METHODS OF CHARACTERIZING CLOUD DROP SPECTRA SPATIAL VARIATION
Charles C. Ryerson*, Rae A. Melloh*, and George G. Koenig* U. S. Army Corps of Engineers Engineer Research and Development Center
Cold Regions Research and Engineering Laboratory Hanover, NH 03755-1290 U.S.A.
ABSTRACT
Cloud liquid water content (LWC), particle concentration, median volume droplet diameter (MVD), mixed-phase conditions, and spectra of drop and particle sizes vary in space and time. Aircraft traversing clouds accumulate ice as a function of the spatially changing microphysical conditions within the clouds. Remote sensing system performance also varies as cloud microphysical conditions change, thus potentially affecting information provided to pilots. Since remote sensing devices operating in millimeter wave (MMW) frequencies have some ability to operate within or through clouds, it is essential that a better understanding be developed of the fluctuation of cloud microphysical properties for modeling and simulation. This report investigates and evaluates three methods of quantifying the spatial variation of cloud drop spectra: 1) Drop Size Correlation, 2) Constrained Clustering, and 3) Spectral Shape Analysis. The three methodologies were evaluated against time-series of natural cloud measurements. The three techniques show promise as methods of quantitatively evaluating spatial fluctuation of drop size spectra. However, they vary considerably in ease of use and interpretation, and in the type and quality of information provided. Theory and examples of each are demonstrated and evaluations of the potential of each are presented.
INTRODUCTION
Quantifying the spatial variation of cloud is
critical to evaluating how remote sensing systems will perform when attempting to detect _______________ * Physical Scientist
icing conditions1 and assessing the impact of natural icing clouds on aircraft performance.2 The structure of clouds, especially the spatial variation of the drop and ice crystal sizes in clouds, impacts MMW radar and radiometer signatures of in-cloud icing conditions by changing scattering and attenuation relationships, which can cause misinterpretation of cloud signatures.3, 4
Ryerson et al.1, 5 explored the spatial fluctuation of cloud liquid water content (LWC) in natural icing conditions using techniques presented by Jameson and Kostinski.6 Though cloud physicists and remote sensing specialists have long understood that the spectra of cloud drop sizes vary in space and time, the spatial variation of spectra are not typically expressed when describing cloud properties. The objective of this paper is to demonstrate and evaluate three methods of assessing time-series variations, which are spatial variations, of cloud drop spectra; 1) Drop Size Correlation, 2) Constrained Clustering, and 3) Spectral Shape Analysis.
BACKGROUND
Cloud drop size spectra are usually
expressed as mathematical distributions over a measurement interval, with little regard for the actual space and time variation of spectra within that interval. A single mathematical distribution representing the entire interval averages or blurs the true physical distributions within the interval. While the total LWC of the mathematical distribution may reflect the true LWC, radar and radiometer cloud measurements are also controlled to some degree by variations of the drop spectrum. Cloud physicists typically do not quantify the variation of cloud drop sizes over space, remote sensing specialists do not have mathematical models of spatial variations of physical drop size distributions necessary for modeling radar backscatter or radiometer brightness temperatures accurately, and icing specialists cannot reliably model or simulate the
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42nd AIAA Aerospace Sciences Meeting and Exhibit5 - 8 January 2004, Reno, Nevada
variation in cloud conditions causing ice accretion on aircraft.
Cloud drop size spectra may be broad at one location, and narrow at another. The number of drops within any one portion of the size spectrum can vary considerably from place to place in a cloud. As a result, expression of the variation of the drop size spectrum is a complex problem, and one of the challenges is to determine the physical distributions rather than the artificial mathematical distributions.
