## 1. Introduction

This study follows the “Cellular Statistical Models of Broken Cloud Fields” series of papers (Alexandrov et al. 2010a,b, hereinafter CSM1 and CSM2, respectively; Alexandrov and Marshak 2017, 2019, hereinafter CSM3 and CSM4, respectively). In these series we developed statistical parameterization and modeling algorithms for cloud structure in binary-mixture framework. The approach adopted there is based on cloud-mask statistics of 1D or 2D broken cloud fields derived from observations [in the 2D case made along 1D linear transects (chords)]. Such observations consist of the lengths of cloudy and clear intervals in each transect. The cloud statistics in this approach are always finite, so it works equally well for cumulus and stratocumulus cloud fields with a smooth transition between these two types (which are symmetrical to each other under cloud–clear interchange). This is an advantage compared to, e.g., area-based characterization which yields infinite cloud areas in Sc cases and infinite clear areas in Cu cases. In CSM2 the analytical expressions derived in CSM1 were demonstrated to adequately describe the statistics of shallow, broken cloud fields generated using a realistic large-eddy simulation (LES) model. In CSM3 the results of CSM1 were interpreted in terms of the theory of Markov processes (e.g., Kulkarni 2011; Ibe 2013). This interpretation is based on the assumption that each 1D sample consisting of subsequent cloudy and clear segments is a realization of a binary Markov process, which can take only two (generally nonnumerical) values: occupied (cloudy, “•”) or empty (clear, “∘”). In CSM4 the binary Markov processes framework was used for quantitative evaluation of the effects of low resolution of idealized observations on the statistics of the retrieved cloud masks.

In the above described approach cloud mask statistics depend on the nominal threshold in continuous-value field [such as cloud optical thickness (COT)] separating “clear” from “cloudy,” which is chosen rather subjectively by the investigator. In this study we want to overcome this subjectivity by turning our attention from Markovian cloud mask model to this of the underlaying continuous-value cloud fields themselves. The continuous-value fields used in this study are these of COT which can take values ranging from zero to (formally) infinity. They will be characterized in our model by four statistical parameters: the absolute (zero-threshold) cloud fraction, the autocorrelation (scale) length, and the two parameters of the normalized probability density function of nonzero COT values (this PDF is assumed to have gamma-distribution functional shape).

Our approach to COT modeling is substantially simpler than those developed in the past, in particular, compared to the following two types of stochastic models of cloud horizontal structure. The first is the bounded cascade model (Marshak et al. 1993, 1994; Cahalan 1994) which is a nonstationary two-parameter generalization of Meneveau and Sreenivasan’s *p*-models for the intermittent dissipation rate field in turbulence (Meneveau and Sreenivasan 1987). The second class, suggested by Schertzer and Lovejoy (1987), consists of fractionally integrated cascade models where power-law filtering in Fourier space brings the spectral exponent to any prescribed value.

We will show that the binary Markovian model described in CSM3 can be derived from our continuous-value Markovian model by attribution of the COT values to binary classes using a selected cloud/clear separation threshold. The data points with values higher than this threshold are attributed to the class • (cloudy), while those with values lower than the threshold are attributed to the class ∘ (clear). The corresponding binary statistics (cloud fraction, spatial scale, transition probabilities, etc.) can be derived from the continuous Markovian model by integration from zero to the threshold for ∘ and from the threshold to infinity for •.

Realizations of the continuous-value model are piecewise constant 1D functions, which, nevertheless, look quite realistic. We will demonstrate this by generating synthetic COT fields, which imitate those actually observed by the airborne Research Scanning Polarimeter (RSP) during two field campaigns. The examples will include various cloud-field types: Cu, Sc, and As.

## 2. Binary Markovian model of broken cloud field

The theoretical framework for Markovian parameterization of broken cloud field statistics was presented in CSM3. Here we outline the main points.

### a. Cloud mask statistics

*L*

_{•}and

*L*

_{∘}of cloudy and clear intervals, respectively. The pair of numbers (

*L*

_{•},

*L*

_{∘}) provides complete parameterization of the model in the infinite space. This parameterization, however, is not unique and not the most convenient. An alternative set of two independent parameters can be chosen as (cf. Levermore et al. 1988; Pomraning 1989; CSM3)

*L*

_{*}is the double of the harmonic mean of

*L*

_{•}and

*L*

_{∘}:

*L*

_{*}can be considered as a universal scale length of the cloud field. This parameter is also called “autocorrelation length” (cf. Levermore et al. 1988; Pomraning 1989) since it enters the exponent of the corresponding autocorrelation function (Morf 1998, 2011) (see also section 6 below). Note, however, that in order to define autocorrelation function, “•” and “∘” should be assigned with numerical values (e.g., 1 and 0).

In addition to infinite-length realizations of the binary Markovian model, the ensemble of finite-length samples extracted from these realizations was considered in CSM1. It was demonstrated in CSM2 that the same analysis is valid for finite 1D transects extracted from a 2D cloud field (in that case LES-simulated cloud masks). The sampling procedure is specified by the length *L* of the sample(s) and the probabilities of its initial point to be cloudy or clear. If the samples are chosen at random, their initial states are • with the probability

### b. Idealized observations

The image of the cloud field obtained by an idealized airborne or satellite sensor is the result of sampling with finite sample size *L* and finite resolution *l*. Multiple images with the same parameters constitute the observational dataset. A sequence of consecutive • points is considered a cloud, while a sequence of consecutive ∘ points is considered a gap. The mean observed lengths of such “clouds” and “gaps” [

*i*= • or ∘; thus,

*L*

_{*}. This effect was in the focus of CSM4 where the following correction was introduced:

*l*< 5

*L*

_{*}). Below both corrections will be used sequentially: first Eq. (5), and then Eq. (6). However, our inability to address both effects simultaneously may result in some small biases in estimation of

*L*

_{*}.

