A fractal-multifractal approach to groundwater contamination. 2. Predicting conservative tracers at the Borden site

 In a companion paper (Puente et al., this issue), the capability of a deterministic fractal-multifractal (FM) approach to faithfully and compactly describe the geometry of chloride and bromide tracers gathered at the Borden site was illustrated. As trends in surrogate parameter space were found for successive plume contours (i.e. linear growth in coordinates by which fractal interpolating functions pass, nearly constant rotations and fairly high scalings), this article reports usage of a variety of prediction schemes, based on linear regressions and the aforementioned trends, in order to study the evolving plumes. It is shown that the FM representation leads to plausible non-Gaussian plume evolutions and yields predictions that closely approximate records for a period of time that extends even beyond one year. It is illustrated that such predicted geometries are also consistent with predictions made via stochastic theories (i.e. Dagan, 1984).     

A physical interpretation of the deterministic fractal–multifractal method as a realization of a generalized multiplicative cascade

In this study, we attempt to offer a solid physical basis for the deterministic fractal–multifractal (FM) approach in geophysics (Puente, Phys Let A 161:441–447, 1992; J Hydrol 187:65–80, 1996). We show how the geometric construction of derived measures, as Platonic projections of fractal interpolating functions transforming multinomial multifractal measures, naturally defines a non-trivial cascade process that may be interpreted as a particular realization of a random multiplicative cascade. In such a light, we argue that the FM approach is as “physical” as any other phenomenological approach based on Richardson’s eddies splitting, which indeed lead to well-accepted models of the intermittencies of nature, as it happens, for instance, when rainfall is interpreted as a quasi-passive tracer in a turbulent flow. Although neither a fractal interpolating function nor the specific multipliers of a random multiplicative cascade can be measured physically, we show how a fractal transformation “cuts through” plausible scenarios to produce a suitable realization that reflects specific arrangements of energies (masses) as seen in nature. This explains why the FM approach properly captures the spectrum of singularities and other statistical features of given data sets. As the FM approach faithfully encodes data sets with compression ratios typically exceeding 100:1, such a property further enhances its “physical simplicity.” We also provide a connection between the FM approach and advection–diffusion processes.     

Encoding daily rainfall records via adaptations of the fractal multifractal method

A deterministic geometric approach, the fractal–multifractal (FM) method, already found useful in modeling storm events, is adapted here in order to encode, for the first time, highly intermittent daily rainfall records gathered over a water year and containing many days of zero rain. Through application to data sets gathered at Laikakota in Bolivia and Tinkham in Washington, USA, it is demonstrated that the modified FM approach can represent erratic rainfall records faithfully, while using only a few FM parameters. It is shown that the modified FM approach, by capturing the rain accumulated over the season, ends up preserving other statistical attributes as well as the overall “texture” of the records, leading to FM sets that are indistinguishable from observed sets and certainly within the limits of accuracy of measured rainfall. This fact is further corroborated comparing 20 consecutive years at Laikakota and a modified FM representation, via common statistical qualifiers, such as histogram, entropy function, and inter-arrival times.     

Deterministic simulation of highly intermittent hydrologic time series

Application of a deterministic geometric approach for the simulation of highly intermittent hydrologic data is presented. Specifically, adaptations of the fractal-multifractal (FM) method and a Cantorian extension are advanced in order to simulate rainfall records measured at the daily scale and encompassing a water year. It is shown, using as case studies 2 years of rainfall sets gathered in Laikakota, Bolivia and Tinkham, Washington, USA, that the FM approach, relying on only at most 8 parameters, is capable of closely preserving either the whole record’s histogram (therefore including moments), the whole data’s Rényi entropy function and/or the maximum number of consecutive zero values present in the sets, resulting in suitable rainfall simulations, whose overall features and textures are similar to those of the observed sets. The study hence establishes the possibility of simulating highly intermittent sets in time in a deterministic and holistic way as a novel parsimonious methodology to supplement available stochastic frameworks.     

Nonlinear extensions of a fractal–multifractal approach for environmental modeling

We present the extension of a deterministic fractal geometric procedure aimed at representing the complexity of patterns encountered in environmental applications. The procedure, which is based on transformations of multifractal distributions via fractal functions, is extended through the introduction of nonlinear perturbations in the generating iterated linear maps. We demonstrate, by means of various simulations based on changes in parameters, that the nonlinear perturbations generate yet a richer collection of interesting patterns, as reflected by their overall shapes and their statistical and multifractal properties. It is shown that the nonlinear extensions yield structures that closely resemble complex hydrologic spatio-temporal datasets, such as rainfall and runoff time series, and width-functions of river networks. The implications of this nonlinear approach for environmental modeling and prediction are discussed.     

