpreted as “the same horizontal irradiance.”

## 3. Results

We applied the proposed method to a dataset that was purposely selected. Specifically, data were taken from the ARM Southern Great Plains Central Facility (36.61°N, 97.49°W, 320 m MSL), where concurrent measurements from an MFRSR and a microwave radiometer (MWR) are available, along with sky condition characterization from a sky camera, the Total Sky Imager (TSI). At this station, when both the MFRSR and MWR instruments provide measurements, an estimation of the cloud optical thickness is provided. This COD is obtained by applying the MH96 algorithm, while an estimation of *r _{e}* is also produced by using temporal averages of the liquid water path measurements. These estimations, which are publicly available at the ARM Web site and that Turner et al. (2004) consider to be correct for “horizontally homogeneous stratiform clouds with optical depth larger than approximately 7,” have been used in the present study as the reference to compare our proposed, simple method. Based on in situ measurements performed during the ARESE-II campaign and on the closure comparison between the broadband irradiance obtained by modeling (using the estimated cloud properties) and the irradiance measured by pyranometers [this is considered a suitable way to validate the retrieved cloud optical properties; Barnard et al. (2008)], Min et al. (2003) assigned a 5% of error to COD, while the uncertainty for

*r*was better than 5.5%. They suggested that the main source of error on the estimation is the uncertainty attached to MWR measurements of LWP.

_{e}The comparison of the proposed method has been performed over a set of cases selected from a whole year (2006) of instantaneous MFRSR measurements of the atmospheric transmittance at 415 nm, besides the corresponding ARM estimations of COD (which are obtained by applying the MH96 method). Some conditions were applied to select the suitable cases. First, in order to assure overcast conditions, a minimum fractional sky cover (based on the TSI estimations) of 99% was required. Second, liquid water path measurements, which show large variability, were averaged over 5-min intervals centered on the time of the available transmittance measurements. Moreover, and following Morris (2006), cases with LWP > 1 mm were removed from the dataset.

Figures 1a and 1b show frequency distributions in the datasets of COD and *r _{e}*, respectively. The median of COD is 31.9, the most frequent value is around 20, and 87% of cases are within the range 10–100. The corresponding characteristics of the

*r*distribution are median 7.8

_{e}*μ*m, most frequent value 7

*μ*m, and 96% of cases have

*r*< 20

_{e}*μ*m. As has been mentioned previously, only cases within the range of COD values between 10 and 100 were considered in the computation of statistical parameters for assessing and discussing the agreement between the results of our method and the ARM dataset. In addition, cases with

*r*> 20

_{e}*μ*m (about a 3.5% of the total) were not considered in the analysis. These retrieved high values might correspond to conditions with possible light drizzle; another possibility is that three-dimensional effects in non-plane-parallel conditions might enhance the measured transmittance, thus leading to an overestimation of the effective radius retrieval (and correspondingly an underestimation of the cloud optical depth). In any case, this would correspond to conditions outside the aim of this study. In addition, effective radii above 20

*μ*m are considered rare in the literature, except for cases of cumulonimbus. With all these restrictions, the comparison dataset still has 16 170 cases, corresponding to 114 days during 2006.

On the other hand, the surface albedo has been calculated as the ratio between upwelling irradiances measured by a multifilter radiometer installed at 25 m AGL and downwelling irradiances measured simultaneously at the ground level by another multifilter radiometer. These albedo values were simultaneous to MFRSR irradiance measurements used to obtain the required transmittance values. Figure 1c shows the distribution of the albedo values, which is centered at 0.03 and has a standard deviation of 0.01.

Figures 2a and 2c show the results of the first estimation of COD, obtained directly from Eq. (3), ignoring the liquid water path measurements and fixing *r _{e}* = 8

*μ*m. A fixed value of AOD = 0.11 at 550 nm was initially used; this value comes from the Cimel photometer of the National Aeronautics and Space Administration’s Aerosol Robotic Network (AERONET) installed at the ARM Cloud and Radiation Test Bed (CART) site (36.61°N, 97.49°W, 318 m MSL), near the other radiometric instruments at the SGP central facility. This value is the average for 2006, and was obtained from the interpolation between the AODs at 500 and 675 nm, corresponding to cloudless conditions between January and September (the available months with level 2.0 data). Our estimations of COD are presented against the corresponding ARM (MH96) values in Fig. 2a. The agreement is qualitatively good and the correlation coefficient between both estimations is high (