### c. Markovian properties

*x*>

*x*

_{0}to be in one of the two states depend only on the state of

*x*

_{0}itself (and not on the states of the previous points

*x*<

*x*

_{0}). These probabilities are combined into the transition probability matrix of the form

*P*is the probability of transition from the state

_{ij}*i*at

*x*

_{0}into the state

*j*at

*x*>

*x*

_{0}(

*i*and

*j*can be either • or ∘). By definition of transition probabilities, each row of

*u*and ∘ with the complimentary probability

*υ*= 1 −

*u*. In this case the state of

*x*

_{0}can be characterized by the vector

**u**at

*x*

_{0}to the state

**u**′ at

*x*>

*x*

_{0}is the result of matrix multiplication:

**u**with the “state matrix”

We assume the model to be spatially homogeneous, so the transition matrix depends only on the distance (lag) *L* = *x* − *x*_{0}, rather than on *x* and *x*_{0} themselves: *L*). We denote both sample length and transition lag by the same letter *L* since the formalism described in CSM3 provides a unified description of the cloud-field statistics in the sample and the probabilities of transition between its ends.

*L*

_{1}and

*L*

_{2}obey the two group properties:

*L*→ ∞ when correlation between the states at points

*x*and

*x*+

*L*is lost:

*w*plays the role of relative weight between the two matrices corresponding to the respective asymptotic regimes in Eqs. (17) and (19). Note that the weights

_{L}*w*obey the following group properties:

_{L}**c**and the matrix

## 3. Formulation of continuous-value Markovian model

Our tests on realistic LES-generated cloud masks (presented in CSM2) showed very good agreement between the data statistics and the binary Markovian model predictions. This allows us to assume that the underlaying continuous-value fields (COT, LWP, etc.) can be also described by a Markovian statistical model. In this section we will describe general properties of such a model and present a specific form of it consistent with binary models of cloud masks derived from continuous-value fields.

### a. General properties

In continuous-value Markovian model the stochastic 1D-field *τ* is defined on real line *τ*_{max}], where the upper bound *τ*_{max} can be chosen arbitrary large but is naturally finite (*τ*_{max} = ∞ can be used to simplify analytical computations). The generalization to 2D fields can be performed, as in the binary model, using the statistical ensemble of 1D transects. The field *τ* can be sampled on intervals of a finite length *L*, and the corresponding *L*-dependent statistics can be derived.

*u υ*), which can be considered as a probability function defined on the binary set, is replaced in the continuous-value model by a normalized probability density function

*η*(

*τ*) defined on [0,

*τ*

_{max}] and associated with the location

*x*. The condition

*u*+

*υ*= 1 corresponds to the normalization condition for

*η*(

*τ*):

*τ*

_{0}the probability density has a

*δ*-function form:

*ρ*(

*τ*) normalized by the condition

*ρ*

_{cld}(

*τ*) and the singular part

*δ*(

*τ*) corresponding to gaps between clouds (where

*τ*= 0):

*ν*is the “absolute” cloud fraction (independent of cloud/clear separation setup). We assume that

*ρ*

_{cld}is normalized by the same condition Eq. (27) as

*ρ*(

*τ*). The moments of

*ρ*

_{tot}(

*τ*) and

*ρ*

_{cld}(

*τ*) are obviously related:

*L*) from Eq. (7) is replaced in continuous-value model by the integral operator

*P*(

_{L}*τ*

_{1},

*τ*

_{2}) which determines the probability of the transition from the interval [

*τ*

_{1},

*τ*

_{1}+

*dτ*] at the point

*x*to that of [

*τ*

_{2},

*τ*

_{2}+

*dτ*] at

*x*+

*L*. We assume homogeneity of the model, so

*x*and depends only on the lag

*L*. The transition between the state vectors

*η*

_{1}at

*x*and

*η*

_{2}at

*x*+

*L*is defined as

*τ*

_{1}at

*x*to some value

*τ*

_{2}at

*x*+

*L*with unit probability; thus,

*η*

_{2}(

*τ*

_{2}) is normalized to unity as in Eq. (25).

*L*→ ∞ the expression for the transition operator kernel is the generalization of Eq. (19):

*η*

_{2}(

*τ*

_{2}) =

*ρ*(

*τ*

_{2}) meaning that the value of

*τ*

_{2}at infinity is independent from

*τ*

_{1}and is just randomly chosen from the general PDF.

### b. Specific form of the transition operator

*L*→ 0 and random value choice at

*L*→ ∞). The resulting expression of the transition operator kernel is

*w*= exp(−

_{L}*L*/

*L*

_{*}) are the same as in the binary model’s Eq. (21). Here

*L*

_{*}is the scale (or autocorrelation) length of the continuous-value model. We will show below that the scale lengths of all binary cloud-mask models derived from a single continuous-value model are the same and coincide with

*L*

_{*}of this model. This means that the continuous-value model’s

*L*

_{*}can be obtained from any derived binary model according to the procedures outlined in section 2, or independently from the autocorrelation function (see section 6 below).

Note that in the binary case the transition matrix, Eq. (20), is uniquely determined by the group and limit case requirements in Eqs. (17)–(19) (see appendix A). However, we cannot currently say the same about the transition operator kernel, Eq. (35), and the conditions (32)–(34). We will examine the possibilities to generalize Eq. (35) in our future work.

*w*

_{0}is set to 1 and 0, respectively, in Eq. (35). The second group property, Eq. (33), is also satisfied:

*ρ*(

*τ*) and the characteristic scale

*L*

_{*}.

*x*, selection of the value at point

*x*+

*L*falls into two cases with the probabilities

*w*and (1 −

_{L}*w*), respectively. In the first case

_{L}*τ*(

*x*+

*L*) remains the same as

*τ*(

*x*), while in the second case it is randomly taken from the global probability distribution

*ρ*

_{tot}(

*τ*).

