{"@context":{"@language":"en","Affiliation":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","AggregatedSourceRepository":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","Campus":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","Creator":"http:\/\/purl.org\/dc\/terms\/creator","DateAvailable":"http:\/\/purl.org\/dc\/terms\/issued","DateIssued":"http:\/\/purl.org\/dc\/terms\/issued","Degree":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","DegreeGrantor":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","Description":"http:\/\/purl.org\/dc\/terms\/description","DigitalResourceOriginalRecord":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","Extent":"http:\/\/purl.org\/dc\/terms\/extent","FileFormat":"http:\/\/purl.org\/dc\/elements\/1.1\/format","FullText":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","Genre":"http:\/\/www.europeana.eu\/schemas\/edm\/hasType","GraduationDate":"http:\/\/vivoweb.org\/ontology\/core#dateIssued","IsShownAt":"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt","Language":"http:\/\/purl.org\/dc\/terms\/language","Program":"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline","Provider":"http:\/\/www.europeana.eu\/schemas\/edm\/provider","Publisher":"http:\/\/purl.org\/dc\/terms\/publisher","Rights":"http:\/\/purl.org\/dc\/terms\/rights","ScholarlyLevel":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","Title":"http:\/\/purl.org\/dc\/terms\/title","Type":"http:\/\/purl.org\/dc\/terms\/type","URI":"https:\/\/open.library.ubc.ca\/terms#identifierURI","SortDate":"http:\/\/purl.org\/dc\/terms\/date"},"Affiliation":[{"@value":"Arts, Faculty of","@language":"en"},{"@value":"Geography, Department of","@language":"en"}],"AggregatedSourceRepository":[{"@value":"DSpace","@language":"en"}],"Campus":[{"@value":"UBCV","@language":"en"}],"Creator":[{"@value":"Szczodrak, Malgorzata","@language":"en"}],"DateAvailable":[{"@value":"2009-06-02T20:15:54Z","@language":"en"}],"DateIssued":[{"@value":"1998","@language":"en"}],"Degree":[{"@value":"Doctor of Philosophy - PhD","@language":"en"}],"DegreeGrantor":[{"@value":"University of British Columbia","@language":"en"}],"Description":[{"@value":"Measurements made by the AVHRR (Advanced Very High Resolution Radiometer)\r\non board of five NOAA polar orbiting satellites were used to retrieve cloud\r\noptical depth (\u03c4) and cloud droplet effective radius (r[sub eff]) for marine boundary layer\r\nclouds over the Pacific Ocean west of California and over the Southern Ocean near\r\nTasmania. Retrievals were obtained for 21 days of data acquired between 1987 and\r\n1995 from which over 300 subscenes ~ 256 km x 256 km in size were extracted. On\r\nthis spatial scale cloud fields were found to have mean \u03c4 between 8 and 32 and mean\r\nr[sub eff] between 6 and 17 \u03bcm. The frequency distribution of \u03c4 is well approximated by\r\na two parameter gamma distribution. The gamma distribution also provides a good\r\nfit to the observed r[sub eff] distribution if the distribution is symmetric or positively\r\nskewed but fails for negatively skewed or bi-modal distributions of r[sub eff] which were\r\nalso observed.\r\nThe retrievals show a relationship between \u03c4 and r[sub eff] which is consistent with\r\na simple \"reference\" cloud model with reff ~ r[sup 1 \/ 5]. The proportionality constant\r\ndepends on cloud droplet number concentration N and cloud subadiabaticity \u03b2\r\nthrough the parameter N[sub sat] = N\/ [sq rt. \u0392]. Departures from the reference behaviour\r\noccur in scenes with spatially coherent N[sub sat] regimes, separated by a sharp boundary.\r\nAVHRR imagery is able to separate two N[sub sat] regimes if they differ by at least 30%\r\nin most cases.\r\nSatellite retrievals of \u03c4 and r[sub eff] were compared with in situ aircraft measurement\r\nnear Tasmania. The retrievals overestimated r[sub eff] by 0.7 to 3.6 \u03bcm on\r\ndifferent flights, in agreement with results from earlier comparison studies. The\r\nr[sub eff] overestimation was found to be an offset independent of \u03c4. The reference cloud\r\nmodel and the N[sub sat] retrieval were tested on aircraft data and yield results consistent\r\nwith direct in situ measurements of N and 8.\r\nSpectral and multifractal analyses of the spatial structure of cloud visible\r\nradiance, \u03c4 and r[sub eff] fields in 34 satellite scenes revealed scale breaks at 3 to 2 km in all analysed scenes in agreement with some earlier observations (Davis et al.\r\n(1996a)) but in contrast with other work (Lovejoy et al. (1993)). The nonstationarity\r\nH(1) and intermittency C(1) parameters were computed for the 34 scenes, stratified\r\nusing the reference cloud model and according to mean \u03c4 and r[sub eff]. Similar values\r\nof H(1) and C(1) were found in all these categories.\r\nThese measurements of the frequency distribution and spatial variability of \u03c4,\r\nr[sub eff], liquid water path (Iwp), and N[sub sat] can be used to place constraints on mesoscale\r\nmodels of layer clouds.","@language":"en"}],"DigitalResourceOriginalRecord":[{"@value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/8647?expand=metadata","@language":"en"}],"Extent":[{"@value":"16624918 bytes","@language":"en"}],"FileFormat":[{"@value":"application\/pdf","@language":"en"}],"FullText":[{"@value":"Variability of cloud optical depth and cloud droplet effective radius in layer clouds: satellite based analysis by Malgorzata Szczodrak M.Sc , Jagiellonian University, 1984 A DISSERTATION S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F Doctor of Philosophy in T H E F A C U L T Y O F G R A D U A T E STUDIES (Department of Geography) We accept this dissertation as conforming to the required standard T H E U N I V E R S I T Y O F BRITISH C O L U M B I A January 1998 \u00a9 Malgorzata Szczodrak, 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my writ ten permission. Department of (rOD^ycL The University of British Columbia Vancouver, Canada Date \/ 9 . 0 3 , 179$ DE-6 (2\/88) Abstract Measurements made by the A V H R R (Advanced Very High Resolution Ra-diometer) on board of five N O A A polar orbiting satellites were used to retrieve cloud optical depth (T) and cloud droplet effective radius (rejj) for marine boundary layer clouds over the Pacific Ocean west of California and over the Southern Ocean near Tasmania. Retrievals were obtained for 21 days of data acquired between 1987 and 1995 from which over 300 subscenes ~ 256 km x 256 km in size were extracted. On this spatial scale cloud fields were found to have mean r between 8 and 32 and mean reff between 6 and 17 (j,m. The frequency distribution of r is well approximated by a two parameter gamma distribution. The gamma distribution also provides a good fit to the observed reff distribution if the distribution is symmetric or positively skewed but fails for negatively skewed or bi-modal distributions of rejf which were also observed. The retrievals show a relationship between T and refj which is consistent with a simple \"reference\" cloud model with reff ~ r 1 \/ 5 . The proportionality constant depends on cloud droplet number concentration iV and cloud subadiabaticity 8 through the parameter Nsat = N\/y\/]3. Departures from the reference behaviour occur in scenes with spatially coherent Nsat regimes, separated by a sharp boundary. A V H R R imagery is able to separate two Nsat regimes if they differ by at least 30% in most cases. Satellite retrievals of r and rejf were compared with in situ aircraft mea-surement near Tasmania. The retrievals overestimated refj,by 0.7 to 3.6 fim on different flights, in agreement with results from earlier comparison studies. The reff overestimation was found to be an offset independent of r. The reference cloud model and the Nsat retrieval were tested on aircraft data and yield results consistent with direct in situ measurements of N and 8. Spectral and multifractal analyses of the spatial structure of cloud visible radiance, r and reff fields in 34 satellite scenes revealed scale breaks at 3 to 20 n km in all analysed scenes in agreement with some earlier observations (Davis et al. (1996a)) but in contrast with other work (Lovejoy et al. (1993)). The nonstationarity H(l) and intermittency C(l) parameters were computed for the 34 scenes, stratified using the reference cloud model and according to mean r and reff . Similar values of H(1) and C(l) were found in all these categories. These measurements of the frequency distribution and spatial variability of r, reff, liquid water path (Iwp), and Nsat can be used to place constraints on mesoscale models of layer clouds. in Contents Abstract ii Contents iv List of Tables ix List of Figures xii List of Symbols xvi List of Acronyms xxi Acknowledgements xxii Chapter 1 Introduction 1 1.1 Role of boundary layer clouds in Earth's radiation budget 1 1.2 Cloud optical depth and cloud droplet effective radius 3 1.2.1 Basic definitions 3 1.2.2 Review of sensitivity studies 4 1.2.3 Observations 8 1.2.4 Summary 10 1.3 Implications of cloud inhomogeneity for radiative transfer calculations on G C M grid size scale 11 iv 1.4 Quantifying spatial inhomogeneity in cloud optical depth and cloud droplet effective radius 12 1.5 Previous satellite and aircraft observations of cloud optical depth and cloud droplet effective radius 15 1.6 Thesis outline 17 Chapter 2 Retrieval of cloud parameters from AVHRR radiance mea-surements 20 2.1 Basic concept 20 2.2 The forward model 26 2.3 The retrieval 30 2.4 Theoretical estimates of errors 33 2.4.1 Errors due to approximations in the retrieval method . . . . 33 2.4.2 Errors due to model assumptions 34 2.4.3 Errors due to measurement uncertainty 35 2.4.4 Errors introduced by the independent pixel approximation . . 36 Chapter 3 The relationship between cloud optical depth and cloud droplet effective radius 38 3.1 A simple model of boundary layer cloud 39 3.2 Previous observations of reff and r or reff and Iwp 42 3.3 Observations of T and r e yj 47 3.3.1 Examples of power law dependence between r and rejf . . . 48 3.3.2 Departures from a power law behaviour 50 3.3.3 Special case: thick clouds 55 3.3.4 Bi-modal joint probability distributions of r and r e jy . . . . 57 3.3.5 Summary 59 Chapter 4 Quantitative treatment of the relationship between cloud optical depth and cloud droplet effective radius 61 4.1 General power law fit to the observations of T and reff 62 4.1.1 Scene selection for the general power law fit 62 4.1.2 Transformation of variables 66 4.1.3 Bivariate linear regression with errors in both variables . . . 69 4.2 Mesoscale frequency distributions of r and r e \/ \/ 74 4.2.1 Clouds with power law relationship between r and reff . . . 76 4.2.2 Clouds with a bi-modal joint distribution of r and r e \/ \/ . . . 80 4.2.3 Special case: thick clouds 80 4.2.4 Parameters of the fit gamma distribution 80 4.3 Inference of the Nsat based on the relationship between cloud optical depth and cloud droplet effective radius 86 4.4 Summary 91 Chapter 5 Validation of the satellite retrievals by in situ aircraft measurements 93 5.1 Previous studies comparing satellite retrievals and in situ measure-ments of r and reff 94 5.2 Outline of the analysis . . . 97 5.3 Aircraft cloud sampling during S O C E X 98 5.4 Flight average soundings, estimation of (3 and r 101 5.5 Comparison of satellite and aircraft observations in the reff \u2014 r plane. 108 5.6 Summary 115 Chapter 6 Spatial structure of stratocumulus clouds I: Basic concepts and definitions 116 6.1 Questions 117 6.2 Stochastic processes and geophysical data sets 118 6.2.1 Scale invariance in stochastic processes 118 6.2.2 Statistical stationarity and stochastic continuity 119 vi 6.2.3 Scale invariance and stationarity in Fourier space 121 6.2.4 Scale invariant nonstationary process with stationary incre-ments 123 6.3 Previous work: Fourier analysis approach 125 6.4 Multifractal analysis 127 6.4.1 Motivation 127 6.4.2 Concept of multifractals 128 6.4.3 Formalism of multifractal analysis 129 6.5 Previous work: multifractal approach 137 6.6 Summary . . 138 Chapter 7 Spatial structure of stratocumulus clouds I I : Results from multifractal analysis 141 7.1 Transects of satellite fields and ensemble averages 142 7.2 Flow of the analysis 145 7.3 Analysis results 156 7.3.1 Nonstationarity and intermittency parameters 157 7.3.2 Type of scaling 168 7.3.3 Nonstationarity and intermittency in the bifractal plane. . . 182 7.4 Summary of the results of multifractal analysis . 186 Chapter 8 Conclusions 189 8.1 Summary 189 8.2 Discussion 193 8.2.1 The success of the simple \"reference\" cloud model 193 8.2.2 Overestimation of cloud droplet effective radius by remote sensing 194 8.2.3 Spatial structure of cloud fields 195 8.3 Future Considerations and new research 197 vii 8.3.1 Spatial structure of marine stratocumulus 197 8.3.2 G C M scale cloud reflectivity and cloud droplet effective radius 199 8.3.3 Cloud subadiabaticity 200 Bibliography 202 Appendix A Parametrisation of the relationship between cloud opti-cal depth and cloud droplet effective radius 215 Appendix B Data 219 B.l Synoptic conditions 220 B.2 Lists of scenes analysed in the thesis 224 Appendix C Removing auto-correlation in spatial data 230 Appendix D Correlation coefficient between r and refj 237 Appendix E Cloud subadiabaticity 0 241 Appendix F Concepts related to fractal sets and fractal measures 244 Appendix G Tables of nonstationarity and intermittency parameters249 vm Lis t of Tables 2.1 Wavelength bands of the 5 channels of the AVHRR on board of NOAA 9 21 2.2 The grid system of the look-up tables 29 4.1 Parameters intercept a and slope b and their uncertainties from the linear regression for 10 test scenes (see Table B.l) 69 4.2 Parameters of gamma distributions fitted to cloud optical depth and cloud droplet effective radius frequency distributions 80 4.3 Estimates of Nsat and the Nsat uncertainty intervals for each cluster in scenes 8 and 9 85 5.1 SOCEX flight missions and coordinated satellite overpasses 96 5.2 Results of the general power law fit to the SOCEX aircraft data. . . 