ESTIMATING THE CORRELATION DIMENSION OF ATMOSPHERIC TIME-SERIES

The correlation dimension D is commonly used to quantify the chaotic s tructure of atmospheric time series. The standard algorithm for estima ting the value of D is based on finding the slope of the curve obtaine d by plotting ln C(r) versus ln r, where C(r) is the correlation integ ral and r is the distance between points on the attractor. An alternat ive, probabilistic method proposed by Takens is extended and tested he re. This method is based on finding the sample means of the random var iable (r/rho)(p)[ln(r/rho](k), expressed as the conditional expected v alue E((r/rho)(p)[ln(r/rho)](k): r < rho), for p and k nonnegative num bers. The sensitivity of the slope method and of the extended estimato rs D-pk(rho) for approximating D is studied in detail for three ad hoc correlation integrals and for integer values of p and k. The first tw o integrals represent the effects of noise or undersampling at small d istances and the third captures periodic lacunarity, which occurs by d efinition when the ratio C(x rho)/C(rho) fails to converge as rho appr oaches zero. All the extended estimators give results that are superio r to that produced by the most commonly used slope method. Moreover, t he various estimators exhibit much different behavior in the two ad ho c cases: noise-contaminated signals are best diagnosed using D-11(rho) , and lacunar signals are best studied using D-0k(rho), with k as larg e as possible in magnitude. Therefore, by using a wide range of values of p and k, one can infer whether degradation arising from noise or a rising from lacunarity is more pronounced in the time series being stu died, and hence, one can decide which of the estimates most efficientl y approximates the correlation dimension for the series. These ideas a re applied to relatively coarse samplings of the Henon, Lorenz convect ion, and Lorenz climate attractors that in each case are obtained by c alculating the distances between pairs of points on two trajectories. As expected from previous studies, lacunarity apparently dominates the Henon results, with the best estimate of D, D = 1.20 +/- 0.01, given by the case D-03(rho). In contrast, undersampling or noise apparently affects the Lorenz convection and climate attractor results. The best estimates of D are given by the estimator D-11(rho) in both cases. The dimension of the convection attractor is D = 2.06 +/- 0.005, and that of the climate attractor is D = 14.9 +/- 0.1. Finally, lagged and emb edded time series for the Lorenz convection attractor are studied to i dentify embedding dimension signatures when model reconstruction is em ployed. In the last part of this study, the above results are used to help identify the best possible estimate of the correlation dimension for a low-frequency boundary layer time series of low-level horizontal winds. To obtain such an estimate, Lorenz notes that an optimally cou pled time series must be extracted from the data and then lagged and e mbedded appropriately. The specific kinetic energy appears to be more closely coupled to the underlying low-frequency attractor, and so more nearly optimal, than is either individual wind component. When severa l estimates are considered, this kinetic energy series exhibits the sa me qualitative behavior as does the lagged and embedded Lorenz convect ive system time series. The series is either noise contaminated or und ersampled, a result that is not surprising given the small number of t ime series points used, for which the best estimate is given by D-11(r ho). The obtained boundary layer time series dimension estimate, 3.9 /- 0.1, is similar to the values obtained by some other investigators who have analyzed higher-frequency boundary layer time series. Althoug h this time series does not contain as many points as might be require d to accurately estimate the dimension of the underlying attractor, it does illustrate the requirement that in any estimate of the correlati on dimension, a function of the measured variables must be chosen that is strongly coupled to the attractor.