Search

Recent studies have shown that a useful scheme to characterize subsurface heterogeneity is to use fractional Brownian motion (fBm) and fractional Gaussian noise (fGn). Commonly used techniques to detect these fractal distributions include rescaled range, spectral, and semivariogram analyses. Although rescaled range analysis is considered to be superior to spectral or semivariogram techniques, it cannot be used to clearly discriminate fGn and fBm from a given data set in some cases. On the basis of the fact that both adjusted range and rescaled range analyses, for a true fGn data set, should yield the same slope value in the relevant log-log plots, a simple scheme is developed to discriminate fGn and fBm using a combination of the two analyses. The usefulness of the proposed scheme is evaluated with one-dimensional computational experiments and by an application to a data set of observed air permeability logs. Lack of clear discrimination may have contributed to inconsistent past results, and it is recommended that old data sets be reanalyzed.

After more than 40 years the so-called Hurst effect remains an open problem in stochastic hydrology. Historically, its existence has been explained either by preasymptotic behavior of the rescaled adjusted range R*n, certain classes of nonstationari ty in time series, infinite memory, or erroneous estimation of the Hurst exponent. Various statistical tests to determine whether an observed time series exhibits the Hurst effect are presented. The tests are based on the fact that for the family of processes in the Brownian domain of attraction, R*n/((0n))1/2 converges in distribution to a nondegenerate random variable with known distribution (functional central limit theorem). The scale of fluctuation 0, defined as the sum of the correlation function, plays a key role. Application of the tests to several geophysical time series seems to indicate that they do not exhibit the Hurst effect, although those series have been used as examples of its existence, and furthermore the traditional pox diagram method to estimate the Hurst exponent gives values larger than 0.5. It turned out that the coefficient in the relation of/?* versus n, which is directly proportional to the scale of fluctuation, was more important than the exponent. The Hurst effect motivated the popularization of I//noises and related ideas of fractals and scaling. This work illustrates how delicate the procedures to deal with infinity must be.

[1] Rescaled range analysis is one of the classical methods used for detecting and quantifying long-term dependence in time series. However, rescaled range analysis has been shown in several studies to give biased estimates of the Hurst exponent in finite samples. A new estimator based on a modified expression of the expected value of the rescaled range is proposed. A comparison of the modified estimator with other alternatives shows that the modified estimator offers a great improvement over the classical rescaled range estimator in terms of bias and root mean square error, which makes it comparable to some of the leading estimators. The application of the proposed modified rescaled range estimator to a group of temperature, rainfall, river flow, and tree-ring time series in the Midwest USA demonstrates the extent to which classical rescaled range analysis can give misleading results. Based on a statistical test of significance, the number of time series exhibiting the Hurst effect is reduced from 36 to only 11 out of 56 temperature series, and from 23 to 7 out of 60 rainfall series. On the other hand, the number of time series exhibiting the Hurst effect marginally increased from 8 to 9 out of 49 river flow series, and from 49 to 54 out of 88 tree-ring series.

This note illustrates that certain shifting level processes possess long-term dependence and preserve the Hurst effect for the rescaled adjusted range. Although such processes belong to a stable domain of attraction, and not that of fractional Brownian motion, they do have asymptotic behavior of the rescaled adjusted range identical to that of fractional Brownian noise; further, they are easily simulated. Other shifting level processes belonging to the same stable domain of attraction preserve the Hurst effect for the raw and adjusted ranges but not for the rescaled adjusted range.

Turbulent open channel flows are studied in a laboratory by using ‘Lego’ blocks at various spacings on the bed to examine the effect of different shear Reynolds numbers on the fine-scale structure. Measurements of dissipation rates, computed by various methods, are compared and the high values shown to be reasonable when viewed against local mean shear values. The fine-scale structure and the intermittent nature of the small-scale motions are examined spectrally and by studying the moments. The ‘rescaled range,’ a technique developed in communications work (Mandelbrot and Van Ness, 1968) and in geophysical time series analysis (Klemes, 1974), is utilized to investigate the possible scale interactions among the various processes of production and dissipation in a turbulent boundary layer.

The expected value of the rescaled range R/S, which is a measure of long-term persistence in hydrology, has analytically been obtained by making use of t distribution. The application of the model has been performed for various hydrological processes. Finally, the asymptotic equivalence for expectations of the range and the rescaled range has been proven.

Records of turbulent velocity fluctuations from both laboratory flumes and rivers exhibit the Hurst phenomenon, R/S ∽ sH, where R is the range of the cumulative sum of departures from the mean for a time series of length s, S is the standard deviation of the time series, and 0.5 ≤ H ≤ 1. The Hurst phenomenon suggests long-term persistence in a time series. The implications in turbulent flows are that the integral time scales do not exist and, rather, that the turbulent velocity fluctuations have the properties of fractional Gaussian noise. Some indirect evidence from dispersion studies supports these implications.

Continuing the preceding paper, we shall report on computer experiments concerning the ‘rescaled range’ and the spectrum of fractional Gaussian noise.

The rescaled range R(t, s) / S(t, s) is shown by extensive computer simulation to be a very robust statistic for testing the presence of noncyclic long run statistical dependence in records and, in cases where such dependence is present, for estimating its intensity. The processes examined in this paper extend to extraordinarily non-Gaussian processes with huge skewness and/or kurtosis (that is, third and/or fourth moments).

Experimental observation of a long-established theoretical result for the behavior of the rescaled range is reported. Suitable dimensional scaling is found to collapse all available geophysical and laboratory data for the rescaled range in turbulent flows.

Nordin et al. [1972] have analyzed a number of records of turbulent velocity fluctuations in several open channels and have concluded that within the range of their observations the rescaled range of the velocity deviation increases faster than the square root of time. This implies that turbulent fluctuations have a longer memory, or correlation time, than the period of the observations. This letter suggests a possible explanation. When the writer was beginning his Ph.D. dissertation experiments [Fischer, 1966], he made some visual experiments in which a line source of a small quantity of intense blue dye was dropped onto the surface of a uniform flow. The flow was approximately 6 cm deep in a rectangular, smooth-bottomed channel 110 cm wide. The dye was dropped onto the water surface to form an initially straight line perpendicular to the flow direction. Invariably, the line did not remain straight. Soon after hitting the water surface some of the dye was carried forward of the line in the shape of a series of tongues. Between the tongues some of the dye lagged behind the line and formed a series of tails. Figure 1a is a sketch of the appearance of the line shortly after entering the water. The striking observation was that each time the experiment was repeated the tongues and tails were separated by a distance of approximately twice the depth of flow.