A considerable amount of research has addressed the fluctuations of drop size and LWC at the sub-meter scale within clouds to assess turbulence, cloud evolution, and the radiative properties of clouds.7, 8, 9, 10 However, this work is at the tens-of-meters scale because that is the scale that is most relevant to remote sensing devices detecting aircraft icing conditions, such as the typical meters to tens-of-meters resolution of cloud radars.3, 4 Jameson and Kostinski have presented the most comprehensive ideas about quantifying the clustering of cloud properties in a series of papers beginning in 1997.11, 12 They describe clustering of drop sizes in rainfall and clouds using coherence length and a cluster parameter.
Clustering is the statistical correlation of cloud microphysical characteristics in one volume to those in another volume such that bunching of similar conditions occurs.12 Jameson and Kostinski13 explored methods of representing the size and intensity of cloud liquid water and drop size clusters using data they generated by making theoretical assumptions and applying them to drop size measurements. They defined the spatial distribution of cloud microphysical characteristics through clustering intensity parameters and coherence lengths, but did not address the shape of the entire spectrum, only the variance of specific drop sizes. We used this approach to analytically describe LWC clustering of actual icing clouds measured during NASA research flights.1 Unlike LWC, a single variable in time and space, cloud drops cover ranges of size that vary in time and space and cannot be readily represented by a single metric. In relation to cloud drop size spectra, Jameson et al.12 demonstrated how the texture of clouds could be expressed by computing two-point correlation functions of single-size drops. Though they used single drop sizes to express cloud texture, they suggest that all drop sizes could and need to be assessed.
TEST AND EVALUATION DATA
Our three methodologies were evaluated against synthetic drop size spectra series, not described in this report, and against natural cloud measurements from the NASA Glenn Research Center's (NASA GRC) Supercooled Large Drop Research Program (SLDRP).14 Cloud drop spectrum measurements were obtained from a Particle Measuring Systems (PMS) Forward Scattering Spectrometer Probe (FSSP) which typically operated in range 0 (2-47 µm diameter) or range 1 (5-95 µm diameter). The FSSP counts drops, by size, in 15-bins that are 3 or 6 µm wide. This study used 1-s counts from the FSSP uncorrected for sampling volume. We ignored drops larger than those measured by the FSSP. Though not accurately representing the full possible range of drop sizes, the measurements are sufficient to illustrate methodology. Measurements were selected from segments of three flights with time series of counts per FSSP bin that have different characteristics. Series 970306f2x is discontinuous, with many bins filled with particles (Figure 1). Series 970311f3x is a continuous series with broad spectra occupying many bins (Figure 2). Flight segment 980126f2x is composed of narrow spectra, though the bin width is 6 µm rather than 3 µm as with the other two series (Figure 3).
METHOD 1 - DROP SIZE CORRELATION
How can clusters, where drop size
distributions are sufficiently homogeneous to obey Poisson-like statistics, be identified? Kostinski and Jameson argue that an initially homogeneous “blob” will be distorted by turbulence and will, after some time, exhibit clustering or clumping.11 Unlike a random series, clustering exhibits correlations between clumps or clusters. However, identifying specific segments of a series that obey the Poisson probability law is a more difficult task. A physical interpretation of a cluster in terms of the drop-size concentrations can provide some insight on how to extract clusters from the overall series.
If we assume in a region that obeys the Poisson probability law that the physical processes responsible for generating the drop-size distribution do not change the shape of the distribution, then we can use this concept to define techniques to identify clusters. The fact that the shape of the distribution does not
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change does not mean the liquid water in these regions is constant. Since the liquid water is the
area under the curve defined by the drop-size distribution, the area under the curve can change by changing the shape of the curve, or by increasing (or decreasing) the drop count at each radii in a manner that retains the shape of the distribution. In clusters, variations in the ratio of the drop concentration for any two-drop radii should only exhibit Poisson distribution-like variations. This will not be the case in a region where the shape of the distribution is continually changing.
Starting with the Jameson and Kostinski15 approach, techniques were explored to extract regions that represent clusters using synthetic cloud series and the NASA flight series. The first step involved generating the accumulative drop
concentration for two radii (usually FSSP bin 0 and bin 1). The accumulative drop concentration essentially smooths the data since the bin counts are typically large numbers (accumulative concentrations) with embedded small changes in the drop concentration.
