The roughness of Martian topography: A metre-scale fractal analysis of six selected areas

Available online 23 May 2022Studies of the roughness of natural surfaces (landscapes) provide useful information for planetary geology. This paper covers the mapping and analysis of the spatial variability of the surface roughness of Martian topography at a high spatial resolution (metre-scale). The methodology provides new images of the Martian surface texture at the metre-scale that can assist in the interpretation of geological events, processes and formations. It can also assist in geological mapping and in the evaluation of sites that merit further exploration. Digital elevation models, generated by stereo-pair HiRISE images, of six different terrains (aeolian, volcanic, hydrated, cratered, reticulate and sublimated) were used to characterize the metre-scale terrain roughness of representative test sites on Mars. Surface roughness was evaluated by using the local fractal dimension and the results show that the mean of the local fractal dimension ranges from 2.17 in reticulate terrain to 2.71 in sublimated terrain in the southern polar cap. The roughness of the sublimated terrain is significantly higher than the roughness of typical terrains on Earth. Basically, the roughness of the Martian terrain at the metre-scale depends on the rugosity of the landscape, which can be quantified as the number of metric-scale closed depressions and mounds present on the terrain. The information provided by the spatial variability patterns of metre-scale roughness maps provides a significant resource for local planetary geology research at high resolution scale.E. Pardo-Igúzquiza, P.A. Dow

The Many Forms of Co-kriging: A Diversity of Multivariate Spatial Estimators

OnlinePublIn this expository review paper, we show that co-kriging, a widely used geostatistical multivariate optimal linear estimator, has a diverse range of extensions that we have collected and illustrated to show the potential of this spatial interpolator. In the context of spatial stochastic processes, this paper covers scenarios including increasing the spatial resolution of a spatial variable (downscaling), solving inverse problems, estimating directional derivatives, and spatial interpolation taking boundary conditions into account. All these spatial interpolators are optimal linear estimators in the sense of being unbiased and minimising the variance of the estimation error.Peter A. Dowd, Eulogio Pardo-Igúzquiz

Geostatistical methods to map the probability of hydrogeotoxic risk by high As concentrations in groundwater. Case study in Ávila province ( Spain)

[EN] The presence of As in groundwater is a priority public health issue because it imposes serious restrictions on drinking water. Mapping probabilities of exceedance of the threshold permitted by the World Health Organization, WHO (10 μg/L) allow delimiting the most vulnerable areas. The existing geostatistical techniques are a common tool for the evaluation of these maps, though, there is no agreement on which of the methods is the best. In this study different comparison criteria are illustrated. Seven non-parametric kriging methods are used to estimate the map of probability of exceeding the As concentration the limit of 10 mg/L in groundwater at the province of Ávila. Performed validation reveals that one the best results correspond to the simplicial indicator kriging, never before compared in studies of presence of geogenic As in groundwater.[ES] La presencia de As en las aguas subterráneas es un problema prioritario de salud pública e impone serias restricciones en el agua de consumo. Los mapas de probabilidad de superar el umbral permitido por la Organización Mundial de la Salud, OMS (10 μg/L) permiten delimitar las áreas que más riesgo presentan en relación con este parámetro. Las técnicas geoestadísticas constituyen una herramienta de uso común para elaborar estos mapas, aunque lamentablemente no hay un acuerdo sobre qué técnica es la más adecuada. El presente estudio recopila distintos criterios para decidir qué método presenta resultados más robustos. Se utilizan siete métodos de kriging no paramétrico en la estimación del mapa de probabilidad de que la concentración de As en manantiales de la provincia de Ávila supere el límite de 10 μg/L. La validación revela que uno de los mejores resultados es del simplicial indicator kriging, nunca antes tenido en cuenta en estudios sobre presencia de As geogénico en aguas subterráneas.Los autores agradecen a la Obra Social de Caja de Ávila el apoyo a la investigación, al financiar el proyecto “Manantiales de la provincia de Ávila (2006-2007)” y a los revisores anónimos por los comentarios realizados.Guardiola-Albert, C.; Pardo-Igúzquiza, E.; Giménez-Forcada, E. (2017). Métodos geoestadísticos para la elaboración de mapas de probabilidad de riesgo hidrogeotóxico (HGT) por altas concentraciones de As en las aguas subterráneas. Aplicación a la distribución de HGT en la provincia de Ávila (España). Ingeniería del Agua. 21(1):71-85. doi:10.4995/ia.2017.6798.SWORD7185211Aragonés Sanz, N., Palacios Diez, M., Avello de Miguel, A., Gómez Rodríguez, P., Martínez Cortés, M., Rodríguez Bernabeu, M.J. 2001. 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Oxford University Press, New York, USA.Goovaerts, P., AvRuskin, G., Meliker, J., Slotnick, M., Jacquez, G., Nriagu, J. 2005. Geostatistical modeling of the spatial variability of arsenic in groundwater of southeast Michigan. Water Resources Research, 41(7), W07013. doi:10.1029/2004WR003705Goovaerts, P. 2009. AUTO-IK: A 2D indicator kriging program for the automated non-parametric modeling of local uncertainty in earth sciences. Computers & Geoscienes, 35(6), 1255-1270. doi:10.1016/j.cageo.2008.08.014Guardiola-Albert, C., Pardo-Igúzquiza, E. 2011. Compositional Bayesian indicator estimation. Stochastic Environmental Research and Risk Assessment, 25(6), 835-849. doi:10.1007/s00477-011-0455-yIsaaks, E.H., Srivastava, R.M. 1989. An Introduction to Applied Geostatistics. Oxford University Press, New York, USA.Journel, A.G. 1983. Non-parametric estimation of spatial distributions. Mathematical Geology, 15(3), 445-468. doi:10.1007/BF01031292Journel, A., Kyriakidis, P.C., Mao, S. 2000. 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Comparison of statistical methods for testing the hypothesis of constant global mean in spatial statistics

