Introduction
The cloud amount, its distribution, intensity, and organization over an area depends on the dynamic and thermodynamic factors interacting on several scales. Model results and observations have shown the importance of such interactions for the complexity of cloud patterns. On the other hand, cloudiness features, including cloud amount, top height, anvil extension, and intensity of convective and mesoscale updrafts and downdrafts, can act upon the evolution of meteorological variables and on the dynamic patterns themselves (Betts 1976; Ogura and Cho 1973; Albrecht et al. 1986;Frank and Cohen 1985; Johnson 1976).
One dramatic example of the importance of cloud organization upon its evolution can be found during the mesoscale convective system (MCS) genesis. The merging of anvils in the first stage of an MCS leads to the formation of elevated stratiform precipitation, an upward shift in the mean vertical motion (Gamache and Houze 1982; Dudhia and Moncrieff 1987), and an associated upward shift of the level of maximum heating (Esbensen and Wang 1984; Silva-Dias et al. 1984), favoring its upscale growth to mesoscale proportion (Anthes and Keyser 1979; Hack and Schubert 1986), which is characteristic of the development of a mature, midlatitude mesoscale convective complex (MCC).
The objective of this paper is to propose a new method of analysis of cloud geometry distribution obtained from satellite infrared (IR) images that is able to recognize mesoscale patterns. Weather forecasters and climatologists have classified different mesoscale systems from satellite IR images in a subjective way. We discuss an automatic discriminating method applied to just one channel (the infrared) to detect different organizations of clouds over a limited area.
The fractal geometry (Mandelbrot 1977, 1982) provides a model to investigate cloud complex shapes and complex cloud distribution over an area. One general characterization of fractals is that they are the end result of physical processes that modify shape through global, local, and internal action. Therefore, the fractal dimension of a surface corresponds quite closely to our intuitive notion of roughness and can be used to resolve the problems of texture segmentation and classification (Peleg et al. 1984; Pentland 1984).
Scaling properties in clouds have been found from satellite images, indicating self-similarities in several scales (Lovejoy 1982; Machado et al. 1992). However, the studies of Cahalan and Joseph (1989) and Sengupta et al. (1990), using enhanced data resolution with the Landsat multispectral scanner, have shown different power laws (fractal dimensions), depending on the nature of clouds. Lovejoy et al. (1987) and Gabriel et al. (1988), using an analysis technique called “functional box counting,” have shown that both radar rain and satellite cloud fields are multifractal over various ranges in scale. Schertzer and Lovejoy (1987), Lovejoy and Schertzer (1990), Olsson et al. (1993), Tessier et al. (1993), Cahalan et al. (1994), and Marshak et al. (1997) present some examples of multifractal properties applied to observed rain and cloud fields. The fractal structure of clouds with a simple atmospheric model is examined in Yano et al. (1996), who show that the fractality of simulated, large-scale cloud systems depends on cumulus parameterization. Yano et al. thus show the necessity of quantifying the relationship between fractal properties and mesoscale cloud system dynamics.
In this paper, clouds viewed by satellite IR images are considered in a mesoscale context. Multiple dimensions are found, and the relationship between the fractal box dimension and the patterns of satellite cloud images is investigated. Within the analyzed spatial scale we will show that the distribution of clouds is scaling.
To interpret the multiple dimensions of cloud distribution, we focus our analysis on two main aspects. First, we analyze the box dimensions of actual IR satellite images of clouds in a limited area. Second, we compute box dimensions for the simulated distributions of “well-behaved” structures, such as circles. The simulated structures give us a basis for the discussion of the fractal behavior of the cloud distribution over the area. The main aspect highlighted by the fractal box dimension is its ability to identify different structures according to their level of aggregation, the circular patterns, and the isotropy of their pixel distribution. As discussed in Pentland (1984), the fractal box dimension is related to the complexity of the objects over an area, but the concept of roughness of a homogeneous surface (i.e., a single cloud) attributed to fractal dimensions can be broadened for the distribution of a set of fractal structures (several clouds).
This paper is organized as follows. Section 2 describes two procedures to compute box dimensions. Section 3 discusses the differences in the behavior of those dimensions and the possibility of using the dimensions to identify cloud patterns in two series of satellite IR images that show some of the main mesoscale systems in the continental area during different periods of the summer. Section 4 proposes a method to identify patterns in satellite images by using box dimension and multivariate statistical methods such as principal components and clustering analysis. We show that by combining both methods a large variety of geometrical patterns of mesoscale systems are identifiable, among them the different evolutionary stages of MCCs.
The fractal box-dimension definition
The fractal dimension as defined by Eq. (3) is usually identified with the Hausdorff–Besicovitch dimension and is known as the capacity dimension DO (see, e.g., Tsonis 1992). Nevertheless, there is a fine distinction between the Hausdorff–Besicovitch dimension and the capacity dimension: while the former is obtained by covering the set minimally with hypercubes that may be different in size, the latter is obtained with the same process except that the hypercubes are the same size (Essex and Nerenberg 1990).
