Multifractal modeling and spatial point processes

The multifractal model can be applied to spatial point processes. It provides new, approximately power-law type, expressions for their second-order intensity and K (r) functions. The box-counting and cluster dimensions are different but mutually interrelated according to multifractal theory. This approach is used to describe the underlying spatial structure of gold mineral occurrences in the Iskut River area, northwestern British Columbia. The box-counting and cluster dimensions for the example are estimated to be 1.335±0.077 and 1.219±0.037, respectively. The relatively strong clustering of the gold deposits is reflected by the fact that both values are considerably less than the corresponding Euclidean dimension (=2).     

Numerical Method for Conditional Simulation of Levy Random Fields

Stochastic simulations of subsurface heterogeneity require accurate statistical models for spatial fluctuations. Incremental values in subsurface properties were shown previously to be approximated accurately by Levy distributions in the center and in the start of the tails of the distribution. New simulation methods utilizing these observations have been developed. Multivariate Levy distributions are used to model the multipoint joint probability density. Explicit bounds on the simulated variables prevent nonphysical extreme values and introduce a cutoff in the tails of the distribution of increments. Long-range spatial dependence is introduced through off-diagonal terms in the Levy association matrix, which is decomposed to yield a maximum likelihood type estimate at unobserved locations. This procedure reduces to a known interpolation formula developed for Gaussian fractal fields in the situation of two control points. The conditional density is not univariate Levy and is not available in closed form, but can be constructed numerically. Sequential simulation algorithms utilizing the numerically constructed conditional density successfully reproduce the desired statistical properties in simulations.     

Improved ocean-geoid resolution from retracked ERS-1 satellite altimeter waveforms

The ocean geoid can be inferred from the topography of the mean sea surface. Satellite altimeters transmit radar pulses and determine the return traveltime to measure sea-surface height. The ERS-1 altimeter stacks 51 consecutive radar reflections on board the satellite to a single waveform. Tracking the time shift of the waveform gives an estimate of the distance to the sea surface. We retrack the ERS-1 radar traveltimes using a model that is focused on the leading edge of the waveforms. While earlier methods regarded adjacent waveforms as independent statistical events, we invert a whole sequence of waveforms simultaneously for a spline geoid solution. Smoothness is controlled by spectral constraints on the spline coefficients. Our geoid solutions have an average spectral density equal to the expected power spectrum of the true geoid. The coherence of repeat track solutions indicates a spatial resolution of 31  km, as compared to 41  km resolution for the ERS-1 Ocean Product. While the resolution of the latter deteriorates to 47  km for wave heights above 2  m, our geoid solution maintains its resolution of 31  km for rough sea. Retracking altimeter waveform data and constraining the solution by a spectral model leads to a realistic geoid solution with significantly improved along-track resolution.     

Bias in estimating fractal dimension with the rescaled-range (R/S) technique

Fractal geostatistics are being applied to subsurface geological data as a way of predicting the spatial distribution of hydrocarbon reservoir properties. The fractal dimension is the controlling parameter in stochastic methods to produce random fields of porosity and permeability. Rescaled range (R/S)analysis has become a popular way of estimating the fractal dimension, via determination of the Hurst exponent (H). A systematic investigation has been undertaken of the bias to be expected due to a range of factors commonly inherent in borehole data, particularly downhole wireline logs. The results are integrated with a review of previous work in this area. Small datasets. overlapping samples, drift and nonstationariry of means can produce a very large bias, and convergence of estimates of H around 0.85–0.90 regardless of original fractal dimension. Nonstationarity can also account for H>1, which has been reported in the literature but which is theoretically impossible for fractal time series. These results call into question the validity of fractal stochastic models built using fractal dimensions estimated with the R/Smethod.     

Stochastic behaviour and scaling laws in geoelectrical signals measured in a seismic area of southern Italy

We explore the inner dynamics of daily geoelectrical time series measured in a seismic area of the southern Apennine chain (southern Italy). Autoregressive models and the Higuchi fractal method are applied to extract maximum quantitative information about the time dynamics from these geoelectrical signals. First, the predictability of the geoelectrical measurements is investigated using autoregressive models. The procedure is based on two forecasting approaches: the global and the local autoregressive approximations. The first views the data as a realization of a linear stochastic process, whereas the second considers the data points as a realization of a deterministic process, which may be non-linear. Comparison of the predictive skills of the two techniques allows discrimination between low-dimensional chaos and stochastic dynamics. Our findings suggest that the physical systems governing electrical phenomena are characterized by a very large number of degrees of freedom and can be described only with statistical laws. Second, we investigate the stochastic properties of the same geoelectrical signals, searching for scaling laws in the power spectrum. The spectrum fits a power law P (  f )∝  f  ^−α , with the scaling exponent α a typical fingerprint of fractional Brownian processes. In this analysis we apply the Higuchi method, which gives a linear relationship between the fractal dimension D _Σ and the spectral power law scaling index α : D _Σ=(3− α )/2. This analysis highlights the stochastic nature of geoelectrical signals recorded in this seismic area of southern Italy.     

The Relationship between Fe Mineralization and Magnetic Basement Faults using Multifractal Modeling in the Esfordi and Behabad Areas (BMD), Central Iran

Multifractal modeling is a mathematical method for the separation of a high potential mineralized background from a non-mineralized background. The Concentration-Distance to Fault structures (C-DF) fractal model and the distribution of the known iron (Fe) deposits/mines seen in the Esfordi and Behabad 1:100,000 sheets from the Bafq region of central Iran are used to distinguish Fe mineralization based on their distance to magnetic basement structures and surface faults, separately, using airborne geophysical data and field surveys. Application of the C-DF fractal model for the classification of Fe mineralizations in the Esfordi and Behabad areas reveals that the main ones show a correlation with their distance from magnetic basement structures. Accordingly, the distances of Fe mineralizations with grades of Fe higher than 55% )43% < Fe ≤ 60%) are located at a distance of less than 1 km, whereas for surfacial faults with grades of 43% ≤ Fe ≤ 60%, the distances are 3162< DF ≤ 4365 m from the faults. Thus, there is a positive relationship between Fe mineralization and magnetic basement structures. Also, the proximity evidence of Precambrian high-grade Fe mineralization related to magnetic basement structures indicates syn-rifting tectonic events. Finally, this C-DF fractal model can be used for exploration of magmatic and hydrothermal ore deposits.