## 1. Introduction

In this paper, an algorithm is presented that derives the statistics of the horizontal distribution of the cloud optical depth over a large region (a grid cell) based on the statistics of the horizontal distribution of the visible reflectance and the mean values of other channel observations. In this context, a large region is an area consisting of many pixels such as grid cells used for numerical weather prediction (NWP) models that have resolutions ranging from 20 to 100 km. In addition to the statistics of the horizontal distribution of optical depth, this algorithm also estimates the mean cloud-top effective particle radius and cloud-top temperature within each grid cell.

There are three main reasons why this type of algorithm is appealing. The first reason is computational efficiency. The production of real-time pixel-level global cloud data from imagers with resolutions of 1–4 km is challenging. The reprocessing of imager datasets at full resolution for climate studies presents even more computational challenges. The computational efficiency of this algorithm is achieved by requiring only one retrieval per cloud layer per grid cell. In contrast, pixel-level algorithms perform a retrieval for each cloudy pixel. For a 50-km grid cell composed of 1–4-km pixels, this results in potentially a two to three order of magnitude reduction in required retrievals. Simulations will demonstrate that this reduction in computations preserves a meaningful representation of the cloudiness.

The second reason is that the radiance distributions required for this algorithm already exist and continue to be compiled as standard products. For example, a standard International Satellite Cloud Climatology Project (ISCCP) product includes the mean and standard deviation of the cloudy radiances within each grid cell. In addition, the Advanced Very High Resolution Radiometer (AVHRR) Pathfinder Atmospheres (PATMOS) also provides this information from the afternoon polar orbiter AVHRR data from 1981 to 2001 (Stowe et al. 2002). The advantage of this approach is that once the radiance distributions are made, there is no need to reprocess the full-resolution data. The reprocessing capability is crucial to cloud climate studies because improvements in the radiative modeling of clouds are ongoing. For example, as new and more accurate models of cirrus scattering properties become available (Key et al. 2002), this algorithm would allow for relatively rapid recomputation of cloud climatologies consistent with these improvements.

Last, the product of this algorithm, the horizontal distribution of cloud optical depth, has been shown to be sufficient for accurate modeling of the solar radiative heating within a grid cell. Barker et al. (1996) demonstrated this using three-dimensional radiative transfer modeling applied to high-resolution cloud fields. Oreopoulos and Barker (1999) developed a computationally efficient two-stream radiative transfer solution that directly incorporates the grid cell distribution of optical depth. The capability of remotely sensing the optical depth distribution within a grid cell may therefore be a critical tool for the efficient assimilation of relevant cloud properties by numerical weather prediction models.

In summary, the goal of this paper is to present an algorithmic approach to the daytime estimation of cloud properties that operates on the radiance distributions from cloud layers over a large area or grid cell. The cloud properties provided by this approach are the statistics of the horizontal distribution of the vertically integrated optical depth, the mean cloud-top particle effective radius, and the mean cloud-top temperature. The purpose of this paper is to demonstrate that the assumptions necessary for this algorithm are sufficiently valid and robust to allow for meaningful estimation of grid cell cloud properties. This demonstration will be conducted using both simulations and actual satellite observations.

The algorithm presented in this paper is currently being applied to data from the Advanced Very High Resolution Radiometer and serves as the daytime cloud property retrieval module in the Clouds from AVHRR (CLAVR) system. The CLAVR system will serve as the National Oceanic and Atmospheric Administration's (NOAA) operational AVHRR cloud-processing system. The CLAVR products are generated at a resolution of 50 km and will be produced for the AVHRR data from the morning, midmorning, and afternoon polar-orbiting satellites. A separate but conceptually similar approach is applied for the nighttime generation of cloud properties. In addition to the cloud retrieval algorithms, the CLAVR system is composed of pixel-level cloud masking (Stowe et al. 1999; Heidinger et al. 2002) and cloud phase determination as well as the cloud-layering algorithm used to derive the radiance distributions used for the cloud property retrievals. The evolution of this algorithm can be traced in the conference proceedings of Heidinger and Stowe (1999) and Heidinger and Liu (2001).

