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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112, D10204, doi:10.1029/2006JD007371, 2007
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Statistics on the macrophysical properties of trade wind cumuli
over the tropical western Atlantic
Guangyu Zhao1 and Larry Di Girolamo1
Received 5 April 2006; revised 14 December 2006; accepted 17 December 2006; published 19 May 2007.
[1] This study presents a comprehensive statistical overview of the macrophysical
properties of trade wind cumulus clouds over the tropical western Atlantic using 152
scenes taken from the Advanced Spaceborne Thermal Emission and Reflection
Radiometer (ASTER) between September and December 2004. The size distribution,
shapes, and spatial distribution of cumulus clouds were examined with ASTER nearinfrared data at 15 m resolution. The height distribution of these cumulus clouds was
derived from ASTER thermal infrared data at 90 m resolution. The size distribution of
cumuli exhibited a power law form and an exponent of 2.19 with a correlation coefficient
of 0.99 using a direct power law fit method. The total cloud fraction of trade wind cumulus
was 0.086, half of which was contributed from clouds smaller than 2 km in equivalent
area diameter. An area-perimeter power law was observed with a dimension of 1.28 and a
correlation coefficient of 0.87. The majority of cloudy pixels had cloud top altitudes
around 1 km and increasing altitude with increasing cloud equivalent area diameter.
Seventy-five percent of clouds have a nearest neighbor within a distance of 10 times their
area-equivalent radius. Our results are compared to other studies of small cumulus taken
over different parts of the world observed using different instruments. The statistics of
cumuli observed in this study are poorly related to synoptic scale meteorological
conditions from reanalysis data.
Citation: Zhao, G., and L. Di Girolamo (2007), Statistics on the macrophysical properties of trade wind cumuli over the tropical
western Atlantic, J. Geophys. Res., 112, D10204, doi:10.1029/2006JD007371.
1. Introduction
[2] Trade wind cumuli’s role in the climate and global
energy cycle has prompted numerous model studies that
attempt to characterize the dynamic and radiative interactions between trade wind cumuli and their environment (see
Zhao and Austin [2005] for a review on modeling studies).
Evaluating these cloud models requires quantitative information of cumuli macrophysical properties such as size
distribution, morphology, cloud top height and spatial
distribution [e.g., Siebesma and Cujipers, 1995; Zhao and
Austin, 2005]. In addition, these macrophysical properties of
cumulus clouds from observations can be used to synthesize
cloud fields as input into three-dimensional (3-D) radiative
transfer models [e.g., Evans and Wiscombe, 2004; Zuidema
et al., 2003]. Although these macrophysical properties can
be measured from ground-based or in situ instruments, only
satellites can provide large enough spatial and temporal
coverage to effectively sample these clouds. However,
satellite studies of trade wind cumuli have been difficult
because the ground instantaneous field of view of typical
meteorological satellite instruments tend to be larger than
1
Department of Atmospheric Sciences, University of Illinois at UrbanaChampaign, Urbana, Illinois, USA.
Copyright 2007 by the American Geophysical Union.
0148-0227/07/2006JD007371$09.00
the typical horizontal extent of individual clouds. A proper
study of these clouds demands high-resolution satellite data.
[3] Past studies of cumulus macrophysical properties
using high-resolution 2-D images taken from aircraft, space
shuttle and satellite instruments are summarized in Table 1.
Results from these studies are compared with results from
our study. Table 1 does not include past studies that only
focus on cumulus spatial distributions, since they are not
extensively compared to our study (see section 8 for further
explanation). The spatial resolution of the data used in the
studies listed in Table 1 ranged from 28.5 to 57 m.
However, the statistics on the macrophysical properties of
cumulus were based on only a handful of scenes. Furthermore, all the scenes analyzed were subjectively selected
subsets (subscenes) of larger scenes and manually cut to
show only the cumuli-dominated area. It is probable that the
statistics derived from limited, subjectively sampled clouds
could be biased. In addition, several macrophysical properties of cumuli (e.g., spatial distribution) depend on scene
size, making statistics derived from scenes of different sizes
difficult to compare. Cumulus properties can also be spatially dependent owing to variation in the metrological
conditions from one region of the world to another. Therefore it may not be appropriate to merge the statistics on
cumuli properties from different regions. To generate robust
statistics of cumuli for a particular region not only requires
high-resolution data but also long-term observations. We
accomplish this in this study using a 15 m resolution data
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1 of 10
2 of 10
TM
ASTER
30
28.5
15
110 ! 145
65 ! 65
60 ! 60
50
c
b
Studies focusing on only cumulus spatial distribution are not included.
Power law defined by equation (1).
Power law defined by equation (7) for area perimeters.
d
NR means not reported.
e
LANDSAT Multispectral Scanner (MSS).
f
LANDSAT Thematic Mapper (TM).
g
Moderate Resolution Imaging Spectrometer (MODIS) Airborne Simulator (MAS).
h
Single-line least squares fit.
i
Double-line least squares fit with a scale break.
j
Direct power law fit.
a
Gotoh and Fujii [1998]
This study
space shuttle
37 ! 37
Sengupta et al. [1990]
MASg
57
28.5
57
Benner and Curry [1998]
57
170 ! 185
170 ! 185
65 ! 65
170 ! 185
Florida Coast
NRd
16 ! 32
United States, tropical western Atlantic,
western Arkansas, Gulf of Mexico
Pacific, South America, Florida coast
Pacific, South America, Florida coast
tropical Atlantic, Gulf of Mexico,
United States, France
tropical western and central Pacific,
Maldives, Somali coast, Coral Sea,
Caribbean Sea
tropical western and central Pacific,
Maldives, Somali coast, Coral Sea,
Caribbean Sea
Japan
tropical western Atlantic
Location
Spatial
resolution, m
Domain,
km2
MMS
TMf
MMS
camera on
aircraft
MMSe
Instrument
Cahalan and Joseph [1989]
Wielicki and Welch [1986]
Plank [1969]
Reference
Data Description
Table 1. Past Studies on Cumulus Macrophysical Properties Using 2-D High-Resolution Imagesa
1
152
5
17
16
19
10
4
12
Number
of Scenes
yes
no
yes
yes
yes
yes
yes
yes
yes
Subscenes?