110 7.1 Least square estimates of \u00a3(q) an it uncertainties cr^q^ calculated for 3 different assumptions of the scaling 145 B . l List of scenes: uni-modal (1-6), bimodal (7-10) and thick cloud (11-14).218 B.2 List of 10 randomly chosen uni-modal scenes 219 B.3 Scenes of F87 data set included in the F87 ensemble average 220 B.4 Scenes of P94 data set included in the P94 ensemble average 221 B.5 Scenes of P95 data set included in the P95 ensemble average 222 B.6 Scenes of 595 data set included in the S*95 ensemble average 223 ix D . l Pearson's (rp) and Spearman's (rs) linear correlation coefficients for 10 test scenes 231 G . l Multifractal parameters of cloud visible wavelength radiance field for the 15 example scenes: (1-6, and 15) uni-modal, (7-10) bi-modal, (11-14) thick clouds 242 G.2 Multifractal parameters of cloud optical depth field for the 15 example scenes: (1-6, and 15) uni-modal, (7-10) bi-modal, (11-14) thick clouds. 243 G.3 Multifractal parameters cloud droplet effective radius field for the 15 example scenes 244 G.4 Multifractal parameters of cloud visible wavelength radiance field for FS7 data set 245 G.5 Multifractal parameters of cloud optical depth field for the F I R E data set 246 G.6 Multifractal parameters cloud droplet effective radius field for the F I R E data set 247 G.7 Multifractal parameters of cloud visible wavelength radiance field for the Pacific Ocean 1994 data set 248 G.8 Multifractal parameters of cloud optical depth field for the Pacific Ocean 1994 data set : : 248 G.9 Multifractal parameters of cloud droplet effective radius field for the the Pacific Ocean 1994 data set 249 G.10 Multifractal parameters of cloud visible wavelength radiance field for the Pacific Ocean 1995 data set 249 G . l l Multifractal parameters of cloud optical depth field for the Pacific Ocean 1995 data set 250 G.12 Multifractal parameters of cloud droplet effective radius field for the Pacific Ocean 1995 data set 250 x G.13 Multifractal parameters of cloud visible wavelength radiance field for the SOCEX 1995 data set xi List of Figures 2.1 Sun-satellite geometry 22 2.2 Satellite geometry 23 2.3 Model cloud reflected solar radiance in A V H R R channel 1 and 3 as a function of r and reff 24 2.4 Flow chart of the iteration in the retrieval process 31 3.1 Sketch of a cloud layer 38 3.2 Lines of constant cloud droplet concentration in parameterisation ac-cording to (3.2) 40 3.3 Scatter plots of reff and r from F I R E scene for July 10 1987 after Nakajima and Nakajima (1995) 43 3.4 Scatter plots oireff and r from A S T E X scene for June 13 1992 after Nakajima and Nakajima (1995) 44 3.5 Scatter plots of r e \/ \/ and r from 6 cloudy scenes of approximate areas ranging from 128x256 to 256x256 km 47 3.6 Scatter plots of ref f and r from 4 cloudy scenes of approximate areas of 256x256 km 49 3.7 Lines of constant cloud droplet concentration and lines of constant cloud liquid water path 50 3.8 Contours of a joint probability distribution of r and reff correspond-ing to the scatter plots of figure 3.6. 52 xii 3.9 Contours of a joint probability distribution of T and reff for thick clouds 54 3.10 Spatial separation of the two clusters in the scatter plots of T and reff. 56 4.1 Pitfalls of fitting a power law curve to data points from a narrow range 62 4.2 Scatter plots of cloud optical depth and cloud droplet effective radius for 4 test scenes 64 4.3 Scatter plots of cloud optical depth and cloud droplet effective radius (in the log-log scale) and their linear regression 70 4.4 Geographical location of scenes 1-25. . 72 4.5 Frequency distribution of cloud optical depth for 9 uni-modal scenes 74 4.6 Frequency distribution of cloud droplet effective radius for uni-modal scenes 75 4.7 Scatter plot of mean cloud optical depth and mean cloud droplet effective radius for the uni-modal scenes 76 4.8 Frequency distribution of cloud optical depth (a\u2014d) and cloud droplet effective radius (e \u2014 h) for bi-modal scenes 78 4.9 Frequency distribution of cloud optical depth (a\u2014d) and cloud droplet effective radius (e \u2014 h) for thick cloud scenes 79 4.10 Scatter plot of mean cloud optical depth and mean cloud droplet effective radius for all scenes 82 4.11 Contours of cloud droplet number concentration N labeled in [cm - 3 ] as a function of Nsat and j3 88 5.1 Flight path of S O C E X flight no. 10 overlayed on channel 1 A V H R R image 96 5.2 Aircraft vertical cloud sampling pattern (horizontal stacks) 97 5.3 Flight average liquid water profiles 98 xiii 5.4 Flight average cloud droplet number concentration profiles 101 5.5 Flight average cloud optical depth profiles 103 5.6 Scatter plots of satellite retrieved T and reff and in situ aircraft measurements in S O C E X 108 5.7 Plots of satellite retrieved r and reff averaged over intervals AT=1 and in situ aircraft measurements in S O C E X 109 7.1 A V H R R channel 1 radiance field for scene no 1 141 7.2 Power spectrum for channel 1 radiance field of scene no 1. 142 7.3 Second order structure functions for channel 1 radiance field of scene 1.143 7.4 Structure functions of order 1 to 5 for channel 1 radiance field of scene 1 144 7.5 Scaling exponent ((q) of the moments of structure function as a func-tion of the moment order for channel 1 radiance of scene no 1. . . . 146 7.6 Power spectrum of the channel 1 radiance gradient field in scene no 1. 147 7.7 Singular measures moments plotted against the scale r for channel 1 radiance in scene no 1 148 7.8 Exponent function K(q) for channel 1 radiance in scene no 1 149 7.9 Information dimension D(q) for channel 1 radiance in scene no 1 . . 150 7.10 Power spectra of cloud visible radiance (chl), cloud optical depth (T) and cloud droplet effective radius r e \/ \/ for scene no 11 155 7.11 Second order structure functions of cloud visible radiance (chl), cloud optical depth (r) and cloud droplet effective radius r e \/ \/ for the uni-modal scenes 1 to 6 (see Figure 3.5 for scatter plots of r and r e \/ \/ ) . . 157 7.12 Second order structure functions of cloud visible radiance (chl), cloud optical depth (r) and cloud droplet effective radius reff for the bi-modal scenes 7 to 10 (see Figure 3.6 for scatter plots of r and reff). 158 xiv 7.13 Second order structure functions of cloud visible radiance (chl), cloud optical depth (r) and cloud droplet effective radius rejj for thick clouds in scenes 11 to 14 (see Figure 3.9 for contour plots of r and reff) 159 7.14 False colour image of cloud droplet effective radius [\/J.m] in scene 8. . 161 7.15 Second order structure functions for regions \"A\" and \"B\" of cloud droplet effective radius in scene 8 162 7.16 Exponent function ((q) of cloud visible radiance field, cloud optical depth , and cloud droplet effective radius for uni-modal scenes 1 to 6. 164 7.17 Exponent function ((q) of cloud visible radiance field, cloud optical depth , and cloud droplet effective radius for bi-modal scenes 7 to 9 165 7.18 Exponent function ((q) of cloud visible radiance field, cloud optical depth , and cloud droplet effective radius for thick clouds, scenes 11 to 14 166 7.19 Weak multiscaling signal built by superimposing Heaviside function and ordinary Brownian motion . 168 7.20 Strong multiscaling signal build by superimposing Heaviside function and ordinary Brownian motion 169 7.21 Demeaned and mean normalised transects of cloud optical depth and cloud droplet effective radius from scene -F87I6. . 170 7.22 Exponent functions for the transects of cloud optical depth and cloud droplet effective radius shown Figure 7.21 172 7.23 Demeaned and mean normalised transects of cloud optical depth and cloud droplet effective radius from scene 8 173 7.24 Exponent functions for the transects of cloud optical depth and cloud droplet effective radius shown Figure 7.23 174 xv 7.25 Information dimension D(q) of cloud visible radiance field, cloud op-tical depth , and cloud droplet effective radius for scenes uni-modal scenes 1 to 6 176 7.26 FIRE cloud field visible radiance, cloud optical depth and cloud droplet effective radius multifractal parameters in the bifractal plane. 178 7.27 The bifractal plane location of ensemble averages of F87, P94, P95 and 595 data sets for cloud visible wavelength radiance field, cloud optical depth and cloud droplet effective radius 180 B . l North Pacific summer circulation 215 B . 2 Southern Ocean summer circulation 217 C. l Schematic semivariogram 227 C.2 The sampling procedure, a) the scatter plot of cloud optical depth and cloud droplet effective radius of the full data set; b) semivar-iograms of r in four directions (N-S, NE-SW, W-E, and NW-SE) showing autocorrelation range 7-10 km; c) scatter plot of T and rejj of an autocorrelation free sample; d) plot of the residuals of the linear least square fit to log(re\/j) and log(r) as a function of r 229 E. l Scatter plot of cloud subadiabaticity 6 and cloud droplet number concentration 235 F. l Construction of the classical Koch curve 238 xvi List of Symbols a intercept in the linear least squares regression ao constant in eq. 3.2 and 3.2 Ac cloud fraction Ag ground albedo b exponent in the general power law B(Temp) Planck function B(x) ordinary Brownian motion c, cadiab rate of increase of cloud liquid water content with height chl visible wavelength cloud reflected radiance C(q) hierarchy function of exponents K(q) CCN cloud condensation nuclei DG Holder dimension of graph g DQ generalised dimension DH Hausdorf dimension E(k) power spectrum E Euclidean dimension FQ solar flux gq(r) structure function of order q G(r, x) autocorrelation function H,H(q) scaling exponent (eq. 6.1), hierarchy function of exponents ((q) k wavenumber K(q) exponent function for scaling of singular measures Iwc cloud liquid water content Iwp cloud liquid water path L domain of the stochastic process in chapter 6 xvii L, L0bs cloud reflected radiance and satellite received radiance, chapter 2 n(r) distribution of cloud droplets size in eq. 1.1, 1.2 and 2.1 N cloud droplet number concentration Nsat N\/VP N-, N+ lower and upper limits of cloud droplet number concentration uncertainty interval q moment order of structure functions and singular measures in chapter 6 9) Qadiab cloud liquid water content and adiabatic cloud liquid water content in Appendix A Qext extinction efficiency factor for a water droplets pressure, saturation level pressure p(r) frequency distribution of r in G C M pixel r cloud droplet radius in eq. 1.1, 1.2 and 2.1 plane albedo of the cloud layer in eq. 2.3 and 8.1, log(r e\/j), distance lag for structure functions and degradation scale for singular measures in chapter 6 rp percentage of variability of Nsat explained by variability of cloud subadiabaticity 6 reff cloud droplet effective radius fef\/,< reff > mean reff refjf reff in equation 4.11 rpp,rs plane albedo in PPH approximation in eq. 8.1, and spherical albedo of the cloud layer in eq. 2.3 T*JV percentage of variability of Nsat explained by variability of cloud droplet number concentration N ro mode radius of droplet size distribution in eq. 2.1 xvin rr Pearson's linear correlation coefficient rrs Spearman's rank order correlation coefficient R bidirectional reflection in chapter 2, upper limit of scaling range in chapter 6 t transmissivity of the cloud layer in eq. 2.3, log(r) tc transmissivity of cloud layer in A V H R R channel 4 T bidirectional transmission Tc cloud top temperature Tg ground temperature (vi, Vj) values of measured field at locations i and j z, Az height in the atmosphere, cloud geometrical thickness a, ao constant in the general power law, constant denned in eq. A.7 Holder order of singularity in section 6.4.2 (3 cloud subadiabaticity defined in eq. A.2, scaling exponent in the power spectrum eq. 6.11 [3B Betts' mixing parameter 7,7(h) atmospheric lapse rate, semivariogram r the gamma function e(r,x) singular measure ((g) exponent function in structure function scaling 77 lower limit of scaling range in chapter 6 9 cosine of satellite viewing angle $o cosine solar zenith angle A wavelength, change of scale ratio in chapter 6 \\x cosine of satellite viewing angle cosine solar zenith angle v parameter of gamma distribution in eq. 4.6, number of degree of freedom in chapter 4 x i x uT, Vreff parameters of gamma distribution fitting histograms of r and refj p density of water a log standard deviation of droplet size distribution in eq. 2.1 aa uncertainty of intercept from the least squares regression ab uncertainty of slope from the least squares regression ae standard deviation of regression residuals (log), eq. 4.3 areff standard deviation of ref \/ op uncertainty of intercept of cloud subadiabaticity Q estimated in chapter 5 ar standard deviation of T r cloud optical depth, total optical depth of the atmosphere in eq. 2.2 and 2.3 r\/ r in equation 4.11 f, < r > mean < r > T C , TU cloud optical depth, optical depth of the atmosphere above the cloud layer in eq. 2.2 and 2.3 4> sun-satellite azimuth angle (pn AVHRR response function in channel n (p(x), Aip(x,r) a generic stochastic process and its increment over a distance r at point x xx List of Acronyms AVHRR Advanced Very High Resolution Radiometer ASTEX Atlantic Stratus Transition Experiment CSIRO The Commonwealth Scientific and Industrial Research Organisation ERBE Earth Radiation Budget Experiment FIRE First ISCCP Regional Experiment F87 FIRE 1987 data set GCM Global Circulation Model IP A Independent Pixel Approximation ISCCP International Satellite Cloud Climatology Project MCR Multi-spectral Cloud Radiometer MODIS Moderate Resolution Imaging Spectroradiometer NASA National Aeronautics and Space Administration NOAA National Oceanic and Atmospheric Administration NCAR National Center for Atmospheric Research PPH Plane Parallel Homogeneous P94,P95 Pacific Ocean 1994 and 1995 data sets SOCEX Southern Ocean Cloud Experiment SSMI Special Sensor Microwave Imager 595 SOCEX 1995 data set TOA Top of Atmosphere TM Thematic Mapper VAS Vertical