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CCN, Vertical Velocity and Cloud Droplet Concentration Multiple Regression Analysis

Since cloud droplet concentrations (N_{c}) depend on CCN concentrations (N_{CCN}) and vertical velocity (W), multiple regression analysis (MRA) is applicable. In MRA a dependent variable, Y, is influenced by, X_{1}, X_{2},_{…}X_{P } that are multidimensional, i.e.,

(1) Y = a + b_{1}X_{1} + b_{2}X_{2} +…+ b_{p}X_{p}

coefficients a, b_{1}, b_{2},…, b_{p} and R^{2}, are obtained by solving multiple equations based on multiple data (n) consisting of parameters Y, X_{1}, X_{2}…X_{p}. Data (n) must exceed inputs, p. Y is N_{c }and X are N_{CCN} at various supersaturations, S, (e.g., N_{1%}, N_{0.6%}, etc.) and eventually W. Coefficients (a, b_{1,} b_{2},..., b_{p}) then go into (1) for each datum; predictions of Y are calculated and compared to measured Y and linear regression R^{2} is calculated.

Adjusted R^{2} (R^{2}_{adj}) reveal whether usual R^{2 }increases with p are real or random

(2) R^{2}_{adj} = 1 – (1-R^{2}) (n-1)/(n-p-1)

Higher regression improvements are not as good for additional inputs that are correlated with variables already applied.

Single, double and higher regressions are illustrated in a and b of Fig. for wintertime Caribbean small cumuli (RICO) where R^{2}_{adj} of N_{CCN} (cumulative; critical S, S_{c} < S) and W with N_{c} are plotted against S. Black are single regressions, N_{CCN}-N_{c}. Red are double regressions that add W as input. Green are regressions of N_{CCN} at all S between 0.02% and abscissa S. At S = 0.02% green are single regressions identical to black (N_{0.02%}-N_{c}). Toward the right green show increasing N_{CCN} regressions; at S = 0.04% green are double regressions (N_{0.02%} and N_{0.04%} with N_{c}); at S = 0.06% green are triple regressions (N_{0.02%}, N_{0.04%} and N_{0.06%} with N_{c}); at S = 1.5% green include 10 N_{CCN} with N_{c}. Blue correspond to green as red corresponds to black; i.e., blue adds W to corresponding greens. At 0.02% blue are double regressions (N_{0.02%} and W with N_{c}) identical to red. Blue includes increasing quantities of N_{CCN} toward the right that also include W. At far right blue are 11-fold regressions. Contrasts between a and b demonstrate differences between more adiabatic cloud parcels (b) and averages of all low-altitude clouds (a). N_{0.6%} has the highest single and double R^{2}_{adj} (black and red). Results: (1) lack of R^{2}_{adj} improvement by addition of W (compare red with black and blue with green), (2) significantly higher R^{2}_{adj} of multiple regressions over single and double regressions (compare green with black and blue with red), (3) limited improvement for b over a indicates that entrainment only marginally perturbed N_{CCN}-N_{c }relationships. This extends the effect of N_{CCN} on cloud microphysics, which bolsters the indirect aerosol effect. Analogous figures for ICE-T (summertime Caribbean cumuli) and POST (California stratus) show similar results.

MASE (polluted California stratus) suffered from negative N_{CCN}-N_{c} R (black panel c). Nevertheless, when N_{CCN} are combined with standard deviation of W, σ_{w} (R= 0.51 for σ_{w}-N_{c}) there are positive regressions with maximum R^{2}_{adj} of 0.43 at S = 0.20%, red. That higher S N_{CCN} do not improve R^{2}_{adj} is consistent with the low MASE cloud S due to lower W of stratus and high MASE N_{CCN} that further suppress cloud S to where CCN spectra are steeper, which provides greater importance to W variations than N_{CCN} variations (Twomey 1959).

Lack of W effect on ICE-T and RICO regressions is consistent with Twomey (1959); higher cumuli W cause higher cloud S where typical CCN spectra are less steep, which favors N_{CCN} variations over W variations for determining N_{c}. Nonetheless, W is not irrelevant because within flights, where N_{CCN} is less varying, N_{c} is positively related to W. But N_{CCN} variability among flights was greater than W variability among flights; sd/mean of N_{1% }0.49 compared to 0.36 for sd/mean of W in ICE-T and 0.34 compared to 0.22 for RICO. This is consistent with the greater N_{CCN}-N_{c }than W-N_{c} correlations and the limited improvements when W is included in the multiple regressions of cumuli flight averages.

Twomey, S., 1959: *Geophys. Pure. Appl.*, **43**, 243-249.