NR
2.85h
1.88i
2.19j
0.94
1.98
NR
0.89
1.39
NR
NR
l1
NR
2.85
3.18
2.19
2.91
3.06
NR
2.76
2.35
NR
NR
l2
NR
none
0.6
none
0.6
0.9
NR
0.5
1.0
NR
NR
Dc,
km
Size Distributionb
1.364
1.28
1.10
1.23
1.27
1.34
1.20 – 1.27
NR
NR
d1
1.677
1.28
1.34
1.374
1.55
1.47
1.50 – 1.73
NR
NR
d2
Fractal Dimensionc
Statistical Properties
0.7
NR
0.5
0.5
0.5
0.5
0.5
NR
NR
dc,
km
NR
0.086
0.090
0.0925
0.45
0.55
NR
0.15 – 0.19
0.25
Average Cloud
Fraction
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ZHAO AND DI GIROLAMO: MACROPHYSICAL PROPERTIES OF CUMULI
D10204
D10204
ZHAO AND DI GIROLAMO: MACROPHYSICAL PROPERTIES OF CUMULI
D10204
experiment (details of the RICO project can be found at
http://www.joss.ucar.edu/rico/). The time period and location of the RICO experiment was chosen to best represent
the maritime trade wind cumuli of the Western and WestCentral Atlantic. The ASTER data used in this study were
Version 4 Level 1B calibrated radiance. In total, there are
403 scenes from 29 separate days. Any scene visually
identified as contaminated by cirrus, dominated by stratus
clouds, or filled with poor quality data (e.g., striping) were
wholly discarded from our analysis, reducing the number to
152 scenes (60 ! 60 km2 each) from 24 separate days. The
remaining scenes are dominated by cumuliform clouds,
where we simply refer to them as trade wind cumuli. The
number of the scenes used were 77, 30, 23, and 22 for the
months of September, October, November, and December,
respectively. Figure 1 shows a histogram of the cloud
fraction in each scene for the 152 ASTER scenes calculated
from the cloud mask described in section 3.
Figure 1. A histogram of cloud fraction per Advanced
Spaceborne Thermal Emission and Reflection Radiometer
(ASTER) scene for the 152 ASTER scenes examined in this
study.
set from the Advanced Spaceborne Thermal Emission
Reflection Radiometer (ASTER). Using several months of
ASTER data over the tropical southwestern Atlantic Ocean,
we provide a comprehensive examination of the macrophysical properties of trade wind cumuli and derive robust
statistics.
[4] The paper is organized as follows. ASTER data are
described in section 2. Section 3 presents the procedure for
cloud masking and labeling. Sections 4 – 8 cover the statistics on trade wind cumuli properties including cloud size,
cloud fraction, cloud area – perimeter relationship, cloud
height, and spatial distribution. Section 9 summarizes and
discusses our results.
2. ASTER Data
[5] ASTER is onboard the EOS Terra spacecraft, which
crosses the equator around 10:30 local time in a 705 km
Sun-synchronous orbit. Details of the ASTER instrument
and its performance can be found in the work of Abrams
[2000]. In brief, ASTER has two cameras. One camera,
which points at nadir, has three visible and near-infrared
(VIR) spectral bands (0.5 to 1.0 mm) with 15 m spatial
resolution, six shortwave infrared (SWIR) spectral bands
(1.0 to 2.5 mm) with 30 m spatial resolution, and five
thermal infrared (TIR) spectral bands (8 to 12 mm) with
90 m spatial resolution. The other camera, which points
backward in the along-track direction, has only one spectral
band (0.78 to 0.86 mm) with 15 m spatial resolution. VIR
and SWIR data are 8 bits, and TIR data are 12 bits. ASTER
produces about 650 scenes per day, with each scene having
a spatial coverage of 60 ! 60 km2.
[6] Although ASTER data are primarily collected over
land, the instrument was tasked to acquire data over the
tropical western Atlantic Ocean (20!– 12!N latitude, 66! –
55!W longitude) between September and December 2004 to
overlap with the Rain In Cumulus over the Ocean (RICO)
3. Cloud Masking and Labeling
[7] The initial step was to generate cloud masks, which
classify satellite instantaneous fields of view (pixels) as
either clear or cloudy, for each ASTER scene. The quality of
the cloud masks directly impacts the accuracy of the
statistics. ASTER does not have cloud products available
to the public and we had to derive the cloud masks on our
own. Although there are numerous cloud detection algorithms (see, e.g., Goodman and Henderson-Sellers [1988]
for a review), a single threshold approach was appropriate
for this study, given that the variation of clear radiance
within an ASTER scene was small and the radiative and
spatial contrast between bright clouds and the dark ocean
was large. A single threshold was manually selected for
each scene using Channel 3N (0.78 " 0.8 mm) at 15 m
resolution, given that the Sun-view geometry, aerosol concentration, sea surface roughness, etc., varied from one
scene to the next. Channel 3N was chosen because of its
high spatial resolution, low atmospheric scattering and
absorption, and low surface reflectance. A pixel was flagged
cloudy if its digital number, which can be converted to
radiance, was larger than the threshold. Otherwise, it was
flagged clear.
