Comparative Fractal Dimensions for Daytime and Nocturnal Surface Layer Turbulence

Presented at the 11th Symposium on Boundary Layers and Turbulence, American Meteorological Society, 27-31 March 1995, Charlotte, NC

1. Introduction

The term fractal dimension includes many calculation methods such as capacity, correlation, and information dimensions. The differences between these measures of fractal dimension can be illustrated through a simple example. Imagine a line of finite length, L, consisting of uniformly distributed points. Cover the line with a minimum number of squares, N(e), with sides of length e. If the sides of the squares were reduced by half, twice as many squares would be required to cover the same line. If a D-dimensional object were used in place of the line, a relationship of N(e) = 1/ed is observed, where d is the capacity or box-counting dimension for non-integer d. The capacity dimension lacks information on frequency of orbit visits to each square or box; it is only a geometric measure of complexity (Theiler, 1990). The information dimension quantifies the non-uniformity of points on an attractor by weighing the boxes proportional to the number of data points contained in each; this gives the probability of finding a point in a box. If all data points were distributed uniformly among boxes, the information and capacity dimensions would be equal. Correlation dimension is based on pairwise distances. One method calculates distances between every pair of data points to determine the number of pairs less than a distance r. A second method constructs spheres of radius r at each point and counts the number of points in each sphere (Moon, 1992).

The present study is potentially more comprehensive than previous studies for several reasons:

1. Multiple days/data series (four days and four nights) were used instead of the usual single day time series.
2. The length of each data series is 108,000 data points at 10 Hz (3 hours) instead of under 10,000 data points as commonly used.
3. Unlike most studies which focus only on daytime measurements, this study compares daytime and nocturnal results. Data measurement times were around noon and midnight when the boundary layer was strongly convective and stably stratified, respectively, and in a quasi-steady state.
4. Multiple levels, 10, 22, 50, and 150 meters (from the Boulder Atmospheric Observatory tower) were used for comparison unlike the commonly used single level in most studies.
5. Most studies use the Grassberger and Procaccia method to find only the correlation dimension; here, we have used a new algorithm by Sarraille and DiFalco (1992), FD3, to calculate the capacity, information, and correlation dimensions.

2. Methodology

Sarraille and DiFalco (1992) use a generalized fractal dimension formula by which the capacity, information, and correlation dimensions may be calculated in N(logN) time:

I(q,e) = (1-q)-1 log [sum [ P(m,e)^q ]]

where the sum is from m=1 to N(e). Here P(m,e)^q is the q power of the proportion of the points of the set that lie in the ith box of a minimal covering by boxes of size e. N(e) is the number of boxes in such a covering. These boxes may be cubes, disks, or spheres. The generalized dimension is then defined as D(q) = I(q,e)/log e. D(q) represents the capacity dimension for q=0, information dimension for q=1, and the correlation dimension for q=2. Initial testing of the algorithm gave excellent results when compared to actual fractal dimensions of the Henon system, Cantor set, Koch curve, and Lorenz system, yielding results with less than about 5% error (Table 1).

To determine the fractal dimension, first the data series had to be embedded into higher dimensional space, using a time delay scheme devised by Packard et al. (1980) to reconstruct the state space. The goal is to find an embedding dimension, M, greater than the actual dimension, D, of the system. Strictly, we must have M > 2D+1 to reconstruct the dynamical state space; however, the reconstructed set with M > D will almost always have the same dimension as the attractor (Eckmann and Ruelle, 1985). The fractal dimension is computed for at least M embedding dimensions and D versus M is plotted as in Figure 1. A graphical plateau should be distinguishable in the true fractal dimension region, but the final choice of the fractal dimension is subjective.

The quality of the reconstructed state space depends on the choice of the time delay, t (Fraser and Swinney, 1986). For noise-free data, the tme delay can in theory be chosen arbitrarily, but geophysical data contain noise. The time delay is defined in terms of decorrelation time of the data series and the choice of computation for this study is t = w/(M-1), where w is the data point at a specified target autocorrelation value. However, this process measures only linear dependence; although linear dependence methods are most commonly used, the consensus is the new method of mutual information is more efficient since it measures general dependence.

In addition to the fractal dimension, the power spectrum was calculated and spectral frequencies were averaged and graphed. Vertical velocity spectra were indicative of the inertial subrange structure, displaying the -5/3 slope for day and night data. The temperature spectra resembled typical boundary-layer as those published by Kaimal and Finnigan (1994); daytime spectra display a -5/3 slope, while nocturnal spectra show a -5/3 slope with perhaps some influence from gravity waves.

3. Results

The FD3 program used to determine the fractal dimensions suggests the limitation of N > 2^4D - one of the most stringent limitatison in the literature, where N is the total number of data points and D the fractal dimension. Therefore, the dimension estimates did not converge quickly enough to give numeric results for embedding dimensions greater than about 4.5. However, nighttime fractal dimensions were consistently smaller than for daytime. The data series length used in most of the previous studies (< 10,000) would not satisfy this criterion. Greater convergence to the actual fractal dimension was observed for nocturnal results. Varying degrees of convergence were observed for daytime results such as in Figure 1, suggesting a fractal dimension limit well under 15, but greater than daytime results reported.