## 2. Modeling *h*(*τ*) with a gamma distribution

*h*(

*τ*), for each cloud layer in a grid cell can be modeled using a gamma distribution. With this assumption, the

*h*(

*τ*) can be expressed as where

*τ*is the optical depth,

*τ*

*ν*is the width parameter. The standard deviation of

*τ,*

*σ*

_{τ}, is given by

*τ*

*ν*

*h*(

*τ*) refers to the probability of occurrence within a cloud layer of a vertically integrated optical depth of value

*τ.*

To illustrate the behavior of the gamma distribution, Fig. 1 shows plots of *h*(*τ*) for *τ**ν* = 1 and 10. As the gamma distribution width parameter, *ν,* increases, the distribution narrows. For values of *ν* ≪ 1, the gamma distribution takes on the shape of an exponential distribution. For values of *ν* > 10, the gamma distribution approaches the form of a Gaussian distribution. The distributions of the 0.63-*μ*m reflectances corresponding to the optical depth distributions shown in Fig. 1 are given in Fig. 2. These reflectances were computed assuming overhead illumination, an oceanic surface, and a cloud composed of water droplets with an effective radius of 10 *μ*m. Because of the asymptotic behavior of the variation of reflectance with optical depth, the distributions of reflectance tend to be more narrow than the distributions of optical depth. If reflectance varied linearly with optical depth, the two distributions would have the same shape. This retrieval does not assume any particular shape of the reflectance distribution. The only assumption is that knowledge of the statistics of the reflectance distribution is sufficient to uniquely retrieve *h*(*τ*). Results shown later will support this assumption.

Modeling *h*(*τ*) with a gamma distribution has been proposed by Barker et al. (1996). In this study, the distributions of optical fields from marine stratocumulus were derived from high-resolution (28 m) Landsat data. The results indicated a gamma distribution accurately modeled the observed *h*(*τ*). To test these assumptions for a wider range of cloudiness and for observations with the resolution of AVHRR GAC (4 km), an orbital segment of *NOAA-16* AVHRR GAC data was analyzed. The left panel in Fig. 3 shows the 0.63-*μ*m reflectance of an orbital segment of *NOAA-16* AVHRR GAC that stretches from the equator to roughly 35°N. The coast of California is evident in the upper-right corner of this image. This scene was chosen for analysis because it contains multiple types of cloudiness. Present within this image are stratus, convective, and cirrus clouds. The radiance distributions are compiled separately for water and ice cloud pixels in each grid cell. The decision as to which pixels enter into the radiance distribution is critically dependent on the cloud mask and cloud phase algorithms. The center panel in Fig. 3 shows the CLAVR cloud mask for this scene. The CLAVR cloud mask discretely classifies each pixel as being clear (0), partly clear (1), partly cloudy (2), or cloudy (3). The current version of CLAVR allows for one ice and one liquid water cloud layer in each grid cell. The right panel in Fig. 3 shows the cloud type or phase derived for this scene with the dark gray pixels being the noncloudy, the gray pixels representing the liquid water clouds, and the white pixels representing the ice phase clouds. Both the cloud mask and cloud typing algorithms use a multispectral approach.

To analyze the pixel-scale distribution of cloud properties, a pixel-scale retrieval of *τ* and *τ*_{e} was used. This retrieval uses only 0.63- and 1.6-*μ*m reflectances and does not retrieve the cloud-top temperature *T*_{c}. Values of the 11-*μ*m brightness temperature are used for surrogates of *T*_{c} for this analysis. The forward models used in the retrieval for this section are identical to the ones described in the next section except that each pixel is modeled as a plane-parallel (horizontally uniform) cloud. Once the pixel values of *τ* and cloud-top effective radius, *r*_{e}, were computed for this image, the pixels were grouped into grid cells with each grid cell being a 20 × 20 array of pixels. The analysis was found to be insensitive to the size of the grid cells; therefore, the results apply to a wide range of grid resolutions. The mean and standard deviation computed from observed *h*(*τ*) were used to define a gamma distribution for each grid cell. A reduced *χ*^{2} test (Taylor 1982) was performed to determine how well the gamma distribution fit the observed *h*(*τ*). If the reduced *χ*^{2} value is of the order or less than unity, the gamma distribution is a good model of the observed distribution. The bin size used for computing *h*(*τ*) was 1 and the allowable range was 0 to 100. For most grid-cell-reduced *χ*^{2} computations, the number of degrees of freedom was much larger than 10. Therefore, this analysis should be able to test the validity of the gamma distribution for modeling *h*(*τ*).

The distributions of the values of reduced *χ*^{2} for this scene are given in Fig. 4 and are scaled to integrate to unity. Distributions were constructed for both ice and water cloudy pixels as well as for all cloudy pixels combined. Because the cloud mask can classify a pixel as either cloudy or partly cloudy, the distribution for cloudy and partly cloudy pixels combined is also shown. The values in parentheses in the legend of Fig. 4 are the mean values of reduced *χ*^{2} for each distribution. These results indicate that the *h*(*τ*)'s derived from cloudy pixels with and without cloud typing are well modeled by the gamma distribution. The inclusion of the partly cloudy pixels does raise the mean reduced *χ*^{2} but not to the point of indicating a failure of the gamma distribution assumption. Even without cloud typing, the gamma distribution appears to be valid for most grid cells in this scene. Only for less than 1% of the grid cells, does reduced *χ*^{2} exceed 1.5 indicating an inability of the gamma distribution to accurately model *h*(*τ*).