[8] An accepted way to evaluate the performance of a
cloud mask is to visually compare it with its corresponding
radiance image using interactive visualization software
[e.g., Ackerman et al., 1998; Berendes et al., 2004; Zhao
and Di Girolamo, 2004]. Hence we manually tuned the
threshold for each scene until the resulting cloud mask
passed visual inspections (see Wielicki and Welch [1986]
and Wielicki and Parker [1992] for further discussions on
manually setting thresholds). Table 2 lists the thresholds for
each ASTER scene used in this study. Figure 2 shows an
example of an ASTER subscene (20 ! 20 km2) along with
its cloud mask collected on 9 December 2004.
[9] Once pixels were classified as either clear or cloudy,
they were grouped into individual clouds (details of the
procedure can be found in the work of Zhao [2006]). Two
cloudy pixels that share one edge but not one vertex, belong
to the same cloud (in the computer vision literature, this is
called ‘‘4-connected’’ [e.g., Shapiro and Stockman, 2001]).
The total number of the clouds found within the 152 scenes
3 of 10
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ZHAO AND DI GIROLAMO: MACROPHYSICAL PROPERTIES OF CUMULI
Table 2. List of the Thresholds for Each ASTER Scene Used in
This Studya
ASTER File Name
Threshold
AST_L1B_00309152004140557_09282004120748.hdf
AST_L1B_00309152004140606_09282004120827.hdf
AST_L1B_00309152004140614_09282004120842.hdf
AST_L1B_00309152004140623_09282004121146.hdf
AST_L1B_00309152004140632_09282004121306.hdf
AST_L1B_00309152004140641_09282004121401.hdf
AST_L1B_00309182004143752_09302004114635.hdf
AST_L1B_00309202004142242_10022004122438.hdf
AST_L1B_00309202004142250_10022004122511.hdf
AST_L1B_00309202004142259_10022004122534.hdf
AST_L1B_00309202004142308_10022004122556.hdf
AST_L1B_00309202004142317_10022004122651.hdf
AST_L1B_00309202004142326_10022004122657.hdf
AST_L1B_00309202004142335_10022004122708.hdf
AST_L1B_00309202004142344_10022004122804.hdf
AST_L1B_00309202004142352_10022004122811.hdf
AST_L1B_00309202004142401_10022004123250.hdf
AST_L1B_00309202004142410_10022004123314.hdf
AST_L1B_00309202004142419_10022004123320.hdf
AST_L1B_00309202004142428_10022004123335.hdf
AST_L1B_00309202004142437_10022004123403.hdf
AST_L1B_00309202004142445_10022004123539.hdf
AST_L1B_00309202004142454_10022004123446.hdf
AST_L1B_00309202004142530_10022004123547.hdf
AST_L1B_00309222004141139_10042004140939.hdf
AST_L1B_00309222004141148_10042004141131.hdf
AST_L1B_00309222004141157_10042004141240.hdf
AST_L1B_00309222004141223_10042004141453.hdf
AST_L1B_00309222004141232_10042004141522.hdf
AST_L1B_00309222004141241_10042004141702.hdf
AST_L1B_00309222004141250_10042004141627.hdf
AST_L1B_00309222004141259_10042004141726.hdf
AST_L1B_00309222004141307_10042004141803.hdf
AST_L1B_00309222004141316_10042004141831.hdf
AST_L1B_00309232004145335_10052004133522.hdf
AST_L1B_00309232004145343_10052004134107.hdf
AST_L1B_00309232004145352_10052004134335.hdf
AST_L1B_00309232004145401_10052004134142.hdf
AST_L1B_00309232004145410_10052004134258.hdf
AST_L1B_00309232004145419_10052004134236.hdf
AST_L1B_00309232004145428_10052004134307.hdf
AST_L1B_00309252004144147_10072004115751.hdf
AST_L1B_00309252004144156_10072004110938.hdf
AST_L1B_00309252004144205_10072004110939.hdf
AST_L1B_00309252004144214_10072004110851.hdf
AST_L1B_00309252004144223_10072004110918.hdf
AST_L1B_00309252004144232_10072004111002.hdf
AST_L1B_00309252004144241_10072004110905.hdf
AST_L1B_00309252004144249_10072004111119.hdf
AST_L1B_00309252004144307_10072004112530.hdf
AST_L1B_00309252004144316_10072004112532.hdf
AST_L1B_00309272004142858_10092004100929.hdf
AST_L1B_00309272004142925_10092004101248.hdf
AST_L1B_00309272004142934_10092004101419.hdf
AST_L1B_00309272004142943_10092004101336.hdf
AST_L1B_00309272004142951_10092004101419.hdf
AST_L1B_00309272004143000_10092004101439.hdf
AST_L1B_00309272004143009_10092004101522.hdf
AST_L1B_00309272004143018_10092004102011.hdf
AST_L1B_00309272004143027_10092004102215.hdf
AST_L1B_00309272004143045_10092004102206.hdf
AST_L1B_00309272004143053_10092004102222.hdf
AST_L1B_00309272004143102_10092004102157.hdf
AST_L1B_00309272004143111_10092004102329.hdf
AST_L1B_00309292004141645_10102004115140.hdf
AST_L1B_00309292004141653_10102004115323.hdf
AST_L1B_00309292004141720_10102004115646.hdf
AST_L1B_00309292004141729_10102004115737.hdf
AST_L1B_00309292004141738_10102004115852.hdf
AST_L1B_00309292004141747_10102004115940.hdf
AST_L1B_00309292004141831_10102004120158.hdf
AST_L1B_00309292004141840_10102004120235.hdf
27
26
26
27
29
30
41
36
36
37
37
37
39
40
40
41
42
43
48
48
53
56
58
99
23
24
26
26
27
27
27
29
33
33
35
36
38
39
42
42
43
40
39
40
49
43
50
53
54