In this work, we present two methods to compute the box dimension. In the first, each image is considered a set S(Tc) = {x, y|T(x, y) ≥ Tc}, where Tc is a threshold in counts. The image is covered by square boxes that vary their scale from 2 to 24 pixels. This upper limit is defined as the break point relative to the length of the side of each image (98 pixels) in order to obtain just one value of D from Eq. (3). The box dimension obtained from this method will be referred to as Db. This method is similar to that Lovejoy et al. (1987) apply to radar reflectivity for the dimension they call D2(T), where T is a threshold.
Here Cmax and Cmin are the maximum and minimum value of counts, respectively, and n is the number of boxes in the scale Δs. Since different units are used in the z direction [z = C(x, y), where C is the count value, equivalent to temperature], the box-counting dimension computed from Eqs. (4) and (5) is equivalent to the“graph” dimension (Mandelbrot 1977) and is referred to in this paper as Dg. Its value ranges between 2 (the topological dimension of the plane) and 3 (the topological dimension of the cube). It should be emphasized that the proposed method of computing Dg includes all boxes inside the studied area, not just those that include“clouds” or simulated circles.
Application of the box dimension to satellite IR images
Lovejoy (1982), Wielicki and Welch (1986), Gabriel et al. (1988), Lovejoy and Schertzer (1990), Tessier et al. (1993), Cahalan and Joseph (1989), Machado et al. (1992), and Mapes and Houze (1993) are some examples of studies that have applied scaling properties to satellite images of clouds. In this section we will show how the scaling properties of mesoscale cloud fields inferred by box dimension can be applied to the recognition of fractal mesoscale patterns in satellite IR images.
We will show that the fractal box dimensions are sensitive to different distributions of cloudy pixels over an area. Mesoscale convective systems appear in different spatial organizations, according to the dynamic and thermodynamic behavior of meteorological variables such as wind field, temperature, humidity, convective available potential energy, and so on. We will attempt to identify the cloud distribution organization from a few parameters extracted only from satellite IR images, among them the box dimensions.
The satellite dataset
To carry out this study, some areas were chosen in the South American tropical and subtropical zones. The IR images were obtained from the Meteosat-3 satellite in two different periods during the summer of 1994 with a 1-h time step. The images correspond to two periods of consecutive days 20–21 and 24–25 January and were found to be enough to sample several MCSs in the Tropics.
During summer, several convective systems, such as MCCs, squall lines, and isolated convective cells, develop in tropical continental areas. We investigate such MCSs from satellite IR images, sampled over four regions, corresponding to 98 × 98 satellite pixels, whose coordinates are given in Table 1. These regions were named based on the region covered by most pixels in each area. The samples cover the Amazon Basin and the regions of north-northeast, south, and southeast Brazil, southern Bolivia, northern Argentina, Paraguay.
For each of those areas, a time series of images was available for the two periods. To develop the best method to recognize mesoscale patterns we chose as the“control series” the sample series of 24 and 25 January 1994. These images were chosen with the purpose of sampling the most common patterns of cloudiness in summer over the continental tropical and subtropical area. One of the most important mesoscale systems found in this sample was an MCC at the latitude of Paraguay and northern Argentina.
The time series of 20 and 21 January was used as a“test series” to verify the results obtained with the control series in independent satellite images. It includes the samples over Bolivia (Table 1). These images were obtained in the same areas as the control series but with no previous knowledge of the characteristics of the developing MCSs. The control series contains 174 images, while the test series contains 178. Each sequence focuses on the evolution of a given cloud scene.
The algorithm used to compute the box dimension—and therefore all of the variables used to analyze the satellite data—is based on count value, not temperature. This choice was made in order to generalize the method to other possible fractal, noncloud structures observed in images. The equivalence between temperature and count value for the Meteosat satellite can change slightly from day to day. This is one reason to limit the time series of satellite data to only two consecutive days.
For all of these time series the following variables were computed: dimensions Db and Dg, and the percentage of area covered by values greater than count 185 considered as cloud scenes, or simply the cloud fraction for the satellite resolution pixel (hereafter referred to as Ac). Note that the original count value (Cv) of Meteosat was transformed to 255 Cv.
Count 185 was chosen to represent the cloudy areas since the corresponding temperature in the studied days is about 240 K—that is, 239.9 K (day 24), 240.0 K (day 25), 240.3K (day 20), and 240.4 (day 21)—and therefore corresponds to cloud scenes. These differences in the temperature threshold can be neglected for our purposes.