## 4. Functional form of the COT PDF

We adopted gamma distribution as the functional form of non-zero-value COT PDF *ρ*_{cld}(*τ*). This assumption is based on observations of COT in marine boundary layer clouds (Barker 1996; Barker et al. 1996; Pincus et al. 1999) and our own analyses of airborne remote sensing measurements presented in section 8 below. Gamma distribution also provides good parameterization for COT histograms from LES-modeled trade wind cloud fields used in CSM2. In GISS GCM ModelE3 gamma distribution has been used as PDF for in-cloud water content following Morrison and Gettelman (2008).

*a*> 0, 0 <

*b*< 1/2, and Γ(

*z*) is the gamma function:

*a*and

*b*have the respective meanings of the effective COT

*τ*

_{eff}and the effective variance

*υ*

_{eff}of the PDF:

*a*,

*b*) and (

*τ*

_{eff},

*υ*

_{eff}) interchangeably since they are equivalent.

While the parameterization in Eq. (40) may not look familiar to someone outside the radiation modeling or remote sensing community, we refer our readers to appendix A of Alexandrov et al. (2018) where *a* and *b* are related to the parameters more common in other fields.

*b*after

*a*is computed:

*β*> 2), instead of

*b*. For example, this simplifies Eq. (40):

## 5. Cloud masks

A binary cloud-mask Markovian model can be derived from a continuous-value one by selecting a cloud/clear separation threshold *τ*_{thr} and attributing the values of *τ* ≥ *τ*_{thr} to the class • (cloudy), while the values of *τ* < *τ*_{thr} to the class ∘ (clear). For analysis of the dependence of cloud-mask statistics on the (brightness) threshold value for real satellite observations, see, e.g., Wielicki and Welch (1986).

### a. Cloud fraction

*ρ*(

*τ*) of the form in Eq. (28) the global cloud fraction

*ρ*

_{cld}(

*τ*) has gamma-distribution functional shape in Eq. (47) the integral in Eq. (48) can be computed analytically yielding the following expression for

*τ*

_{max}= ∞ is assumed for simplicity and

*ν*with the increase of

*τ*

_{thr}, since Γ(

*z*,

*x*) is a decreasing function of

*x*(with maximum at

*x*= 0).

### b. Markovian model of cloud mask

**u**= (

*u υ*) can be obtained from the continuous-value state

*η*(

*τ*) by means of integration:

*u*+

*υ*= 1 follows from the normalization condition in Eq. (25) of

*η*(

*τ*). Any continuous-value state function

*η*(

*τ*) can be expressed as

*η*

_{•}(

*τ*) and

*η*

_{∘}(

*τ*) are continuous-value representations of the binary states • and ∘, respectively. As any state they are normalized to unity. Explicitly,

*η*

_{•}(

*τ*) is defined as

*η*

_{∘}(

*τ*) is defined as

*x*to another point

*x*+

*L*corresponds in the underlaying continuous-value model to a transition from the states

*η*

_{•}or

*η*

_{∘}, respectively. The result of this transition

*i*= ∘, •) is not a “pure” • or ∘ state and can be decomposed according to Eq. (53) with the coefficients

*u*and

*υ*playing the roles of the transition probabilities between the initial and final binary states:

*i*= ∘, •. The row-sum property in Eq. (8)

*L*) are computed according to Eqs. (58) and (59) in appendix B resulting to the expression

*τ*

_{thr}, the scale

*L*

_{*}is inherited from Eq. (35) through the weight

*w*and, thus, is independent from

_{L}*τ*

_{thr}value.

We have demonstrated that the Markovian statistics of cloud masks derived from continuous-value model described by Eq. (35) are consistent with the binary Markovian model described in section 2 and CSM3.

## 6. Structure and autocorrelation functions

*L*is common in atmospheric sciences (see, e.g., Davis et al. 1994; Lovejoy and Schertzer 2012) including our own works on atmospheric aerosols (Alexandrov et al. 2004, 2016b). Here we will use the second-order SF that is defined in 1D case as

*x*and over all realizations in the ensemble. To derive the analytical form of SF from Eqs. (28), (35), and (40) we need to compute statistics 〈(

*τ*

_{2}–

*τ*

_{1})

^{2}〉, where

*τ*

_{1}=

*τ*(

*x*) is randomly taken from the global total distribution

*ρ*

_{tot}(

*τ*)(and can be zero), while

*τ*

_{2}=

*τ*(

*x*+

*L*) is the result of application of the transition operator to

*τ*

_{1}. The averaging is then made over both

*τ*

_{1}and

*τ*

_{2}within their respective distributions.

*τ*

_{1}is distributed according to

*ρ*(

*τ*), while

*τ*

_{2}is distributed according to

*η*(

*τ*) from Eq. (39). This means that

*τ*

_{2}=

*τ*

_{1}[thus,

*S*

_{2}(

*L*) = 0] with the probability

*w*, while in the opposite case (with the probability 1 −

_{L}*w*)

_{L}*τ*

_{2}gets a random value taken from

*ρ*(

*τ*). The former case corresponds to the limit

*L*= 0, while the latter corresponds to

*L*→ ∞. This allows us to write the expression for SF in the following form:

*S*

_{2}(0) = 0, while in the limit of infinite

*L*the values of

*τ*

_{1}and

*τ*

_{2}are independent:

*ρ*(

*τ*) with the mean 〈

*τ*〉. Thus, using Eq. (64) we obtain

*ρ*

_{tot}(

*τ*). Then the expression Eq. (63) for the SF takes the following form:

*ρ*

_{cld}(

*τ*) using Eq. (29):

*ρ*

_{cld}(

*τ*) having gamma-distribution form in Eq. (40) are given by Eqs. (43) and (44), respectively. They yield the following expression for COT variance:

*S*

_{2}∝

*L*corresponding to the Hurst exponent

*H*= 1/2 (generally,

*S*

_{2}∝

*L*

^{2}

*). This value of*

^{H}*H*corresponds to classical Brownian motion, which is a Markovian random process, so this result is expected as being derived from a Markovian COT model.