58
51
20
22
22
22
18
20
22
23
23
28
28
28
28
20
26
18
18
22
22
36
30
D10204
Table 2. (continued)
ASTER File Name
Threshold
AST_L1B_00309292004141848_10102004120332.hdf
AST_L1B_00309292004141857_10102004120753.hdf
AST_L1B_00309292004141906_10102004120803.hdf
AST_L1B_00309292004141915_10102004120836.hdf
AST_L1B_00309292004141924_10102004120858.hdf
AST_L1B_00310012004140550_10122004103315.hdf
AST_L1B_00310012004140559_10122004103332.hdf
AST_L1B_00310012004140617_10122004103331.hdf
AST_L1B_00310022004144755_10132004123153.hdf
AST_L1B_00310022004144804_10132004123305.hdf
AST_L1B_00310022004144812_10132004123325.hdf
AST_L1B_00310022004144821_10132004123520.hdf
AST_L1B_00310082004141223_10222004113452.hdf
AST_L1B_00310082004141232_10222004113357.hdf
AST_L1B_00310182004144742_10312004094830.hdf
AST_L1B_00310182004144751_10312004094733.hdf
AST_L1B_00310182004144800_10312004094915.hdf
AST_L1B_00310182004144809_10312004094918.hdf
AST_L1B_00310182004144818_10312004094845.hdf
AST_L1B_00310182004144928_10312004095131.hdf
AST_L1B_00310202004143612_11012004110326.hdf
AST_L1B_00310202004143629_11012004121958.hdf
AST_L1B_00310202004143647_11012004121704.hdf
AST_L1B_00310242004141235_11052004104453.hdf
AST_L1B_00310242004141244_11052004104422.hdf
AST_L1B_00310242004141253_11052004104549.hdf
AST_L1B_00310252004145412_11062004101058.hdf
AST_L1B_00310252004145430_11062004101307.hdf
AST_L1B_00310272004144335_11072004092756.hdf
AST_L1B_00310292004142953_11092004112753.hdf
AST_L1B_00310292004143002_11092004112826.hdf
AST_L1B_00310292004143020_11092004112845.hdf
AST_L1B_00310292004143029_11092004112856.hdf
AST_L1B_00310292004143038_11092004113032.hdf
AST_L1B_00310292004143055_11092004113244.hdf
AST_L1B_00311092004141221_11212004101131.hdf
AST_L1B_00311092004141229_11212004101606.hdf
AST_L1B_00311092004141238_11212004101723.hdf
AST_L1B_00311092004141247_11212004101749.hdf
AST_L1B_00311142004143057_11252004104426.hdf
AST_L1B_00311142004143106_11252004104422.hdf
AST_L1B_00311212004143456_12152004102110.hdf
AST_L1B_00311212004143504_12152004101456.hdf
AST_L1B_00311212004143513_12152004102210.hdf
AST_L1B_00311212004143531_12152004102328.hdf
AST_L1B_00311212004143540_12152004102521.hdf
AST_L1B_00311232004142249_12032004184139.hdf
AST_L1B_00311232004142258_12032004184228.hdf
AST_L1B_00311232004142307_12032004184113.hdf
AST_L1B_00311282004144111_12112004125113.hdf
AST_L1B_00311282004144120_12112004125143.hdf
AST_L1B_00311282004144147_12112004125400.hdf
AST_L1B_00311282004144155_12112004125411.hdf
AST_L1B_00311282004144204_12112004125617.hdf
AST_L1B_00311282004144257_12112004130507.hdf
AST_L1B_00311282004144306_12112004130410.hdf
AST_L1B_00311282004144315_12112004130546.hdf
AST_L1B_00311282004144333_12112004130849.hdf
AST_L1B_00312022004141642_12152004110719.hdf
AST_L1B_00312022004141651_12152004110804.hdf
AST_L1B_00312022004141700_12152004110757.hdf
AST_L1B_00312022004141709_12152004111047.hdf
AST_L1B_00312022004141718_12152004110839.hdf
AST_L1B_00312092004142226_12242004184845.hdf
AST_L1B_00312092004142235_12242004184743.hdf
AST_L1B_00312092004142244_12242004184730.hdf
AST_L1B_00312092004142253_12242004184817.hdf
AST_L1B_00312092004142302_12242004184826.hdf
AST_L1B_00312092004142311_12242004185036.hdf
AST_L1B_00312092004142319_12242004185103.hdf
AST_L1B_00312092004142328_12242004184919.hdf
AST_L1B_00312092004142337_12242004185229.hdf
AST_L1B_00312092004142346_12242004185003.hdf
30
28
31
32
32
29
30
28
24
26
26
26
30
24
18
22
22
18
23
30
31
30
35
23
24
27
20
16
20
26
23
19
24
25
27
10
9
11
11
20
24
11
8
11
14
17
12
8
8
13
17
8
13
13
17
18
13
13
5
5
6
5
6
11
11
10
9
10
9
9
9
9
9
4 of 10
ZHAO AND DI GIROLAMO: MACROPHYSICAL PROPERTIES OF CUMULI
D10204
Table 2. (continued)
ASTER File Name
Threshold
AST_L1B_00312092004142355_12242004185015.hdf
AST_L1B_00312092004142404_12242004185045.hdf
AST_L1B_00312092004142412_12242004185143.hdf
AST_L1B_00312092004142421_12242004185205.hdf
AST_L1B_00312092004142430_12242004185612.hdf
AST_L1B_00312092004142439_12242004185524.hdf
AST_L1B_00312092004142457_12242004185805.hdf
9
10
7
12
12
19
16
D10204
which is a linear equation. Thus l can be equal to the slope
of the line using a simple least squares fit to the data on a
ln n(D) versus ln D plot. The solid step line in Figure 3
shows the histogram of all the clouds smaller than 7 km in
diameter binned to a 100 m diameter increment in
logarithmic coordinates. The bins with D > 7 km start to
a
The Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) file name for each scene is the name given in the ASTER
archive at the NASA Land Processes Distributed Active Archive Center
(LPDAAC). Thresholds are set for ASTER Channel 3N (0.78 " 0.8 mm)
digital numbers.