The other assumption in this retrieval is that it is sufficient to estimate only the mean values of *r*_{e} and *T*_{c} for each layer in each grid cell. To test this assumption, the distributions of *r*_{e} and *T*_{c} were also computed. The bin sizes used were 0.5 *μ*m for *r*_{e} and 0.5 K for *T*_{c}. The mean value of the width parameter for *h*(*τ*) was 4. In corresponding mean width, parameters for the distribution of *r*_{e} and *T*_{c} were 10 and 30. The narrow distributions observed for *r*_{e} and *T*_{c} lend support to the assumption that only the mean values of these parameters need to be retrieved. This work does not imply that there is no information in these distributions but only that a meaningful representation of cloudiness for this application is provided by their mean values.

## 3. Retrieval algorithm

To retrieve the cloud parameters, a one-dimensional variational (1DVAR) technique is used. In this retrieval, four parameters are being estimated from five observations. A 1DVAR approach was selected because more traditional iterative approaches were found to offer no advantage in speed or in convergence: A 1DVAR approach also offers the benefits of ensuring the retrievals are consistent with the uncertainties in both the forward models and the measurements.

### a. Required observations

As mentioned previously, the data used for this algorithm comes from NOAA's AVHRR. The observations used are the channel 1 (0.63 *μ*m) reflectance *R*_{1}, the channel 4 (11 *μ*m) brightness temperature *T*_{4}, and the channel 5 (12 *μ*m) brightness temperature *T*_{5}. The near-infrared reflectance offered by the AVHRR is obtained from either channel 3a (1.6 *μ*m), *R*_{3a}, or channel 3b (3.75 *μ*m), *R*_{3b}. The AVHRR instruments on *NOAA-15* and later can only record channel 3a or channel 3b. Before *NOAA-15,* all AVHRR's provided only ch3b. Because the current operational afternoon-orbiting AVHRR (*NOAA-16*) records channel 3a during daylight operation, results for retrievals using *R*_{3a} are emphasized. Simulations will be presented that offer insight into the differences in retrieval performance for both channel configurations. In this algorithm, both the solar and thermal components of channel 3b are combined and expressed as a reflectance. Separate solar and thermal models, described later, are combined to serve as the forward model of *R*_{3b}.

The required observations to this algorithm are the mean and standard deviation of *R*_{1} and the mean values of *R*_{3a/b}, *T*_{4}, and *T*_{4} − *T*_{5}. As stated before, these observations are compiled separately for ice and water cloud layers within each grid cell. The use of *R*_{1} and *R*_{3a/b} to simultaneously derive cloud optical depth and cloud-top effective radius has been well documented (i.e., Nakajima and King 1990). In addition, many algorithms have incorporated *T*_{4} to estimate cloud-top temperature. The use of *T*_{4} − *T*_{5} is not so common in cloud retrievals. Figure 5 shows the modeled variation of *T*_{4} − *T*_{5} as a function of *τ* and *r*_{e} for a cirrus cloud viewed at nadir with *T*_{c} = 240 K over a surface with a temperature of 300 K. The width parameter *ν* was set to 20 for these simulations and these clouds are therefore relatively spatially uniform in optical depth or emissivity. As this figure shows, for thin clouds (*τ* < 4), values of *T*_{4} − *T*_{5} constrain the allowable values of *r*_{e} and *τ.* For thick clouds, *T*_{4} − *T*_{5} provides little information except for a small dependence on *r*_{e}. In addition, use of *T*_{4} − *T*_{5} aids in achieving continuity between the cloud products with the nighttime algorithm. As Parol et al. (1991) demonstrated, *T*_{4} − *T*_{5} can exhibit a strong sensitivity to particle shape. Since a fixed particle shape is assumed here, this potential error will be included in the uncertainty assigned to estimates of these observations.

### b. 1D-VAR retrieval

*y,*for this algorithm is given by

*y*

*R*

_{1}

*σ*

_{1}

*R*

_{3a}

*T*

_{4}

*T*

_{4}

*T*

_{5}

*σ*

_{1}is the standard deviation of

*R*

_{1}. The bar superscript denotes the mean for all cloudy pixels of the same phase within a grid cell.