*τ*〉 and the variance Var(

*τ*) the autocorrelation function (AF) is defined as

*ρ*

_{tot}(

*τ*). Expectedly,

*W*→ 0 as

*L*→ ∞, while

*W*→ 1 as

*L*→ 0. Equation (72) demonstrates that

*L*

_{*}is indeed the autocorrelation length of the model and gives a new meaning to the weights

*w*entering the expressions for the binary transition matrix in Eq. (20) and the continuous-value transition operator’s kernel in Eq. (35).

_{L}## 7. Construction of examples

*ν*,

*L*

_{*}, and

*ρ*

_{cld}(

*τ*), one should first specify the resolution

*l*of the realization and compute the probability

*l*. Then, the initial value

*τ*

_{0}of the realization is chosen by picking it at random from the distribution

*ρ*(

*τ*). To do this we first determine if this point is cloudy or clear by “flipping a coin” with probability

*ν*for cloudy. If the result is “clear” then

*τ*

_{0}= 0; otherwise, we chose

*τ*

_{0}from the distribution

*ρ*

_{cld}(

*τ*)of the positive COT values. A simple way to do this starts with generation of two random numbers

*z*and

_{τ}*z*on the unit interval (0, 1). Then the first is used to get random

_{ρ}*τ*

_{0}=

*z*

_{τ}τ_{max}, and the second is used to get a random value in the distribution dimension

*ρ*

_{0}=

*z*max(

_{ρ}*ρ*

_{cld}). Then, if

*ρ*

_{0}≤

*ρ*

_{cld}(

*τ*

_{0}) [i.e., if the point (

*τ*

_{0},

*ρ*

_{0}) is below the curve

*ρ*

_{cld}(

*τ*)] the random value

*τ*

_{0}is declared the result of the trial; otherwise, the trial is discarded and the procedure is repeated again.

Given the value *τ _{i}* at the

*i*th step of the procedure, the next value

*τ*

_{i}_{+1}is chosen in the way outlined in Fig. 1. First, we “flip a coin” returning 1 with the probability

*w*(

*l*) and 0 with the probability 1 −

*w*(

*l*). If the result is 1, then

*τ*

_{i}_{+1}=

*τ*[corresponding to the first term in Eq. (39)]. If the result is 0, then

_{i}*τ*

_{i}_{+1}is randomly taken from the distribution

*ρ*

_{tot}(

*τ*) in the same way as it was described above for choosing of the initial value

*τ*

_{0}. This corresponds to the second term in Eq. (39).

The described algorithm produces stationary piecewise constant curves which, unlike real COT patterns, do not exhibit any trends. While in most of the cases our model is physically adequate, in some instances (see section 8) a trend has to be added manually. Among future developments of our model we consider its merger with the classical Brownian motion (which is also Markovian) in order to accommodate for random trends and replace constant-COT intervals with something more realistic.

*L*= 9.6 km and

*l*= 100 m, respectively.

The parameterization in Eq. (74) was used to initiate the process described in the beginning of this section to produce an ensemble of realizations of the corresponding Markovian COT model. Hereafter we will refer to our statistically generated dataset as “ATEX imitation” to distinguish it from the dynamical model’s “simulation.” The same sample size and spatial resolution as in the LES dataset were used. While realizations of the Markovian model can be generated only in 1D, the number of independent 1D samples in the imitation dataset was made equal to the total number of the transects in the LES dataset. Figure 2 shows a randomly chosen COT segment of the ATEX imitation dataset.

An example of COT segment from the ATEX imitation dataset generated using our Markovian model. The parameters of the model are defined by Eq. (74).

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

An example of COT segment from the ATEX imitation dataset generated using our Markovian model. The parameters of the model are defined by Eq. (74).

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

An example of COT segment from the ATEX imitation dataset generated using our Markovian model. The parameters of the model are defined by Eq. (74).

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

Statistical comparison of our imitations with the LES datasets themselves and discussion of their differences will be presented in the upcoming Part II of this series. Here we are only interested in how well the statistics of the generated dataset agree with the theoretical results described earlier in the paper. This algorithm performance evaluation was conducted in the following three ways. First, we checked how well the COT PDF derived from the dataset agrees with the gamma distribution used to generate it. Second, we compared the dataset’s cloud-mask statistics corresponding to four COT thresholds (0.1, 1, 10, and 30) to the theoretical laws based on Eq. (50) for CF and Eq. (3) for *L*_{•} and *L*_{∘}, while *L*_{*} values from these cloud masks should be threshold independent and close to 448 m from Eq. (74). Third, we compared the structure and autocorrelation functions derived from the generated data to these theoretically computed using Eqs. (69) and (72), respectively.

The COT PDF derived from the ATEX imitation dataset is shown in Fig. 3 (with its exponential “tail” shown in log scale in right panel). The black curve represents the gamma-shaped distribution, Eq. (40), with the parameters *τ*_{eff} = 31.2 and *υ*_{eff} = 0.38 derived from the COT moments using Eqs. (42) and (45). These parameters are very close (within 2.5%) to the corresponding Eq. (74) values, and the COT PDF derived from the generated dataset appears to be practically identical (with some inevitable random noise) to the gamma-distribution PDF that was used for its generation.

COT probability distribution functions for the ATEX imitation dataset. (right) The *y* axis is in a logarithmic scale to show exponential asymptotic behavior. The gamma-function model PDF is shown in black.

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

COT probability distribution functions for the ATEX imitation dataset. (right) The *y* axis is in a logarithmic scale to show exponential asymptotic behavior. The gamma-function model PDF is shown in black.

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

COT probability distribution functions for the ATEX imitation dataset. (right) The *y* axis is in a logarithmic scale to show exponential asymptotic behavior. The gamma-function model PDF is shown in black.