Box dimensions and fractal cloud patterns
The control series of days 24 and 25
It is well known that in satellite IR images both the pattern and the cloud fraction inside an area change. Two questions arise: Does the box dimension change when the fraction of objects or pixels is increased (decreased), split off (compacted), or spread out (constituting a well-defined circle) over an area? Is there any particular behavior of the box dimension related to such patterns? In this section we will discuss how box dimensions can distinguish fractal patterns of clouds and how they depend on the cloud fraction in the images.
Figures 1a and 1b show the plot of Db and Dg against Ac, respectively. There is a tendency of increasing the value of box dimension as Ac increases. Figure 1a shows a 2° polynomial curve as the best fit between Db and Ac. The value of the coefficients and the percentage of variance explained by the polynomial fit are listed in Table 2. In Fig. 1, Db is plotted by subtracting the polynomial fit.
Figure 1b shows a linear tendency between Dg and Ac. A polynomial fit with degree greater than 1 (not shown) explains only 1% more of the variance, indicating that the straight line can be considered the best fit for this pair of variables. The coefficients of the straight line fitted to the data may be seen in Table 2. The fluctuation values of Db and Dg computed after extracting the line fit will be referred to as
To understand the behavior of fractal box dimensions, idealized pictures were built to represent different spatial distributions of pixels over the area. These pictures were constructed using circles as the basic objects. Although the box-counting technique can be applied to images constituted of circles [a power law can be found according to Eq. (2)], the circle is not self-similar: that is, it cannot be written as an almost-disjoint union of smaller circles. Nevertheless, the circle, a “well-behaved” object, was chosen for two main reasons: 1) the resemblance of circular structures with natural convective clusters in several scales and 2) the understanding of how the box dimension changes, which is due exclusively to the distribution of cloudy pixels over the studied area.
We should emphasize that all experiments performed with circles, and therefore the fluctuations of box dimensions, have validity only for the same area and the same resolution of the satellite IR images. The “circles” are obtained by an integer approximation of the equation r = (x2 + y2)1/2, where r is the radius of the circle and x and y are the coordinates with respect to the center of the circle. The integer approximation implies a nonsmoothed aspect, as shown in Fig. 2. Furthermore, we have limited range of scales allowed in the case of real and simulated images: the limit of Δs → 0 in Eq. (3) is a theoretical limit since satellite images impose this limit due to their resolution.
Figure 2 shows circles of increasing radius, keeping constant the area in which they are embedded. The external circle radius ranges from 10 to 50 pixels, with a step of 5 pixels between images. The side of the square area is equal to 98 pixels. The values of Db and Dg computed according to Eq. (3) and their standard deviation are shown in the same picture.
Note that Db tends to increase as the fraction of pixels, or radius of the circles, increases. Dimension Db ranges from about 1.54 to 1.92, while the percentage of covered pixels ranges from 7% to 78%, indicating that Db tends to the Euclidean dimension of the circle as the circle fills the image. The standard deviation of Db decreases, and the linear correlation coefficient increases as the area is filled by the circle.
It must be noted that the box dimension Db, according to the definition, does not depend on the internal gray level since the box-counting technique counts only those boxes of side Δs that contain at least one pixel with a value greater than the chosen threshold.
The increasing values of Dg as the circle radius is increased are also seen in Fig. 2. The standard deviation error of Dg increases, and the linear correlation coefficient decreases from 0.9999 to 0.9987 as the area is filled for the circle. This means that it begins to deviate from the straight line on a log–log plot, indicating that the proposed methodology of computing Dg is able to detect a slight tendency of breaking the self-similarity or scaling properties as the picture is filled with a circle. Dimension Dg ranges from 2.02 to 2.16 in this simulation—that is, an increase of about 7%—while the increase in Db is about 25%. It seems therefore that Db is more sensitive to the increase of Ac and tends to the Euclidean topological dimension faster than Dg. This is a direct result of the greater increase of N(Δs) for smaller scales Δs, compared to the larger ones, as the circle fills the studied area [see Eq. (3)].
This property of box dimensions seems to be related to the results pointed out by Lovejoy et al. (1987). They found a systematic decrease in the box dimensions as the threshold T of radar reflectivity increases. As T increases, the fraction of radar pixels greater than or equal to T decreases, which leads to a resulting decrease of the box dimensions. A similar result was found by Gabriel et al. (1988) for IR and VIS radiance.
Hamburger et al. (1996), studying fractal properties of randomly placed n-dimensional spheres (n = 1, 2, 3), have shown that fractal dimensions (the box-counting and Minkowski-sausage techniques were applied) are not universal and depend on density. Using analytical and numerical calculations, they have also shown that in the regime of a low-volume fraction occupied by the spheres apparent fractal behavior is observed for a range of scales between physically relevant cutoffs. Hamburger et al.’s results can explain why the circles considered in our experiments have fractal dimensions if we take into account the range of scales Δs used for the solution of Eq. (1). Nevertheless, even being considered “false fractals,” our simulated pictures have behavior that is similar to true fractals (clouds) for the range of scales considered.