Upon inspection of the NASA data it was found that a number of data points contained no counts in any bin, or very low (and unrealistic) counts even in the small radius bins where normally the higher counts are found. It is not known if these missing or low counts are an artifact of the PMS system or of the aircraft moving in and out of clouds. In any case, points with concentrations less than ten drops in any one of the first two bins were removed from the data series.
Figure 4 is a scatter plot of the accumulative counts for FSSP bin 0 and bin 1. The bold line segments in the plot have been determined visually and represent regions of constant slopes, that is, regions of ‘steady drop size concentrations’. Again, since we are dealing with large numbers, small-scale fluctuations tend to be hidden (note the scale of the x and y axes in Figure 4). Therefore, this procedure will not allow us to unambiguously define the cluster vs. non-cluster regions in the data series.
The ratio of the difference of the accumulative drop concentration curves for two radii can overcome the limitation associated with just using the accumulative drop concentration information. Mathematically, the ratio is defined as (denoted as the Accumulative Difference Ratio-ADR)
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‘steady drop size concentrations’ can be achieved by plotting the ADR as defined in equation 1. Figure 5 is a plot of the ADR as a function of the one-second data. Clusters are regions of nearly constant values of the ADR. The bold, diamond-terminated lines in Figure 5 are segments that correspond to the regions with nearly constant slope in Figure 4. The bold broken line near the top of Figure 5 denotes clusters as determined using an automated procedure consisting of generating a 30-point (30 second) running average and the standard deviation associated with the 30 points. Next, the ratio of the 30-point standard deviation
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divided by its average for each segment was calculated. If the 30-point running standard deviation is small relative to its average the calculated ratio will be small. This implies that the value of the individual points do not vary significantly from the average. A value of 0.2 for the ratio of the standard deviation divided by the average was arbitrarily selected to define potential clusters in the data series. A plot of the 30-point standard deviation to the average is given in Figure 6 for flight segment 980126f2x.
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Figure 4. Scatter plot of accumulative counts applied to flight segment 980126f2x. Bold lines represent cluster locations.
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Figure 5. ADR for flight segment 980126f2x. The upper horizontal broken bold line shows the location of clusters using the standard deviation-to-average procedure.
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970306f2x were preprocessed using the same procedures. A significant number of points had low bin counts and were not included in the analysis. The resulting data series is suspect since this introduces gaps that were essentially ignored by joining the individual segments to form a continuous series. The resulting analysis is presented in Figures 7, 8 and 9. Again, clusters were determined visually and using the
automated procedure outlined above with a threshold of 0.2. There is a fair degree of consistency between the visual and automated techniques used to define cluster locations in the data series. The automated procedure resulted in a higher frequency of short duration cluster segments. This is, in part, due to the inability to visually identify these short segments on the accumulative drop concentration plot.
The analysis was done using only two bins. For most of the NASA data sets there were at least three bins and sometimes as many as five bins with sufficient counts to perform the analysis. One possible enhancement to the analysis would involve performing it for all possible bin pair combinations and restricting clusters to those regions that were identified as clusters on all bin pair combinations (or about 90 percent of the bin pair combinations).
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Figure 6. Plot of 30 - second average, standard deviation, and the ratio of the standard deviation divided by the average for NASA data 980126f2x. The horizontal line defines clusters (values <= 0.2 of the ratio of the STDEV/AVG).
METHOD 2 - CONSTRAINED CLUSTERING
The second method, constrained clustering, has the ability to consider magnitude of drop counts as well as the drop size distribution. The data can be preprocessed so that just the distribution is considered, but that was not done in the example presented here. Clusters are identified as contiguous segments of similar drop size distribution and magnitude separated by transitions. The transitions are made up of small clusters and individual points that are dissimilar in drop size distribution and magnitude.