Atmospheric Sounder xxi A c k n o w l e d g e m e n t s I am enormously grateful to my supervisor, Phil Austin, for introducing me to the fascinating world of clouds and satellites, and the inner workings of cloud remote sensing. I greatly appreciate Phil's guidance and assistance through this project, especially the countless hours he spend reviewing my manuscript, and the many intriguing questions which would stimulate my research. This thesis, is in one way my attempt to answer some of Phil's questions. I wish to thank my supervisory committee Douw Steyn and William Hsieh for their guidance and advice provided during my years in the graduate programme and a thorough review of the thesis draft. Douw Steyn first welcomed me to the department while I was still working as a particle physicist at the neighbourhood T R I U M F , and it was his enthusiasm about the atmosphere and atmospheric research that led me away from smashing nuclei in accelerators to the U B C Department of Geography to study more \"gentle\" atmospheric processes. This thesis would not exist in its current form without the assistance of many people. Teruyuki Nakajima and Takashi Nakajima of the University of Tokyo kindly provided their cloud optical depth and cloud droplet effective radius retrieval code and assisted with my initial tests of the model. Steven Platnick of N A S A Goddard provided assistance with further tests by checking a sample of my retrievals using his code. Robert Pincus of N A S A Goddard provided A V H R R and N M C data for the F I R E period and answered my numerous email inquiries of various nature, rang-ing from satellite navigation to, C programming to, Lagrangian cloud trajectories. Paul Krummel of CSIRO Victoria provided the S O C E X data, satellite and aircraft. Through discussions with Paul I gained a better understanding of the conditions during the experiment which help my interpretation of the results of the satellite and in situ comparisons in S O C E X . xxn Denis Laplante from the Department of Oceanography supervised the collec-tion of the A V H R R data at the U B C Satellite Lab. He also graciously shared with me his computer expertise and knowledge of satellite data formats and helped on numerous occasions with decoding data in the many \"standard\" formats of H R P T . Thanks also to Jim Mintha and Vincent Kujala for the practically round the clock computer assistance. xxm Chapter 1 Introduction 1.1 Role of boundary layer clouds in Earth's radiation budget Boundary layer clouds, through their wide extent, persistent occurrence and ra-diative properties are of fundamental importance to the global energy budget and Earth's climate. Warren et al. (1988) estimate the global area coverage of low level stratiform clouds at 29% (34% over the Earth's oceans). Low layer clouds affect the climate system through the net exchange of radiant energy at the top of the boundary layer. Over the ocean, clouds that are highly reflective in visible wavelengths (reflectivity ~ 60%) obscure the dark ocean (reflectivity ~ 5%) contributing significantly to the planetary albedo, while the warm temperatures of these clouds continue to cool the atmosphere by emission of outgoing long wave radiation. On annual average, boundary layer stratiform clouds contribute 15 Wm~2 net cooling to the radiation balance at the top of the atmosphere (TOA) (Hartmann et al. (1992)). Arking (1991) distinguishes three classes of processes which determine the radiative effect of clouds on climate: 1) the macrophysical structure of clouds - their horizontal extent, cloud fraction, optical thickness, horizontal and vertical inhomo-geneities; 2) the microphysical structure - the size distribution of cloud droplets 1 within the macroscale volume; 3) the cloud ambient environment - temperature and humidity structure of the atmosphere, atmospheric stability, and distribution of the atmospheric aerosols. These three groups of processes are tightly coupled. For example, changes in the aerosol population may affect cloud liquid water content by suppressing or promoting precipitation, while aqueous phase chemistry within the cloud droplets modifies the aerosol size distribution. The number of possible interactions between the three process categories makes estimating the sensitivity of the planetary radi-ation budget to various changes in cloud properties problematic. There have been a large number of modelling studies identifying potentially important feedbacks between aerosols, cloud microphysics, and cloud reflectivity. I review several of these in section 1.2.2 below. In section 1.2.3 I review satellite and in-situ observations showing the impact of cloud microphysics on reflectivity and cloud spatial structure. These observations show that layer clouds are horizontally inhomogeneous. In section 1.3 I review modelling work that suggests that the inho-mogeneous spatial distribution of cloud water may by itself have an important effect on cloud reflectivity. In section 1.4 I discuss several approaches to quantifying spa-tial inhomogeneity that are applicable to satellite measurements. Finally, in section 1.5 I review work in which two cloud properties, the cloud optical depth and cloud droplet effective radius (defined below in section 1.2.1) are measured using aircraft and satellite radiometers. In this thesis I focus on processes identified by Arking (1991) in points 1) and 2). In particular, I use satellite data to infer and quantify the variability of cloud optical thickness as a measure of cloud macro-structure, and cloud droplet size as a measure of cloud micro-structure. Furthermore, I quantify the relationship between these two parameters, which establishes a new observational link between cloud macro and microphysics. 2 1.2 Cloud optical depth and cloud droplet effective ra-dius 1.2.1 Basic definitions This thesis examines couplings between macro- and microphysical cloud properties and deals extensively with two cloud parameters. These parameters are: cloud vis-ible wavelength optical depth (r), which characterises cloud extinction at visible wavelengths, and cloud droplet effective radius (rejf) which is a cloud microphys-ical parameter and represents a measure of the distribution of cloud droplet sizes. Another cloud parameter frequently appearing in this thesis is cloud liquid water path (Iwp), which measures the water content in column of cloudy air. Below, I define these cloud parameters explicitly: Cloud optical depth Cloud optical depth determines cloud reflectivity at visible wavelengths and is defined as where the integration is over the cloud droplet radius r and height within the cloud z. Az is the cloud geometrical thickness, n(r) is the cloud droplet size distribution, and Qext(2-Kr\/\\) is the efficiency factor for extinction for a water droplet of radius r at a wavelength A. For large size parameters (2-irr\/X), Qext asymptotically approaches 2 (Stephens (1978)). For cloud droplets r ~ 10\/um, and for visible radiation A \u00ab 0.7^\/m, thus the size parameter S> 1 and the approximation Qext=2 is justified. (1.1) 3 Cloud droplet efFective radius The cloud droplet effective radius is the mean radius of the cloud droplet size distribution n(r) [cm-3pm~l] weighted by the droplet surface area \"eff JQ\u00b0\u00b0 n(r)r3dr JQ\u00b0\u00b0 n(r)r2dr (1.2) Cloud liquid water path The cloud liquid water path is defined as rAz Iwp = \/ lwc(z)dz (1.3) Jo where Iwc = J0\u00b0\u00b0 ^irpn(r)r3dr is cloud liquid water content , z is height within the cloud layer, and Az is cloud geometrical thickness, p is the density of water. From (1.1), (1.2) and (1.3) one obtains fAz 3 Iwc T = \/ dz. 1.4 Jo 2p reff I will also define cloud droplet number concentration N = J 0\u00b0\u00b0 n(r)dr where r is the radius of cloud droplet and n(r) the size distribution of cloud droplets. The parameters r and rejj control the cloud reflectivity on spatial scales greater than 200-500 m, the photon mean free path (Cahalan and Joseph (1989)). They are also directly related to the physical properties of the cloud, the cloud liquid water path defined above in (1.3) and the drop size distribution n(r). 1.2.2 Review of sensitivity studies The interaction between cloud microphysics and atmospheric aerosol has a poten-tially significant impact on cloud radiative properties and may considerably affect 4 the earth's radiation budget and hydrological cycle. Twomey (1977) and Twomey et al. (1984) modelled the effect of an increased concentration of C C N on cloud optical thickness and found that assuming an increase in the total aerosol concen-tration increases concentration of cloud condensation nuclei (CCN) which in turn activate more droplets at cloud base. The resulting reduction in cloud droplet size could potentially increase cloud reflectivity and thus enhance the planetary albedo. Charlson et al. (1987) estimated that a doubling of C C N concentration could offset the warming induced by doubling of the atmospheric CO2 concentration. This in-crease in T O A planetary albedo due to the increase of cloud reflectivity in response to increased aerosol concentration at fixed Iwp is termed \"the indirect effect\" to distinguish it from the \"direct effect\" of aerosol scattering of solar radiation in a cloud free environment. A modelling study of Jones et al. (1994) concluded that modification of cloud droplet number distribution and concentration by anthropogenic aerosol results in an indirect aerosol radiative forcing (i.e. top of the atmosphere flux difference) whose global annual average of -1.3 Wm~2 exceeds the direct cooling effect of aerosol scattering of solar radiation (-0.3 to -0.9 Wm~2). In related studies, Global Circulation Model (GCM.) simulations indicate large sensitivity of climate to changes in cloud cover and cloud microphysical pa-rameters such as cloud optical depth, cloud droplet size and cloud liquid water path. Several authors have shown that even small perturbations to low cloud fraction or cloud droplet effective radius in cloud schemes within G C M s can greatly affect the simulated impact of greenhouse gases on climate. For example, a 15% reduction in global cloud droplet size at constant Iwp offsets the predicted CO2 doubling induced climate warming of 4 W m~2. The same effect is achieved by increasing the cloud fraction by 20% while keeping droplet size and Iwp constant (Slingo (1990)) . Not only the droplet size but also the cloud liquid water path may be indi-rectly affected by aerosols. Albrecht (1989) proposed that a reduction of the mean 5 cloud droplet size in response to increased aerosol concentration can act to inhibit precipitation and contribute to a prolonged cloud lifetime and increased cloud frac-tion. Baker (1993) suggested that variation in cloud droplet number concentration may play a role in determining cloud lifetimes, cloud cover and large scale cloud albedo, and precipitation rates. She postulated the existence of two steady states in the well-mixed marine cloud-topped boundary layer which represent precipitating and non-precipitating clouds. These steady states are determined by distinctly dif-ferent C C N (or cloud droplet number concentration ) concentrations. The low cloud droplet number concentration N state corresponds to approximately 10 c m - 3 , the high N state corresponds to ~ 1000 c m - 3 . The cloud droplet number concentration is regulated by production of droplets and removal due to droplet growth and pre-cipitation. N also plays a role in controlling the heating profile the cloudy boundary layer. Net positive heating occurs in thin clouds with low cloud droplet number con-centration which makes these clouds particularly prone to break-up (Baker (1993)). The modelling study of Ackerman et al. (1993) showed that if the C C N concentration in the cloud-topped boundary layer is depleted to very low values (about 10 c m - 3 ) , (due for example to droplet growth by collisions), the cloud layer can become so optically thin that cloud top radiative cooling will become too weak to drive the vertical mixing in the boundary layer and the layer collapses to a shallower one. Through this mechanism the marine layer clouds can limit their own lifetimes. In another modelling study Pincus and Baker (1994) investigated changes in cloud thickness and cloud albedo resulting from changes in precipitation rates and solar absorption due to varying cloud droplet number concentration. They found that the sensitivity of cloud albedo to cloud droplet number concentration is in-creased by a factor ranging from 1.5 to 2 if cloud thickness is allowed to vary in response to varying cloud droplet number concentration and argued that the rela-tionship between cloud thickness and cloud droplet number concentration needs to be accounted for in predictions of global albedo by climate models. Their model 6 results showed that, in the case of thin clouds with low cloud droplet number con-centration, injection of aerosol into the boundary layer increases cloud liquid water and cloud albedo by suppressing precipitation. In thick clouds with high albedos adding new aerosols changes droplet radius but has only a small effect on albedo. These results are in qualitative agreement with the observation of ship tracks (Radke et al. (1989), Coakley et al. (1987)) discussed below in section 1.2.3. Boers and Mitchell (1994) proposed a feedback mechanism which modified the enhancement of cloud top albedo expected from an increase in the C C N concen-tration (Twomey (1977)). The feedback was based on the effect that the change in droplet size and concentration has on the absorption of solar radiation within the cloud, which in turn modifies the mixing process between cloud and the overlaying dry air, and hence affects the microphysics of cloud droplets. In the limit of thin clouds (r < 10) this feedback mechanism partly offsets the reflectance increase due to the increase of droplet concentration. In optically thick clouds the reflectance is further enhanced due the feedback. Feingold et al. (1997) recognised that the onset of rain drop growth through collisions transforms the unimodal droplet size spectrum created in the process of condensation into a bimodal spectrum with a secondary mode corresponding to drizzle-size droplets. This transition from uni- to bimodal spectrum reduces cloud optical depth through 1) decrease of cloud droplet number concentration, and 2) increase of mean droplet size, even if cloud liquid water path is held constant. Using a simple box model of stochastic collection (Tzivion et al. (1987)) and an eddy resolving model (Stevens et al. (1996)) Feingold et al. (1997) found that the relation between cloud droplet number concentration and cloud optical depth differs between uni- and bi-modal clouds. As a result the susceptibility of cloud albedo to changes in cloud droplet number concentration (i.e. change in cloud albedo in response to changes in N, at constant cloud liquid water content , S = dA\/dN ) can be up to 2.5 times larger in clouds with active collection than in the uni-modal 7 (non precipitating) clouds. The uncertainties surrounding the aerosol, cloud droplet and cloud reflec-tivity interactions have stimulated much research in this field in recent years. The challenge is twofold: 1) to better understand cloud microphysical and radiative properties and their interactions, and 2) to develop more accurate cloud parametri-sations for use in G C M s that capture the important interactions between cloud microphysical and macrophysical processes. 