Overall, the night means were less than the day means by at least 0.30 for capacity, 1.05 for information, and 1.32 for correlation dimension. The maximum difference between day and night means were 0.947 for capacity, 2.013 for information, and 2.603 for correlation dimension. For the capacity and information dimension the average difference for temperature is larger than vertical velocity, but vice-versa for correlation dimension. The results tend to agree with those of Tsonis and Elsner (1989) and Pool (1989) for daytime wind velocities, and Poveda-Jaramillo and Puente (1993) for both daytime wind velocities and temperatures.

4. Discussion

Using FD3 on data from four days and four nights yielded fractal dimensions showing systematic differences between stratified and convective turbulence. These results underestimate the true difference between day and night since FD3 has the limitation of about N > 2^4D; that is, night time values converged reasonably well to the resulting fractal dimensions, but daytime results did not always converge, implying underestimations. It suffices to say that daytime values must exceed nighttime values. With the stated N > 2^4D estimation, a value of N=108300 yields D < 4.1 - obviously not strictly adhered to by FD3 since results of up to about 5 were obtained. Essex and Nerenberg (1991) proposed N > 2D(D+1) points necessary to reconstruct the attractor and Ruelle (1990) proposed N > 10D/2. The topic is still unresolved.

Other sources of error include the variable choice(s), the choice of t, and the data series length. Lorenz (1991) determined that the quality of the estimated fractal dimension is dependent on the length of the data series and the data series variable. If the length of the data series is not large enough, the estimation of the fractal dimension depends greatly on the variable chosen to model the attractor; that is, different variables may yield different fractal dimension estimates (usually underestimates). The problem is determining the suitable variable and overcoming data series length limitations of most fractal dimension algorithms.

5. Acknowledgements

We would like to thank Carlos Puente and John Sairraille for their willingness and speed in answering our questions; Dr. John Gaynor of NCAR for assisting in acquiring the data sets; and Dr. Albert Sheu for explaining mathematical aspects of time delay and series embedding. This research was funded by NSF grant ATM-9319455 and by Iowa Agriculture and Home Economics Experiment Station project 3245.

6. References

Eckmann, J.P. and D. Ruelle, 1985: Ergodic theory of chaos and strange attractors. Reviews of Modern Physics, 57, 617-656.

Essex, C. and Nerenberg, M.A.H.: 1991, Comment on "Deterministic Chaos: The Science and the Fiction" by D. Ruelle. Proc. R. Soc. Lond. A, 435, 287-292.

Fraser, A.M. and H.L. Swinney, 1986: Independent coordinates for strange attractors from mutual information. Phys. Rev. A, 33, 1134-1140.

Kaimal, J.C. and J.J. Finnigan, 1994: Atmospheric Boundary Flows: Their Structure and Measurement. Oxford University Press, 280pp.

Lorenz, E.N., 1991: Dimension of weather and climate attractors. Nature, 353, 241-244.

Moon, F.C., 1992: Chaotic and Fractal Dynamics: An Introduction for Applied Scientists and Engineers. John Wiley and Sons, Inc.

Packard, N.H., J.P. Crutchfield, J.D. Farmer, and R.S. Shaw, 1980: Geometry from a time series. Physical Review Letters, 45, 712-716.

Pool, R., 1989: Is there something strange about the weather? Science, 243, 1290.

Poveda-Jaramillo, G., and C.E. Puente, 1993: Strange attractors in atmospheric turbulence. Boundry-Layer Meteorology, 64, 175-197.

Ruelle, D.: 1990, Deterministic chaos: the science and the fiction. Proc. R. Soc. Lond. A., 427, 241-248.

Sarraille, J. and DiFalco, 1992: FD3 (program for finding fractal dimension). Computer Science Department, CSU Stanislaus, Turklock, CA 95380.

Theiler, J., 1990: Estimating fractal dimension. Journal of the Optical Society of America A, 7, 1055-1073.

Tsonis, A.A. and J.B. Elsner, 1989: Chaos, strange attractors, and weather. Bulletin of the American Meteorological Society, 70, 14.

Zeng, X., R.A. Pielke, and R. Eykholt, 1991: Estimating the fractal dimension and the predictability of the atmosphere. J. Atmo. Sci., 49, 649-659.

Attractor   Number of     Capacity    Information   Correlation 
             Points      Dimension     Dimension     Dimension

Lorenz       5000          1.82          1.93          2.00
(actual)              (2.05+/-0.01)                (2.06+/-0.01) 

Henon        2500          1.22          1.18          1.14  
(actual) 	        (1.25+/-0.02)                 (1.26) 

Cantor       1000          0.66          0.65          0.64 
(actual)                  (0.63)                     

Koch         3073          1.30          1.31          1.3 
(actual)                  (1.26)