A review of fractals in karst

Many features of a karst massif can either be modelled using fractal geometry or have a fractal distribution. For the exokarst, typical examples include the geometry of the landscape and the spatial location and size-distribution of karst depressions. Typical examples for the endokarst are the geometry of the three-dimensional network of karst conduits and the length distribution of caves. In addition, the hydrogeological parameters of the karst massif, such as hydraulic conductivity, and karst spring hydrographs may also exhibit fractal behaviour. In this work we review the karst features that exhibit fractal behaviour, we review the literature in which they are described, and we propose hypotheses and conjectures about the origin of such behaviour. From the review and analysis, we conclude that fractal behaviour is exhibited at all scales in karst systems.Eulogio Pardo-Igúzquiza, Peter A. Dowd, Juan J. Durán, and Pedro Robledo-Ardil

Non-Parametric Approximations for Anisotropy Estimation in Two-dimensional Differentiable Gaussian Random Fields

Spatially referenced data often have autocovariance functions with elliptical isolevel contours, a property known as geometric anisotropy. The anisotropy parameters include the tilt of the ellipse (orientation angle) with respect to a reference axis and the aspect ratio of the principal correlation lengths. Since these parameters are unknown a priori, sample estimates are needed to define suitable spatial models for the interpolation of incomplete data. The distribution of the anisotropy statistics is determined by a non-Gaussian sampling joint probability density. By means of analytical calculations, we derive an explicit expression for the joint probability density function of the anisotropy statistics for Gaussian, stationary and differentiable random fields. Based on this expression, we obtain an approximate joint density which we use to formulate a statistical test for isotropy. The approximate joint density is independent of the autocovariance function and provides conservative probability and confidence regions for the anisotropy parameters. We validate the theoretical analysis by means of simulations using synthetic data, and we illustrate the detection of anisotropy changes with a case study involving background radiation exposure data. The approximate joint density provides (i) a stand-alone approximate estimate of the anisotropy statistics distribution (ii) informed initial values for maximum likelihood estimation, and (iii) a useful prior for Bayesian anisotropy inference.Comment: 39 pages; 8 figure

AMLE3D: a computer program for the inference of spatial covariance parameters by approximate maximum likelihood estimation

Abstract not availableEulogio Pardo-Igúzquiza and Peter A. Dow

Optimal estimation of areal values of near-land-surface temperatures for testing global and local spatio-temporal trends

This paper provides a solution to the problem of estimating the mean value of near-land-surface temperature over a relatively large area (here, by way of example, applied to mainland Spain covering an area of around half a million square kilometres) from a limited number of weather stations covering a non-representative (biased) range of altitudes. As evidence mounts for altitude-dependent global warming, this bias is a significant problem when temperatures at high altitudes are under-represented. We correct this bias by using altitude as a secondary variable and using a novel clustering method for identifying geographical regions (clusters) that maximize the correlation between altitude and mean temperature. In addition, the paper provides an improved regression kriging estimator, which is optimally determined by the cluster analysis. The optimal areal values of near-land-surface temperature are used to generate time series of areal temperature averages in order to assess regional changes in temperature trends. The methodology is applied to records of annual mean temperatures over the period 1950-2011 across mainland Spain. The robust non-parametric Theil-Sen method is used to test for temperature trends in the regional temperature time series. Our analysis shows that, over the 62-year period of the study, 78% of mainland Spain has had a statistically significant increase in annual mean temperature.HongWanga, EulogioPardo-Igúzquizab, Peter A.Dowdc, YongguoYan