Simplified approach for design of raft foundations against fault rupture. Part Ⅱ： soil-structure interaction

This is the second paper of two, which describe the results of an integrated research effort to develop a four-step simplified approach for design of raft foundations against dip-slip （normal and thrust） fault rupture. The first two steps dealing with fault rupture propagation in the free-field were presented in the companion paper. This paper develops an approximate analytical method to analyze soil-foundation-structure interaction （SFSI）, involving two additional phenomena： （i） fault rupture diversion （Step 3）; and （ii） modification of the vertical displacement profile （Step 4）. For the first phenomenon （Step 3）, an approximate energy-based approach is developed to estimate the diversion of a fault rupture due to presence of a raft foundation. The normalized critical load for complete diversion is shown to be a function of soil strength, coefficient of earth pressure at rest, bedrock depth, and the horizontal position of the foundation relative to the outcropping fault rupture. For the second phenomenon （Step 4）, a heuristic approach is proposed, which ＂scans＂ through possible equilibrium positions to detect the one that best satisfies force and moment equilibrium. Thus, we account for the strong geometric nonlinearities that govern this interaction, such as uplifting and second order （P-△） effects. Comparisons with centrifuge-validated finite element analyses demonstrate the efficacy of the method. Its simplicity makes possible its utilization for preliminary design.     

The 20th September 1999 Chi-Chi Earthquake (Taiwan): a case of study for its aftershock seismic sequence

The purpose of this work is to highlight some methodological aspects related to the observation of possible anomalies in the temporal decay of aftershocks temporal series following a mainshock with magnitude M ≥ 7.0. In this paper we present the results for the Taiwan seismic sequence started on 20 September 1999 (M = 7.7) by tuning some seismic parameters that show considerable variations during the aftershock decay process. In here we also present the results obtained using a fractal approach for the seismic sequence. Earthquakes belong to a class of phenomena known as multifractals. In general it is important to define the fractal dimension D, but sometimes is not useful if we are describing a natural phenomenon; so it is necessary to define D ₀ called box-counting dimension and D ₂ called correlation dimension, usually D ₀ ≥ D ₂. In the elaborations of the fractal dimensions, for this sequence, we have obtained values lesser than 1, with a greater tendency of aftershocks to clusterize in time before a large aftershock. This is coherent with the possible existence of seismic anomalies, that could occur before the large aftershock. We also report the results obtained by using the delta/sigma method described firstly in [Caccamo et al., 2005] and later applied to different seismic sequence. The observed temporal series of the aftershocks per day can be considered as a sum of a deterministic and a stochastic contribution. If the decay can be modeled as a non-stationary Poissonian process, the number of aftershocks in a small time interval Δt is the mean value n(t) Δt, with a standard deviation (δ = √n(t)Δt. Investigating both aftershock behavior and a wide spectrum of parameters may find the key to explain better the mechanism of seismicity as a whole.     

Multifractal objective analysis: conditioning and interpolation

 We investigate various ways of statistically estimating multifractal fields from sparse data. First, the problem is set in the general framework of conditional expectations, and the effect of (multi) fractal sampling on the statistics of the measured process is investigated, showing how analytical expressions describing the statistical properties of the phenomenon should be modified by the sampling. Then, several techniques are introduced, our goal being to estimate the intensity of a field at resolution λ, given samples of the process collected by networks at higher resolutions Λ>λ. The general strategy underlying all the estimating techniques presented is to approximate the unknown field values at resolution λ by means of most likely estimates conditional to the available information at resolution Λ>λ. Finally, the procedures are tested on simulated lognormal multifractal fields sampled by means of fractal networks, and the propagation of the errors in a scaling framework is also discussed. These techniques are necessary for estimating geophysical processes in regions where no monitoring stations are present, a scenario often encountered in practice, and may also be of great help in studying natural hazards and risk assessment.     

Simultaneous analyses and applications of multiple fluorobenzoate and halide tracers in hydrologic studies

An analytical method that employs ion chromatography has been developed to exploit the use of fluorobenzoic acids (FBAs) and halides more fully as hydrologic tracers. In a single run, this reliable, sensitive, and robust method can simultaneously separate and quantify halides (fluoride, chloride, bromide, and iodide) and up to seven FBAs from other common groundwater constituents (e.g. nitrate and sulphate). The usefulness of this analytical method is demonstrated in both field and laboratory tracer experiments. The field study examines the hydrologic response of fractures and the matrix to different flow rates and the contribution of matrix diffusion in chemical transport. Laboratory tracer experiments with eight geologic media from across the USA—mostly from Department of Energy facilities where groundwater contamination is prevalent and where subsurface characterization employing tracers has been ongoing or is in need—reveal several insights about tracer transport behaviour: (1) bromide and FBAs are not always transported conservatively; (2) the delayed transport of these anionic tracers is likely related to geologic media characteristics, such as organic matter, pH, iron oxide content, and clay mineralogy; (3) use of iodine as a hydrologic tracer should take into account the different sorption behaviours of iodide and iodate and the possible conversion of iodine's initial chemical form; (4) the transport behaviour of potential FBA and halide tracers under relevant geochemical conditions should be evaluated before beginning ambitious, large‐scale field tracer experiments. Copyright © 2005 John Wiley & Sons, Ltd.     