If the changes in fractal box dimensions are related to how the cloudy pixels are distributed over the area, the values of
Figure 4 contains eight images with values of
Using Machado et al.’s (1992) method to detect connected areas of cold cloudiness, we obtained the eccentricity e (minor axes/major axes) of the largest cloud. This is shown in Fig. 4 (category 3: Fig. 4c, e = 0.78;Fig. 4d, e = 0.87). The eccentricity of the largest contiguous cloud cell for category 1 is equal to 0.52 (Fig. 4a) and 0.58 (Fig. 4b). Nevertheless, for the smaller cells in all categories the method detects high eccentricity. The number of contiguous cells occupying a fraction of the area Ac > 0.3% (an arbitrary threshold of Ac) is equal to 10 and 15 in the examples shown in Fig. 4a and Fig. 4b, respectively, and to 3 and 1 in the examples shown in Fig. 4c and Fig. 4d, respectively. The general properties of such categories will be discussed based on the simulations of the circles.
Since box dimensions increase as the fraction of covered pixels is increased, it is interesting to understand what happens to Db and Dg if the object is divided but the percentage of covered pixels is kept constant and the number of circles is changed. Figure 5 illustrates how this simulation was performed. The first panel in the upper left-hand corner shows a circle with an external radius equal to 30 pixels. The other panels in Fig. 5 were constructed by splitting the fraction of pixels of the first panel into n parts (n = 2–9). To obtain a close approximation to the fraction of covered pixels in the first panel, some of the other panels in Fig. 5 contain circles that do not all have the same radius. This is due to the integer approximation associated with the number of pixels in each radius. The results for Db and Dg are shown in Fig. 5
Dimension Db tends to decrease as the total fraction of covered pixels is divided into smaller circles. This behavior is opposite that observed when the fraction of pixels of a single object is increased over the area. On the other hand, Dg increases as the number of circles increases—the opposite behavior when compared with Db. The behavior of Dg as the number of circles is increased is explained by the fact that for smaller scales (smaller Δs) N(Δs) is increased because the number of boxes with a nonzero difference of counts increases [see Eq. (4)]. On the other hand, Db increases as the nonempty pixels are concentrated in a single circle because the number of empty squares for larger Δs increases, indicating lower values of N(Δs) for these scales. This results in an increasing slope of the fitted straight line between log[N(Δs)] and log(Δs).
Another useful property of box dimension is illustrated in the last two frames of Fig. 5: Db and Dg increase as the space is filled anisotropically. This behavior can be explained considering that for larger Δs, N(Δs) is smaller when the structures are anisotropically distributed over the area, implying an increase in both Db and Dg.
The images typical of category 4 (Figs. 4e,f) show some centralized structure, with no evident circular shape, shifted from the center of the area. The eccentricity of the largest cell in the two examples is equal to 0.54 and 0.63 (Figs. 4e and 4f, respectively). The images typical of category 5, at the other extreme, though apparently not so different from the category 4 images, seem to have no preferential portions of the area occupied by clouds (Figs. 4g,h). Compared to category 1, these images with sparsely distributed clouds do not include, markedly, more cloudy pixels; that is, the cloud cells in category 5 are smaller and/or more sparse than those in category 1.
In Fig. 3 the images at the center of the cross, in the most densely filled category 2, show a more complex structure compared with the images shown in the other categories (Fig. 6). The shapes found in category 2 range from large structures, probably some of them representing the decaying MCC, to complex structures, whose edges indicate complex fractal structures rather than the well-behaved geometric patterns found in other groups.
These results illustrate four features of fractal box-dimension analysis:
Circular patterns in the range of scales analyzed from these images (see Table 1) are the most readily identifiable patterns that can be obtained from the fractal box-dimension properties since single, perfect circles have the box dimension
D′g D′g As a consequence of feature 1, the spread distribution of cloud cells with no circular organization is well distinguished by this method. This agrees with what was discussed with simulated circles in Fig. 5, in which it is demonstrated that Dg increases as the cloudy pixels are spread throughout the area with fixed cloud fraction.
Anisotropic distribution of cloudy pixels over the area tends to have larger values of Db, which explains the right side of the cross in Fig. 3.
The variance of Db is well explained (about 87%) by Ac according to the second degree polynomial fit for the cloud scenes (see Table 2). Since we removed the tendency on Ac, only about 13% of variance remained is not explained by the polynomial fit. Figure 2 shows that Db is in fact more sensitive to Ac than Dg.
The test series of days 20 and 21
The same variables, Db, Dg, and Ac, were computed for the test series to investigate the fractal properties of other independent series. Similar to the control series, the test series showed increasing values of box dimension with the increase of Ac. The polynomial fit is shown for each dataset, and the corresponding values of the coefficients are presented in Table 2.