*R*

^{2}> 0.9906). The mean value of COD, for this set of considered data, is 37.1; the mean bias deviation (MBD) between both estimates is 0.24 in units of optical thickness and the standard deviation (SD) of the differences is 2.1. The relative differences against cloud optical depth are shown in Fig. 2c. Despite extreme relative differences as large as +20% and from −9%, the standard deviation of the relative differences is only 4.8%. This value is close to the error assigned to the reference method (5%, as noted above). Therefore, Eq. (3) can be considered a reasonable estimator of COD when no information about either the liquid water path or droplet radius is available.

The use of the iterative process described in section 2, which needs LWP as an additional input, allows for improvements in the estimation of COD and also yields as a result the estimation of *r _{e}*. Figures 2b and 2d show results of COD, obtained by our iterative method, compared again with the values from the reference method, and using AOD = 0.11, as above. The correlation coefficient between both estimations is now 0.9998 (

*R*

^{2}= 0.9996). The agreement of our method with respect to the ARM (MH96) results is very good, with an MBD of −0.08 in units of optical thickness, and an SD of only 0.46. Figure 2d shows the relative differences against the cloud optical depth. Differences are always within the range from −4.4% to 4.6%. Therefore, both estimations, one based on a fitting of the SBDART modeling results, and the other (MH96) based on an inversion of the adjoint formulation for radiative transfer, are consistent.

As stated above, the linear relationship between the optical depth and the reciprocal of the transmittance breaks down for nonconservative scattering, even for very low absorption when the optical depths are high enough. Equation (3) has been obtained by fitting to SBDART results for CODs lower than 100, which are more than 92% of the cases (see Fig. 1a). Thus, we expect that Eq. (3), which is the basis of our method, must fail for higher CODs, when the weak absorptivity of the cloud (single-scattering albedo is as high as 0.999 999 7 at 415 nm for an effective radius of 8 *μ*m) causes a curvature in the relationship between COD and 1/*T*. In fact, the addition on the right-hand side of Eq. (3) of a quadratic term in 1/*T* (which can be easily justified from simple treatments, such as the two-stream method) could improve the agreement between both methods. Specifically, the addition of an empirically determined term such as −0.007/*T* ^{2} into the square bracket in Eq. (3) lowers the bias of our iterative method with respect to the MH96 algorithm, in the range of CODs between 150 and 200, from about +7 (4%) to only +2 (1%) units of optical depth without any noticeable change in the agreement for CODs lower than 100.

All of the results already presented correspond to estimations obtained by using the surface albedo from the upward radiation measurements taken by a looking-downward radiometer installed 25 m above the ground. If we use the surface albedo calculated from another radiometer installed only 10 m above the ground, the mean bias with respect to the reference method slightly increases to 0.16, and the standard deviation of the differences grows to 0.56. These values are somewhat higher than those obtained originally by the suggested method (−0.08 and 0.46, respectively). In the absence of local surface albedo measurements, an albedo value based on the land use around the region should be used. To check the effects of using such a fixed albedo value, we tried several values. When the value of 0.03 (the average of the 25-m AGL measured value) is used, we obtain the best agreement: the mean bias deviation is only −0.03 and the standard deviation is 0.46. This agreement is very similar to (actually, slightly better than) the results obtained by using the measured values at 25 m AGL. For values of albedo of 0.036 [a value used in previous studies in the same area; Min and Harrison (1996b); Kim et al. (2003)] and 0.039 (the average of the 10-m AGL measured values) agreement is worse, with mean bias deviations of 0.18 and 0.28, respectively, and standard deviations of 0.56 and 0.61 respectively.

As mentioned above, when concurrent measurements of LWP are available, these can be used along with the transmittance measurements in the iterative algorithm suggested. With this, simultaneous estimations of COD and the effective radius are obtained; both values can be compared with the estimations resulting from the MH96 method applied to the ARM data. Figure 3 shows the agreement between the two estimations of the effective radius. Figure 4 shows the relative difference among them as a function of COD (as derived from the MH96 method). Both estimations are very similar, despite a slight underestimation by the present algorithm. The mean deviation is only −0.13 *μ*m, and the standard deviation of the differences is 0.14 *μ*m (recall that the typical value of the radius is around 8 *μ*m). Relative differences are in the range between −7.5% and 2.3%.