On the fractal dimension of the fallout deposits: A case study of the 79 A.D. Plinian eruption at Mt. Vesuvius

We investigated the existence of a fractal law (power law) distribution of size pyroclastic fragments erupted during the fallout phase of the 79 A.D. Plinian eruption at Mt. Vesuvius. In particular, we performed a particle size distribution analysis on 18 white and grey pumice samples collected in six sites distributed in the SW sector of Mt. Vesuvius. Our measurements show that the fragmentation of samples in the investigated range (from 32 mm to 850 μm) follows a power law, guaranteeing the scale invariance of the process. The relationship frequency-size distribution of the fragments is verified independently from the nature (i.e., pumices and lithics) and stratigraphic height of the considered samples in the pyroclastic deposit. Therefore, the fractal fragmentation theory can be indicated for evaluating the relationship between the intensity of fragmentation (fractal dimension D) and eruption energy. In this way the apparent chaotic distribution of the particles in the fallout deposits hides a self-organized complexity revealed by the retrieved power law distribution. We further remark that a key aspect of our analysis is the founded evidence that the fractal dimension of the lithics is systematically greater than that of the pumices.     

The 982 Ma Re-Os age of copper-nickel sulfide ores in the Baotan area, Guangxi and its geological significance

Re-Os dating on copper-nickel sulfide ores from the Baotan area, Guangxi, yielded an ore-forming age of 982 ± 21 Ma (2σ), which demonstrates that copper-nickel sulfide deposits and their related mafic-ultramafic rocks occurred in the same period of time with the ophiolites in northeastern Jiangxi. Both of them are the products of collision-convergence between the Cathaysian plate and Yangtze plate and the subsequent extensional environment. Calculation of the γ_Os of the 982 Ma copper-nickel sulfide ores and its correlation with Re/Os indicate that injection-type massive ores display lower γ_Os values (-15.6 to -8.2) and lower Re/Os ratios (0.32 to 0.43), while basal liquation-type ores have γ_Os-27.9 to -7.3 and Re/Os=5.36 to 11.24. This suggests that these copper-nickel sulfide ores and their related mafic-ultramafic rocks were derived from a Re-depleted mantle source and that contamination with some crustal material occurred during their intrusion.     

Research on nonlinear R/S method and its application in earthquake prediction

4 Conclusions The preliminary result of this work was obtained in 1992. At Consulting Meeting for Seismic Trends of China in Jan. 1993, we handed in a report and predicted that “Until now, the average magnitude curve haven’t gotten risen again after a fall, and the value ofH is under 0.4, so no alarm is made by each of these two. We predicted that, seismicity of the Chinese mainland in 1993 may be at the same level as in 1991–1992, or a little higher than it, but mustn’t be very high; this is to say that no earthquake withM _s⩾7.2 will occur in the Chinese mainland in 1993.” The real situation is: No earthquake withM _s⩾7.2 occurred in the Chinese mainland, and it shows that this real prediction is true. To state succinctly: (1)In ths paper nonlinearR/S fractal method is applicated in earthquake prediction, and two algorithms RSH and RSHM are proposed. (2)R values of this two algorithms reached a notable value, and pass confidence test, so a certain efficiency is presented. But RSHM is some better. (3) It is presented by the successful prediction example that there is a good prospect in predicting with fractal method. Contribution No. 95A0056, Institute of Geophysics, SSB, China. This study is supported by the Chinese Joint Seismological Science Foundation.     

Is a chaotic multi‐fractal approach for rainfall possible?