The polynomial coefficients fitted for each curve are different from those of the control series, as well as the percentage of residuals explained by each fit (Table 2). We found that a 2° polynomial fit between Dg and Ac is more adequate than the linear one. These modifications led to differentiated averages and standard deviations for
We verify that the test series shows the overall behavior found for the control series. Figure 8 shows one typical image for categories 1, 3, 4, and 5, illustrating such similarities. Note that categories 6 and 7 have been added in this case (see Fig. 7). The lowermost values of
It is clear that for meteorological purposes it would be more useful to identify and separate those images that correspond to enhanced convection instead of identifying only similar fractal patterns regardless of how much total cloud flux of mass, heat, moisture, and so forth they represent. Therefore, the fraction of cloudy pixels in the area should be included in order to cluster fractal patterns of interest to meteorological studies.
Fractal patterns and convective activity
The results from the previous section suggest that the information of cloud fraction in the area should be included for the definition of fractal patterns in order to elucidate the differences among the convective intensities as well as their patterns. This information is included in both the fractal dimensions Db and Dg and in a more straightforward property, the fraction of cloudy pixels (Ac). Another variable that may be included to investigate the intensity of the convection is the lowermost temperature (equivalent to the higher count, hereafter referred to as “Cmax”), as well as the fraction of area occupied for these higher tops (hereafter referred to as “Pmax”). Several authors (see, for example, Adler and Fenn 1979; Scofield 1987) have shown the importance of these parameters in the determination of convective activity and the likelihood of precipitation on the ground. In this section, we will discuss two statistical methods to classify similar patterns in the cloudiness based on such variables and in the fractal box dimensions: the principal components (PC) analysis and the cluster analysis.
The PC analysis
Each image of the four time series defined in the control series can be thought of as an individual observation that defines some patterns that could be related to others in the same time series and even in other time series. Our major premise is that such patterns can be defined by the variables
A correlation matrix is generated from the matrix
The greatest absolute weights (those values corresponding to at least 70% of the major weight found) are boldface in Table 3 in order to identify the variables that have the greatest contribution in each PC. This procedure indicates that the first PC has its variance explained mainly by the relative high weight on Ac and Pxcm. The positive sign of those variables indicates that they increase (or decrease) in the same “direction” in the first component. The second PC, on the other hand, has its greatest weight on
The third PC has the main contribution in variables
Therefore, we can conclude that the first three PCs can define six main patterns, if we assume that the images can have a high magnitude on some of these PCs, with a positive or negative value. We can define two opposite patterns with high relative magnitude on the first PC, two opposite patterns with high relative magnitude on the second PC, and so forth. Thus, the verification of the PC analysis should be carried out through the analysis of the subset of images of the control series corresponding to the six main patterns found. This was performed through an automated clustering analysis.
The clustering model
The construction of clusters from PC analysis
The clustering method provides the most adequate process of grouping similar data in a subset (see, e.g., Kaufman and Rousseuw 1990). Based on Euclidean distances, with standardized data, the clustering method is able to find, from an objective procedure, the similar patterns represented by the eigenvectors obtained from the PC analysis. The following steps were performed from the results of the control series.
From the matrix of expansion coefficients of the PC analysis, we chose the six main “seeds” through the relative value of each of the first three PCs, shown in Table 4. Only the first three PCs were considered because they represent almost 97% of the accumulated explained variance.
From those seeds, a clustering analysis was applied to the whole set of images for the control series in order to find the best correlated images according to their patterns, as represented by the PCs.
The subsets of the images obtained from the clustering analysis were investigated according to the average properties of the variables included in the PC analysis and according to the patterns they represent.
Figure 9 illustrates the average values for each variable included in the PC analysis, according to the clusters defined before. Clusters 1 and 5 presented the greatest values of Ac, Pxcm, and Db. Note that Dg has its greatest value in cluster 1, decreasing toward cluster 6, with a peak in cluster 5. This behavior is similar to those found for Ac and Pxcm, though the peak in cluster 5 exceeded the one in cluster 3. Dimension
Figure 10 displays the results of clustering analysis for the control series showing some typical images belonging to each cluster. The layout of the images in Fig. 10 was chosen to emphasize the differences between the clusters seeded with opposite signs of PC. The properties of such images can be summarized as follows.
Cluster 1: The patterns of this cluster resemble those of the most enhanced convection, spreading out for the studied area (Figs. 10a,b). Some decaying but massive MCC images were found in this group. The shape defined by count 185 shows noncircular patterns. The high gray levels indicate the intensity of convection in the area.
Cluster 2: The pattern of the images in this cluster was defined mainly by the high values of
D′g D′b D′g D′b Cluster 3: (Compared to the images in clusters 1 and 2). The images in this cluster show fewer cloud pixels, and the pixels tend to group together (Figs. 10i,j). These images present an anisotropic distribution, revealing the behavior discussed before, in which such distribution of cloudy pixels in the area is described for high values of
D′b D′g D′b D′g Cluster 4: This cluster represents the images with properties that are asymmetric to cluster 1 (see Table 4). It grouped preferably those images that had low fraction of clouds, were sparsely distributed, or had no clear organization over the area (Figs. 10c,d). It is a result of the lower weight on Ac and Pxcm (Table 3).
