Throughout this analysis, overlap of clouds is treated as a linear combination of maximum and random overlap [Hogan and Illingworth, 2000; Bergman and Rasch, 2002]. To illustrate, if ck and cl are fractional amounts of cloud in layers k and l at altitudes zk and zl, then total, vertically projected cloud fraction ck,l for the two layers is defined as
where αk,l is the weighting parameter. The maximum component is unambiguous: positions of clouds in one layer are determined by, and overlay, those in the other. The random component is the expectation value for the distribution of total cloud fraction that occurs when the positions of clouds in both layers are completely uncorrelated. When clouds overlap less than expected when cloud positions in one layer are completely independent of those in the other,
which leads to αk,l < 0 in (6). Hence, the smallest values that αk,l can attain are
which occur when clouds do not overlap at all and ck,l = ck + cl ≤ 1. Clearly,
 It has become almost customary, and with fairly good reason [see Astin and Di Girolamo, 2006], to define
where cf is decorrelation length for overlapping fractional cloud which, in general, varies with z. Note that this definition does not admit αk,l < 0 [cf. Mace and Benson-Troth, 2002; Oreopoulos and Khairoutdinov, 2003] and so ck,l is always within [max (ck, cl), ck + cl − ckcl]. Nor does it allow = 1 when max (c(z)) < 1, where c(z) is layer cloud fraction profile; though these cases can, and do, occur.
 Within most conventional GCMs, all the relevant information one has pertaining to cloud structure inside a column is c(z). There are, therefore, infinitely many ways to overlap the fractions c(z) and thus produce corresponding total cloud fractions C that can, in principle, take values within [max(c(z)), 1]. With (6) and (7), a convenient, and potentially useful, way to assess overlap for nonovercast scenes, is to enforce a vertically invariant cf on c(z) [Barker and Räisänen, 2005]. This leads to a unique function C(cf) ∈ [max(c(z)), 1]. This function is, by definition, nonanalytic, but can be evaluated numerically using a stochastic cloud generator. The generator used here was developed originally by Räisänen et al.  for use with McICA [Barker et al., 2002; Pincus et al., 2003]. Unless stated otherwise, all applications of the generator used 25,000 subcolumns.
 When dealing with a 3-D field of cloud or a 2-D cross section, obtained from either a CSRM or observations, one can compute the domain's actual total cloud fraction (along with c(z), and even cf(z)), and thus define an effective decorrelation length cf* as the solution to
Again, however, this cannot be done with GCM data for all one has to work with is c(z).
 One way to solve for cf* is Brent's method [Brent, 1973] which is ideal when computed values of a function are all that are known, and roots are one-dimensional. Brent's method combines the safety of bisection methods with the convergence speed of higher-order methods, provided the root lies between extremes set by the user. The extremes used here were 0 km and 20 km. If cf* > 20 km the algorithm defaults to 20 km, which is often very close to maximum overlap. The convergence criterion was 0.05 km. It was set small to diminish the impact of stochastic fluctuations in returned values of C. The number of iterations required for convergence was typical 7 or 8 but ranged from 5 to 12.
 Figure 1 shows a comparison of roots between those based on the settings just mentioned and a benchmark that used an upper limit of 100 km, a convergence criterion of 0.01 km, and 250,000 subcolumns. The latter required generally 10 to 15 iterations and took about 20 times more CPU time than the operational settings. For cf* < 20 km, the operational settings lead to errors in cf* that are typically less than 0.1 km, well below the resolution of the data and thus adequate for the purpose at hand. While errors for cf* > 20 km are exclusively bias, and of magnitude cf* − 20 km, only 50 of the 11,498 cross sections for L = 1000 km had cf* > 20 km. Moreover, based on experience [Barker and Räisänen, 2005], dFICA/dcf* are usually very small for cf* > 20 km and so errors are negligible. While 1000 km transects were used to produce Figure 1, samples for shorter cross sections yielded virtually identical results.
Figure 1. Error in the estimation of cf* via Brent's root-finding method when the convergence criterion was 0.1 km and 25,000 subcolumns were used in the stochastic cloud generator. The benchmark solution was taken to be a convergence criterion of 0.01 km along with 250,000 subcolumns in the stochastic cloud generator.
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 Before moving to the global analysis, consider an example of the methodology proposed in the previous subsection and the impact of the precipitation screen.
 Figure 2 contrasts (4) with its precipitation-screened counterpart for a typical 1000 km sample drawn from an orbit on 1 January 2007. Clearly, this screening goes too far in some cases and not far enough in others, but generally speaking, it appears to move matters in the desired direction. The plot of layer cloud fraction profile in the lower left corner of Figure 2 shows that screening eliminated volumes up to about 3 km above the surface. Since application of (5) never increases , and rarely reduces it (it went from 0.876 to 0.875 in this case), sizable reductions in layer cloud fractions mean that there is less cloud in a profile needed to make-up . In general this means that screening will reduce cf*; in this instance, solving (8) for the unscreened and screened profiles yielded cf* of 2.0 km and 1.5 km, respectively. When these values of cf* are used in the stochastic generator they produce fields that by design, had the correct profile of layer cloud fraction and associated , but because cf* was set constant with height, profiles of cumulative cloud fraction were slightly incorrect, as seen in the central plot in Figure 2. The rightmost plot in Figure 2 shows, however, that the generated fields have fractional amounts of cloud exposed to space, m, that not only approximate the observed amounts well but that are impacted only slightly by alteration of cf* induced by precipitation screening.
Figure 2. (top) A 1000 km long cloud mask cross section produced by a merger of CloudSat and CALIPSO data. The four shades are defined as follows: (1) white corresponds to radar volumes with 99% or more of their lidar volumes identified as containing cloud though the radar did not register cloud; (2) light gray represents radar volumes in which only the radar registered cloud; (3) dark gray is when both conditions 1 and 2 are satisfied; and (4) black corresponds to no cloud detected. This information is straight from the CloudSat database. (middle) The same as the upper except the precipitation mask was applied (see (5)). Plots along the bottom show profiles pertaining to the two cross sections. Leftmost plot shows layer cloud fractions; shaded area indicates the reduction in cloud cover due to application of (5). Center plot shows downward cumulative cloud fractions for the cross sections, as well as for fields produced by the stochastic cloud generator using cf* = 2.0 km and cf* = 1.5 km, which are applicable to the unscreened and screened cases, respectively.
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Figure 4. (left) Cumulative distribution of cf* obtained via Brent's method for the unscreened curve shown in Figure 3. These are for convergence criterion of 0.1 km and 25,000 stochastic subcolumns in the cloud generator. Using this distribution of cf* in the stochastic generator, with 100,000 subcolumns, yields (right) the cumulative distribution of total cloud fractions C.
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