1.2.3 Observations Modelling studies postulate numerous feedbacks between cloud properties and cli-mate. Observations which could confirm the modelling results are fewer and often contradictory. Martin et al. (1994) observed an increase in cloud droplet number concentra-tion with increasing aerosol concentration in warm stratocumulus clouds confirming the predictions of Twomey (1977). Kim and Cess (1993) reported satellite based observations of increased albedos in low level marine clouds near coastal boundaries where aerosol concentrations were large. The ship track phenomena, where cloud reflectivity is increased in trails of ship effluents which act as C C N and lead to an increase in cloud droplet number concentration, provides observational evidence of the modification of cloud prop-erties by atmospheric aerosol (Radke et al. (1989), Coakley et al. (1987), Platnick et al. (1997)). Suggestions that precipitation may be an important modulator of cloud thick-ness and cloud albedo are particularly difficult to verify observationally. Austin et al. (1995) observed complex spatial variations in precipitation rates and optical depth of marine boundary layer clouds. The observed clouds exhibited localised regions of intense precipitation large enough to cause cloud water depletion. These precip-itation events did not seem to have an impact on hourly satellite retrieved cloud 8 fraction or cloud optical depth in the region, where both remained high throughout the day. Additional satellite observations of Pincus et al. (1997) also do not indicate that precipitation greatly affects cloud fraction or cloud optical depth variability. Pincus et al. (1997) used cloud optical depth retrievals from geostationary satellites to study the Lagrangian evolution of cloud fraction and cloud optical depth in North Pacific clouds in response to varying environmental parameters (sea surface temper-ature, lower troposphere stratification and others). They found that in the principal component decomposition the environmental components explain only a fraction of the variability in cloud fraction and cloud optical depth. The cloud optical depth at sunrise turned out to be the best predictor of the diurnal cycle of cloud fraction and cloud optical depth (a threshold value seems to exists for the sunrise cloud op-tical depth which determines whether or not the cloud breaks-up in the afternoon). Pincus et al. (1997) conjectured that cloud droplet number concentration can be an-other major factor in determining cloud properties, as large concentrations increase cloud optical depth even in the absence of changes in the cloud liquid water, but had no observations of either cloud droplet number concentration or cloud droplet effective radius to examine this question in detail. Drizzle depletion of cloud water was also observed in situ by Boers et al. (1996), who reported aircraft observations of marine clouds over the Southern Ocean where horizontal variability in cloud liquid water content appeared to be driven primarily by variations in cloud droplet number concentration. Their calculations showed that on days with intense drizzle, cloud optical depth was possibly reduced (at the same Iwp) by as much as 50% due to a shift to larger values of cloud droplet effective radius associated with precipitation. In another study Boers et al. (1997) sampled a layer of stratocumulus clouds embedded with linear convective elements. They found a factor of two difference in cloud droplet number concentration between the convective line and the surrounding 9 stratocumulus deck. This contrast between convective and quiescent regions was also apparent in cloud liquid water content. In other in situ studies, aircraft observations (Paluch and Lenschow (1991), Austin et al. (1995)) revealed large variability in cloud base height in marine stra-tocumulus and relatively less variable cloud top height in precipitating layer clouds. Collectively, all of these postulated feedbacks and interactions raise questions about the links between cloud fraction, liquid water path, cloud droplet effective radius , cloud optical depth and cloud reflectivity. 1.2.4 Summary Both observational evidence and modelling results point to potentially complex in-teractions between cloud processes on scales ranging from microscale (size of cloud droplets [fim]) to macroscale (geometrical cloud thickness ~ 100 m, cloud horizontal extent ~ 100 km). These interactions can affect cloud spatial structure and cloud radiative properties, and thus can help to determine the role of clouds in global energy budget and their effect on Earth's climate. We need to understand the couplings between cloud fraction, liquid water path, cloud droplet effective radius, cloud optical depth and cloud reflectivity, and the spatial distribution of these cloud parameters to resolve the uncertainties sur-rounding postulated cloud-climate feedbacks, and improve cloud parametrisations in global circulation models. As I discuss next in section 1.3, the inhomogeneous distribution of liquid water even in fully cloudy layers has a particularly important impact on cloud reflectivity, and is the current focus of significant observational and theoretical work. 10 1.3 Implications of cloud inhomogeneity for radiative transfer calculations on G C M grid size scale Satellite and aircraft observations provide evidence of structural complexity and spatial inhomogeneity of cloud fields on scales ranging from a few centimetres (Baker (1992)) to 1000 km (Cahalan et al. (1982), Davies (1994) for example). This range of scales encompasses those of the G C M gird cell size (approximately 200 kmx 200 km). This contrasts with the assumptions made in GCMs, which treat clouds as plane parallel and homogeneous (PPH). Numerous studies (Welch and Wielicki (1984), Davis et al. (1990) Barker (1992)) have shown that there is a significant difference between radiative transfer within plane parallel homogeneous clouds and in inhomogeneous clouds with the same mean optical properties. In view of this research the P P H assumption of G C M s is clearly unsatisfactory, even for marine stratocumulus, which of all cloud types best fit the description of plane parallel and homogeneous. For typical marine stratocumulus cloud optical depths (~ 10) the P P H assumption can generate an albedo bias (difference between the observed and model values of albedo) of 10-30% (Barker (1996), Cahalan et al. (1993)) when compared to albedos calculated by models which take into consideration the spatial variability of cloud optical depth . There are several methods for computing radiative transfer in horizontally inhomogeneous clouds (Stephens (1988), Evans (1993) for example) but these are too computationally demanding to be of use in GCMs. The alternative is to develop methods of accounting for the sub-grid scale variability of G C M pixels. Barker (1996) proposed a parametrisation for computing G C M grid averaged solar fluxes for inhomogeneous boundary layer clouds based on the independent pixel approximation (IPA) (Cahalan et al. (1993), Cahalan et al. (1994), Chambers et al. (1997b)) and an assumed distribution of cloud optical depth in a G C M cell. Motivated in part by Barker's work, Abdella and McFarlane (1997) proposed 11 a stochastic cloud scheme for use in G C M based on a statistical approach in the formulation of the sub-grid scale condensation process. In this approach, first used by Sommeria and Deardorff (1977) and Bougeault (1981) in simulations of the trade wind boundary layer, the liquid water content is modelled as a sum of the large scale mean and local fluctuations induced by sub-grid scale condensation. These fluctuations are expressed as a departure of the actual thermodynamic state from the saturation curve of the mean state and are distributed according to a prescribed probability density function. For a given probability distribution function the cloud fraction and the liquid water content are functions of the mean state departure from saturation and the variance of the fluctuations. The cloud fraction and the cloud liquid water path are then used to compute the radiative fluxes in a cloudy boundary layer. Both the methods of Abdella and McFarlane (1997) and Sommeria and Dear-dorff (1977) require knowledge of the distribution of cloud liquid water path, and in future versions of the model, could benefit from the knowledge of correlations be-tween cloud liquid water path, cloud optical depth and cloud droplet effective radius on scales below the G C M cell size. Satellite imagery provided by the Advanced Very High Resolution Radiometer (AVHRR), which has a nadir resolution of 1.1 tax 1.1 km, can provide distributions of cloud optical depth and cloud droplet size within G C M cells for use with these schemes. The measurement and parametrisation of these distributions is the topic of section 4.2 of this thesis. 1.4 Quantifying spatial inhomogeneity in cloud optical depth and cloud droplet effective radius A V H R R imagery provides information about the spatial distribution (spatial struc-ture) of cloud parameters on scales from 1-1000 km , and thus provides ways of quantifying the inhomogeneity of the fields of cloud parameters within a typical 12 G C M grid cell. Given the variety of potentially important links between aerosols, cloud droplet size and cloud liquid water discussed above, new observations of cloud micro and macrophysical characteristics may provide constraints on some of the proposed sensitivities reported in section 1.2. In this thesis I report the distribution of r and reff on the scale of G C M cell. I also use various forms of texture analysis to address the following questions: \u2022 what is the minimum sample size for meaningful spatial averages of cloud parameters? \u2022 what is the scale distribution of variance (the power spectrum) of the T and reff fields? Texture analysis methods can be applied to images of cloud parameters in order to quantify the spatial structure of cloud fields. In a broad sense, texture is defined as a set of statistical measures on the spatial distribution of gray levels in an image. There are a number of statistical approaches to image analysis (see Welch et al. (1988)). In this thesis I consider two of them: 1) spectral analysis and 2) fractal (or multifractal) analysis. Spectral analysis (the decomposition of the total signal variance into con-tributions from various scales) is the standard procedure for characterising spatial correlation in data sets, but this approach has serious limitations for extracting in-formation which would aid cloud modelling. For example, signal intermittency (the occurrence of sudden bursts of intense variability), characteristic of many geophysi-cal signals such as the spatial distribution of cloud reflectivity, cannot be determined by this type of analysis. For processes that are scale invariant (at least over some range of scales), spectral analysis can identify scales of stationary and nonstationary regimes. I will use this approach in this thesis to establish the scale invariant regime in fields of 13 cloud optical depth , cloud droplet effective radius and cloud radiance, and find the minimum averaging scale required to insure statistical stationarity of the inferred statistics. In random field theory stationarity refers to statistical invariance under trans-lation in time and homogeneity to statistical invariance under translation in space. In cloud analysis and modelling literature it is conventional to use the term \"station-ary\" to denote either space or time invariance while \"(in)homogeneity\" is reserved to designate (non-)constant (\"(non-)triviaP) fields. The issue of nonstationarity is particularly important, yet rarely address directly. Most atmospheric fields are nonstationary, at least over some range of scales. When parametrisations or generalisations are based on statistical inference from observations it is important to establish that the observations come from the stationary regime otherwise the results could be artifacts of the details of the aver-aging procedure. Therefore, it is important that distributions of T and reff which I measure in later chapters of this thesis come from stationary regimes. In the case of cloud liquid water fluctuations Marshak et al. (1997) showed that using datasets shorter than 20-40 km to infer statistical properties (means and variances) of clouds yields questionable results since the data are nonstationary on these scales. The