Cluster 5: With characteristics opposite those of cluster 2, this cluster identifies the MCC cycle over the Paraguay area very well (Figs. 10g,h). This cluster grouped images whose cloud patterns showed a shape that is quite circular occupying a large area. This is in agreement with the observation of decreasing values of
D′g D′b D′b D′g Cluster 6: This cluster has properties asymmetric to cluster 3 (Table 4). Therefore, it represents both images that have fewer clouds (low values of PC1) and those that have clouds that are not grouped (low values of PC3). This cluster, of course, shares some properties with cluster 4, but this was considered important because it is able to select the less expressive subset of convective activity in the area from the others.
Figure 12 shows the distribution of the number of contiguous cells with Acl > 0.3% for clusters 1, 2, and 5, illustrating how the total Act is distributed in individual, contiguous cells over the area. Figures 11 and 12 emphasize the differences in mesoscale cloud patterns shown before and how the box dimensions can summarize properties, such as the number of cells, the total cloud area, and the cloud patterns distribution and shape.
These results suggest the following:
Clusters 1 and 5 represent the most intense phenomenon regarding the convective activity over the area, which can be differentiated from their shape.
Cluster 2 could be considered the middle of an idealized scale of convective intensity. It is between the highly intensive homogeneous distribution of cloudy pixels in cluster 1 and the random and sparse distribution in cluster 6.
Cluster 3, whose images show some kind of grouped structures, can be considered the middle of the same idealized convective intensity scale but between the highly circular and massive shapes in cluster 5 and, perhaps, the small amount of cloudiness in cluster 4. It should be noted, however, that large-scale and mesoscale meteorological fields related to these clusters should be considerably different, and therefore, could define differentiated patterns as well.
The generalization of the clustering model
Since the control series samples the most common patterns of cloudiness in the summer over the tropical and subtropical continental areas, we may expect that there is a high probability of finding images that belong to the clusters defined above for other samples obtained close to, or in, the same areas.
Based on such observations, a generalized clustering model was defined by seeding the clusters with the average value of the variables in each cluster previously obtained (Table 5). In this way, the clustering analysis is simplified into a single step.
Results of the generalized clustering model for the test series
The result of the generalized clustering model for the test series was obtained assuming the same six seeds as defined in Table 5 and seen in Fig. 13.
It can be verified from the typical images in each cluster that the same patterns found in the control series also appear in the test series. 1) Cluster 1 includes images that represent the enhanced convection that covers a larger fraction of the area. Cluster 2 includes images with cloud cells that are distributed sparsely and uniformly over the entire area. Cluster 3 includes images with high values of
It is important to comment that the clustering analysis for the control series from the seeds described in Table 5, according to our expectations, resulted in clusters quite similar to those obtained from the PC analysis (not shown due to redundancy).
The life cycle of cloudiness
Figure 15 summarizes the time evolution of the classification into the six clusters for each time series studied for the control and test series. The differences between the control and series over the Paraguay area can be seen in this figure. During days 24 and 25, the MCC life cycle is described by the predominance (∼64% of the images) of cluster 5 (the most circular and massive shape) and cluster 1 (massive, noncircular, enhanced convection). The time series in days 20 and 21 for the same area showed the predominance (∼32%) of grouped anisotropic cloudiness (cluster 3). However, at the end of the studied period, the MCC signature was found in this region. The lack of data from 1100 to 1700 UTC in this area was due to the absence of clouds defined for count 185.
Note that in the afternoon and early evening the convection in the tropical area is dominated by clusters 1 and 2, indicating the link between the diurnal cycle of the convection over the tropical areas (Silva-Dias et al. 1987) and the observed patterns of clouds. For the MCC life cycle on days 24 and 25 note that most circular patterns prevail late at night and in the early morning, in agreement with the observations of Velasco and Fritsh (1987).
Summary and conclusions
The analysis of the properties of box dimensions obtained from satellite cloud images from two different periods in several areas during the South American summer and elucidated from the simulated circles can be summarized as follows.
Dimensions Db and Dg tend to increase as the fraction of objects increases over a fixed area.
Dimension Db tends to increase, while Dg tends to decrease, as the cloudy pixels are grouped with fixed cloud fraction in the area.
Dimension Dg is very sensitive to the way the cloudy pixels are distributed. For some cloudy pixels occupying a single circular area, Dg assumes smaller values than if the cloudy pixels were distributed into several cells.
Anisotropic distribution of cloudy pixels over the area yields higher values of Db and Dg. No conclusive behavior of Dg is found if new gray levels are added to a single cell.
