Journal of the International Association for Mathematical Geology

Long sedimentary cycles with periods of 400 thousand to two million years are difficult to establish. The main reasons are a lack of stationarity in stratigraphic time series and unknown changes in sedimentation rates. Filtering, trend removal and time scale transformations can help to interpret stratigraphic sequences, but the application of any such method has to be justified by geological arguments. The effects of filtering, scale transformation and trend removal are explored using sets of field data from the Carboniferous (NW Ireland), the Cretaceous (Gubbio, Italy) and Pliocene (Sicily). Their location and stratigraphy are described.

Present-day observed downhole quantities, which a dynamical model of basin evolution should account for, include: total depth drilled, formation thicknesses, variations of porosity, permeability and total fluid pressure with depth, and depths of unconformities. Following a line of logic previously employed with multiple thermal indicators, it is shown how the observed quantities can be used in a nonlinear inverse sense to determine, or at least constrain, parameters and functions entering quantitative models of dyanmical sedimentary evolution. A procedure is given so that the inverse methods can be used: (a) with single well data; (b) with multiple well data; and (c) simultaneously with thermal indicator data, which have already been previously successfully inverted using a tomographic procedure. Parameters that can be evaluated using the dynamical indicator inversion (dynamical tomography) include, but are not limited to, values dealing with geological events (such as unconformity timing and amount of material eroded, the “openness” or “shutness” of faults; critical fracture pressure, etc.), as well as values dealing with intrinsic, or assumed, lithologic equations of state (such as power law values in connections between permeability and void ratio, or between frame pressure and void ratio). The dynamical tomography procedure can be used with or without weighting the data and/or the dynamical indicators; is guaranteed to produce a closer correspondence between predicted and observed behaviors at each nonlinear iteration; and is guaranteed to keep all parameters within any chosen domain. When used in a multiple well setting, the dynamical tomography method enables an assessment to be made of the assumed invariance to spatial location of parameters in equations of state, as well as allowing geologic process parameters to vary with well location. The procedure also automatically incorporates the ability to determine precision, resolution, sensitivity, and uniqueness of any or all parameters, both associated with equations of state and associated with geological processes. Thus, a sharper understanding is achieved of the trustworthiness and uncertainty of quantitative basin analysis models in respect of: (i) intrinsic assumptions of a model; (ii) implicit or explicit parameter dependences for both geological events and imposed functional dependences of variables; (iii) resolution with respect to finite sampling and measurement error or uncertainty in the quality and quantity of observed data.

We give an elementary introduction to some ideas and methods in the qualitative theory of differentiable dynamical systems, emphasizing the geometrical description of certain simple bifurcations. As an example of the use of such methods we review two models for the reversal phenomenon exhibited by the earth's magnetic field. The second model displays surprisingly rich dynamical behavior that has only recently been studied in detail. In closing we show that recent work on periodically forced weakly dissipative systems occurring as models of magneto-elastic interactions may be relevant to the geomagnetic reversal question.

Philip and Watson have made many valid criticisms of the theory and application of geostatistics. Journel and Srivastava, in their turn, have defended the basis of geostatistics and highlighted a few of its unique advantages as an approach to analyzing spatial data without, however, answering most of the criticisms. The controversy remains unresolved; but a possibility of resolution does exist, however, given the answers to two particular questions: These questions are not of mere academic interest but have major economic implications. In the mining industry, for example, the best possible estimates of tonnage, grade, and error bounds are vital for economic decisions; so long as disagreement exists on validity of different estimation techniques, the choice among conflicting estimates will continue to be made on extraneous (nontechnical and perhaps irrational) criteria.

Journel (1974) developed the turning-bands method which allows a three-dimensional data set with specified covariance to be obtained by the simulation of several one-dimensional realizations which have an intermediate covariance. The relationship between the threedimensional and one-dimensional covariance is straightforward and allows the one-dimensional covariance to be obtained immediately. In theory a dense uniform distribution of lines in three-dimensional space is required along which the one-dimensional realizations are generated; in practice most workers have been content to use the fifteen axes of the regular icosahedron. Many mining problems may be treated in two dimensions, and in this paper a turning-bands approach is developed to generate two-dimensional data sets with a specified covariance. By working in two dimensions, the area on which the data is simulated may be divided as finely as desired by the lines on which the one-dimensional realizations are first generated. The relationship between the two-dimensional and one-dimensional covariance is derived as a nontrivial integral equation. This is solved analytically for the onedimensional covariance. The method is applied to the generation of a two-dimensional data set with spherical covariance.

In geostatistics, factorial kriging is often proposed to filter noise. This filter is built from a linear model which is ideally suited to a Gaussian signal with additive independent noise. Robustness of the performance of factorial kriging is evaluated in less congenial situations. Three different types of noise are considered all perturbing a lognormally distributed signal. The first noise model is independent of the signal. The second noise model is heteroscedastic; its variance depends on the signal, yet noise and signal are uncorrelated. The third noise model is both heteroscedastic and linearly correlated with the signal. In ideal conditions, exhaustive sampling and additive independent noise, factorial kriging succeeds to reproduce the spatial patterns of high signal values. This score remains good in presence of heteroscedastic noise variance but falls quickly in presence of noise-to-signal correlation as soon as the sample becomes sparser.

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.