The procedure is most easily visualized as a 2- or 3-dimensional plot, but the analysis extends to n-dimensions. In 3-dimensions each of the three axes correspond to drop counts for
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three different bin sizes, for example FSSP bins 0, 1 and 2. At the initial clustering step drop counts for each flight second are represented by a single point plotted in 3-dimensional space with coordinates x, y, and z corresponding to the counts in FSSP bins 0, 1 and 2. In subsequent steps, the most similar points are joined into successively larger clusters of points. The
Figure 7. Scatter plot of accumulative counts applied to flight segment 970306f2x. Bold lines represent cluster locations. joinings are constrained to points or clusters immediately preceding or following one another in the flight time series by the rules of the minimum variance method.16 The sum of the squared Euclidean distances
E (a, b) = ∑ i (ai - bi)2 (2) between the points within each potential new cluster are then calculated. The two clusters that will result in the least increase in within-cluster variance are joined into one larger cluster. The process, if not stopped, continues until all points are contained in one large cluster. The process of combining can be stopped after a given number of combinations have occurred or a given similarity level (within-cluster variance) has been exceeded. Specifying a stopping point sets the number and size of clusters, and conversely the number and size of transitions that result. The analyst must decide what similarity level, or number of joinings, is most appropriate for the data set and purpose of the analysis. This may require trial-and-error or some objective criteria such as the inflection point of the dissimilarity versus number of joinings plot (Figure 10). For our purposes, the inflection point appears not to have enough joinings, and roughly the half-way mark between the inflection point and process end gave a more
reasonable separation of larger clusters and transitions.
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Figure 8. Ratio of the difference of the accumulative drop concentration for flight segment 970306f2x. Broken horizontal lines show the location of clusters in Figure 7. The upper horizontal bold line shows the location of clusters using the standard deviation to average procedure.
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Figure 9. Thirty second average, standard deviation, and the ratio of the standard deviation divided by the average for flight segment 970306f2x. The horizontal line defines clusters (values <= 0.2 of the ratio of the STDEV/AVG).
Clusters and transitions for a segment of the 980126f2x flightline (Figure 11) are identified by the presence or absence of vertical lines that mark the boundaries of clusters. Segments without vertical lines are the clusters, and segments with many vertical lines are the transitions. A Constrained Clustering determination compared to the other methods presented in this paper shows that the methods agree, in some cases, and not in others (Figure 11). Some of the differences in this preliminary comparison of the methods are because the constrained clustering method considers magnitudes, and the others do not. A close-up view of the cluster boundaries (Fig. 12) for flight 980306f2x show transitions located at the edges
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of intermittent cloud segments where count magnitudes are changing.
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Figure 10. Distances between clusters increase for incremental nodal combinations.
The minimum variance method can be used
to separate the time series into true clusters and transitions, and can provide frequency distributions of cluster sizes. The time series is partitioned into discrete clusters, some large, and many small. The large clusters represent contiguous areas of similar drop size distribution and magnitude and are considered to be the true clusters. Transition zones are represented by contiguous areas of numerous small clusters where drop size distributions or magnitudes vary rapidly. The frequency distribution of cluster sizes can be presented as a histogram with a threshold size separating clusters and transitions (Figure 13). A histogram showing the frequency of occurrence of cluster sizes can be
used to select a threshold size separating clusters and transitions, and also to show the distribution in size of the true “larger” clusters. METHOD 3 - SPECTRAL SHAPE ANALYSIS
For this paper, we developed cloud drop size spectra by counting the number of drops in each FSSP size bin per second of flight. The shape of the drop size spectrum created by drop counts per bin can be represented using well-established mathematical functions that describe the shape of the entire spectrum with a few terms of one equation. The equation terms, or shape parameters, change as the shape of the distribution changes. Therefore, the terms of the fit equations can be investigated to assess the location and magnitude of clusters, and cluster-to-cluster variation. However, the challenge lies in determining which equation terms, or shape parameters, are most fundamental in describing the distribution shape, and then how to analyze the variation of the terms. We assumed all distributions to be unimodal. Curve Fit Procedure
The method was initially explored by creating synthetic time series of known clustering conditions using clouds defined by a modified gamma distribution.17 Synthetic
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Figure 11. Squared Euclidian distance versus number of joinings (nodal combinations) for flight segment 980126f2x compared to Drop Size Correlation method identifying clusters with standard deviation divided by average method.