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

The cloud-mask statistics of the ATEX imitation dataset were derived following the algorithm outlined in CSM2. First, we used the data samples to collect the lengths of cloudy and clear intervals (for given *τ*_{thr}). Then, the mean cloud and gap lengths were derived from these collections. These two values were subsequently corrected according to Eq. (4) for finite-sample size *L* resulting in the “infinite-sample” values of *L*_{•} and *L*_{∘}, respectively. These parameters then were used to compute the scale length *L*_{*} according to Eq. (1). On the next step, *L*_{*} was corrected for the model’s finite resolution *l* according to Eq. (6), and its adjusted value was used in Eq. (3) to obtain the resolution-corrected *L*_{•} and *L*_{∘}. The cloud fraction

Figure 4 shows plots of *L*_{*}, *L*_{•}, and *L*_{∘} as functions of the COT threshold *τ*_{thr}. The corresponding theoretical dependances are shown by green lines: from Eq. (50) for *L*_{•} and *L*_{∘} in Fig. 4 (bottom, left and right, respectively). Diamonds there depict the statistics corresponding to the thresholds of 0.1, 1, 10, and 30 (green: theoretical; blue: derived from the imitation dataset). We see that in all four panels the values from the imitation dataset are practically identical to the corresponding theoretical values. In particular, the imitation-dataset values of *L*_{*} in Fig. 4 (top right) are indeed threshold independent.

Cloud-mask statistics from the ATEX imitation dataset as functions of COT threshold: (top left) cloud fraction *L*_{•}, and (bottom right) mean gap length *L*_{∘}. The theoretical dependances are shown by green lines. Diamonds depict statistics corresponding to the cloud–clear separation thresholds of 0.1, 1, 10, and 30 (green: theoretical; blue: derived from the imitation dataset).

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

Cloud-mask statistics from the ATEX imitation dataset as functions of COT threshold: (top left) cloud fraction *L*_{•}, and (bottom right) mean gap length *L*_{∘}. The theoretical dependances are shown by green lines. Diamonds depict statistics corresponding to the cloud–clear separation thresholds of 0.1, 1, 10, and 30 (green: theoretical; blue: derived from the imitation dataset).

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

Cloud-mask statistics from the ATEX imitation dataset as functions of COT threshold: (top left) cloud fraction *L*_{•}, and (bottom right) mean gap length *L*_{∘}. The theoretical dependances are shown by green lines. Diamonds depict statistics corresponding to the cloud–clear separation thresholds of 0.1, 1, 10, and 30 (green: theoretical; blue: derived from the imitation dataset).

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

Figure 5 demonstrates that the autocorrelation and structure functions of the imitation dataset also coincide with their theoretical predictions in Eqs. (72) and (69), respectively. The above comparisons demonstrate the robustness of the algorithm described in this section.

(left) Autocorrelation functions in Eq. (70) for ATEX imitation dataset (blue) and the theoretical function in Eq. (72) (green). (right) As in the left panel, but for second-order structure functions, Eq. (62); the theoretical curve is from Eq. (69).

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

(left) Autocorrelation functions in Eq. (70) for ATEX imitation dataset (blue) and the theoretical function in Eq. (72) (green). (right) As in the left panel, but for second-order structure functions, Eq. (62); the theoretical curve is from Eq. (69).

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

(left) Autocorrelation functions in Eq. (70) for ATEX imitation dataset (blue) and the theoretical function in Eq. (72) (green). (right) As in the left panel, but for second-order structure functions, Eq. (62); the theoretical curve is from Eq. (69).

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

## 8. Examples from real-life RSP datasets

In this section we will demonstrate the ability of our Markovian model to parameterize a variety of real COT datasets and to generate synthetic data with the same statistical properties. For this purpose we use the COT retrievals made from the RSP measurements during two field campaigns. The observed cloud fields include Cu, Sc, and As.

The RSP is an airborne along-track scanner with high angular resolution of 14 mrad FOV. It makes measurements at 0.8° intervals within ±60° from nadir. The scanning is continuous with 0.8 s per scan. The translation of these parameters into spatial resolutions of the RSP datasets depends on the speed and the altitude of the aircraft (Alexandrov et al. 2016a). It can range from about 200 m for NASA’s high-altitude ER-2 flying with the speed of 210 m s^{−1} at 20 km above ground down to less than 100 m for more conventional airplanes (P3-B, B-200, UC-12, C-130) usually making about 120 m s^{−1} at 5–7-km altitude. The resolution of RSP’s cloud products may change during the flight with changing speed, altitude, and the cloud-top height beneath the aircraft.

The RSP has been deployed in numerous NASA field campaigns during the past decades (see, e.g., Alexandrov et al. 2015, 2016a,c, 2018; Sinclair et al. 2017, 2019).

The COT retrievals from RSP-measured total reflectances at nadir view are made using a modification of the widely used bispectral technique (Nakajima and King 1990). In the modified algorithm no absorbing spectral channels are used while the droplet effective radius is derived from the polarized reflectance in the rainbow (cloud bow) scattering range between 137° and 165° (Alexandrov et al. 2012). Then, the COT value is determined from lookup table built for nonabsorbing 863-nm channel using plain-parallel radiative transfer computations. As the original bispectral technique, this method can produce biases in COT values in the presence of 3D radiative effects (such as light escape from broken cloud sides or shadowing at low sun angles). However, in this study we do not address such biases and consider the RSP-derived COTs as true records subject to statistical parameterization and imitation.

A problem common to most statistical analyses using real-life data is in keeping the balance between homogeneity of the dataset (so it should be small enough for its statistical properties not to change within it) and the data volume sufficiently large for derivation of reliable statistics. While we have not yet implemented a rigorous procedure for estimation of statistical homogeneity of the sample, we currently relay on visual analysis of the COT pattern and also on how well the COT PDF can be fitted by a relatively narrow gamma distribution. A prospective algorithm for quantification of the sample homogeneity can be based on comparison between the statistics of different subsamples.