concept of a \"scaling fractal\" (Mandelbrot (1977)) has proven useful in application to analysis of the structure of inhomogeneous cloud fields (Davis et al. (1994), Cahalan and Joseph (1989), Cahalan et al. (1993) for example). A scaling fractal is defined as an object or set which is very irregular on all scales, yet at the same time statistically invariant under certain transformations of scale. The fractal dimension determines the scaling properties of the fractal object. The simplest fractals are geometrical objects. The geometrical fractals are self-similar i.e. their magnified subsets look like or are identical to each other and to the whole object (Barnsley et al. (1989)). Non-geometrical fractals objects may scale differently in different coordinates. Such objects are called self-affine. The most 14 general scaling fractals are the multifractals, which exhibit a spectrum of fractal dimensions. Multifractals were introduced by Hentschel and Procaccia (1983) and Parisi and Frisch (1985) and today constitute a general framework for statistical analysis and stochastic modelling of natural phenomena. In the atmospheric sciences multifractals have been used in several areas. The better known applications involve rainfall analysis and modelling (Gupta and Waymire (1990) and Gupta and Waymire (1993)), analysis of the spatial variability of the earth's radiation field (Tessier et al. (1993)), interpretation of satellite im-agery (Gabriel et al. (1988)), and analysis of cloud liquid water fluctuations from aircraft measurements (Marshak et al. (1997)). The energy cascade in fully devel-oped turbulence was successfully described by a multifractal model in Meneveau and Sreenivasan (1987). Fractal properties of cloud fields were explored among others by Cahalan and Joseph (1989), Cahalan et al. (1993). Recent theoretical studies of cloud radiation used fractal (Barker and Davies (1992)) and multifractal (Cahalan et al. (1994)a,b; Marshak et al. (1995a),b) cloud models. Marshak et al. (1994) explored multifractal properties, nonstationarity and intermittency of bounded cas-cade models used in simulation of cloud inhomogeneity. I will employ spectral analysis and multifractal analysis in my investigation of the spatial structure of cloud optical depth and cloud droplet effective radius in chapter 7. 1.5 Previous satellite and aircraft observations of cloud optical depth and cloud droplet effective radius The connection between macro and microphysical cloud properties means that to interpret changes in cloud radiative processes one has to monitor many cloud charac-teristics together. On the global scale satellite imagery provides one means by which such a task might be accomplished. Early satellite observations of cloud properties 15 concentrated on determination of cloud amount and cloud top temperature. Rossow (1989) and Rossow and Lacis (1990) extended the satellite measurements of cloud properties to retrievals of cloud optical depth. The inference of cloud optical depth from satellite observation relies on the dependence of cloud visible wavelength re-flectivity on cloud optical depth. Extensive measurements of global cloud amount, cloud top temperature and cloud optical depth are the mandate of the International Satellite Cloud Climatology Project (ISCCP). The ISCCP analysis of satellite ra-diances for the inference of the cloud optical depth assumes' a constant droplet size for all clouds (cloud droplet effective radius of 10 fim, where the effective radius is a parameter which characterises the distribution of cloud droplets sizes). This assumption leads to uncertainties of 15% - 25% in the retrieved optical depth for water clouds (Rossow et al. (1989), Nakajima et al. (1991)). Inclusion of information from the near infrared and infrared spectral bands permits the inference of cloud droplet size from satellite radiance measurements. Arking and Childs (1985), Nakajima and Nakajima (1995), and Platnick and Valero (1995) utilised the visible and near infrared band radiances for simultaneous retrieval of cloud optical depth and cloud droplet effective radius from the A V H R R flown on board of a series of N O A A polar orbiting satellites. Similar techniques were employed in the retrieval of cloud droplet size and cloud optical depth form airborne remote sensing platforms (Nakajima and King (1990), Rawlins and Foot (1990), Nakajima et al. (1991)). Han et al. (1994) used the A V H R R near-infrared channel measurements to produce global retrievals of cloud droplet effective radius for the ISCCP data and revise the ISCCP retrievals of cloud optical depth. The possibility of utilising other wavelengths in cloud droplet size retrievals was explored by Lin and Coakley (1993) who used two A V H R R thermal channels to retrieve the droplet size for semi-transparent clouds. The advantage of this technique is its applicability to nighttime observations, however its accuracy is not as good as that of the methods employing the visible and near infrared channels. 16 To date, the A V H R R and Landsat Thematic Mapper are the only satel-lite instruments whose measurements allow for a simultaneous retrieval of cloud optical depth and cloud droplet effective radius. In field experiments, the First ISCCP Regional Experiment (FIRE) and the Atlantic Stratus Transition Experi-ment (ASTEX) aircraft observations during satellite overpasses were used to validate the A V H R R estimates of cloud optical depth and cloud droplet effective radius by comparison with the in situ aircraft measurements. These studies (Nakajima and Nakajima (1995), F I R E and A S T E X ; Platnick and Valero (1995) A S T E X ) found that the remotely and in situ measured cloud optical depth agreed within 3 for most part in clouds with mean optical depths between 10 and 40, although Nakajima and Nakajima (1995) reported some large excursions of ~ 10. They attribute these large differences to the satellite measurements encompassing a large field of view (~ 1 x 1 km) compared to the in situ measurements. Remote sensing consistently retrieves larger droplet sizes than measured in situ by about 2 \/im for effective radii around 10 fim (Nakajima and Nakajima (1995), Platnick and Valero (1995)). The cause of this overestimation has not been determined. However, very good spatial correla-tions between in situ and remotely measured r and r e \/ \/ were reported in Nakajima and Nakajima (1995). 1.6 Thesis outline In this thesis I present satellite derived observations of cloud optical depth (r) and cloud droplet effective radius {reff) and interpret them using a simple cloud model in which the relationship between r and reff is parametrised in terms of cloud droplet number concentration. Furthermore, I quantify the spatial variability of cloud optical depth, cloud droplet effective radius and cloud visible wavelength radiance fields within the formalism of multifractal analysis. The thesis will address the following questions: 17 \u2022 Are the retrievals of cloud optical depth and cloud droplet effective radius valid? (i.e. do they agree with in situ aircraft measurements considered in chapter 5.) \u2022 Can the observed variability in r and reff be stratified in terms of other cloud parameters? (cloud droplet number concentration, cloud liquid water path, geographical location.) To assist in this stratification, and in the in situ validation I will introduce a simple \"reference cloud model\" in chapter 3, and present data sets that conform to and depart from this reference model in chapters 3 and 4. \u2022 Is there a characteristic spatial structure in fields of r and r e yj? Does it depend on factors such as cloud droplet number concentration or cloud thickness? Chap te r 2 contains a description of the cloud optical depth and effective radius retrieval technique developed of Nakajima and Nakajima (1995) which was use in this study. C h a p t e r 3 introduces the \"reference cloud model\" which leads to a power law relationship between r and reff parameterised by cloud droplet number con-centration and gives examples of cloud scenes which are representative of the simple model and examples of scenes which mark a departure form the simple behaviour. In Chap te r 4 the relationship between cloud optical depth and cloud droplet effective radius is explored with statistical methods. Bivariate linear regres-sion with errors in both variables is used to fit a power law to the r and reff data for 325 cloud scenes of approximate areas of 256 km x 256 km. On this spatial scale, I find that the reference cloud model provides an accurate description of r and reff correlations for over 55% of examined cloud scenes. I also show how a measure of cloud droplet number concentration concentration can be derive from satellite retrievals with help of the reference cloud model. Further in Chap te r 4 I present mesoscale frequency distribution of cloud 18 optical depth and cloud droplet effective radius and parametrise these distributions in terms of a gamma function (Barker (1996)). I discuss the relationship between mean mesoscale r and reff. In Chapter 5 I analyse satellite and aircraft observations acquired during the Southern Ocean Cloud Experiment (SOCEX). The aircraft observation provide a validation of the remote sensing retrievals of cloud optical depth and cloud droplet effective radius and confirm the relationship between r and reff observed in the satellite data as consistent with the reference cloud model. Chapter 6 introduces the formalism of multifractal analysis, and the con-cepts of nonstationarity and intermittency in geophysical data following Davis et al. (1994) and Marshak et al. (1997). Chapter 7 presents the results of spectral and multifractal analysis of 34 A V H R R fields of cloud visible wavelength radiance, cloud optical depth and cloud droplet effective radius. The scenes come from four differ-ent data sets: 1) the F I R E 1987 data set, 2) Pacific Ocean 1994 data set, 3) Pacific Ocean 1995 data set, and 4) S O C E X 1995 data set. Also included in Chapter 7 is a practical guide to this spectral and multifractal analysis. Chapter 8 summarise the results, discusses the implication of the thesis findings to remote sensing and the climate modelling and outlines the possibilities for future research. 19 Chapter 2 Retrieval of cloud parameters from A V H R R radiance measurements The retrievals of cloud optical depth and cloud droplet effective radius in this study were obtained with the technique developed by Nakajima and Nakajima (1995). This chapter describes radiative transfer basis and some details of this technique. Section 2.1 presents the principal idea behind the simultaneous retrieval of cloud optical depth and cloud droplet effective radius from satellite radiance mea-surements. Sections 2.2 and 2.3 describe the details of Nakajima and Nakajima (1995) forward model and retrieval algorithm respectively. In section 2.4 I discuss sources of errors and estimates of uncertainty in retrievals of cloud optical depth and cloud droplet effective radius from satellite measurements. 2.1 Basic concept The retrieval technique of Nakajima and Nakajima (1995) relies on the reflectance of solar radiation by cloud droplets. Techniques based on solar reflectance use visible wavelengths which are scattered by water droplets without absorption' and near-infrared wavelengths absorbed by cloud droplets for the simultaneous retrieval of cloud optical depth and cloud droplet effective radius . The visible range cloud reflectance is sensitive primarily to cloud optical depth , while the near-infrared re-20 fleeted radiance depends mostly on cloud droplet effective radius . This wavelength dependent sensitivity is due to the different absorption of visible and infrared wave-length by water droplets. Cloud droplets absorb strongly in infrared. As a result infrared radiation which emerges back from the cloud and reaches the satellite sensor consist mostly of photons which underwent just a few scattering events near the cloud top. These photons having scattered only a few times carry the 'memory' of the size of the droplet they interacted with. The visible wavelength radiation propagates through a cloud practically without absorption. It can penetrate deep into the cloud before it is reflected back to space in a sequence of scattering events. In the process of multiple scattering the photons lose the information about the size of cloud droplets ,but being able to sample deep into the cloud they gather information of the column extinction (cloud optical depth ). The A V H R R makes measurements in 5 channels. Table 2.1 list the spectral band width of the A V H R R channels for the instrument on board on the NOAA9 satellite (NOAA Polar Orbiter Data User's Guide http:\/\/www2.ncdc.noaa.gov\/POD \/ podug\/index.htm). channel band width [\/im] 1 0.58-0.68 2 0.725-1.10 3 3.55-3.93 4 10.3-11.3 5 11.5-12.5 Table 2.1: Wavelength bands of the 5 channels of the A V H R R on board of N O A A 9. 21 For my application the visible and near-infrared wavelength ranges corre-spond to A V H R R channel 1 (visible) and 3 (near-infrared) respectively. At a given sun-satellite geometry, a layer cloud of cloud optical depth r and cloud droplet effective radius reff reflects a specific amount of solar radiation in A V H R R channel 1 and 3. The sun-satellite geometry can be expressed in terms of solar zenith angle 6>o, the satellite zenith angle 9 and the relative sun-satellite azimuth cj> which are all shown in Figure 2.1. Figure 2.2 defines the satellite viewing angle 9S. Sun Satellite Figure 2.1: Angles defining the sun-satellite geometry: OQ sun zenith angle, <\/>o sun azimuth angle, 9 satellite zenith angle,
