#### 4.1. Methodology

[15] 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 *c*_{k} and *c*_{l} are fractional amounts of cloud in layers *k* and *l* at altitudes *z*_{k} and *z*_{l}, then total, vertically projected cloud fraction *c*_{k,l} for the two layers is defined as

where *α*_{k,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 *α*_{k,l} < 0 in (6). Hence, the smallest values that *α*_{k,l} can attain are

which occur when clouds do not overlap at all and *c*_{k,l} = *c*_{k} + *c*_{l} ≤ 1. Clearly,

[16] 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 *α*_{k,l} < 0 [cf. *Mace and Benson-Troth*, 2002; *Oreopoulos and Khairoutdinov*, 2003] and so *c*_{k,l} is always within [max (*c*_{k}, *c*_{l}), *c*_{k} + *c*_{l} − *c*_{k}*c*_{l}]. 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.

[17] 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.* [2004] for use with McICA [*Barker et al.*, 2002; *Pincus et al.*, 2003]. Unless stated otherwise, all applications of the generator used 25,000 subcolumns.

[18] 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*).

[19] 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.

[20] 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], *dF*_{ICA}/*d*_{cf}^{*} 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.

#### 4.2. Example

[21] Before moving to the global analysis, consider an example of the methodology proposed in the previous subsection and the impact of the precipitation screen.

[22] 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.