*x,*is defined as

*x*

*τ*

*r*

_{e}

*T*

_{c}

*ν*

*τ*

*r*

_{e}is the mean effective radius,

*T*

_{c}is the mean cloud-top temperature, and

*ν*is the width parameter of

*h*(

*τ*). Logarithmic values of

*τ*

*ν,*and

*r*

_{e}are retrieved to improve the convergence characteristics.

In addition, this retrieval method requires the definition of a priori estimates of the retrieved parameters, *x*_{ap}, that are derived based on representative values for each cloud phase. The final solution is an optimal estimate of *x* that balances the uncertainties in the measurements, the forward models, and the a priori parameters.

*x*

_{ap}. In many applications, model or climatologic values can be used but for this application, this data is generally not available for these properties. The a priori values of

*τ*

*r*

_{e}are taken from simple relationships derived from analysis of simulations of clouds over the ocean using the forward models described in a following section. These relationships are meant to offer only rough estimates of

*τ*

*r*

_{e}. The expression used to derive the a priori of

*τ*

*τ*

_{ap}

*R*

_{1}

^{0.9}

*R*

_{1}is a reflectance expressed as a percentage. To derive a priori estimates of

*r*

_{e}, separate expressions for ice and water clouds are necessary. When using

*R*

_{3a}, the ratio of

*R*

_{3a}to

*R*

_{1}was found to be the dominant driver in the retrieval of

*r*

_{e}. For water clouds, the a priori value of

*r*

_{e}is computed as and for ice cloud the a priori value of

*r*

_{e}is computed as When

*R*

_{3b}is used, the a priori values of

*r*

_{e}are set to 10

*μ*m for water clouds and to 20

*μ*m for ice clouds. To estimate the a priori value of

*ν,*the width parameter derived from the mean and standard deviation of

*R*

_{1}is used (see Figs. 1 and 2). The reduction of the reflectance width parameter in half accounts for the tendency for optical depth distributions to be wider than reflectance distributions (see Figs. 1 and 2). For the a priori value of

*T*

_{c}, the value of

*T*

_{4}is used. The above values can be inaccurate estimates under many conditions. The proper setting of the uncertainties of these estimates will prevent them from overconstraining the retrieval.

Another requirement of the 1DVAR retrieval not typically levied on other cloud algorithms is the requirement to specify the uncertainties of *x*_{ap}, the measurements *y,* and in the forward model's estimate of the measurements *f*. Because little information is available to refine the value of *x*_{ap} their uncertainties are set to high values to prevent improper constraints on the solution. For example, the uncertainty for the a priori value of log*τ**ν* and log*r*_{e} are both set to 0.5. In general, these values imply greater than 100% relative uncertainty in terms of the actual values of *τ**r*_{e}, and *ν.* The uncertainty in a priori value of *T*_{c} is set to 20 K. These values are admittedly large but it is difficult to justify reducing these uncertainties without the use of other ancillary data sources. If the prescribed uncertainties in *x*_{ap} are too low, the retrieval will ignore the observations and will never deviate far from *x*_{ap}. If the uncertainties in *x*_{ap} are too great, the stabilizing effect of using a priori information is lost. In general, the values of *x*_{ap} will affect the accuracy of the retrieval only where the observations offer incomplete or nonunique information.

*y,*and the forward model,

*f*, is also a difficult task. Setting these uncertainties too large will cause the retrieval to deviate little from

*x*

_{ap}. Setting the uncertainties too small will cause the retrieval to force agreement with all observations and could lead to excessive retrieval iterations and lack of retrieval convergence. In this retrieval, the uncertainty of the observations and forward model for each channel are combined into one term,

*δy*

_{i}as demonstrated in (6) and (7). The sources of uncertainty accounted for are calibration (

*ε*

_{cal}), forward model (

*ε*

_{fm}), and the plane-parallel modeling errors (

*ε*

_{pp}). Plane-parallel modeling error is separated from the forward model error so that it can be made a function of the spatial uniformity. For the reflectance observations (

*R*

_{1},

*R*

_{3a},

*R*

_{3b}), where

*ε*

_{cal}is set to 5%,

*ε*

_{fm}is set to 5% for water clouds and to 10% for ice clouds, and

*ε*

_{pp}is set to 30%. When

*R*

_{3b}is used,

*ε*

_{cal}is cut in half to reflect the presence of onboard calibration of this channel. For the thermal observations (

*T*

_{4},

*T*

_{4}−

*T*

_{5}), where

*ε*

_{cal}is set to 1 K,

*ε*

_{fm}is set to 1 K for water clouds and to 2 K for ice clouds, and

*ε*

_{pp}is set to 5 K. The forward model error is set to a higher value for ice clouds than water clouds to reflect the higher uncertainty in prescribing the scattering properties of ice crystals. These values are admittedly not based on rigorous analysis. The attempt here is to include all relevant errors and to adjust these uncertainties when knowledge warrants it.