is 1,097,165, which is at least 2 orders of magnitude larger
than any other previous study.
4. Cloud Size Distribution
[10] The cloud size distribution represents the fraction of
total clouds within a finite range of sizes. The cloud size
distribution is used to calculate mass flux, energy transport,
and other quantities in cloud models, and it is one of the key
parameters used to evaluate cloud models against observations (see Neggers et al. [2003] for a review), and has
therefore been reported in numerous studies (Table 1). Early
studies showed cloud size was exponentially distributed
[e.g., Plank, 1969; Wielicki and Welch, 1986]. It has now
been well accepted (see Benner and Curry [1998] for an
excellent review) that the cloud size distribution can be best
represented in a power law form:
nð DÞ ¼ aD&l ;
ð1Þ
where D is the cloud area – equivalent diameter, and a and l
are constants. Many studies on cumulus clouds reported that
the cloud size distribution have a double power law form:
n(D) / D&l1 (D < Dc); n(D) / D&l2 (D > Dc), where Dc is
called a scale break. The values of l1, l2 and Dc varies from
one study to another (Table 1). Despite the large
discrepancy in the value of Dc in Table 1, the natural
question that may be asked is why a scale break exists? The
most popular answer is that the scale break may be caused
by differences in cloud dynamics between small clouds and
large clouds (see Sengupta et al. [1990] for a detailed
discussion). Nevertheless, under or subjectively sampling
clouds may artificially create a scale break in the cloud size
distribution as well, and this cannot be decoupled from the
cloud dynamical argument. From the point of view of
finding the function that best represents the data, a doublepower law fit will always be equal to or better than a singlepower law fit, a triple-power law fit will always be equal to
or better than a double-power law fit, and so on. Thus our
analysis below focuses on how to best represent the data
objectively.
[11] The widely used approach to calculate l is to take
the natural logarithm of both sides of equation (1),
ln nð DÞ ¼ a & l ln D;
ð2Þ
Figure 2. (a) Subscene (20 ! 24 km2) of an ASTER
Channel 3N image taken on 2 December 2004 and (b) the
subscene’s cloud mask. For the cloud mask, white
represents clouds, and black represents clear.
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bound of each bin in Figure 3 gives l = 3.07). Without the
detail information of a fitting process, it is not appropriate to
directly compare the results amongst different studies.
[14] To avoid the limitation of the line-fit method, we
used the mean of all the cloud sizes, D, to estimate l:
D¼
n
1X
Di ;
n i
ð3Þ
where n is the total number of clouds. From equation (1),
the probability density function of D can be written as: f (D) =
(l & 1)D&l. The expected value of D, E(D) is simply
ZDu
E ð DÞ ¼
Figure 3. Normalized distribution of cloud equivalent
diameter with a 100 m bin width, using clouds smaller than
7 km in diameter from 152 ASTER scenes. The solid
histogram is from observations. The dashed histogram is
from direct power law fit, and the straight line represents a
single power law fit.
have zero or one cloud indicating that clouds larger than
7 km in diameter might be poorly sampled and therefore
were not included in the analysis of cloud size distribution.
It is not difficult to conceive how the bin width can alter
the appearance of a histogram. Although there are
numerous algorithms for choosing an optimal bin width,
none of them is superior to the others. Details on how to
construct histogram algorithms are beyond the scope of this
paper (see Wand [1997] for a review). Given the finite
number of clouds and the cloud size range that have been
examined, a 100 m bin width was a proper choice based on
the discussions in the work of Wand [1997]. To our
knowledge, cloud size distributions using 100 m bin width
is the smallest bin width ever used, and is the bin width
used by Benner and Curry [1998].
[12] A single least squares fit to the center of each bin in
Figure 3 gives l = 2.85. Double power law fits give l1 =
1.88, l2 = 3.18, and Dc = 0.6 km, which has the least
residual. If a histogram is plotted for each scene, two or
more apparent scale breaks appeared for some scenes. For
most of the scenes, however, it was difficult to visually
identify any apparent scale break. When l from a single
least squares fit is calculated for each day, the daily
averaged l varied between 2.58 and 3.55. Averaging the
daily average l over the 24 days of data gave 3.01.