Another practical issue comes from the fact that whereas the microwave radiometer measures the liquid water path and is not sensitive to the ice path, the atmosphere transmittance can be strongly affected by ice layers. In the case that the ice-cloud optical depth is nonnegligible in the total cloud optical depth, use of Eq. (5) in the liquid COD retrieval could lead to important errors. This kind of “contamination” by ice layers could make the interpretation of retrievals from both our proposed method and the MH96 algorithm difficult.

The conditions applied to the database from ARM, limiting the liquid water path (coming from the microwave radiometer) and the cloud cover (coming from the TSI), do not guarantee against the possibility that some ice-cloud layer may be present above the water-cloud layers. Thus, we tried to assess to what extent this kind of ice contamination may be present in our database. For this, we used an estimation of the cloud top from the Active Remotely Sensed Cloud Locations (ARSCL) Value-Added Product from ARM, which uses measurements from ceilometer, lidar, and the Millimeter Wavelength Cloud Radar (MMCR; Clothiaux et al. 2000). In particular, we used the profiles of the radar reflectivity to assign a cloud-top altitude for the highest layer for each period considered in the database. Results indicate that the probability of the presence of an ice layer would be notably high, thus leading to a reduced meaningfulness in the COD retrievals presented here. Retrievals of the effective radius would also be affected: lower values are obtained than if the ice layer were not present. We have found that about 72% of the cases (corresponding to 84% of the days selected) have cloud tops above 2000 m, about 63% (75% of days) above 4000 m, and 50% (60% of days) above 6000 m, with a high probability that ice is formed. Another rough criterion that we could use is the temperature at the altitude of the highest cloud top. If we use the surface temperature (taken as half-hour averages) and estimate the top-cloud temperature with a lapse rate of −6 K km^{−1}, we find that for only 21% of cases (in 23% of days) is the temperature above 0°C. For the cases with an estimated highest cloud top lower than 2000 m, we find that the mean bias deviation between our method and that of MH96 is −0.11 and the standard deviation is 0.42 (very close to the values from the whole dataset, which were −0.08 and 0.46, respectively, despite the probable ice contamination).

The possibility of introducing an estimation of the aerosol optical depth is an extra feature of our method and a noticeable difference with respect to the MH96 algorithm. To check the effects that the estimation of the AOD may have on the agreement between the suggested iterative method and the reference ARM (MH96) dataset method, several fixed values of the aerosol load were tested. Thus, for an AOD (at 550 nm) of 0.05, and for the COD range of 10–100, the MBD is 0.56 and the SD is 0.65 in units of optical depth. If a higher value for the aerosol load is used (AOD = 0.25), MBD changes to −1.04 and SD is 0.39. The best agreement is obtained with AOD around 0.10 (specifically, for AOD = 0.11, MBD is as low as −0.08 and SD is 0.46), which is fully coherent with the average value during 2006 from the AERONET measurements (see above). Figure 5 shows the pattern of behavior for the differences (MBD ±2SD) between CODs from our suggested method and the ARM (MH96) dataset as a function of COD values themselves (clustered into several bins), and for three different values of AOD. As expected, for AOD = 0.11 (i.e., the average for 2006) the error bars (±2SD) contain the null deviation for all COD bins, despite a slight trend of our method to overestimate the highest COD values.

In Fig. 6 a month-by-month comparison of the agreement of our method with the MH96 algorithm is presented for two cases. First, when a yearly representative value of AOD is introduced, this reveals the inherent differences between both methods. Second, when a monthly value of AOD is introduced (only in our method), this leads to differences that also reveal the aerosol effects. Note that, for this set of comparisons, we introduce as representative values of AOD for October, November, and December, the values obtained from level 2.0 data for 2005. We assume that these values are representative enough for the aerosol conditions during the period October–December. During the period January–September, we used the representative level 2.0 data available for year 2006, which is synchronic with the other measurements. Fig. 6a shows the evolution of AOD throughout the year prior to the aerosol effect on COD retrievals. It can be seen that there are important differences in the aerosol load between winter and summer.