In the case of aerial overpass above a rapidly changing cloud field we can obtain a conclusive analysis only if the field’s statistics do not change or change much slower than the field itself (e.g., clouds move but do not disappear all together). On the other hand, in the case of geostationary satellite observations (e.g., Seelig et al. 2021) 2D spatial COT fields can be derived independently for any moment of the cloud system evolution. These 2D snapshots (if sufficiently large and homogeneous) can be used for derivation of all four parameters of our model. This is a rather simple statistical procedure outlined for 1D case in the next paragraph, so unlike Seelig et al. (2021) we do not need to track individual clouds. The model parameters then can be combined into time series reflecting the evolution of the cloud field. For example, in a simple case of evaporating cloud system such time series are expected to behave similarly to the curves in Fig. 4 (if the threshold in COT on *x* axis is replaced by the COT value lost due to evaporation): CF declines, clouds become smaller on average (some probably disappear) and the gaps between them increase in size. The effective COT should also decline.

Below we present examples of analysis and imitation of the RSP COT retrievals from selected cases with long flight legs over statistically homogeneous (in the above-described sense) cloud fields. The parameterizations of these cloud fields were made using the methodology similar to that applied to the imitated ATEX dataset at the end of section 7. The PDF parameters *τ*_{eff} and *υ*_{eff} were derived from the COT moments using Eqs. (42) and (45). The cloud fraction *ν* was computed as the ratio between the number of pixels determined as cloudy by the RSP algorithm (basing on the presence of cloud bow in the polarized reflectance) to the total number of pixels in the selected interval. The scale length *L*_{*} was derived from the COT’s autocorrelation function assuming that it has the form in Eq. (72). The Markovian model’s parameters derived from the four datasets described in this section are presented in Table 1. They were used for generation of the corresponding imitations according to the algorithm described in section 7.

Cloud field parameters for four RSP examples.

### a. RACORO: Cu-to-Sc transition

The first couple of examples are from the 2009 field campaign Routine Atmospheric Radiation Measurement (ARM) Aerial Facility (AAF) Clouds with Low Optical Water Depth (CLOWD) Optical Radiative Observations (RACORO) coordinated by the AAF (Vogelmann et al. 2012). The campaign was held in Oklahoma’s southern Great Plains from January to June 2009. The RSP was on board NASA’s B-200 aircraft. At this part of the year there are plenty of “popcorn” cumulus clouds in this area forming vast and statistically homogeneous fields suitable for our analysis. Both examples presented here are from the same long leg flown on 5 June 2009. While the two selected intervals are adjacent to each other (1418:58–1429:45 and 1435:06–1458:44 UTC) they have different cloud field structures: the small-size Cu clouds in the first part (Fig. 6) later gave way to a Sc field with frequent gaps (Fig. 7). The cloud-top heights in both cases were between 400 and 500 m. The cloud-field parameters presented in Table 1 show that the Cu-to-Sc transition results in small decrease in the mean and effective COTs by about 1 (from 7 to 6), while the COT PDF remains narrow (*υ*_{eff} = 0.02–0.04) and well-represented by gamma distribution (see bottom-right panels of Figs. 6 and 7). The autocorrelation length *L*_{*} moderately increased during the transition (from around 300 to 350 m), while the most dramatic change occurred in the cloud fraction *ν*: an increase from 7% to 79%. The results of Markovian model’s imitations are presented in top-right panels of Figs. 6 and 7. They are compared to the actual COT measurements (from top-left plots) in bottom-left panels (within shorter intervals for better clarity). As a result of both actual and simulated datasets having the same statistics, these comparisons show visual similarities in the structure of COT curves (while no point-by-point matching can be expected).

(top left) Cu COT derived from the RSP measurements made during RACORO field campaign between 1418:58 and 1429:45 UTC 5 Jun 2009. (top right) Imitated COT based on the statistics of the field in the top-left panel. (bottom left) Zoom-in on both datasets at the middle of the interval (note that the simulated COT is expected to be similar to the original only statistically, not point by point). (bottom right) PDF of the nonzero COT (red) and its fit with the gamma distribution (blue). Here *a* and *b* stand for *τ*_{eff} and *υ*_{eff}, respectively; see Eq. (42).

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

(top left) Cu COT derived from the RSP measurements made during RACORO field campaign between 1418:58 and 1429:45 UTC 5 Jun 2009. (top right) Imitated COT based on the statistics of the field in the top-left panel. (bottom left) Zoom-in on both datasets at the middle of the interval (note that the simulated COT is expected to be similar to the original only statistically, not point by point). (bottom right) PDF of the nonzero COT (red) and its fit with the gamma distribution (blue). Here *a* and *b* stand for *τ*_{eff} and *υ*_{eff}, respectively; see Eq. (42).

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

(top left) Cu COT derived from the RSP measurements made during RACORO field campaign between 1418:58 and 1429:45 UTC 5 Jun 2009. (top right) Imitated COT based on the statistics of the field in the top-left panel. (bottom left) Zoom-in on both datasets at the middle of the interval (note that the simulated COT is expected to be similar to the original only statistically, not point by point). (bottom right) PDF of the nonzero COT (red) and its fit with the gamma distribution (blue). Here *a* and *b* stand for *τ*_{eff} and *υ*_{eff}, respectively; see Eq. (42).

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

As in Fig. 6, but for the RSP measurements made between 1435:06 and 1458:44 UTC 5 Jun 2009.

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

As in Fig. 6, but for the RSP measurements made between 1435:06 and 1458:44 UTC 5 Jun 2009.

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

As in Fig. 6, but for the RSP measurements made between 1435:06 and 1458:44 UTC 5 Jun 2009.

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

### b. PODEX: Marine stratocumulus and altostratus clouds

The next two examples had been selected from the RSP dataset obtained during the Polarimeter Definition Experiment (PODEX) held between 14 January and 6 February 2013. The NASA ER-2 airplane carrying the RSP was based at NASA Dryden (now Armstrong) Aircraft Operation Facility in Palmdale, California, north from Los Angeles. The retrievals of cloud properties from the RSP measurements were performed for both the California Valley and the coastal waters (Alexandrov et al. 2015).