Empirical maximum likelihood kriging: The general case

Although linear kriging is a distribution-free spatial interpolator, its efficiency is maximal only when the experimental data follow a Gaussian distribution. Transformation of the data to normality has thus always been appealing. The idea is to transform the experimental data to normal scores, krige values in the “Gaussian domain” and then back-transform the estimates and uncertainty measures to the “original domain.” An additional advantage of the Gaussian transform is that spatial variability is easier to model from the normal scores because the transformation reduces effects of extreme values. There are, however, difficulties with this methodology, particularly, choosing the transformation to be used and back-transforming the estimates in such a way as to ensure that the estimation is conditionally unbiased. The problem has been solved for cases in which the experimental data follow some particular type of distribution. In general, however, it is not possible to verify distributional assumptions on the basis of experimental histograms calculated from relatively few data and where the uncertainty is such that several distributional models could fit equally well. For the general case, we propose an empirical maximum likelihood method in which transformation to normality is via the empirical probability distribution function. Although the Gaussian domain simple kriging estimate is identical to the maximum likelihood estimate, we propose use of the latter, in the form of a likelihood profile, to solve the problem of conditional unbiasedness in the back-transformed estimates. Conditional unbiasedness is achieved by adopting a Bayesian procedure in which the likelihood profile is the posterior distribution of the unknown value to be estimated and the mean of the posterior distribution is the conditionally unbiased estimate. The likelihood profile also provides several ways of assessing the uncertainty of the estimation. Point estimates, interval estimates, and uncertainty measures can be calculated from the posterior distribution.Eulogio Pardo-Igúzquiza and Peter A. Dow

A new method for reconstructing past-climate trends using tree-ring data and kernel smoothing

Mediterranean high-relief karst areas are very vulnerable to changes in temporal patterns of precipitation and temperature. Understanding climate change in these areas requires current climate trends to be assessed within the context of the variability of rainfall and temperature trends in the recent past. A major difficulty is that the instrumental record in these high-relief areas is very limited and the use of data from paleoclimatic proxies, such as tree-ring data, is required to infer past climate variability. Furthermore, for complex relationships between tree-ring data and climatic variables, it is almost impossible to infer past inter-annual variations in temperature or precipitation, and the inference is limited to the reconstruction of low-frequency variability (i.e., the trend). To do so, in this work, we propose a new method based on detecting trends (by kernel smoothing) in tree variables that show maximum correlation with the trends (also estimated by kernel smoothing) of climate variables. This enables a standard regression framework to be established to reconstruct past climate. We have used tree-ring proxy data from Abies pinsapo to evaluate past climate trends in the Sierra de las Nieves karst massif in Southern Spain. Our analysis has found that during the last three hundred years the smoothed mean annual rainfall steadily decreased until the beginning of the 20th century and thereafter it remained more or less constant until the end of the century. On the other hand, the smoothed mean annual temperature has steadily increased since the beginning of the 18th century until recent times. These trends are also suggested by the climate projections for the latter part of the current 21st century. As the study area is a high-relief karst massif of significant hydrologic and ecologic interest, the implications of these trends should be taken into account when formulating effective action plans to mitigate the impact of climate change.JoséSanchez-Moralesa, EulogioPardo-Igúzquiza, Francisco Javier, Rodríguez-Tovar, Peter A.Dow

A parsimonious distributed model for simulating transient water flow in a high-relief karst aquifer

A mathematical model of a highly heterogeneous functioning karst aquifer is described. The aquifer is in a high-relief karst massif and, as is common for such locations, data are scarce and there are no borehole, piezometer or pumping-test data. The scarcity of data in this case required a parsimonious approach to ensure that the level of complexity of the model was commensurate with the amount, type and quality of the available data. Parsimony also requires the model to include the minimum essential components that account adequately for the data, which in this and similar cases are the functional dualities of the karst system: duality in recharge, flow and discharge. The model is three-dimensional (3D) in the sense that the aquifer is discretized into 3D voxels, although the flow is one-dimensional (1D) and vertical in the vadose zone, and horizontal and two-dimensional (2D) in the saturated zone. The parsimonious model was designed by coupling a 1D unsaturated gravity-driven flow along the vertical (along each column of voxels that discretize the aquifer) and a 2D unconfined Darcy flow in the saturated zone. In the context of this type of aquifer, preferential recharge through the network of karst conduits implies a rapid rise in the water table, the location and extension of which are model parameters. The karst springs are simulated by drains. The methodology, which is completely general, is illustrated by application to the karst aquifer in the Sierra de las Nieves mountains in southern Spain.Eulogio Pardo-Igúzquiza, Peter Dowd, Antonio Pulido Bosch, Juan A. Luque-Espinar, Javier Heredia, Juan J. Durán-Valser