We can see in Fig. 6b that aerosol effects introduce a range of variations of 1.5 in the monthly estimations of COD (not considering the inherent deviations between methods). We can compare this range with the global parameters assessing the differences between methods (MBD = −0.08 and SD = 0.46) to conclude that taking into account the AOD variations has a real consequence in the COD retrievals. This conclusion is robust against the possibility of having ice-cloud layers above the liquid layers, as demonstrated in Fig. 6c, which shows the evolution of the mean bias on a monthly basis, for the cases with an estimated highest cloud top that is lower than 2000 m.

Finally, it must be pointed out that the comparisons presented only demonstrate that AOD has an effect in the retrieval of cloud properties, but do not prove that introducing AOD improves the agreement between the retrieved properties and the true (and unknown) values of COD and effective radius. Barnard et al. (2008) argued that closure analysis on modeled and measured surface irradiances would be the reliable way to validate retrieved cloud optical properties.

## 4. Conclusions

In this paper, we have presented a method for inverting measurements of atmospheric transmittance at 415 nm with the goal of estimating some optical properties of stratiform clouds. The proposed method is computationally efficient and very easily implemented in a spreadsheet. Specifically, Eqs. (3) and (4) combined with Eqs. (5) and (7), and using a simple iterative method, allow us to determine the cloud optical thickness and the effective radius of the cloud droplet size distribution if liquid water path measurements are available. For a selected, but large, dataset, the comparison of this method with that of Min and Harrison (1996a), taken as reference, results in mean bias differences of −0.08 and a standard deviation of 0.46 in cloud optical depth units. When the available liquid water path measurements are averaged for 5-min time intervals, the droplet effective radius derived from the present method differs from the reference estimations with a mean bias of −0.13 *μ*m and a standard deviation of 0.14 *μ*m. In addition, the notable effects of the aerosol optical depth have also been shown, and we have demonstrated that the best estimations, compared with the ARM dataset using the MH96 method, are obtained when the annual course of the aerosol optical depth is derived from measurements taken from a close-by AERONET station. A rural aerosol has been assumed for all tests. Moreover, the agreement between the present method and the reference method depends slightly on the assumed value of the surface albedo. A brief discussion about the probability of having ice-cloud layers present and its possible effects on retrievals has also been presented.

The proposed method has been developed, applied, and evaluated for optically thick clouds (COD > 10) and assuming overcast sky (fractional sky cover of at least 99%) and a horizontally homogeneous ceiling. The ability of this method away from these conditions is uncertain. The method has not been applied to conditions of high albedo (i.e., a snow-covered surface).

## Acknowledgments

We thank Dr. Dave Turner from the University of Wisconsin—Madison for his help with the ARM data and Dr. Qilong Min, from the Atmospheric Sciences Research Center, who suggested several ideas about the use of the liquid water path measurements and kindly provided us a version of the MH96 code. Data were obtained from the Atmospheric Radiation Measurement Program (ARM) sponsored by the U.S. Department of Energy Office of Science, Office of Biological and Enviromental Research, and Climate and Environmental Sciences Division. The Cimel sun-photometer measurements of aerosol optical depth were collected by the U.S. Department of Energy as part of the ARM Climate Research Facility (ACRF) and processed by the National Aeronautics and Space Administration’s Aerosol Robotic Network (AERONET). We thank Rick Wagener for his efforts in establishing and maintaining the AERONET Cart_Site data. This study was partially financed by the Spanish Ministry of Science and Innovation, through Project NUCLIEREX (CGL2007-62664/CLI). Finally, we are also very grateful to the comments of the reviewers, which lead to a great improvement of this paper.

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## APPENDIX

### Parameterization of Radiative Transfer Inversion

As explained in the paper, we followed a semiempirical approach in developing the parameterization that reproduces the inversion of the radiative transfer results, with the goal of retrieving the cloud optical depth from transmittance measurements and, if available, the cloud effective radius.