\u2014 4>Q \u2014 (\/>'. 22 Satellite Zenith Figure 2.2: Angles defining the satellite geometry: 0 satellite zenith angle, 9S satellite viewing angle, RE radius of the Earth, h height of the satellite above the ground. Figure 2.3 shows a plot of modelled solar radiance reflected by cloud in AVHRR channel 3 against the radiance reflected in AVHRR channel 1 computed by the code of Nakajima and Nakajima (1995) (see section 2.2) for a particular sun-satellite geometry (expressed in terms of the solar zenith angle (#o=60\u00b0), satellite zenith angle (#=40\u00b0) and the relative sun-satellite azimuth angle (0=50\u00b0). The radiances were computed for a range of values of cloud optical depth (r) arid cloud droplet effective radius (reff). Solid lines are lines of constant cloud optical depth and dash lines represent constant cloud droplet effective radius . 23 0.7 0 50 100 150 200 250 Channel 1 radiance [W m~2 sr\"1 nm\"1] Figure 2.3: Model cloud reflected solar radiance in A V H R R channel 1 and 3 as a function of cloud optical depth and cloud droplet effective radius calculated with the code of Nakajima and Nakajima (1995) . Solid and dashed lines represent radiances at constant r and constant reff respectively. Sun-satellite geometry is fixed at # 0 = 6 0 \u00b0 , 0 = 4 0 \u00b0 and 0=50\u00b0 . Figures similar to figure 2.3 can be computed for all interesting sun-satellite geometries. Figure 2.3 illustrates the idea behind the simultaneous retrieval of cloud optical depth and cloud droplet effective radius from the visible and near-infrared radiance measurements. If the cloud reflected radiances in channels 1 and 3 are known along with the sun-satellite geometry, one can look-up the figure with the specified geometry, enter the values of channel 1 and 3 radiances and from the r-ref f grid read off the corresponding values of cloud optical depth and cloud droplet effective radius . This approach to the retrieval of r and reff dictates a two step 24 procedure. First, the reflected radiances in channel 1 and 3 have to be simulated for a range of cloud optical depth and cloud droplet effective radii for all sun-satellite geometries in the problem. This step is often referred to as forward modelling. Results of the simulation are stored in tables which are used in the second phase, the actual retrieval, to look-up values of r and ref j corresponding to measure values of channel 1 and channel 3 reflected radiances at given sun-satellite geometry. The process of the retrieval of cloud optical depth and cloud droplet effec-tive radius from satellite measured radiances is complicated by the presence of a radiatively active atmosphere and the Earth's surface. As a result, the radiance measured by satellite is a combination of the radiance reflected from clouds (which are a function of r and J\"e\/\/)> radiances contributed by the surface and atmosphere below and above the cloud, and by the cloud's own emission in the near-infrared channel. The cloud reflected radiance must be decoupled from the other radiation components before one can make use of the look-up tables to retrieve the values of cloud optical depth and cloud droplet effective radius . A large percentage of the channel 3 radiance is contributed by the thermal emission from the Earth's surface and from the cloud tops (20 - 90% depending on cloud thickness). Nakajima and Nakajima (1995) estimate the thermal emission in A V H R R channel 3 by making use of the measurements in one of the A V H R R thermal channels (channel 4). Effectively, measurements from three A V H R R channels are required to retrieve of cloud optical depth and cloud droplet effective radius with this technique. In the following sections I describe the forward model used in Nakajima and Nakajima (1995) (2.2) and the flow of the second phase of the retrieval (2.3). 2.2 The forward model The forward model is built on a 4 layer plane parallel atmosphere with interfaces at z \u2014 Az , z , and 12 km and the top at 120 km, where z is the top of the cloud layer and Az is the geometrical thickness of the cloud (see table 2.2 page29 for the grid values 25 of z and Az other model parameters). A homogeneous cloud layer is assumed that consists of Mie scattering water droplets whose sizes follow a lognormal distribution i s , N r (Inr \u2014 Inrn)2 , , . n ( r ) = ~7^7reXP[ 2a2 ] ( } where r is the droplet radius, n(r)dr is the number of droplets with radii between [r, r+dr] per unit volume, TV is the total number of droplets per unit volume, r# is the mode radius which is related to the effective radius by reff \u2014 r o e 3 5 < j 2 , and a is the log standard deviation of the droplet size distribution. For the marine stratocumulus the model assumes a=0.35. The effect of assuming vertical homogeneity in the cloud droplet distribution is discussed in section 2.4. The underlying surface is assumed to be a Lambertian reflector. After Naka-jima and Nakajima (1995) I assume a ground albedo Ag of 0.05 for the ocean surface. The L O W T R A N - 7 midlatitude summer atmosphere (MLS) (Kneizys et al. (1988)) is assumed in our version of the model as representative of the climatic regions considered in this study. The profiles of atmospheric gases which include 7 princi-pal gases and 21 trace gases come from L O W T R A N - 7 atmospheric absorption and transmission package (Kneizys et al. (1988)). The radiative transfer theory for plane parallel layers with an underlying Lambertian surface leads to the following equations for the the satellite received radiance in the visible (AVHRR channel 1) and near-infrared (AVHRR channel 3) spectral range 1. visible wavelengths L0bs(T,reff;iJ,,no,(f)) = L{T,reff,fj,,fj.o,(p) Mo-Fb + ^ ^ \/ ; M ) 1 _ r a ( T | ; e \/ \/ ) A B t ( r > r e \/ \/ ; W , ) 7T (2.2) 26 2. near infrared wavelengths L0bs(T,reff,fj,,iJ,o,(f)) = L(r,r e\/\/;\/\/,\/x0 )<\/ )) +t(Tu,\/j,)[l -t{Tc,reff,iJ.) -r(Tc,reff;n)]B(Tc) (2.3) where L(T,reff] [i, noi4>) IS the cloud reflected radiance, L0bs(T, reff\\ \/j,, noi4>) IS the satellite received radiance, \\i and \/io are cosines of the satellite zenith angle 8, and the solar zenith angle 9Q respectively, Fo is extraterrestrial solar flux, and Ag is the ground albedo, r, r c , and r u are the total optical depth of the atmosphere, cloud optical depth and optical depth of the atmosphere above the cloud. B(T) is the Planck function, Tg is the ground temperature and Tc is the cloud top tempera-ture. The variables t, r and rs are respectively the transmissivity, plane albedo and spherical albedo of the cloud layer defined as where T ( r , r e y j ; \/\/,\/\/o>0) a n d R(r,reff, fi', fi,(f>) are bidirectional transmission and reflection functions (2.4) (2.5) and T{T,reff-n,fi0,(f) = L(T,reff; -fj,,fj.0,4>)\/F0 (2.7) 27 R{T,reff,fx',\/!,(\/)) = L{T,reff,n',ti, 0, H 6 R (6.1) Scaling in this distribution sense is known as strict \u2014 sense scaling as opposed to wide \u2014 sense scaling which refers to scaling in the covariance function (moments of the distribution). The process in (6.1) represents simple scaling (monoscaling) where variability of the process is scale independent. It follows from (6.1) that the moments E((pq) of the process, where q is the order of the moment, if they exist, satisfy E[ = G{r) (6.3) where < \u2022 > denotes ensemble average, is independent of the position x depending only on the distance r between two points. Equation (6.3) is the so-called broad sense definition of stationarity based on only two-point statistics. The narrow sense definition uses all n-point statistics. 115 Stochasic continuity relates to correlations between neighbouring points. If for a small separation r, (p(x + r) and tp(x) are highly correlated then the increment \\ip(x + r) \u2014 ip(x)\\ is usually small and the data is stochastically continuous. If this increment is large then the data is discontinuous. In short, stochastic continuity requires that (Papoulis (1965)) < [ip(x + r) - ip(x)]2 > ->\u2022 0 as \\r\\ -> 0 (6.4) For stationary processes < [ = 2[G(0) - G(r)] (6.5) thus ip(x) is stochastically continuous if G(r) is continuous at r = 0. Processes which are not continuous are intermittent (or singular). The following three processes are often presented to illustrate the concepts of (non)stationarity, (dis)continuity and scaling (scale invariance) and nonscaling (Davis et al. (1996)). 1. White noise. White noise is a sequence of independent random numbers. Its autocorrelation function is G{r) oc 5{r) (6.6) where 6(r) is the Dirac 6. It can be seen from (6.6) that white noise is sta-tionary (satisfies condition of (6.3)) but discontinues process (does not satisfy condition of (6.4)). 2. Brownian motion. Brownian motion is the integral of white noise. The one-point variance of Brownian motion depends on position (is proportional to \\x\\) and the two-point autocorrelation function depends on both x and r and has the form: 116 G(x,r) oc \\x\\ + \\x + r\\ \u2014 \\r\\. (6.7) It follows from (6.7) that Brownian motion is a nonstationary process. At the same time it is a stochastically continuous process since < [(p(x+r)\u2014 < x r. 3. Ornstein-Uhlenbeck processes. Contrary to white noise and Brownian motion Ornstein-Uhlenbeck process has a nonscaling autocorrelation function G(r) oc exp(-^) (6.8) where R is the integral scale of the process. Although not scale invariant, Ornstein-Uhlenbeck processes are both stochastically stationary and continu-ous (< [ cx [1 \u2014 exp(-^j-)] which goes to zero with |r| \u2014> 0). 6.2.3 Scale invariance and stationarity in Fourier space Let (p(k),\u2014oo < k = 1\/r < oo be the Fourier transform of the stochastic process ip(x) introduced in section 6.2.1. The energy spectrum E(k) (wavenumber spectrum, power spectrum, variance spectrum and power spectral density are equivalent terms) of the process (p(x) is defined as E(k) = j < \\y{k)\\2+ \\(p{-k)\\2 >, A;>0 (6.9) Li where < \u2022 > denotes ensemble average (average over all possible realizations of ip{x)). The Wiener-Khinchine theorem (Monin and Yaglom (1965)) guarantees that the autocorrelation function of stationary process G(r) (defined in (6.3)) and the power spectrum E(k) (defined in (6.9)) are Fourier transforms of each other. The 117 autocorrelation function of a stationary, scale invariant process is given by a power law G(r) oc | r | - \" (6.10) where the exponent p has to be positive since the autocorrelation is expected to de-crease as \\r\\ increases. Notice, that the singularity at r = 0 implies that stationary, scale invariant processes are necessarily stochastically discontinuous. The Fourier transform of (6.10) leads to a power spectrum E(k) in form of E(k)ock~f) (6.11) where O<0=l-p 0. Equation (6.13) defines a process whose increments are scale invariant and stationary. Notice that obtaining increments of a process in physical space (taking a gradient or nearest neighbour differences) is equivalent to changing the slope of the power spectrum by 2 in the Fourier space (f3 \u2014> (3 \u2014 2). Thus, the spectral criterion for a nonstationary process with stationary increments 119 is 1 < 0 < 3 . Conversely, one can obtain a nonstationary process with stationary increments by an integration of a regular stationary process (0 < 1). Many geophysical fields are, over some range of scales, nonstationary with stationary increments. For example, the turbulent velocity field in the inertial sub-range is nonstationary (0 \u00ab 5\/3) but the squared velocity gradients which represent energy dissipation filed are stationary although highly intermittent. In the next section I discuss observations which indicate that the spectral exponents for one dimensional transects of cloud liquid water content (live) fluctuation generally fall within the range 0 = 1.3 - 2.0. From the standpoint of the stationarity analysis Iwc fluctuations constitute moderately nonstationary process with stationary incre-ments. This moderate nonstationarity in the Iwc field can be a disadvantage when gradient fields are to be considered. Processes with 0 < 2 yield irregular gradient fields with 0 < 0. The cloud optical depth and cloud radiance fields considered in this study corresponds to vertically integrated Iwc. In a sense, r is a smoothed version of Iwc. With spectral exponents between 2 and 3 (see Barker and Davies (1992), section 7 of this thesis) transects of cloud optical depth (cloud radiance) should be better suited for the analysis involving gradient fields than transects of Iwc. I should mention that there are ways of circumventing the difficulties of dealing with 0 < 0 gradient fields (Schmitt et al. (1992)) by carrying out a fractional differentiation. This, of course, complicates the analysis. 