The first example from PODEX (presented in Fig. 8) is a marine stratocumulus deck observed off the California coast between 1852:11 and 1909:00 UTC 3 February 2013. This was an overcast (*ν* = 0) low-altitude cloud with an average cloud-top height of only 200 m. Figure 8 (top left) shows a pronounced mostly linear trend in COT, which cannot be characterized or simulated within the current framework of our statistical model (see section 7). This is an example of nonhomogeneous sample where the local mean COT (and therefore the local PDF) is changing with the location leading to a wide nongamma global PDF. Thus, we approximated this trend by a linear function and subtracted it from the COT data before performing statistical analysis. Figure 8 (bottom right) showing how closely the real (while detrended) COT PDF is approximated by gamma distribution with *τ*_{eff} = 21 and *υ*_{eff} = 0.02. The autocorrelation length *L*_{*} ≈ 1300 m in this case is quite large compared to the RACORO examples presented above. We used these statistics of the detrended dataset to imitate the stationary COT pattern. After this we added the linear trend back yielding the curve presented in Fig. 8 (top right). It looks quite similar to the actual COT pattern while (by construction) having a simpler trend structure. A zoom-in in Fig. 8 (bottom left) presenting both real and imitated COTs in the same plot shows statistical similarity in their small-scale variability.

As in Fig. 6, but for the RSP observations of Sc deck made during PODEX between 1852:11 and 1909:00 UTC 3 Feb 2013.

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

As in Fig. 6, but for the RSP observations of Sc deck made during PODEX between 1852:11 and 1909:00 UTC 3 Feb 2013.

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

As in Fig. 6, but for the RSP observations of Sc deck made during PODEX between 1852:11 and 1909:00 UTC 3 Feb 2013.

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

Figure 9 presents our second example from PODEX: an altostratus (As) cloud observed between 1955:41 and 2014:04 UTC 1 February 2013. This high-altitude cloud with top height of 8.2 km is quite different from those in the previous examples of boundary layer clouds. There are only few gaps in this cloud (*ν* ≈ 90%) and it is rather thin (*τ*_{eff} = 3.1, 〈*τ*〉 = 2.2). At the same time, it has a wide COT PDF shown in Fig. 9 (bottom right) with *υ*_{eff} = 0.17 (4–8 times larger than in the Cu and Sc examples). This reflects the volatile nature of the COT seen in Fig. 9 (top left). This PDF still can be approximated by gamma distribution reasonably well (while not as well as in the other examples). Another distinctive feature of this cloud is the large autocorrelation length of 6.6 km (substantially exceeding not only *L*_{*} ∼ 300 m in the RACORO examples, but also *L*_{*} ∼ 1.3 km in the PODEX Sc case). This makes the COT curve in Fig. 9 (top left) look more “continuous.” This curve also shows a degree of nonstationarity common to Brownian motions, while not represented in our statistical model. This explains the visible difference in variability between the actual COT pattern in Fig. 9 (top left) and the imitated one in Fig. 9 (top right). On smaller spatial scales the agreement is better, as shown in Fig. 9 (bottom left), while the imitation curve lacks small-scale fluctuations present in the measurements. This example indicates that the performance of the current statistical cloud model depends on cloud type and further efforts are needed to adapt it to a wider range of different cloud fields.

As in Fig. 6, but for the altostratus (As) cloud observed by the RSP during PODEX between 1955:41 and 2014:04 UTC 1 Feb 2013.

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

As in Fig. 6, but for the altostratus (As) cloud observed by the RSP during PODEX between 1955:41 and 2014:04 UTC 1 Feb 2013.

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

As in Fig. 6, but for the altostratus (As) cloud observed by the RSP during PODEX between 1955:41 and 2014:04 UTC 1 Feb 2013.

Citation: Journal of the Atmospheric Sciences 79, 12; 10.1175/JAS-D-22-0125.1

## 9. Discussion of cloud scale

While cloud fraction is a standard data product for many satellite instruments, and COT histograms are being stored (e.g., on 1° × 1° grid in MODIS data collection; King et al. 2013), the cloud-scale statistics still deserve more attention by scientific community. The existing observational studies of cloud scale fall into two main categories: derivation of cloud size (area or chord length) distributions and autocorrelation/power-spectrum analysis of COT fields. Various types of data were used: ground based, airborne, and satellite; optical or radar measurements. Our model covers both approaches since we describe both (threshold-dependent) cloud mask statistics and autocorrelation and structure functions. Note that the small-scale behavior of the latter is related to that of the power spectrum (see, e.g., Davis et al. 1994; Alexandrov et al. 2004).

A number of studies have been published describing cloud chord statistics based on ground-based measurements (e.g., Lane et al. 2002; Berg and Kassianov 2008) as well as aircraft and satellite data (e.g., Plank 1969; Cahalan and Joseph 1989; Rodts et al. 2003). Some of these studies report exponential cloud and gap chord length distributions (Astin and Latter 1998; Lane et al. 2002), while others find power-law distributions (Cahalan and Joseph 1989; Koren et al. 2008; Wood and Field 2011; Dorff et al. 2022). Joseph and Cahalan (1990) found the gap length (“nearest neighbor spacing”) distribution to be close to a combination of Weibull and lognormal functional shapes. In our approach (see CSM1) the distributions of both cloud and gap chord lengths are exponential with the mean values *L*_{•} and *L*_{∘}, respectively [related to the model parameters CF and *L*_{*} via Eq. (3)]. However, we demonstrated in CSM2 that in the case of a diverse sample (where the parameters *L*_{•} and *L*_{∘} are themselves statistically distributed) the cloud and gap length distributions have a power-law shape. This is particularly characteristic for long-term observations on global scale (e.g., Wood and Field 2011).