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Figure 12. Closeup view of cluster boundaries intermittent cloud reach of flightline 970306f2x.
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Figure 13. Histogram of cluster sizes extracted from flight 980126f2x using Constrained Clustering methodology. series were used to assess if changing shape parameters of selected curve fits could discriminate where clusters were located. Only tests using the NASA natural cloud series, however, are presented here.
Cloud drop spectra are typically represented by the beta, gamma, and lognormal distributions.17 Five curve types were fitted to the synthetic and natural drop distributions; the beta, gamma, lognormal, extreme value, and inverted gamma. Since the primary interest of the Spectral Shape Analysis was to represent changes in curve shape, and not changes in curve magnitude, all distributions were normalized against the bin with the largest drop count in that distribution. Normalization also allowed creation of zero intercept equations reducing the number of equation terms.
The purpose of the Spectral Shape Analysis was to fit all distributions with the same fundamental type of curve fit allowing the shape parameters to be analyzed for change of shape. Therefore, each flight was fitted to each of the listed curve types. Curves for each 1-sec distribution were fitted automatically using Systat's TableCurve 2D software. Curves were fitted until a minimum least square fit was
obtained. Figure 14 shows example fits for each of the curve fit types for the natural spectra for flight 980126f2x. Flight 980126f2x typically had large counts in only a few bins, so curve fits occasionally provided small r2 values. Flight 970306f2x was not analyzed because it contained periods with few or no cloud drops to which curves could not be fit. This is a potential weakness of this method - cloud sequences with short segments of clear air cannot generate curve fits and thus prevent a full analysis.
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Cluster identification involved analysis of curve equation terms as they changed with spectrum shape change. This assessment was accomplished subjectively by changing the coefficients of each term, holding others constant, and observing the effects on curve shape. Seven terms were selected from the five curve types for cluster analysis (Table 1). Cluster Analysis
Two approaches were used to assess clustering from the resulting time-series of curve fit equation terms. The first method, the standard deviation to average ratio method, involved applying techniques similar to those developed in the Drop Size Correlation method, described above, dividing the 30-point running standard deviation by the 30-point running average. However, instead of calculating the ratio of the running standard deviation and average of drop
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counts, ratios of running standard deviations and averages of equation terms were computed.
The second approach used methods, described by Jameson and Kostinski6 for assessing clustering of cloud supercooled liquid water content to assess clustering of individual equation terms. This provides an indication of cluster intensity.
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Figure 15. Cluster locations indicated by horizontal lines for each of the seven curve fit terms for flight segment 970311f3x. Standard Deviation to Average Ratio Method
The running standard deviation to average ratio method was used to discriminate cluster locations by thresholding the ratio at 20% of its maximum value during each flight segment. This method was effective for equation terms responding with all positive values over a small numerical range. However, four of the equation terms for flight segment 970311f3x, and one term for flight segment 980126f2x, created series with large ranges of positive and negative values. In these cases, the absolute value of all terms was taken, and cluster thresholds were arbitrarily chosen when ratios were less than 4.0.