Dimensions Db and Dg depend on anisotropy due to the objects’ distribution over the area. Some tests with other Euclidean shapes, such as ellipses (not included), showed that box dimensions can distinguish between circles and ellipses with equivalent area only if the ellipses have an eccentricity (minor axes/major axes) of less than 0.2. In this case the ratio Dbcircle/Dbellipse is definitively greater than 1.1. Otherwise, the ratio fluctuates around 1 ± 0.05.
Six main fractal cloud patterns became evident from a PC analysis since the first three components explained almost 97% of the variance of the patterns in the control series. These patterns were found from the positive and negative high weight on the first, second, and third PC.
The two main sets of images related to enhanced convection have been classified into the well-organized circular structures of the MCCs (cluster 5) and the most widespread noncircular convection (cluster 1), based on the high negative weight on the second PC and the high positive weight on the first PC, respectively. The identification of the MCC life cycle from the clustering model was a result of the differentiation between the geometry of the massive circular structures, with fractal dimension tending to the topological dimension DT = 2, and the massive, widespread structures related to higher values of Ac and fractal dimensions Dg and Db, indicating increasing complexity of the shape. The fraction of cloudiness distributed in a larger number of cells and/or the presence of cells with complex edges increases the values of fractal dimensions and can be considered the “opposite” geometry of the well-organized circular structures of the MCCs (cluster 2).
Less expressive convective activity was also classified (clusters 3, 4, and 6) and is necessary to ensure the separation between moderate and light convective activity. One of the patterns represents the anisotropy of cloud distribution over the area (cluster 3).
The generalization of the clustering model was performed throughout the investigation of the existence of the same six clusters in the test series. Similar patterns were found in the test series, validating the results found for the control series. Nevertheless, the new clusters revealed a differentiated proportion of images compared to the control series, indicating that the proposed clustering model can be generalized to any time series of images for the studied areas. The life cycle of convection can be captured from the clustering analysis. It shows that massive circular shapes require dynamically organized structures not common for most time series analyzed. Furthermore, the clustering analysis captured the diurnal cycle of convection and of cloud patterns.
From this analysis we demonstrated that the fractal box dimension can be a powerful tool in analysis of satellite images since it can discriminate a large range of fractal patterns in mesoscale convective systems. The signature of massive circular patterns in the behavior of the fractal box dimension can improve the diagnostics of such systems in an automatic way, without any other source of information. This may permit an improvement in climatological analysis over a wide range of time-and space scales since this analysis can be applied to large areas to monitor tropical thunderstorms and hurricanes.
Radiation versus cloud parameterizations have an implication on the surface energy budget and consequently on the life cycle of modeled precipitation systems (Betts and Ridgway 1988). Therefore, the patterns defined by Db and Dg and the other parameters used to describe convective activity in the cluster method may also be a useful way to address the problem of subgrid-scale distribution of cloudiness in numerical models.
Acknowledgments
We acknowledge the comments of D. Avinir, A. A. Tsonis, S. Lovejoy, and L. A. T. Machado, as well as the financial support of FAPESP and CNPq.
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(a) Dimensions Db and
Citation: Journal of Applied Meteorology 37, 10; 10.1175/1520-0450(1998)037<1265:AAOFBD>2.0.CO;2
Circles with external radius increasing from 10 to 50 pixels with a step of 5 pixels. Beneath each picture are the values of the external radii Rex, Ac, Db, and Dg and the standard deviation error.
Citation: Journal of Applied Meteorology 37, 10; 10.1175/1520-0450(1998)037<1265:AAOFBD>2.0.CO;2
Diagram of
Citation: Journal of Applied Meteorology 37, 10; 10.1175/1520-0450(1998)037<1265:AAOFBD>2.0.CO;2
Some examples of satellite IR images belonging to categories 1 [(a) and (b)], 3 [(c) and (d)], 4 [(e) and (f)], and 5 [(g) and (h)] for the control series: (a) southeast Brazil, day 25, 2200 UTC; (b) Amazon Basin, day 24, 2000 UTC; (c) Paraguay, day 25, 0100 UTC; (d) Paraguay, day 25, 0400 UTC; (e) southeast Brazil, day 24, 1100 UTC; (f) northern Brazil, day 25, 1500 UTC; (g) southeast Brazil, day 25, 1500 UTC; and (h) northern Brazil, day 24, 1000 UTC.
Citation: Journal of Applied Meteorology 37, 10; 10.1175/1520-0450(1998)037<1265:AAOFBD>2.0.CO;2
Increasing number of circles keeping constant the same equivalent area of a single circle with r = 30 pixels. The values of Db and Dg and the respective standard deviation error are presented beneath each picture. The last two pictures show the same nine circles with different configurations in the space: the picture on the left shows circles distributed isotropically, and the one on the right shows the circles grouped in the left corner of the area.