Another type of studies of COT inhomogeneity is based on autocorrelation functions and power spectrum analysis. Schäfer et al. (2017) computed 1D and 2D autocorrelation functions of COT fields derived from ground-based and airborne observations. They obtained the autocorrelation length estimates assuming that the autocorrelation function has exponential form [as in Eq. (72) in this paper]. Study of 2D AF allowed Schäfer et al. (2017) to characterize anisotropy of the cloud field, e.g., when clouds are elongated in the wind direction (cf. Dorff et al. 2022).

Note that at a fixed CF a decrease in *L*_{*} results in smaller clouds and smaller cloud-free areas, thus, causing extension of the total length of cloud boundaries as well as closer proximity of these boundaries to clouds interiors and exteriors. Both effects facilitate intensification of interaction between clouds and the air surrounding them. This includes entrainment of dry air and aerosol particles into the cloud causing both evaporation of cloud droplets and formation of new ones (Yang et al. 2019). On the other hand, in oceanic regions more than half of all aerosol measurements by passive satellite instruments come from near-cloud areas, where cloud-related processes may significantly modify aerosol optical depth and particle size (Várnai et al. 2017; Várnai and Marshak 2018). Cloud scale also influences the magnitude of radiative 3D effects affecting the accuracy of COT retrievals based on plane-parallel RT assumptions.

## 10. Concluding remarks

We introduced a parameterization of COT fields which can be potentially useful for automatic determination of cloud type in airborne and satellite observations, and also may serve as a common language connecting observations and models. This continuous-value COT model is a generalization of the binary cloud-mask model developed in the “Cellular Statistical Models of Broken Cloud Fields” series of papers which are referred to in this study as CSM1, CSM2, CSM3, and CSM4, respectively. In both models cloud fields have Markovian properties. In addition of being an “inspiration” for the COT model, its binary “ancestor” can be derived from it by setting a threshold in optical thickness to be used for separation between “cloudy” and “clear” pixels in the cloud mask (this was demonstrated in section 5 and appendix B). The parameterization of our continuous-value COT model consists of the absolute cloud fraction *ν*, the scale (autocorrelation) length *L*_{*}, and the COT PDF. The latter is assumed to have gamma-distribution functional shape characterized by two parameters: the effective COT *τ*_{eff} and the effective variance *υ*_{eff}.

Basing on the theoretical formulation of the model, we developed a simple algorithm for generation of synthetic COT fields with given parameters. This algorithm was successfully tested by comparison between the statistics of the generated dataset and their theoretical values computed using the model’s parameters. Such statistics include the COT PDF, the structure and autocorrelation functions, as well as threshold-dependent cloud-mask characteristics (cloud fraction *L*_{*}, mean cloudy interval length *L*_{•}, and mean clear interval length *L*_{∘}).

The parameters used in our test model were derived from an LES dataset previously used in CSM2, so we in some sense tried to imitate that dataset using our statistical technique. However, the comparison between the statistics of our imitation and those of the LES dataset itself showed some discrepancies for which we currently do not have sufficient explanation. So we do not present any of such comparisons in this paper, while continue working on our model and look forward to including the LES discussion into the upcoming Part II of this series.

As one of the ways to make our model more realistic, we look for some kind of merger of the current model with that based on exponential classical Brownian random walks (which are also Markovian). The latter model of optical thickness has been introduced (in application to aerosol fields) in our previous study (Alexandrov et al. 2016b). We hope that such a merger will result in replacement of the constant COT segments in the model’s realizations with those having a more realistic small-scale behavior, and also may allow for large-scale trends.

We demonstrated that the choice of the four model’s parameters determines the cloud field type. To do this we took several examples of COT fields observed by the Research Scanning Polarimeter during two field campaigns, derived their parameters and “imitated” them using our statistical model. While rather simple cases of Cu, Sc, and As clouds were presented in this paper, we expect that further development of our model will allow us to describe more complex cloud systems. In particular, we anticipate using this methodology in application to the RSP data from Aerosol Cloud Meteorology Interactions Over the western Atlantic Experiment (ACTIVATE) and the upcoming Arctic Radiation-Cloud-Aerosol-Surface-Interaction Experiment (ARCSIX).

## Acknowledgments.

This research was funded by the NASA Radiation Sciences Program managed by Hal Maring, NASA Aerosols/Clouds/Ecosystems (ACE) project, Aerosol Cloud Meteorology Interactions Over the western Atlantic Experiment (ACTIVATE), and the Science of the *Terra*, *Aqua*, and *Suomi NPP* Program managed by Paula Bontempi.

## Data availability statement.

The datasets generated for or used in this study will be available from the authors upon request. All the RSP data are available from https://data.giss.nasa.gov/pub/rsp/.

## APPENDIX A

### Derivation of Transition Matrix from Group Properties

*L*) satisfies the group and limit-case properties in Eqs. (17)–(19). We will also use another simple condition, Eq. (24), following from Eqs. (18) and (19) when

*L*

_{1}= ∞ [thus,

*L*

_{1}) =

*L*

_{2}=

*L*:

*L*= ∞ on the rhs.

*L*) in a very general form satisfying the row-sum condition in Eq. (8):

*ξ*

_{•}(

*L*) and

*ξ*

_{∘}(

*L*) are independent functions of the lag and may include

^{2}=

*L*

_{*}> 0 is a constant. It coincides with Eq. (21) for the coefficients

*w*in Eq. (20). Note also that Eq. (A13) with

_{L}*ω*(

*L*) =

*w*is equivalent to Eq. (20).

_{L}## APPENDIX B

### Derivation of Cloud Mask Transition Matrix from Continuous-Value Model

*L*) can be derived from Eqs. (58) and (59) using convenient change of integration order:

*P*(

_{ij}*L*) in the case of the continuous-value transition operator with the kernel described by Eq. (35). First, we introduce the following convenient notations:

*w*=

*w*and

_{L}*w*′ = 1 −

*w*to make the expressions more compact. Similarly,

_{L}*η*

_{•}and

*η*

_{∘}and their respective normalization conditions in Eqs. (55) and (57), we can compute the transition probabilities in Eq. (B1):

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