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Chapter 5. Cloud free statistics 52 for the log-likelihood tests itself. For small p (i.e., p ≤ 0.1) it is very likely that the ratio is an indicator for the better fit and contrary for large p, the ratio is not unambiguous. As already mentioned and demonstrated, the best possible power law fits are ˆα 1 , ˆα 3 , ˆα 5 and ˆα 7 . To check the correctness of the assumption of a power law fit, the estimated distributions are compared against the log-normal and truncated power law distribution with a log-likelihood ratio test (see Table 9). power law vs log-normal power law vs trunc. power law log-normal vs trunc. power law ‖R‖ p ‖R‖ p ‖R‖ p α 1 1.1909 0.2336 0.6392 0.3548 -1.2078 0.2270 α 3 -0.1459 0.8839 2.8826 0 3.6977 0.0002 α 5 0.4121 0.6802 0.1232 0.9602 -0.4048 0.6856 α 7 -1.1804 0.2378 -8.4586 4.44*10 −16 -10.9631 5.74*10 −28 Table 9: The results of the log-likelihood ratio test for selected power law cases and three possible distributions with its log-likelihood ratio R and the respective probability p. While the model distributions ˆα 3 and ˆα 7 are failing the test and are revealing the likelihood of a different distribution, both CALIPSO data fits ˆα 1 and ˆα 5 are in favor of a power law distribution. Therefore the log-likelihood ratio test finally confirms the correctness of a power law distribution with α = 1.51 and σ = 0.01. 5.3 Global cloud gap distribution Taking into account the various geographical variations in the cloud free fraction led to the question whether α = 1.51±0.01 is valid for each point on the Earth. For this purpose the discrete power law fit with an threshold of σ ≤ 0.1 was computed for the cloud gap lengths distribution that was assigned to each 2 ◦ x 2 ◦ grid point (see Figure 22) by the algorithm described in Chapter 3. The global distribution of individual cloud gaps is not equally distributed (see Figure 23) as most cloud gaps occur over the oceans from 45 ◦ S to 45 ◦ N, despite the fact, that there are more measurements in polar regions (see Figure 9). The high concentration of individual cloud gaps combined with the above average cloud free fraction present over subtropic oceans increases the likelihood of larger cloud gaps dominating the grid point which is confirmed by the low power law exponent seen in Figure 22. However, with 82.2 % of all grid points having at least 500 individual cloud gaps (see Figure 23), a power law fit can still be performed.
Chapter 5. Cloud free statistics 53 (a) 60°N 30°N 0° 2.40 2.25 2.10 1.95 1.80 30°S 60°S 1.65 1.50 1.35 180° 135°W 90°W 45°W 0° 45°E 90°E 135°E 180° (b) (c) 0.5 0.4 Non valid area above the threshold 0.1 = 72.6% of all values Median α = 1.65 1.0 0.8 Global mean cloudless fraction 40.5% Median α = 1.65 σ 0.3 0.2 Mean cloudless fraction 0.6 0.4 0.1 0.2 0.0 1.4 1.6 1.8 2.0 2.2 2.4 2.6 α 0.0 1.4 1.6 1.8 2.0 2.2 2.4 2.6 α (d) (e) Mean cloud gap length [km] 250 200 150 100 50 Mean cloud gap length 7.41km Median α = 1.65 Total number of cloud gaps 4500 4000 3500 3000 2500 2000 1500 1000 Mean number of cloud gaps = 924 Median α = 1.65 500 0 1.4 1.6 1.8 2.0 2.2 2.4 2.6 α 0 1.4 1.6 1.8 2.0 2.2 2.4 2.6 α Figure 22: (a) is the best fit power law exponent α computed for each 2 ◦ x 2 ◦ grid point (complete 2007 data set) and its significant error σ (b), where all power law exponents that exceed a threshold of σ = 0.1 are marked as red. Non valid power law fits can mainly be seen in regions with high cloud cover, while the relationship of the cloud free fraction and the power law exponent α is spread over the whole domain as indicated in (c). Only for regions where the mean cloud gap length is above average the power law exponents exceeds α > 2 as indicated in (d). The significant error most likely exceeds the threshold (deviation from power law) for grid points where the total measurement points for the fit lay below average, as seen in (e).
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Technische Universität München Ma
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”Science never solves a problem w
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Acknowledgements I would like to th
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Contents xii 3.1 Introduction . . .
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List of Figures xiv 33 Global cloud
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Abbreviations ABL ATBD BSP CALIOP C
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SI Units and Abbreviations deg degr
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Dedicated to the ones I love and to
Page 26 and 27: Chapter 1. Introduction and motivat
Page 36 and 37: Chapter 2. Background information a
Page 48 and 49: Chapter 3 Methodology behind cloud
Page 50 and 51: Chapter 3. Methodology behind cloud
Page 60 and 61: Chapter 4. Case studies 36 4.1 Typi
Page 62 and 63: Chapter 4. Case studies 38 4.3 Mult
Page 64 and 65: Chapter 5 Cloud free statistics 5.1
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Page 82 and 83: Chapter 6. Cloud flatness occurrenc
Page 88 and 89: Chapter 7 Cloud top height distribu
Page 90 and 91: Chapter 7. Cloud top height distrib
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Page 94 and 95: Chapter 8. Conclusion 70 the existi
Page 96 and 97: Chapter 8. Conclusion 72 the fact t
Page 98 and 99: Appendix. Detailed results 74 CALIP
Page 100 and 101: Appendix. Detailed results 76 Logar
Page 102 and 103: Appendix. Detailed results 78 (a) J
Page 104 and 105: Bibliography J. Alstott, E. Bullmor
Page 106 and 107: Bibliography 82 A. Klaus, S. Yu, an
Page 108 and 109: Bibliography 84 E. P. White, B. J.
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