The correlation dimension D is commonly used to quantify the chaotic structure of atmospheric time series. The standard algorithm for estimating the value of D is based on finding the slope of the curve obtained by plotting ln C(r) versus ln r, where C(r) is the correlation integral and r is the distance between points on the attractor. An alternative, probabilistic method proposed by Takens is extended and tested here. This method is based on finding the sample means of the random variable (r/rho)(p)[ln(r/rho](k), expressed as the conditional expected value E((r/rho)(p)[ln(r/rho)](k): r < rho), for p and k nonnegative numbers. The sensitivity of the slope method and of the extended estimators 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 two integrals represent the effects of noise or undersampling at small distances and the third captures periodic lacunarity, which occurs by definition when the ratio C(x rho)/C(rho) fails to converge as rho approaches zero. All the extended estimators give results that are superior to that produced by the most commonly used slope method. Moreover, the various estimators exhibit much different behavior in the two ad hoc cases: noise-contaminated signals are best diagnosed using D-11(rho), and lacunar signals are best studied using D-0k(rho), with k as large 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 arising from lacunarity is more pronounced in the time series being studied, and hence, one can decide which of the estimates most efficiently approximates the correlation dimension for the series. These ideas are applied to relatively coarse samplings of the Henon, Lorenz convection, and Lorenz climate attractors that in each case are obtained by calculating 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 embedded time series for the Lorenz convection attractor are studied to identify embedding dimension signatures when model reconstruction is employed. 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 coupled time series must be extracted from the data and then lagged and embedded 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 several estimates are considered, this kinetic energy series exhibits the same qualitative behavior as does the lagged and embedded Lorenz convective system time series. The series is either noise contaminated or undersampled, a result that is not surprising given the small number of time series points used, for which the best estimate is given by D-11(rho). 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. Although this time series does not contain as many points as might be required to accurately estimate the dimension of the underlying attractor, it does illustrate the requirement that in any estimate of the correlation dimension, a function of the measured variables must be chosen that is strongly coupled to the attractor.