Figures 15 and 16 show the results of these analyses for each of the seven equation terms used to locate clusters. Though the seven terms do locate areas of clustering and areas that are not clustered, each term varies in sensitivity and thus each presents a generally different pattern of clustering. This is especially true for Figure 15, where four curve fit terms (beta b, gamma b, inverted gamma b, extreme value b) had large ranges with negative values which required using the value of 4.0 described above as a cluster threshold. Figure 16 suggests somewhat
more consistency of pattern among terms than does Figure 15. Jameson and Kostinski Method
The Jameson and Kostinski'6 cluster
intensity is a relative bulk cluster intensity value, with small numbers indicating little clustering and essentially a Poissonian series. Their cluster intensity value cannot be used to locate clusters, it simply indicates the relative clustering intensity of one series relative to another series. Attempts to compute correlation length to indicate the representative size of clusters within a series resulted in occasional negative values and thus are not presented here.
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Figure 16. Cluster locations indicated by horizontal lines for each of the seven curve fit terms for flight segment 980126f2x.
As with the standard deviation to average ratio method, cluster intensities of the terms within the two natural cloud series were considerably different in magnitude (Table 1). In general cluster intensities were larger for series 970311f3x, except for the inverted gamma c term. The widely differing magnitudes of cluster intensities between terms within a cloud series may be attributed, again, to the non-linearity and sensitivity of many of the function terms to changes in drop size spectrum shape.
Spectral Shape Analysis Discussion
Results suggest that the Spectral Shape Analysis method may identify the existence of clusters within a drop size spectra series. However, the method is not as effective as anticipated because it trades efficiencies in
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describing a spectrum using a single equation with complexities related to selecting non-linear shape factors and their interpretation. In addition, the fidelity of the method is limited by the ability of any one curve fit method of faithfully representing drop spectrum shape with accuracy for all spectrum shapes. This is especially true if the spectra are multi-modal, as can occur with drizzle drops present. In addition, analyzing and comparing terms from different curve fit functions within a single flight can cause interpretation difficulties. The standard deviation to average ratio method appears to be the most effective for analyzing equation term series.
Table 1. Cluster intensity for function terms. 970311f3x 980126f2x Beta b 1.93 0.07 Gamma b 69.18 0.06 Lognormal b 0.14 0.015 Invgamma b 11.70 0.05 Invgamma c 0.47 4.23 Extreme value b 22.86 0.44 Lognormalic c 0.15 0.014
Invgamma = Inverted gamma. Lognormalic = lognormal with constant.
EVALUATION
Our goal is to develop methods to quantify spatial variability of cloud drop size spectra, and to show proof of concept. Evaluation of the Drop Size Correlation, Constrained Clustering, and Spectral Shape Analysis methods indicates that each has the potential for identifying the location and duration of clusters. In addition, each method is logically sound. We have yet to complete a thorough comparison of results from each of the methods. Figure 11 compares the capability of our methods for locating and sizing clusters in flight 980126f2x. The horizontal broken lines in Figure 11 indicate where each method shows clusters to exist. Only the beta curve of the seven curve fit methods used in the Spectral Shape Analysis is compared.
Figure 11 does not indicate whether the locations of clusters have been correctly chosen since this is a natural cloud series and the actual locations of clusters is not known. Additional work is needed to determine if any of the methods compare well using synthetic and natural cloud series.
FUTURE WORK
Our next tasks are to perform more thorough comparisons, and to provide more complete information about the logic and capabilities of each technique. We will then demonstrate their capability to identify cluster location, cluster intensity, and cluster duration and make recommendations to the cloud physics community.
ACKNOWLEDGMENTS
Funding originated from Army Corps of
Engineers Engineer Research and Development Center Cold Regions Research and Engineering Laboratory AT-59 work item 0087D2 Spatial Variation of Drop Size Spectra in Icing Clouds and the NASA Aerospace Operations Systems Program managed by NASA-GRC. This research is also in response to requirements and funding by the Federal Aviation Administration (FAA). The views expressed are those of the authors and do not necessarily represent the official policy or position of the Army, NASA, or the FAA. The authors express their appreciation to NASA-GRC for providing cloud data, and to M. Politovich of NCAR and F. Scott of SCA Inc. for thoughtful technical reviews.
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American Institute of Aeronautics and Astronautics
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