An Erratum has been published for this article in Hydrological Processes 15 (12) 2001, 2381–2382. Applications of the ideas gained from fractal theory to characterize rainfall have been one of the most exciting areas of research in recent times. The studies conducted thus far have nearly unanimously yielded positive evidence regarding the existence of fractal behaviour in rainfall. The studies also revealed the insufficiency of the mono‐fractal approaches to characterizing the rainfall process in time and space and, hence, the necessity for multi‐fractal approaches. The assumption behind multi‐fractal approaches for rainfall is that the variability of the rainfall process could be directly modelled as a stochastic (or random) turbulent cascade process, since such stochastic cascade processes were found to generically yield multi‐fractals. However, it has been observed recently that multi‐fractal approaches might provide positive evidence of a multi‐fractal nature not only in stochastic processes but also in, for example, chaotic processes. The purpose of the present study is to investigate the presence of both chaotic and fractal behaviours in the rainfall process to consider the possibility of using a chaotic multi‐fractal approach for rainfall characterization. For this purpose, daily rainfall data observed at the Leaf River basin in Mississippi are studied, and only temporal analysis is carried out. The autocorrelation function, the power spectrum, the empirical probability distribution function, and the statistical moment scaling function are used as indicators to investigate the presence of fractal, whereas the presence of chaos is investigated by employing the correlation dimension method. The results from the fractal identification methods indicate that the rainfall data exhibit multi‐fractal behaviour. The correlation dimension method yields a low dimension, suggesting the presence of chaotic behaviour. The existence of both multi‐fractal and chaotic behaviours in the rainfall data suggests the possibility of a chaotic multi‐fractal approach for rainfall characterization. Copyright © 2001 John Wiley & Sons, Ltd.     

Nonlinear dynamics and chaos in hydrologic systems: latest developments and a look forward

During the last two decades or so, studies on the applications of the concepts of nonlinear dynamics and chaos to hydrologic systems and processes have been on the rise. Earlier studies on this topic focused mainly on the investigation and prediction of chaos in rainfall and river flow, and further advances were made during the subsequent years through applications of the concepts to other problems (e.g. data disaggregation, missing data estimation, and reconstruction of system equations) and other processes (e.g. rainfall-runoff and sediment transport). The outcomes of these studies are certainly encouraging, especially considering the exploratory stage of the concepts in hydrologic sciences. This paper discusses some of the latest developments on the applications of these concepts to hydrologic systems and the challenges that lie ahead on the way to further progress. As for their applications, studies in the important areas of scaling, groundwater contamination, parameter estimation and optimization, and catchment classification are reviewed and the inroads made thus far are reported. In regards to the challenges that lie ahead, particular focus is given to improving our understanding of these largely less-understood concepts and also finding ways to integrate these concepts with the others. With the recognition that none of the existing one-sided ‘extreme-view’ modeling approaches is capable of solving the hydrologic problems that we are faced with, the need for finding a balanced ‘middle-ground’ approach that can integrate different methods is stressed. To this end, the viability of bringing together the stochastic concepts and the deterministic concepts as a starting point is also highlighted.     

Geometric approach to statistical analysis on the simplex

 The geometric interpretation of the expected value and the variance in real Euclidean space is used as a starting point to introduce metric counterparts on an arbitrary finite dimensional Hilbert space. This approach allows us to define general reasonable properties for estimators of parameters, like metric unbiasedness and minimum metric variance, resulting in a useful tool to better understand the logratio approach to the statistical analysis of compositional data, who's natural sample space is the simplex.     

Transport and fate of nonaqueous phase liquid (NAPL) in variably saturated porous media with evolving scales of heterogeneity

 A stochastic simulation is performed to study multiphase flow and contaminant transport in fractal porous media with evolving scales of heterogeneity. Numerical simulations of residual NAPL mass transfer and subsequent transport of dissolved and/or volatilized NAPL mass in variably saturated media are carried out in conjunction with Monte Carlo techniques. The impact of fractal dimension, plume scale and anisotropy (stratification) of fractal media on relative dispersivities is investigated and discussed. The results indicate the significance of evolving scale of porous media heterogeneity to the NAPL transport in the subsurface. In general, the fractal porous media enhance the dispersivities of NAPL mass plume transport in both the water phase and the gas phase while the influence on the water phase is more significant. The porous media with larger fractal dimension have larger relative dispersivities. The aqueous horizontal dispersivity exhibits a most significant increase against the plume scale.     

Parameters of Higuchi’s method to characterize primary waves in some seismograms from the Mexican subduction zone

Higuchi’s method is a procedure that, if applied appropriately, can determine in a reliable way the fractal dimension D of time series; this fractal dimension permits to characterize the degree of correlation of the series. However, when analyzing some time series with Higuchi’s method, there are oscillations at the right-hand side of the graph, which can cause a mistaken determination of the fractal dimension. In this work, an appropriate explanation is given to this type of behaviour. Using the seismogram as a time series and the properties of the P and S waves, it is possible to use the properties of Higuchi’s method to previously detect the arrival of the earthquake shacking stage, some seconds in advance, approximately 30–35 s in the case of Mexico City. Thus, we propose the Higuchi’s method to characterize and detect the P waves in order to estimate the strength of the forthcoming S waves.