_{x}is the error covariance matrix of

*x*and is given by

^{−1}

_{x}

^{−1}

_{a}

^{T}

^{−1}

_{y}

*f*to

*x*(i.e.,

*df*/

*dx*). In (9), 𝗦

_{a}is the error covariance matrix of the a priori parameters and 𝗦

_{y}is the error covariance of the measurements and the forward model. Here 𝗦

_{a}and 𝗦

_{y}are treated as diagonal matrices with each diagonal term being the square of the uncertainties given above. In an ideal observing system, 𝗔 would be the identity matrix indicating no reliance of the retrieval on the a priori parameters. As will be shown later, the diagonal terms of 𝗔 are useful for predicting conditions where the retrieval relies significantly on the a priori information.

### c. Forward modeling

The goal of the forward models put forth here is to simulate the behavior of cloud fields within grid cells well enough to allow for meaningful estimation of their properties. Since global, faster than real time processing is required of this algorithm, true radiative transfer models are approximated by lookup tables. The forward models described here are similar to the forward models used in pixel-scale retrievals except that the width of the *τ* distribution for a grid cell, *ν,* is an additional free parameter. The forward models described below provide the vector *f* and the kernel matrix 𝗞, where each element contains the terms of the Jacobean matrix (∂*f*/∂*x*). To compute the scattering properties of clouds, the cloud particles were assumed to be spherical droplets. While this is a well accepted practice for water clouds, for ice clouds it can lead to large uncertainties. Finding methods for proper specification of ice crystal scattering properties is an area of ongoing research (Baran et al. 1999; Yang et al. 2000). In future validation studies, more appropriate ice crystal models will be implemented. The use of spherical ice particles does not directly impact the relative findings of this study.

#### 1) Modeling solar reflectance

*R*

_{c}. Once the lookup tables for single-layer plane-parallel clouds are made, lookup tables with the additional dimensions of

*ν*and the Lambertian surface reflectance,

*a*

_{s}, are made. To compute the top of atmosphere reflectance for a plane-parallel cloud,

*R*

_{pp}, above a Lambertian reflecting surface, the following expression taken from Chandrasekhar (1960) is used: where

*α*

_{sph}is the spherical albedo of the cloud layer,

*t*

_{ac}is the nadir transmission from the top of atmosphere to cloud top,

*m*is the airmass factor (1/

*μ*+ 1/

*μ*

_{o}),

*μ*

_{o}is the cosine of the solar zenith angle, and

*μ*is the cosine of the viewing zenith angle. The dependencies on the solar/viewing geometries and other parameters are not included in (10) for simplicity. The terms

*T*(

*μ*) and

*T*(

*μ*

_{o}) are the flux transmissions through the cloud layer (direct and diffuse) for a solar beam incident at zenith angles defined by

*μ*and

*μ*

_{o}. The modified surface albedo,

*a*

^{′}

_{s}

*a*

^{′}

_{s}

*t*

^{m}

_{bc}

*a*

_{s}

*t*

_{bc}is the nadir transmission from the cloud to the surface. Because this study deals with clouds over an oceanic surface,

*a*

_{s}was set to 3% for all reflectance computations. In the global application,

*a*

_{s}is determined for each channel as a function of the surface type and the solar zenith angle.

*R*

*τ*

*ν*) is found by integrating (10) over the assumed

*h*(

*τ*) to give with the standard deviation

*σ*being computed as

The result of these computations are lookup tables for the mean and the standard deviation in reflectance as a function of the cloud properties, *τ**r*_{e}, *ν*; the surface reflectance *a*_{s}; the viewing geometry defined by *μ,* *μ*_{o}; and the relative solar azimuth angle (*ϕ*_{o} − *ϕ*). As mentioned earlier, the dimensions of the reflectance lookup tables are composed of equally spaced vectors of log*τ**r*_{e}), and log(*ν*).

#### 2) Modeling thermal emission

*ε*

_{c}, was computed as the ratio of the radiance emanating out of the top of cloud divided by the radiance emanating from a blackbody at the same temperature with no contribution from radiance below the cloud. The transmissivity,

*t*

_{c}, was computed by turning off all cloud emission and computing the ratio of the radiance emanating out of the top of the cloud to that incident at the base of the cloud. These computations therefore result in nearly temperature-independent quantities that include effects of scattering. Values of the mean emissivity

*ε*

_{c}and mean transmissivity

*t*

_{c}are computed using the plane-parallel lookup tables for all values of

*τ*and an integration over

*h*(

*τ*)

*dτ*as shown below:

*τ*

*ν*), log(

*r*

_{e}), and

*μ,*the cloud layer is embedded in a nonscattering atmosphere over a nonscattering surface to model to the top-of-atmosphere radiance using the following relation: where

*E*

_{clear}is the clear-sky radiance,

*E*

_{ac}is the emitted radiance from the layer above the cloud, and

*m*is air mass for emitted radiation (1/

*μ*). To account for the lapse rate within the cloud, the cloud emissivity is redefined. Using a linear-in-depth variation of the emission through the cloud, a modified mean emissivity of the cloud,

*ε*

^{′}

_{c}

*B*

_{n}and

*B*

_{o}are the blackbody-emitted radiances at the cloud top and cloud base and

*τ*in this expression is defined as −ln(

*t*

_{c}).

## 4. Simulated performance of retrievals

In this section, simulations are used to illustrate the performance of the retrieval over a wide range of conditions. These simulations focus only on the retrieval over an ocean surface. The results for retrievals over bright surfaces show larger errors but are qualitatively similar. The goal here is to diagnose which grid cell distributions of optical depth, *h*(*τ*), will present difficulties to this retrieval approach. Because significant differences exist, results using both *R*_{3a} and *R*_{3b} are shown.

*h*(

*τ*) is modeled with a gamma distribution to compute the reflectance and radiance distributions used as input to the retrieval algorithm. These results will therefore not directly test the validity of using a gamma distribution to model

*h*(

*τ*). The analysis of the previous section indicates this is not warranted. These results do, however, test the assumption of using the mean values of

*R*

_{3a/b},

*T*

_{4}, and

*T*

_{5}by allowing

*r*

_{e}and

*T*

_{c}to vary with

*τ*when computing the simulated observations. Based on the analysis of marine stratocumulus clouds by Szczodrak et al. (2001),

*r*

_{e}is assumed to vary linearly with

*τ*

^{0.2}. The actual form used was

*r*

_{e}

*τ*

^{0.2}

*r*

_{e}= 10

*μ*m for

*τ*= 10. After Minnis et al. (1993), the variation of cloud geometrical thickness in kilometers,

*H,*is given by

*H*

*τ*

^{2/3}

*r*

_{e}and

*T*

_{c}that is correlated with

*τ,*as would be expected for most cloud fields. Other specified parameters include an oceanic surface, a cloud base of 500 m, and an atmosphere modeled using the standard midlatitude summer profile. The solar zenith angle was 30°, the satellite zenith angle was 30°, and the relative azimuth was 150°. These simulations are meant only to illustrate the performance of the retrieval under realistic calibration errors and the error incurred by assuming uniform values of

*r*

_{e}and

*T*

_{c}. Retrievals were run on the simulated data using the above relations and the retrieval parameters as described above. Random errors of 5% were applied to

*R*

_{1}and

*R*

_{3a}2% to

*R*

_{3b}, and 0.5 K to

*T*

_{4}and

*T*

_{5}. These errors are conservative estimates of the calibration errors of the AVHRR. To simulate uncertainties in the knowledge of the atmosphere and surface, the skin temperature of the ocean surface was randomly varied by 2 K while the total precipitable water amount was randomly varied by 30%.

Figure 6 shows the results for simulated retrievals for an AVHRR using *R*_{3a} as function *τ**ν* for a water cloud with the properties described above. The results in Fig. 6 are the mean values computed from 100 simulations with errors described above randomly varied. The 1DVAR retrieval approach automatically gives estimates of the error in the retrievals, 𝗦_{x}, and the relative weight of the observations in the retrieval, 𝗔. To show the performance of these simulated retrievals, the values of 𝗦_{x} were not used in favor of the actual relative differences between the retrieved and true parameters. The contours of 𝗦_{x} and 𝗔 were found to be highly correlated and the presentation of the actual errors along with the observation weights was found to provide a more insightful diagnosis of the retrieval performance.

The plots in the left column are the relative errors in the retrieved parameters relative to the true values used to compute the radiance distributions. The right column contains the contour plots of the weight of the retrieval of each parameter on the observation taken from the diagonal elements of 𝗔. In general, the errors are largest where the reliance on the observations is least. The interpretation of these simulated results depends ultimately on the required accuracy of these retrievals. For the interpretations given here, only when errors in *τ**r*_{e}, or *ν* exceed 20% or the errors in *T*_{c} exceed 1 K will a potential weakness with the retrieval performance be noted.