[13] Following a similar discussion in the work of Fraile
and Garcia-Ortega [2005], one can easily prove that this
traditional method of least squares line fitting to equation (2),
called the ‘‘line-fit’’ method, generates more weights to larger
clouds, which can be clearly seen in Figure 3 (see Zhao
[2006] for a detailed derivation of this point). However, large
clouds tend to be poorly sampled, hence the fitting generates
larger errors for small clouds than large clouds. In addition,
the fitting result is sensitive to a binning strategy including
the choice of bin width, the locations of the first and last bins,
and the location within the bin (e.g., the bin center) one
chooses to fit to (e.g., the least squares fit to the upper
D0
ZDu
D0
ZDu
Df ð DÞdD
¼
f ð DÞdD
D1&l dD
D0
ZDu
D&l dD
!
"
ð1 & lÞ D2&l
& D2&l
u
0
!
";
¼
ð2 & lÞ D1&l
& D1&l
u
0
D0
ð4Þ
where D0 and Du are the smallest and largest cloud sizes
among all the clouds, respectively. If the number of samples
is sufficiently large, then D ffi E(D), and l can be solved by
combining equations (3) and (4). Using this method, we
obtained l = 2.19. We name this method the ‘‘direct power
law’’ fit method. Unlike the ‘‘line-fit’’ method, this method
is statistically unbiased with an equal weight assigned to
each data point (see Zhao [2006] for a detailed derivation of
this point). In order to test the goodness-of-fit of the direct
power law fit, we calculated the number of clouds in each
100 m bin from equation (1) with l = 2.19 and then plot the
results as the dashed step line in Figure 3. The correlation
coefficient of this fitting is 0.996. Again, l, obtained from
the direct power law fit can be directly applied to the
modeling studies with no binning procedures required.
5. Cloud Fraction Distribution
[15] In this study, cloud fraction is defined as the ratio of
the number of cloud pixels to the total number of pixels.
Figure 4 gives the cloud fraction and cumulative cloud
fraction as a function of cloud size using bin intervals of
100 m. The total cloud fraction of all 152 scenes is 0.086.
Half of the total cloud fraction is contributed from clouds less
than 2 km in diameter. Note the cloud fraction distribution is
no longer smooth for cloud diameters larger than 3 km. This
is simply because few clouds larger than 3 km were sampled.
The majority of the bins between 10 km to 30 km only
contain one cloud. Since cloud area is proportional to cloud
diameter, the cloud fraction increases within this bin range.
[16] Figure 4 also shows a peak in cloud fraction at cloud
diameters between 400 and 500 m. However, the peak
should not exist if clouds have a size distribution as
prescribed by equation (1). Using equation (1), the total
fraction of clouds having a size D can be expressed as
F¼
p
nð DÞD2 :
8
ð5Þ
Therefore
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dF p
¼ ð2 & lÞaD1&l < 0;
dD 8
when l > 2:
ð6Þ
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Figure 4. Cloud fraction and cumulative cloud fraction as
a function of cloud equivalent diameter with a 100 m
increment.
If l = 2.19, F should decrease monotonically with D, and
no peak should exist. However, deriving equations (5) and
(6) from equation (1) was done under the assumption that
within each bin, cloud sizes are also continuously
distributed and follow the same distribution as prescribed
in equation (1). However, the assumption is invalid since
clouds are measured by finite resolution data and the
observed clouds may not follow this rule in each finite bin.
This explanation holds true for the double power law fit. If
dF
dF
< 0 when D < Dc and
>
l1 < 1.88 and l2 > 3.18, then
dD
dD
0 when D > Dc. Therefore, theoretically, the peak in Figure 4
should be located at Dc, which was measured 0.6 km using
the double power law fit method. This is very close to the
observed peak of 0.4– 0.5 km. The differences may be due
to the true underlying distribution not being a double power
law or issues dealing with sampling.
6. Cloud Area–Perimeter Relationship
[17] Lovejoy [1982] was the first to report a scaling
relationship between cloud perimeter and cloud area:
P/
pffiffiffiffiffi
Ad ;
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Although there are different ways to define a cloud perimeter, they all produce identical results (see Cahalan and
Joseph [1989] for further discussion). Using a least squares
fit, the slope of the fit line is d = 1.28 and the correlation
coefficient is 0.87. Note, d is smaller than most other studies
listed in Table 1, indicating that the cumulus clouds we
sampled have smoother shapes. Figure 5 shows no apparent
scale break.
[19] Figure 5 only shows clouds larger than 12 pixels
(" 2700 m2), since the shapes of clouds smaller 12 pixels
become sensitive to pixel shape (see Cahalan and Joseph
[1989] for further discussion). Since the pixel shape is
regular, d becomes smaller when more small clouds are
included in the analysis. When the clouds smaller than
12 pixels were included (68% of the total cloud population),
d dropped to 1.24 and the correlation coefficient was 0.91.
The drop in the valued of d was small, because sufficient
amounts of large clouds were sampled in this study.