*τ*) and the reciprocal of the total atmospheric transmittance (1/

_{c}*T*) (Petty 2006; Barnard and Long 2004). In this linear relationship, a factor of the form 1/(1 −

*g*) arises. As we are interested in the direct dependency on effective radius, and not on the asymmetry parameter, we first obtained a parameterization of 1/(1 −

*g*) in terms of

*r*. We used the precalculated values of the asymmetry parameter in the SBDART code, which correspond to a size distribution described by a modified gamma function with width

_{e}*p*= 7 (Ricchiazzi et al. 1998). The

*g*values are tabulated for 413- and 420-nm wavelengths among others, and correspond to several values of the effective radius in the range 2–128

*μ*m; from these values, we performed an interpolation to estimate

*g*at 415 nm. Then, we calculated 1/(1 −

*g*) and obtained that this factor can be fitted, with an error lower than 0.5%, by the following explicit function of the effective radius (Fig. A1):

Equation (A2), however, is derived considering a homogeneous medium. To check the linear relationship between COD and 1/*T* for more realistic conditions (a vertically inhomogeneous atmosphere) than those described by Eqs. (2) or (A2), we performed a set of radiative transfer simulations with SBDART. Results of these simulations were also used to develop our parameterization. In the SBDART runs, conditions that should more significantly affect the relationship between COD and 1/*T* were varied. In particular, SZA was set to 10°, 30°, 50°, and 70°; *r _{e}* was set to 2, 8, 20, and 50

*μ*m, being the size distribution described by a modified gamma function of width

*p*= 7 (Ricchiazzi et al. 1998). This latter, very large value (which is clearly above the usual range for stratiform clouds) was included to better illustrate the dependencies on the effective radius. Surface albedo

*a*was set to 0, 0.04, 0.08, and 0.12. Some geometrical characteristics of the cloud layer (which is always considered to be horizontally homogeneous) were also analyzed: cloud-base height was varied between 200 and 3000 m, and cloud thickness was varied between 200 and 1500 m. For the aerosol (considered to be rural) the AOD at 550 nm was set to 0, 0.05, 0.10, 0.12, and 0.20. We performed the simulations for cloud optical depths ranging from 1 to 300.

Based upon the SBDART results, the linearity of the relationship between COD and 1/*T* is quite remarkable. A feeling for this relationship can be gained from Fig. 1b in Kim et al. (2003). For the runs we performed, the correlations found were very linear (with *R ^{2}* always higher than 0.9999) in the range of cloud optical depths between 10 and 150. In Fig. A2 we show the dependence of the regression parameters (slope and intercept) on the cosine of the solar zenith angle, for some selected cases that differ in either the effective radius or the cloud geometry. The linear dependence of the slope on

*μ*

_{0}is very apparent. Contrarily, as expected from simple treatments, the intercept is almost independent of the solar zenith angle. It is also shown how the slope and the intercept vary with the effective radius, and their weaker dependency on the cloud geometry. Note that the cases shown produce the maximum deviations of the slope with respect to the reference case when the geometry was changed.

The relatively minor influence of the cloud geometrical characteristics on the regression parameters led us to neglect these variables (cloud-base height and thickness) in our further analyses. Thus, a representative cloud layer was considered, with the base at 1000 m and a geometrical thickness of 600 m. On the other hand, the results of the performed simulations indicate (not shown here) that the role of the surface albedo matches the pattern of behavior that is derived from Eqs. (1) or (2).

*P*

_{1}–

*P*

_{5}must be adjusted to the SBDART results in order to minimize the squared residuals by a nonlinear curve-fitting procedure. We performed this fitting in the range of COD from 1 to 100. For simulations corresponding to AOD = 0.1, we found

*P*

_{1}= 3.6234,

*P*

_{2}= 2.0121, and

*P*

_{3}= −5.8142, close to the values in Eq. (A2);

*P*

_{4}= 0.4125, as expected from Eq. (A2); and

*P*

_{5}was very close to one, as expected from simple treatments [Eqs. (1) and (2)]. When changing the AOD, the new fittings led to significant changes in parameters

*P*

_{1}–

*P*

_{3}, but not in

*P*

_{4}and

*P*

_{5}, which are further considered as being fixed values. It turned out that parameters

*P*

_{1}–

*P*

_{3}can be described as a linear function of AOD (with

*R*

^{2}> 0.9999), leading to Eq. (4).

Equations (A3) [Eq. (3) in the text] and (4) constitute the parameterization that inverts the SBDART results in order to retrieve the cloud optical depth. Figure A3 shows the cloud optical depth calculated with this parameterization against the corresponding values that were used to perform the SBDART simulations as described above. Further comments and discussions about the performance of this parameterization are given in the main body of this paper.