## 1. Introduction

[2] Knowledge of cloud properties, including their spatial and temporal variability, is needed for understanding and quantifying the role of clouds in climate variability and for modeling clouds and their effects in climate and weather models. Many techniques have been developed for cloud property retrievals from airborne and ground-based remote sensing measurements of reflected irradiance/radiance in the shortwave region. Most of the current retrieval techniques are based on using precomputed radiative transfer model simulations (commonly referred to as the look-up table) and a procedure for evaluating the minimum distance (the best fit) between the modeled and measured cloud reflectance or transmittance in the visible and near-infrared regions of the spectrum (*Twomey and Cocks* [1989], *Nakajima and King* [1990], and *Foot* [1988], among many others). The techniques differ in the selection of the radiative transfer model used to compute the simulated measurements for clouds of varying optical depth and effective radius, distance function (e.g., the choice of weights used for spectral regions), measurement type (e.g., irradiance and radiance) and the spectral bands used in the retrieval. In general, the accuracy of the retrieved cloud properties depends on the accuracy of the radiative transfer model, including knowledge of the uncertainties in independent nonretrieved inputs, modeling assumptions and numerical approach (e.g., the plane parallel assumption and the computation of scattering), and on the accuracy of the measurements and their sensitivity to the cloud properties that are retrieved. The importance of evaluating cloud properties requires that the accuracy and uncertainty of the cloud property retrievals are well characterized to ensure robust estimates of uncertainties in the applications. Additionally, characterization of sources of uncertainty in the retrieval algorithms and the effect of them on the accuracy of the retrieved cloud properties is needed to help with determining observing strategies and approaches for improving the algorithms.

[3] The retrievals are often validated against in situ flight measurements [*King et al.*, 1997, section 3.3.2] to estimate accuracy and errors. This approach may provide a direct measure of the accuracy, but it is limited by the availability and characteristics of the in situ measurements. For example, the in situ measurements are only approximately representative of the cloud properties on a spatial scale of the remote sensing measurement that is used in the retrieval for the same site, unless the clouds are approximately homogenous. In addition, the in situ measurements are not representative of the wide variety of clouds on global scales to which the retrieval algorithms are typically applied. Besides direct comparisons to in situ measurements, uncertainties in the retrieved cloud properties have been evaluated by the method of error propagation [*Coddington et al.*, 2010]. By this method the retrieval algorithm is employed repeatedly with slightly different inputs for the same observed scene. The different inputs are assumed to be representative of the uncertainties in the quantities that are not retrieved such as for example, atmospheric profiles of temperature and humidity, surface optical properties, and aerosol effects. This method requires large numbers of simulations with the retrieval algorithm to obtain a robust estimate of the uncertainties in the retrieved cloud properties. Because of this property the error propagation method is not typically used in standard operational cloud retrievals such as MODIS (Moderate Resolution Imaging Spectrometer) retrievals [*King et al.*, 1997].

[4] The number of computations of error estimates needed for explicit error propagation could be reduced by employing the error estimation theory, where the error of the retrieved cloud properties is modeled by an assumed transformation of the input statistics of the uncertainties into the equivalent output statistics [*Rodgers*, 2000]. Typically, in order to make the modeling of the error statistics tractable it is assumed that both the input and output uncertainties are well represented by normal or Gaussian statistics. This assumption implies that the transformation between the input and output of the retrieval algorithm is linear [*Tarantola*, 2005]. The condition of linear transformation is not satisfied for cloud property retrievals because the radiative transfer model is not linear. Examples of nonlinear effects in the radiative transfer model calculations on the retrieval uncertainties in ice cloud properties retrieved from MODIS [*King et al.*, 1997] measurements are shown by *Posselt et al.* [2008]. In this study a Monte Carlo type technique is used for the retrieval. The retrieval results are provided in terms of the estimates of the probability density function (pdf) of the retrieved quantities, specifically the optical depth and Ice Water Path (IWP). The results by *Posselt et al.* [2008] show that the pdfs of IWP and effective radius are not Gaussian, implying that the associated retrieval error statistics are also not Gaussian. The Monte Carlo type retrievals by design produce the desired uncertainty estimates and include the full nonlinear relationship between the retrieved and input quantities, but are computationally as demanding if not more so than the explicit error propagation method.

[5] In the current study an alternative, computationally efficient, nonlinear method is presented for characterizing the cloud property retrievals. The method is based on the general stochastic inverse problem theory as set forth by *Mosegaard and Tarantola* [2002]. The theoretical basis for the method is presented by *Vukicevic and Posselt* [2008] in the context of diagnostic analysis of inverse problem solutions in atmospheric and oceanic data assimilation problems. A detailed presentation of the mathematical theory is included by *Tarantola* [2005]. The theory is similar to standard Bayesian statistical estimation theory [*Jazwinski*, 1970] in that each quantity that contributes to the total quantitative knowledge of modeled and observed states is treated as a stochastic quantity with an associated pdf. A unique feature of the *Mosegaard and Tarantola* [2002] approach is that contributions from the model (e.g., the radiative transfer model) and the associated modeling errors (e.g., the errors from nonretrieved inputs or model formulation), the observation errors, and the prior information (e.g., the first guess in the retrieval) are each represented by a separate pdf which, when combined together result in a joint posterior pdf. The joint posterior pdf represents the most complete available knowledge of the parameters given the measurements and the (nonlinear) forward model which transforms the parameters into the measurement space. The properties of the joint posterior pdf explicitly determine the accuracy and uncertainties of the inverse solution (e.g., the retrieved cloud properties) and provide the basis for computing the Shannon information content of the measurements that are used in the retrieval [*Rodgers*, 2000; *Posselt et al.*, 2008].

[6] In the work of *Vukicevic and Posselt* [2008] the theoretical approach by *Mosegaard and Tarantola* [2002] is used as the basis for developing a numerical algorithm which is used for diagnosing the impact of the model nonlinearities and model and observation errors on the joint posterior pdf without explicitly computing the propagation of errors through the model. In the current study this diagnostic approach is adapted to the problem of characterizing cloud property retrievals. In this application the nonlinear model in the problem is the radiative transfer model used in the retrieval. The computational efficiency of the new method for characterizing the cloud property retrievals, relative to the explicit error propagation and the Monte Carlo methods, results from using the precomputed radiative transfer model simulations which are already available in the standard retrieval algorithms.

[7] The new method for characterizing cloud property retrievals is presented in section 2. In section 3 this method is applied to retrievals of cloud optical thickness and cloud droplet effective radius from passive remote sensing which were performed by *Coddington et al.* [2010]. *Coddington et al.* [2010] examined the impact of aerosols on cloud optical properties retrieved from airborne measurements of reflected irradiance from marine stratus clouds made by the Solar Spectral Flux Radiometer (SSFR) [*Pilewskie et al.*, 2003]. In the current study it is demonstrated that the equivalent effect could be quantified through the impact of the modeling errors on the posterior pdf of the retrieval. An analysis of Shannon information content of the measurements with the associated measurement errors from a selection of five-wavelength bands spanning the visible to near-infrared used in the SSFR retrievals is also presented. A summary and discussion on the potential broad application of this new diagnostic approach for characterizing cloud property retrievals are included in section 4.