The results for optical depth (Figs. 6a and 6b) indicate good performance and high observation weight over most values of *τ**ν.* Only for large values of *τ**τ**τ*

Figures 6c and 6d give the results for the retrieval of *ν.* Errors in *ν* generally exceed 10% for all simulations. However, errors in *ν* exceed 20% for all cloud layers with exponential distributions (*ν* < 0.5). The largest errors (over 100%) occur for optically thick layers with small values of *ν.* The values of *A* in Fig. 6d are consistent with the simulated errors and show the least reliance on the observations for *ν* < 1.

The performance of the retrieval of *r*_{e} can be taken from Figs. 6e and 6f. Errors in *r*_{e} are greater than 20% for most *ν* when *τ**r*_{e} exceed 50% for *τ**ν* < 2. The corresponding observation weights depart significantly below unity for these large error regions. The reason for these large errors is the lack of significant absorption of *R*_{3a} by water droplets for optically thin clouds. For a given value of *τ**ν* decreases, more and more of the cloud layer optical depth is proportionally distributed among smaller values of optical depth. Because small values of optical depth imply less absorption and less sensitivity to particle size, this contributes to the increase in error for small values of *ν.*

The simulated results for the retrieval of *T*_{c} are shown in Figs. 6g and 6h. Only for optically thin clouds (*τ**T*_{c} exceed 1 K and do the observation weights drop slightly below unity.

Analogous simulations to the ones described above were applied to ice clouds over an oceanic surface and are presented in Fig. 7. The results are qualitatively similar. The main difference in the ice cloud performance is the increase in the uncertainty of the forward model and the increase in the relevance of *T*_{4} − *T*_{5} for optically thin ice clouds. The results for the retrieval of *τ* and *ν* show no dramatic difference. The errors in the retrieved *ν* are generally larger for optically thin ice cloud than water cloud. However, the largest errors in *ν* still occur for layers with exponential distributions (*ν* < 1). Due to an increase in the sensitivity in *R*_{3a} to particle size in the ice cloud simulations, the errors in the retrieved *r*_{e} are correspondingly less than for the water cloud simulations. Because ice clouds are much colder than the surface, there is greater potential for error in *T*_{c} for semitransparent clouds. Correspondingly, the errors in *T*_{c} approach 4 K and the observation weights depart significantly from unity for *τ**ν* < 1.

As mentioned earlier, current AVHRR instruments have the option to measure *R*_{3b} rather than *R*_{3a}. To explore the consequences of this in the context of these simulations, the water cloud simulations described above were repeated substituting *R*_{3b} for *R*_{3a} and the results are given in Fig. 8. The results for the retrieval of *τ**ν,* and *T*_{c} were little affected by the channel choice and are not shown. Dramatic differences were evident in the performance for the retrieval of *r*_{e}. Where the results using *R*_{3a} generally showed errors exceeding 10%, the results using *R*_{3b} show errors of less than 10% for most values for *τ**ν.* Only when *ν* < 1 and *τ**μ*m than at 1.6 *μ*m. In addition, differences in retrieved *r*_{e} using *R*_{3a} or *R*_{3b} can occur due to differing sensitivities at different levels in the cloud between the two channels (Platnick 2000). These effects do indicate a potential discontinuity in the data record of *r*_{e} derived from AVHRR.

The results of these simulations are necessarily dependent on the choices made for the uncertainties used in computing 𝗦_{y} and 𝗦_{a} and in the prescription of the a priori parameters, *x*_{ap}. For example, if the measurements were assumed perfect, the errors in the *r*_{e} retrieval using *R*_{3a} would have been much less. The attempt has been made to assign reasonable values to these parameters to ensure credible simulated performance. The results of these simulations do indicate that retrieval of these four parameters should be successful for most cloudiness conditions. The results for a standard plane parallel retrieval that does not attempt to retrieve *ν* directly can be inferred from the top of each figure where *ν* ≈ 50. In general, the results for *ν* < 20 are not significantly worse than for the standard plane-parallel retrievals. Only for very thin clouds and values of *ν* < 2 do the errors exceed those of plane-parallel retrievals. In defense of this approach, for optically thin cloud fields with *ν* < 2, it can be argued that most 1–4-km pixels are too coarse to allow for plane-parallel retrievals at all. In addition, errors due to horizontal transport, not included here, would also increase the errors in plane-parallel retrievals for these cloud fields.

## 5. Comparison to pixel-level retrievals

Without independent cloudiness observations, the results of this retrieval cannot be verified. A relative validation is possible from the comparison of the grid cell retrievals to the results from composited pixel-scale retrievals. As the simulations showed above, the results from this approach should not be significantly worse than the standard pixel-scale retrievals for many cloudiness conditions.