6.3 Previous work: Fourier analysis approach In the traditional approach one investigates the scaling behaviour of cloud fields by means of Fourier analysis with an objective to find the scaling range, the power law exponent (0 in (6.11)) and possible scale breaks. Scale invariance has been observed in the A V H R R fields of cloud radiance 120 visible and thermal channels by Barker and Davies (1992). They found two scaling regions in the AVHRR images of shallow cumuliform clouds off the east cast of North America. For the smallest scales (less than about 5 km) the wave number spectra of visible and infra-red radiances followed the -3 power law. At larger scales the spectral slope varied between scenes but was in the range of-1 to -5\/3. This change in scaling (spectral slope) was similar to that observed by Cahalan and Snider (1989) in Landsat Thematic Mapper images of California stratocumulus although the scale break (-3 to -5\/3) occurred at about 200 m in this case. Whereas Cahalan and Snider (1989) related the scale break at 200 m to the scale of cloud geometrical thickness, Barker and Davies (1992) relate the scale break at 5 km to the size of typical cloud cells in their images. The 200 m scale break observed by Cahalan and Snider (1989) in Landsat radiance fields was not confirmed by in situ aircraft observations of cloud Iwc (Davis et al. (1996)). Recently, Davis et al. (1997a) have shown, using fractal cloud models and Monte Carlo radiative transfer simulations, that the 200 m scale break in Landsat cloud scenes is caused by radiative smoothing due to horizontal photon transport. The scale break which occurs at 5-20 kilometres is however seen in aircraft Iwc measurements and (Davis et al. (1996), Davis et al. (1997a) and marks the integral scale for cloud variability. Barker and Davies (1992) applied Monte Carlo techniques to photon trans-port to show the similarities between the spectral slopes of cloud vertically integrated optical depth and the corresponding reflected and emitted radiation fields. They found that a the power spectrum slope of -3 for the cloud radiance field also implies a -3 slope for cloud optical depth or the vertically integrated cloud liquid water content. This, in turn, implies that transects of Iwc should have spectral slope close to -2 (under the assumption of isotropic variability of Iwc). Many observa-tions indeed find the nearly -2 (-1.3 to -2) scaling for Iwc transects in cumulus and stratocumulus-like clouds (see King et al. (1981), Marshak et al. (1997)). Serio and Tramutoli (1995) used infrared AVHRR imagery and combined 121 spectral and variogram analysis to study the scaling laws in a cloud system gener-ated by strong baroclinic instability. They found two distinct scaling regions, one extending from 1 km to 15 km with the power slope close to -2, the other stretching from 20 km to 100 km with the slope close to -1.33. Davis et al. (1996) investigated spatial fluctuations of cloud liquid water content in marine stratocumulus measured by the Gerber probe (Gerber (1991)) on board of the C-131A aircraft during the Atlantic Stratocumulus Transition Ex-periment (ASTEX) and the King (King et al. (1981)) probe on the N C A R Electra aircraft in marine stratocumulus over the Pacific Ocean off the coast of California during FIRE. They found scaling regimes of 60 m-60 km in the A S T E X data and 20 m-20 km in FIRE, and scaling exponents between -1.08 and -1.68. Lovejoy et al. (1993) analysed 15 A V H R R images of cloud fields (512 km x 512 km) over the Atlantic Ocean and reported scaling in all five A V H R R channels with no breaks over the entire available range (2 to 512 km) with spectral exponents -1.67 (channels 1 and 2), -1.49 (channel 3) and -1.91 and -1.85 for channels 4 and 5 respectively. They supplemented the ~ 1.1 km resolution A V H R R observations with Landsat MSS data obtained with the resolution of 160 m found that scaling continued through to scales P S 300 m. The above studies indicate that there is no general consensus on the extent of the scaling regime in cloud fields. In particular, the occurrence of scale break between 5 to 60 km is disputed. Scaling regimes can be determined for the A V H R R imagery as it is presented in this thesis. Moreover, the range of scales resolved by the A V H R R captures the interesting scale of the transition (scale break) between nonstationary scaling region and the stationary. In chapter 7 I present results of my analysis of 34 satellite scenes where I observed transitions between stationary and nonstationary regimes occurring at distances between 3 to 20 km. 122 6.4 Multifractal analysis 6.4.1 Motivation The traditional approach to the investigation of the spatial (temporal) variability of geophysical signals employs spectral analysis to determine the scale invariant regimes where the wavenumber spectra follow power laws. Spectral analysis is ap-plicable to both stationary and nonstationary data (Davis et al. (1996), see Flandrin (1989) and Wornell (1990) for the extension of the power spectra formalism to non-stationary processes). In the presence of scaling, the stationary and nonstationary regimes can be empirically detected in the power spectrum of the process. This is of a critical importance since stationarity is a prerequisite to obtaining meaningful spatial statistics. Scale invariance describes the statistical symmetry of the system and is a valuable information for models of the system which should reproduce this symmetry. There are however some unresolved ambiguities which plague conven-tional spectral analysis. Very different stochastic processes can yield very similar wavenumber spectra. For example, Gaussian white noise and randomly positioned Dirac S functions both have flat power spectra S(k) ~ k~@ with 3 = 0. Pure Brow-nian motion and randomly positioned Heaviside functions both have power spectra with 0 = 2. The spectrum of fractional Brownian motion (Mandelbrot (1977)) can be made to coincide with that of the cloud liquid water content (Iwc) fluctuations, although fractional Brownian motion is symmetrical while the fluctuations of Iwc have a negatively skewed probability density function. This ambiguity suggests that, for the accurate modelling of the cloud liquid water distribution, more information is needed than can be obtained through spectral analysis alone. A multifractal analysis finds information about the stochastic process which provides additional constrains on the behaviour of the system. In particular, it quantifies the degrees of stationarity intermittency of the process. 123 Davis et al. (1996a) give two examples of circumstances where two signals with radically different spatial properties but identical power spectra remain unre-solved: 1) in seismic signals the background noise (white) and the interesting events (Dirac ^-functions) both have flat wavenumber spectra; 2) temporal fluctuations of air temperature (Brownian motion-like) and a passage of a front (Heaviside func-tion) both have the same power spectrum (E(k) oc k~2). These intermittent signals do have however unique multifractal statistics. 6.4.2 Concept of multifractals The concept of multifractals was introduced by Parisi and Frisch (1985) to describe the scaling behaviour of the velocity field in fully developed turbulence. Their turbulent velocity example can be translated to an approximately passive scalar such as the cloud reflectivity field R(x), where x denotes horizontal position. Parisi and Frisch (1985) considered scaling oiqth order increments < AR(r)Q >=< (R(x + r) \u2014 R{x))Q > over a distance r (this relation formally defines the qth structure function). They determined that different orders of structure function obey scaling laws (6.2) with different exponents, and related this behaviour to the intermittent nature of R(x) field. The intermittent (singular) character of cloud reflectivity fluctuations is re-lated to cloud internal structure. On scales accessible to satellites, structures (cells) as small as few kilometres in size (AVHRR) or 100s of meters (Landsat) are eas-ily resolved in cloud fields. Aircraft observations reveal intermittent fluctuations of cloud liquid water path down to centimeter scales (Davis et al. (1997b)). These intermittent fields possess singularities defined as points (x) such that lirna;-^ \\R(x) \u2014 R(y)\\\/\\x \u2014 y\\a ^ 0. Here, a > 0 is the order of the singularity. In terms of a satellite image singularity describes a rapid jump in reflectivity between adjacent pixels. If S(a) is a set of points where the field has a singularity of order a then one can find a fractal (Hausdorff) dimension D#(a) of that set (see Appendix 124 F for definitions concerning fractals). Parisi and Frisch (1985) postulated the set Df{(a) can have a nontrivial dependence on a. In the limit of \\x \u2014 y\\ \u2014\u00bb 0 the probability of having a singularity of order a in the field R(x) behaves as \\x \u2014 y \\ E - D H { a ) w h e r e pj i s the Euclidean dimension of space embedding the process R(x) (see Appendix F for more detailed treatment). The qth order structure function can now be written as where dfi(a) is a measure on S(a). It follows from (6.14) that ( q is a function (generally nonlinear) of the order q of structure function and the order a of the singularity. This corresponds to the case of multiscaling in definition (6.2). The numeric technique for computing the dimensions DH{Q) is described in the next section. 6.4.3 Formalism of multifractal analysis The multifractal analysis of nonstationary and intermittent geophysical fields is based on two concepts: structure functions and singular measures ( Davis et al. (1994), Marshak et al. (1997)). In the frame of the multifractal analysis, structure functions quantify and qualify nonstationarity while singular measures quantify and qualify intermittency. Alternatively, in a geometrical sense, structure functions pro-vide a measure of roughness while singular measures quantify sparseness of the field. In this section we introduce the concepts of structure functions and singular mea-sures and their interpretation as quantifiers and qualifiers of nonstationarity and intermittency. Since the process of interest, the cloud optical depth, cloud droplet effective radius or cloud radiance in transects of stratocumulus fields, belong to the class of scale invariant (over a limited spatial scale), nonstationary processes with stationary increments 1 < 0 < 3, they are suited to this type of analysis. (6.14) 125 a. S t ruc ture functions Consider again the random process tp(x) introduced in section 6.2.1. We will re-quire now that the process is scale invariant over some range of scales [77, < R] and nonstationary with stationary increments. Such process matches the description of the cloud liquid water path fluctuation in linear transects through a stratocumulus cloud field. The power spectrum of scale invariant, nonstationary process ip(x) with sta-tionary increments is given by (6.11) with 1 < (3 < 3. The structure function of order q of the process , q>0 (6.15) where < \u2022 > is again the ensemble average and A =< \\A = gq(r) (6.17) Due to the scale invariance of the increments we expect (recall (6.2)) gq(r) =< \\A(p{r)\\q >cxr c (\" ) , q>0 (6.18) in the scaling regime [rj, < R]. In (6.18) proper normalisation requires \u00a3(0) = 0. It can also be shown that the function ((q) is concave (Davis et al. (1994), Parisi and Frisch (1985)) and nondecreasing if the increments Aip(r,x) in (6.15) are bounded 126 (Marshak et al. (1994). For a concave function ((g) one can define a hierarchy (a monotonic function) of the exponents H(q) = C(q) (6.19) which in this case is nonincreasing (Marshak et al. (1994). We have a case of monoscaling if ((g) is linear in g and multiscaling if ((g) is nonlinear (recall the discussion following equation (6.2)). Two special cases of structure functions are usually considered: 1. q =1. The first order structure function is related to the fractal structure of the graph g(ip) of the process (f(x) viewed as a scale invariant geometrical object in the two dimensional Euclidean space (see for example Davis et al. (1994), Mandelbrot (1977) ) . There exist the following relationship between the exponent ((1) and the fractal dimension Dg of graph g(y>) (Holder or roughness dimension) (see Appendix F for definitions of fractals and fractal dimensions) In two dimensional Euclidean space the range of values attainable by Dg ex-tends from 1 (for almost everywhere differentiable functions), to 2 (two di-mensional space filling graphs). 2. q=2. The extension of the Wiener-Khinchine theorem to processes with sta-tionary increments states that for such processes the Fourier duality exists between the second order structure function and the energy spectrum (recall that the duality is between the autocorrelation function and the energy spec-trum for stationary processes). This duality leads to the following relationship between the slope of the power spectrum and ((2) ((1) =H{1) =2-Dg >0 (6.20) 127 0 = ((2) + 1 = 2H{2) + 1 > 1 (6.21) I now discuss how the structure function is related to (non)stationarity. Stationary processes have stationary increments thus (6.21) applies also to stationary processes. From the spectral criterion for stationarity 0 < 1 for a sta-tionary process. The limit 0 \u2014> 1 + implies ((2) \u2014> 0 + in equation (6.21) thus ((g) = 0 (since ((0) =0 and ((g) is nondecreasing, concave function) and H(q) = 0. For stationary processes ((g) = 0 we have a case of trivial scaling (the increments are scale independent). A nontrivial structure function implies a degree of nonstation-arity. (In practice, due to the effects of finite spatial resolution, all measurements and even theoretical models have some small ((g) > 0 (Marshak et al. (1994)). ) It can be seen from the (6.18) that the complete description of nonstationarity of a scale invariant data set is contained in the exponents ((g), or the hierarchy H(q) equivalently. Davis et al. (1994) adopt the H(l) (0 < H(l) < 1) to quantify to first order the degree of nonstationarity of the data. Yet, the entire H(q) function is required to qualify the nonstationarity i.e. determine the type of scaling (mono or multiscaling). If the increments of a process are narrowly (Gaussian-like) distributed then in (6.18) < \\Aip(r)\\q >^< \\A q which immediately implies ((g) = g((l) = qH(l), or equivalently H(q) = constant. Thus processes with narrowly distributed increments (also known as 'short-tailed' processes (Waymire and Gupta (1981)) are monoscaling. The r and reff reported in chapter 4 like typical passive tracer distributions in turbulent flows, have large tails. The extreme located in these tails may produce multiscaling behaviour as we will see below. For a process with stationary increments the autocorrelation function can be computed for the increments of the process (the increments field is stationary so the autocorrelation function is well denned). In can be shown that the correlation coefficient between two successive r increments is 128 r, x + r)Aip(r, x) > = 2^~l - 1 (6.22) A<\/?