Citation: Journal of Applied Meteorology 37, 10; 10.1175/1520-0450(1998)037<1265:AAOFBD>2.0.CO;2
The same as Fig. 4 but for category 2: (a) Paraguay, day 25, 1400 UTC; (b) southeast Brazil, day 24, 0800 UTC; (c) northern Brazil, day 24, 2300 UTC; (d) and Paraguay, day 24, 2300 UTC.
Citation: Journal of Applied Meteorology 37, 10; 10.1175/1520-0450(1998)037<1265:AAOFBD>2.0.CO;2
The same as Fig. 3 but for the test series. Categories 6 and 7 are included in this case.
Citation: Journal of Applied Meteorology 37, 10; 10.1175/1520-0450(1998)037<1265:AAOFBD>2.0.CO;2
The same as Fig. 4 but for the categories in Fig. 7. Panels (e) and (f) represent categories 6 and 7, respectively: (a) northern Brazil, day 20, 2000 UTC; (b) Paraguay, day 21, 2300 UTC; (c) Bolivia, day 20, 1200 UTC; (d) southeast Brazil, day 20, 1600 UTC;(e) Amazon Basin, day 21, 0100 UTC; and (f) Amazon Basin, day 20, 1300 UTC.
Citation: Journal of Applied Meteorology 37, 10; 10.1175/1520-0450(1998)037<1265:AAOFBD>2.0.CO;2
Average value for each cluster obtained from PC analysis for the control series: (a) Db, (b) Dg, (c)
Citation: Journal of Applied Meteorology 37, 10; 10.1175/1520-0450(1998)037<1265:AAOFBD>2.0.CO;2
Some examples of images belonging to each cluster defined from PC analysis for the control series: cluster 1 [(a) and (b)], cluster 4 [(c) and (d)], cluster 2 [(e) and (f)], cluster 5 [(g) and (h)], cluster 3 [(i) and (j)], and cluster 6 [(k) and (l)]: (a) Paraguay, day 24, 1700 UTC; (b) Paraguay, day 25, 1400 UTC; (c) southeast Brazil, day 25, 0700 UTC; (d) amazon Basin, day 25, 1200 UTC; (e) southeast Brazil, day 25, 2000 UTC; (f) Amazon Basin, day 24, 1900 UTC; (g) Paraguay, day 24, 2400 UTC; (h) Paraguay, day 25, 0500 UTC. (i) Southeast Brazil, day 24, 0900 UTC; (j) northern Brazil, day 25, 1800 UTC; (k) southeast Brazil, day 24, 1600 UTC; and (l) northern Brazil, day 24, 0800 UTC.
Citation: Journal of Applied Meteorology 37, 10; 10.1175/1520-0450(1998)037<1265:AAOFBD>2.0.CO;2
Frequency distribution of α for clusters 2 and 5. Numbers at horizontal axes indicate the interval maximum.
Citation: Journal of Applied Meteorology 37, 10; 10.1175/1520-0450(1998)037<1265:AAOFBD>2.0.CO;2
Frequency distribution of the number of cells with Acl > 0.3% for clusters 1, 2, and 5 related to enhanced convection.
Citation: Journal of Applied Meteorology 37, 10; 10.1175/1520-0450(1998)037<1265:AAOFBD>2.0.CO;2
Same as Fig. 10 but for the generalized cluster model for the test series: (a) southeast Brazil, day 20, 2200 UTC; (b) southeast Brazil, day 21, 2300 UTC; (c) Paraguay, day 21, 0300 UTC; (d) southeast Brazil, day 21, 1400 UTC; (e) northern Brazil day 21, 1900 UTC; (f) northern Brazil day 20, 1700 UTC; (g) Paraguay, day 21, 2200 UTC; (h) Paraguay, day 21, 2300 UTC. (i) Paraguay, day 21, 1200 UTC; (j) Amazon Basin, day 21, 0400 UTC; (k) Amazon Basin, day 20, 1100 UTC; and (l) Amazon Basin, day 20, 1600 UTC.
Citation: Journal of Applied Meteorology 37, 10; 10.1175/1520-0450(1998)037<1265:AAOFBD>2.0.CO;2
Effect of increasing the number of internal circles with noncentered axes.
Citation: Journal of Applied Meteorology 37, 10; 10.1175/1520-0450(1998)037<1265:AAOFBD>2.0.CO;2
Time sequence of clusters for each studied area: (a) control series for Paraguay and northern Brazil, (b) control series for southeastern Brazil and the Amazon Basin, (c) same as (a) but for the test series, and (d) same as (b) but for the test series.
Citation: Journal of Applied Meteorology 37, 10; 10.1175/1520-0450(1998)037<1265:AAOFBD>2.0.CO;2
Studied area coordinates.
Coefficients of polynomial fit between Db and Dg vs Ac.
The weights of variables on the eigenvectors of principal components and the percent of variance explained for each component.
Seeds used as centroids of clustering analysis.
Seeds of the generalized clustering model.