7. Cloud Top Height Distribution
[20] Cloud top height is another important property of the
cloud macrophysical structure. ASTER 12 mm data
(channel 14) was used to retrieve the cloud top height
for each 90 m cloudy pixel. Channel 14 was chosen because
it had the least amount of water vapor absorption among
the TIR channels. The height retrieval procedure was as
follows. First, a 90 m pixel in a Channel 14 scene was
flagged cloudy only if all the 15 m subpixels within the
corresponding Channel 3N scene were cloudy on the basis
of the 15 m cloud mask (section 3). Therefore cloud height
is only retrieved for a fully cloud-covered 90 m pixel under
the assumption that all 15 m cloudy pixels are fully cloud
covered. Secondly, the radiance for each cloudy pixel was
then converted into brightness temperature (BT) following
the same procedure documented in the ASTER Level 2
product manual [Alley and Jentoft-Nilsen, 2001], where it is
documented that the error in the BT retrieval is less than
0.2 degree in Kelvin. Finally, the BT was converted into a
cloud top height by equating it with the temperature profile
from a 12Z sounding launched from the nearby island of
Guadeloupe, collected on the day of the ASTER overpass.
ð7Þ
where A is the cloud area, P is the cloud perimeter, and d is
called the fractal dimension. The value of d characterizes
the degree of complexity in cloud shape. If clouds have
regular shapes (e.g., a circle), d will be unity. The more
contorted cloud shapes are, the larger d will be. As a scaling
exponent, d is highly related to turbulent flow behavior, and
has been used as one of the Reynolds number similarity
arguments in cloud models [e.g., Siebesma and Jonker,
2000]. Additionally, scaling relationships can be used to
construct cloud fields for 3-D radiative transfer models
[e.g., Marshak et al., 1995].
[18] Figure 5 gives the log-log scatterplot of cloud
perimeter versus cloud area. The perimeter of each cloud
was defined as the total lengths of all the edges adjacent to
noncloudy pixels and cloud area was the product of the
number of cloudy pixels and the size of each pixel.
Figure 5. Log-log scatterplot of cloud perimeter versus
cloud area. Only clouds larger than 12 pixels are plotted.
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determined by clouds larger than 4 km in diameter, which
only accounts for less than 0.1% of the total number of
clouds but 58% of the total number of pixels. Figure 6
shows a clear positive correlation between cloud diameter
and cloud top height of a pixel. On average, cloudy pixels
from large clouds are higher than those from small clouds.
Only clouds larger than "3 km were able to reach above the
boundary layer that, on average, was at an altitude of
"2.5 km. The shape of the histogram varies only slightly
for clouds between 1 km and 4 km. Note that most of the
contribution to clouds lower than the lifting condensation
level, which ranged from "600 – 800 m for all scenes,
comes from clouds smaller than 1 km in diameter, and is
likely due to partially cloudy pixels or cloud emissivities
less than 1.
Figure 6. Normalized distribution of cloud top heights
with a 100 m bin width.
Only two soundings, 0Z and 12Z were launched per day
from Guadeloupe. The 12Z soundings are used because they
are closer to the ASTER overpass time (about 14Z) than 0Z.
[21] The above retrieval algorithm is based on the
assumption that cloud emissivity is equal to one. However,
this assumption may not be true under certain circumstances. Occasionally, 15 m cloudy pixels may not be fully
cloud covered, and some pixels such as cloud edge pixels
may contain thin clouds having an emissivity much less
than one, hence biasing the resulting height low. Although
we visually filtered out all the scenes with cirrus contamination, it is possible that some cloudy pixels were still
contaminated by subvisual cirrus above, which could not be
visually detected. The cloud heights for these pixels may be
biased high. For Channel 14, some amount of water vapor
absorption above cloud top may still occur, biasing the
heights high as well. This bias can be quantified by
comparing the output of a radiative transfer model with
the observations. Using a typical sounding collected at
Guadeloupe as input into the radiative transfer model,
MODTRAN 4.0, (detailed descriptions of the MODTRAN 4.0
are available at: http://www.vs.afrl.af.mil/ProductLines/
IR-Clutter/modtran4.aspx), the MODTRAN 4.0 cumulus
cloud model was adjusted in altitude until the simulated
radiances matched the observations. By turning water vapor
absorption off, we found that neglecting water vapor
absorption can bias the cloud heights up to 200 m. Errors
in the soundings and the two hours time difference between
soundings and ASTER measurements incurs further errors in
the heights. If we assume a random error of 2!C, this would
translate to a random error in height of "200 m.
[22] Figure 6 shows cloud top height frequency distribution for different range of cloud diameter. The distribution
for each range of cloud diameters was normalized by the
total number of cloudy pixels. The cloud diameter used in
Figure 6 was derived from the 90 m cloud masks following
the same procedure used to calculate cloud size from the
15 m cloud masks. Therefore the cloud diameters are
smaller for 90 m data than the 15 m data, since only the
90 m cloudy pixels that were fully cloud covered were
considered. The shape of the distribution is primarily
8. Spatial Distribution
[23] The earliest study on the spatial distribution of
clouds was conducted by Plank [1969]. Since then, numerous papers on this topic have been published and several
techniques has been developed to classify the organization
of clouds of a cloud field into clustering, randomness, and
regularity (see Nair et al. [1998] for a review). Cloud spatial
distribution is a function of domain size, resolution, the
number of cloud, and cloud morphology. To accurately
determine the spatial distribution requires knowledge of
what this field would look like if its clouds were truly
randomly distributed. To accomplish this, clouds from
observation must be redistributed randomly while preserving their original size and shape. However, this process
requires extensive computer resources and simplifying
assumptions (i.e., circular clouds) that limits its application.
To avoid this limitation, we concentrated on reporting the
statistics of the nearest neighbor distance (NND) in the
observed cloud fields, since it is a property of the spatial
distribution that is easily calculated and can be compared to
other observations (real or modeled) having the same pixel
and domain size. The distance of two clouds is the Euclidian
distance between their mass centers [see Benner and Curry,
Figure 7. Normalized distribution of the nearest neighbor
distance (NND).