(r, x)2 Equation (6.22) yields positive correlations for (1 < \u00a3(2) < 2) or equivalently (1\/2 < H(2) < 1) or (2 < (3 < 3) and negative correlations for (0 < ((2) < 1) or (0 < H(2) < 1\/2) or (1 < \/3 < 2). Processes with positively (negatively) correlated increments are sometimes termed as persistent (antipersistent) (Waymire and Gupta (1981)). For pure Brownian motion \u00a3(2) = 1, which means uncorrelated increments (i.e. the right hand side of (6.22) vanishes). For q=l , it follows from (6.18) that as long as C(l) > 0 the process is stochas-tically continuous (compare equation (6.4)). Thus only stationary processes (C(l) = 0) can be stochastically discontinuous. b. Singular measures Given the nonstationary random process cp(x) (0 < x < L) with stationary incre-ments ((3 < 3) which is scale invariant over the range of scales [r], < R] we can derive a scale invariant, nonnegative stationary process by taking an absolute value of small scale (i.e. 77) differences in the field (f(x) Equation (6.23) takes the following form for the nearest neighbour differences of a discrete series of measurements \\Aip(rj,x)\\ = \\ip(x + rj) - (p{x)\\, 0 < x < L - rj. (6.23) \\Aip(l,Xi)\\ = \\ 129 where < \\ Atp(rj, x)\\ > is the mean of the nonnegative gradient field. Inthe discrete measurement representation 1 L - 1 < | A \u00a5 3 ( l , x ) | > = - ^ | A ^ ( l , 2 ; l ) | . (6.26) The e(r), x) defined by (6.25) are called singular measures. Davis et al. (1994) cite other procedures for deriving a stationary nonnega-tive field. These methods include taking fractional derivatives (Schmitt et al. (1992), second derivatives Tessier et al. (1993), squares rather than absolute values (Mene-veau and Sreenivasan (1987)). However, work of Lavallee et al. (1993) indicates the details of the procedure do not influence the final results of the singularity analysis. In our discrete measurement representation I make an implicit assumption that the smallest scale of interest (??) coincides with the resolution of the measure-ment. This is not necessarily true. If the resolution of the instrument is better than r? the transition to the scaling regime will be observed in the power spectrum and in the physical space in structure functions. In this case one should take the rj scale gradients. If the spatial sampling \/ is insufficient to resolve this transition (\/ > rj) one can take the nearest neighbour differences. After the nonnegative measures e(r?, x) of the gradient field have been defined, the next step in the singularity analysis is to determine the spatial behaviour of these measures. The scale dependence of the measures e(rj, x) is explored through their spatial averages over increasing scales r (the spatial degradation or coarse graining of the measures). The spatially degraded version of the measures at scale r is obtained by computing the average measure on the interval [x, x + r] e(r,x) = - e(M')> 0 < x < L - r (6.27) x'\u00a3[x,x+r \u2014 1] The multiscaling properties of the measures are accessed through the moments < e(r, x)q > of the spatially degraded measures with respect to the scale r. Since we 130 are dealing with a stationary field of measures taking spatial averages of e(r, x) is well defined. In the scaling range (if it exists) we can write (recall equation (6.2)) < e(r, x)q >cx r~KiQ\\ q>0, 0 < x < L - r (6.28) From the normalisation requirement on the probability density distribution of measures e(r, x) we immediately have K(0) = 0. It follows from the definition of measures given in (6.25) that < e(r,x) >= 1 thus K(l) = 0. By analogy to (6.18) (K(q) corresponds to -\u00a3()) K(q) has to be a convex function. From the above, it can be inferred that K(q) < 0 for 0 < q < 1 and K(q) > 0 elsewhere. Furthermore K'(l) > 0 where the prime denotes the first derivative. Similarly as for the structure function one can define a hierarchy C(q) of the exponents K(q) C(q) = ^ (6.29) For q \u2014>\u2022 1 equation (6.29) and the l'Hospital's rule yield C(l ) = K'(l) (6.30) This time the hierarchy C(q) is nondecreasing for q > 1. C(q) is related to the nonincreasing hierarchy of 'generalised dimensions' D(q) (Appendix F) D(q) = l-C(q) = l-j^ (6.31) introduced by Grassberger (1983) and Hentschel and Procaccia (1983) in an investi-gation of strange attractors. As before D(q) = constant corresponds to a monoscal-ing measure while D(q) which varies with q corresponds to a multiscaling measure. D(l) is related to the mean of e(r,x) distribution and is known as the information dimension. Events which contribute most to the mean of e(r,x) (singularities) occur on a set with fractal dimension D(l). 131 Intermittency plays the same role for singular measures as nonstationarity does for structure functions. C(l) is again designated as a quantifier of the in-termittency whereas the entire function C(q) is required to qualify it (mono- or multiscaling). In terms of C(l), if C(l) = 0 the data is nonintermittent whereas any C(l) > 0 implies some degree of intermittency.. It is instructive to consider two examples of intermittency: 1. Weakly intermittent data. In case of weakly variable fields < |e(r,x)\\q >\u00ab< \\e{r,x)\\ >q. It follows from (6.28) and (6.29) that in this case K{q) = qK(l) = 0 thus D(q) = 1 and monoscaling prevails in its 'trivial' form. If D(q) < 1 for q > 0 the distribution of measures is singular (skewed). At small scales the most frequent values are small but occasionally spikes with high values occur. 2. Extreme intermittency. A n example of an extreme case of intermittency is a delta function randomly positioned in the interval 0 < x < L. Let xc denote the location of the delta function in the interval 0 < x < L. In limit rj \u2014> 0 + we have e(x) = lim^Q+e^^x) = 5(x \u2014 xc) . The spatial averaging over an interval [x, x + r] yields 1 rx+r \\ if xc 6 \\x, x + rl s(r,x) = - e(x')dx' = I r L J (6.32) r J x I 0 otherwise If xc is uniformly distributed over the interval [0 L] then for the spatially averaged moments one obtains < e(r, x)q > = -jr e(r, x)qdxc = -^\u2014q oc rl~q (6.33) rL 10 implying K(q) = q \u2014 1 hence C(q) = 1 and D(q) = 0 for q > 0 which indicates that all activity in concentrated in a single point. 132 6.5 Previous work: multifractal approach Davis et al. (1994) postulated the use of two multifractal statistics, the singularity measures and gth order structure functions to describe the spatial variability and scale dependence in geophysical fields. These measures seek to quantify the degrees of nonstationarity and intermittency in a geophysical signal by comparing their statistical properties to those of known random processes. Davis et al. (1994) investigated spatial fluctuations of cloud liquid water content in marine stratocumulus measured by the Gerber probe (Gerber (1991)) on board of the C-131A aircraft during the Atlantic Stratocumulus Transition Experi-ment (ASTEX) . Their analysis yield the ensemble average (H(l), C(l)) w (0.29,0.08) in range of 60 m-60 km. Marshak et al. (1997) found 5 nights ensemble average (#(1),C(1)) PS (0.28,0.10) over the scales 20 m-20 km for the L W C fluctuation measured by the King (King et al. (1981)) probe on the N C A R Electra aircraft in marine stratocumulus over the Pacific Ocean off the coast of California during FIRE. Marshak et al. (1997) point to the close proximity of the A S T E X and F I R E data sets in the bifractal plane and interpret this proximity as a consequence of the common nature of the nonlinear physical processes that determine the internal structure of the marine stratocumulus. The authors link the different scaling range of these two data sets (60 m-60 km for A S T E X and 20 m-20 km for FIRE) to the different boundary layer depth (1.5 km for A S T E X and 0.5 km for FIRE). Davis et al. (1996a) reported (iJ(l) ,C(l)) = (0.28,0.09) and scaling range 5 m to 5 km for an ensemble of aircraft measurements of Iwc (Gerber probe) obtained during S O C E X . The location of both data sets in the bifractal plane indicates that the LWC fluctuation in marine stratocumulus are both nonstationary and intermittent and calls for hybrid stochastic models which combine both the nonstationarity and the intermittency to adequately describe the processes governing the L W C fluctuation in clouds and likely other geophysical fields. 133 Marshak et al. (1997) note a large scatter of the individual spatial averages around the ensemble averages which suggests ergodicity violation and argues in favour of non-ergodic models. They stress the need for data to test their findings. So far, the A S T E X , FIRE and S O C E X Iwc flight data, and one F I R E Landsat Thematic Mapper (channel 2) scene [ (#(1), C(l)) = (0.54,0.06) ] (Davis et al. (1996a)) are the only data sets placed in the bifractal plane. Satellite data are much easier accessible and provide much grater spatial and temporal coverage than any aircraft data. They have the disadvantage of limited resolution towards the small scales (1 km for the AVHRR) thus necessarily shorter cascades of the scaling range. In chapter 7 of my dissertation I find the multifractal characteristics of the A V H R R data sets of cloud radiances, cloud optical depth and cloud droplet effective radius and in particular their location in the bifractal plane. I apply the analysis to four sets of satellite data: 1) F87 - Pacific Ocean the F I R E 1987 data set, 2) P94 - Pacific Ocean 1994 data set acquired by U B C satellite Lab, 3) P95 - Pacific Ocean 1995 data set acquired by U B C satellite Lab and, 4) 595 - Southern Ocean Experiment (SOCEX) 1995 data set acquired over the Indian Ocean. 6.6 Summary The goal of multifractal analysis is to find the scaling behaviour ((g) of the structure functions in (6.18) and the scaling behaviour K(q) of the singular measures in (6.28). The analysis establishes the mono- or multifractal character of the scaling . The scaling exponents of the first order moments of the structure functions H(l) and the singular measures C(l) quantify respectively the degree of nonstationarity and intermittency in the data. Marshak et al. (1997) argue that close proximity the mean multifractal pa-rameters of [7J(1),C(1)] for both their data sets (FIRE and A S T E X ) could be a manifestation of a universal character of the processes determining the structure of 134 marine stratocumulus. Arguing against such universal behaviour is the large num-ber of physical processes that lie between the characteristic scaling behaviour of inertial range turbulence and cloud variables such as r and reff. As the review in section 1.2 indicated, the aerosol population in particular may be injecting its own characteristic scale on r and reff which has nothing to do with the variability of passive tracers in turbulent flow. I have assembled a large number (34) of A V H R R images of cloud optical depth, cloud droplet effective radius and visible cloud radiance corresponding to different micro and macrophysical marine stratocumulus regimes. My scenes corre-spond to uni-modal, bi-modal, and optically thick clouds as discussed in chapters 3 and 4. If the universality argument holds all these regimes should yield similar values of [if (1), C(l)]. In the next chapter I show that this is indeed the case. There is no clear difference in neither H(l) nor C(l) parameters for any of these regimes. There is however a considerable scatter between ensemble averages of H(l) and C(l) for the four major data sets I considered. A trivial dividing line runs between fully cloudy and broken cloud fields which have markedly larger values of C(l) and are often multiscaling in ((q). The [H(l), C(l)\\ set constitutes a test for structural compatibility of cloud models with real clouds. When mapped into the [H(l), C(l)] space, realistic cloud models, whether dynamic or stochastic, should lie as close as possible to the point which represents real data. In another application, the [H(l), C(l)] reference frame provides a test for assumptions about subpixel scale homogeneity made in retrievals of geophysical fields from a remote sensing imagery. The remotely sensed fields can be compared with the in situ measurements in the [H(l), C(l)} plane. The agreement between the remote and in situ [H(l), C(l)] points can serve as a validation of the assump-tions of retrieval technique. Examples of the application of multifractal techniques 135 to evaluation of turbulent cascade models, cloud inhomogeneity models, the validity of remote sensing approximations for retrieval of cloud properties can be found in Meneveau and Sreenivasan (1987), Marshak et al. (1995a), Marshak et al. (1995b) among others. From a satellite and image analysis perspective the parameters H(l) and C(l) may find another application as measures of texture, if it can be established that different cloud types (stratocumulus, cumulus, cirrus, etc.) or different sur-faces (clouds, sea surface, land) are characterised by different sets of [H(l), C(l)\\. However, I do not explore this possibility in this thesis. Chapter 7 of this thesis presents the application of the combined spectral and multifractal analysis to A V H R R imagery fields of cloud radiance, cloud optical depth and cloud droplet effective radius . 136 Chapter 7 Spatial structure of stratocumulus clouds II: Results from multifractal analysis This chapter presents the application of the multifractal analysis technique of chap-ter 6 to the satellite fields of cloud visible radiance, cloud optical depth and cloud droplet effective radius obtained from A V H R R measurements. In earlier chapters I have discussed observational and modelling results which pointed to complex in-teractions between cloud processes on scales ranging from the microscale (size of cloud droplets [fj,m]) to the macroscale (geometrical cloud thickness ~ 300 m, cloud horizontal extent ~ 100 km). My observations in chapters 3 and 4 indicate that many cloud fields can be described in terms of the reference cloud model (3.2) as uni-modal or bi-modal where bi-modality involves variability in Nsat on scales com-parable to the size of the scenes (256 km x 256 km). The goal of the analysis in this chapter is to investigate how (or if) these interactions affect cloud spatial structure. In particular, I will compute and compare: 1. spatial and multifractal properties (nonstationarity and intermittency) of the uni-modal, bi-modal, and thick cloud scenes described in chapters 3 and 4; 2. the ensemble averaged nonstationarity and intermittency parameters for the four data sets (-F87, P94, P95 and