To perform this comparison with the pixel-level retrievals, the data from the image in Fig. 3 is used. The CLAVR system was used to perform the cloud mask and cloud phasing shown in Fig. 3 and to generate the statistics of the AVHRR channel observations for ice and water cloud layers. Any one grid cell can report separate ice and water cloud layers. This analysis uses 0.5° equal area grid with the grid cells having a corresponding spatial resolution of approximately 55 km. This grid corresponds to the CLAVR implementation at NOAA.

The results of the grid cell retrieval for *τ**ν* are shown in Fig. 9 and for *r*_{e} and *T*_{c} in Fig. 10. Though the algorithm retrieves values for ice and water clouds separately, the results in Figs. 9 and 10 show the cloud-fraction weighted average of the ice and water results. The retrieval results are generally consistent with the expected properties for oceanic cloud systems. The water phase clouds have optical depths generally ranging from 4 to 20 and with effective radii ranging from 6 to 20 *μ*m. The ice clouds have greater dynamic range in *τ**r*_{e}. The cloud-top temperature also varies as expected. The retrieved values of *ν* also capture the variation in roughness of the *R*_{1} image shown in Fig. 3 with high values of *ν* being retrieved in uniform stratus region and lower values being retrieved in the more visually rough ice clouds.

To compare the grid cell results to pixel-level results, the same mapping routine used to make the AVHRR pixels was used to map the pixel-level retrievals into the appropriate grid cells. The mean and standard deviation of pixel-level retrievals of *τ* and *r*_{e} were computed separately for the ice and water phase cloudy pixels. The value of *ν* was derived from the pixel-level results from the mean and standard deviation of *τ.* Comparisons were only done when the cloud fraction for ice or water exceeds 5% of the grid cell. To avoid any effects of navigation errors on the retrievals, any grid cell with one or more pixels being classified as land was not used. The effect of these constraints is to ensure that the same pixels that were used to derive the radiance distributions were used to construct the pixel-level retrievals. The resulting grid cells used for this analysis numbered approximately 1600 for the water cloud and 900 for the ice clouds.

The results of the comparison of the grid cell and the pixel-level water cloud values of *τ**ν,* and *r*_{e} are shown in Fig. 11. There are no results for *T*_{c} because the pixel-level retrieval does not estimate it. The results in Fig. 11 are plotted separately for *τ**τ**τ**τ**τ**τ**τ* and *r*_{e} being generally much less than 10%. Only the standard deviation of *r*_{e} for *τ**ν* is small while the inclusion of optically thin clouds causes the mean difference to approach −10%. The standard deviation of *ν* is roughly 20%.

The corresponding results for the comparisons of the pixel-level and grid cell retrievals for the ice clouds are shown in Fig. 12 and Table 2. The results are generally similar to those for water clouds. The most notable change is the mean difference in the retrieval of *τ**ν.*

As noted above, calculations shown in Fig. 12 and Table 2 indicate a systematic difference in the mean optical depth results from the two methods for the optically thick ice cloud retrievals. The reason for this difference is unclear but is allowable given the large uncertainties in the retrieval of *τ**τ*

## 6. Conclusions

A retrieval approach has been developed to rapidly estimate the cloud properties from grid cells composed of many imager pixels. The properties estimated are the mean and width of the horizontal distribution of optical depth and the mean value of the cloud-top effective radius and temperature for each cloud layer in the grid cell. Previous modeling studies of Barker et al. (1996) and Oreopoulos and Barker (1999) indicate that the cloud properties retrieved here are sufficient to accurately compute the mean solar radiative heating within the cloud layer. Simulations indicate that this method should perform well except for very optically thin cloud layers with very broad distributions of optical depth. It is not clear that even pixel-level retrievals perform well for these cloud fields. The differences between the comparisons to the pixel-level comparisons are generally less than the errors predicted by simulations. This finding indicates that there is no significant penalty incurred in the accuracy of the cloud properties when this rapid approach is used.

In the future, this algorithm will be applied globally as part of NOAA's operational CLAVR cloud product system. The current resolution of this product is 0.5°. To present a qualitative look at the application of this algorithm globally, Fig. 13 shows the results of this algorithm applied to all the ascending data (13:30 local time) from *NOAA-16* on day 230 of 2001. The image does not show latitudes poleward of 60° because there is no solar reflectance in this region at this time. Work is ongoing to apply this technique to the radiance statistics of ISCCP or PATMOS to derive new cloud climatologies.

## Acknowledgments

The author appreciates the critical review provided by Sharon Nebuda formerly of NASA's Data Assimilation Office.

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Mean and standard deviation of the relative difference (%) between the grid cell and pixel-scale retrievals for grid cells containing water clouds

Mean and standard deviation of the relative difference (%) between the grid cell and pixel-scale retrievals for grid cells containing ice clouds