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Figure 8. Normalized distribution of the ratio of NND to
cloud area – equivalent radius.
1998]. Figure 7 shows a histogram of the NND distribution
with a peak around 50 m. However, the finite size of a cloud
limits the minimum possible value of its NND. Therefore
we plotted the histogram of the ratio of NND to cloud area –
equivalent radius, half of the cloud area – equivalent diameter, in Figure 8. Approximately, 75% clouds have a nearest
neighbor within a distance of 10 times less than their radius.
The results may not be directly compared with past studies,
since they used different instruments and domain sizes,
which the statistics on the spatial distribution relies on.
9. Summary and Discussion
[24] ASTER measurements over the tropical western
Atlantic were used to extensively examine the macrophysical properties of more than one million cumulus clouds
from 152 scenes collected over 24 days during September
through December 2004. The following summarizes our
results:
[25] 1. Cloud size distribution can be expressed in a
power law form. The exponent index is 3.07 with the
traditional least squares fit method, and 2.19 with the direct
power –law fit method. The double power law fits give
indexes of 1.88 and 3.18 with a scale break at 0.6 km.
[26] 2. The total cloud fraction of trade wind cumulus
cloud fraction is 0.086, half of which is contributed from
clouds less than 2 km in diameter.
[27] 3. An area-perimeter power law was observed with a
dimension 1.28. This is smaller than typically shown in the
past studies, indicating that cumulus clouds we sampled
have more regular shapes than those in other studies. No
apparent scale break was found.
[28] 4. Cumulus cloud top heights were retrieved only for
fully cloudy pixels at 90 m resolutions with an IR technique.
The distribution of cloud top heights peaked "1 km in
altitude. A clear positive correlation between cloud diameter
and cloud top height was also observed.
[29] 5. Approximately, 75% of clouds have a nearest
neighbor distance within 10 times their area-equivalent
radius.
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[30] Our results were compared to those of previous
studies in Table 1. Differences between the different studies
listed in Table 1 may be due to time, location, instrument
spatial resolution, domain size, and sampling issues.
Although our statistical analysis of the macrophysical
properties of trade wind cumuli is the most comprehensive
to date, whether it is representative of other oceanic trade
wind regions or even other time periods remains to be
proven.
[31] Uncertainties in the cloud masks used to generate the
cumuli statistics in this study are difficult to assess. In a few
studies, the sensitivity of the cloud statistics with threshold
used in cloud detection are examined [e.g., Wielicki and
Welch, 1986]. Of the cloud macrophysical properties examined here, cloud fraction has been shown to be the most
sensitive to changes in threshold [e.g., Wielicki and Welch,
1986]. Given that the thresholds (Table 2) used to generate
the cloud masks have been judged to be optimized (see
section 3), we can assume that the uncertainty in cloud
fraction for cumulus clouds comes from pixels near cloud
edge [cf. Di Girolamo and Davies, 1997]. Of the total
cloud fraction of 0.086 derived from the 152 ASTER
scenes, cloud edge pixels only contributed 0.011 to the
cloud fraction. Cloud size distribution has been shown to be
insensitive to threshold, because large clouds will replace
small clouds in the size distribution with decreasing threshold, while small clouds will replace large clouds in the size
distribution with increasing threshold [Wielicki and Welch,
1986]. Similarly, since the value of a threshold in a
reasonable range primarily affects cloud edge pixels not
cloud center pixels, statistics on cloud spatial distribution is
also insensitive to thresholds. In this study, cloud top height
is retrieved for each 90 m pixel. Varying the value of a
threshold only affects the number of 90 m pixels that can be
used to retrieve cloud top height, but not the values of the
cloud top height.
[32] Finally, it is critical to place the cloud statistics
derived in this study within the meteorological conditions
driving the clouds. Although not shown here, we did stratify
the observed cloud properties by selected meteorological
variables, including the average 500 – 700 mbar relative
humidity, 850 mbar wind speed and vertical wind velocity,
and wind shear between lifting condensation level and
boundary layer top. The meteorological data kindly provided
by F. Yang (2004) were derived using the column version of
environmental prediction global forecast system developed
at the National Centers for Environmental Predication.
However, we did not find any relationship between the
meteorological conditions and any of the cloud macrophysical properties (see Zhao [2006] for plots of meteorological conditions against the cloud macrophysical
properties). Our hypothesis is that the forecasted meteorological data simply cannot capture the mesoscale meteorological conditions that drive the clouds. Unfortunately, this
is the best meteorological data set available to us. Forthcoming meteorological conditions from in situ observations
collected during RICO may help, but the lack of space-time
coincidence with ASTER will limit their utility in extending
our analysis, except for providing a summary of the
meteorological conditions under which trade wind cumuli
form.
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[33] Acknowledgments. This research is partially supported by a
grant from the National Research Foundation and from the National
Aeronautics and Space Administration’s New Investigator Program. We
thank Bob Rauber and Pier Siebesma for useful discussions and Eric
Snodgrass for the assistance of acquiring ASTER data. We are grateful to
Fanglin Yang and Steven Krueger for the meteorological data set. We also
thank three reviewers for suggested improvement.
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&&&&&&&&&&&&&&&&&&&&&&
L. Di Girolamo and G. Zhao (corresponding author), Department of
Atmospheric Sciences, University of Illinois at Urbana-Champaign, 105
South Gregory Street, Urbana, IL 61801, USA. ([email protected])
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