Cloud Statistics from Calipso Lidar Data for the ... - espace-tum.de

Appendix. Detailed results 76
Logarithmic binning
By definition a bin of constant logarithmic width means that the logarithm of the upper
edge of a bin (x i+1 ) is equal to the logarithm of the lower edge of that bin x i plus the
bin width b w . That is,
log(x i+1 = log(x i ) + b w
Each binned value x b can easily be calculated by the average of all x i for i=1,..,n such
that i falls in the bin.
x b = x 1 + x 2 + . . . + x n
n
Note that n is the number of integers that fall into each given bin regardless of whether
x b is zero or not. Furthermore, for the last bin all integers, even if they are beyond the
last data point, have to be included in n.
Power law fit uncertainty
60°N
30°N
0.30
0.27
0.24
0.21
0°
0.18
0.15
30°S
60°S
0.12
0.09
0.06
180° 135°W 90°W 45°W 0° 45°E 90°E 135°E
180°
60°N
30°N
10
9
8
7
0°
6
5
30°S
60°S
180° 135°W 90°W 45°W 0° 45°E 90°E 135°E
180°
4
3
2
1
Figure 33: The resulting σ (top) and x min (bottom) values for the discrete best fit
power law exponent α computed for each 2 ◦ x 2 ◦ grid point (complete 2007 data set).
x min is thereby directly correlated to the cloud gap length by x i = i ∗ 0.333km. The
threshold of σ = 0.1 indicate a high uncertainty in the power law distribution.