In spatial statistics in general, and in geostatistics in particular, the choice between a spatial model with drift and a model with constant global mean is often critical, especially when only a small number of samples are available. A statistical test provides an objective means of making this choice. Among the many available statistical tests, a variance-ratio test has been widely used for making this choice because of its good statistical properties but, in addition to a semi-variogram model, it also requires an alternative drift model hypothesis. Another test statistic is the global D-statistic, which is a complementary test in the sense that it does not require an alternative hypothesis model. In this paper, we use sparse data from simulated random fields to evaluate and compare the performances of these two methods for testing the hypothesis of constant global mean in spatial statistics. We do so by considering the influence of four factors: the amount of data, the type of random field, the amount of spatial or temporal correlation and parametric drifts. In addition, we evaluate their performances in time series analysis, in which testing the hypothesis of constant global mean is also of significant interest. The two test statistics are compared in terms of their achieved confidence level and achieved power. The better method is the one that achieves the nominal confidence level and has higher power. We discuss departures from the nominal values and the results are used to highlight the importance of this problem in spatial statistics.Hong Wang, Eulogio Pardo-Igúzquiza, Peter A. Dowd, Yongguo Yan

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

Stochastic simulation of the spatial heterogeneity of deltaic hydrofacies accounting for the uncertainty of facies proportions

The spatial geological heterogeneity of an aquifer significantly affects groundwater storage, flow and the transport of solutes. In the particular case of coastal aquifers, spatial geological heterogeneity is also a major determining factor of the spatio-temporal patterns of water quality (salinity) due to seawater intrusion. While the hydraulics of coastal hydrogeology can be modeled effectively by various density flow equations, the aquifer geology is highly uncertain. A stochastic solution to the problem is to generate numerical realisations of the geology using sequential stratigraphy, geophysical models or geostatistical approaches. The geostatistical methods (two-point geostatistics, Markov chain models and multiple-point geostatistics) have the advantage of minimal data requirements, e.g., when the only data available are from cores from a few sparsely located boreholes. We provide an extension of sequential indicator simulation by including the uncertainty of the hydrofacies proportions in the simulation approach. We also deal with the problem of variogram estimation from sparse boreholes and we discuss the implicit transition probabilities and the connectivity of simulated realisations of a number of categorical variables. The variogram model used in the simulation of hydrofacies significantly influences the degree of connectivity of the hydrofacies in the simulated model. The choice of model is critical as connectivity determines the amount and extent of seawater intrusion and hence the environmental risk. The methodology is illustrated with a case study of the Andarax river delta, a coastal aquifer in south-eastern Spain. This is a semi-arid Mediterranean region in which the increasing use of, and demand for, groundwater is exacerbated by a transient tourist population that reaches its peak in the summer when the demand for the permanent population is at its highest. The work reported here provides a sound basis for designing flow simulation models for the optimal management of groundwater resources. This paper is an extended version of a presentation given at the 2012 GeoENV Conference held in Valencia, Spain.S. Jorreto-Zaguirre, P.A. Dowd, E. Pardo-Igúzquiza, A. Pulido-Bosch and F. Sánchez-Marto

Evolution of the gulf of Cadiz margin and southwest Portugal contourite depositional system : Tectonic, sedimentary and paleoceanographic implications from IODP expedition 339

Acknowledgments This research used samples and data collected through the Integrated Ocean Drilling Program (IODP). The research was partially supported through the CTM 2008-06399-C04/MAR, CTM 2012-39599-C03, CGL2011-26493, CTM2012-38248, SA263U14, IGCP-619, INQUA 1204 and FWF P25831-N29 Projects. Some data were collected with 94-1090-C03-03 (FADO) and MAR-98-0209 (TASYO) Projects. Research was conducted in the framework of the Continental Margins Research Group of the Royal Holloway University of London, People and the Program (Marie Curie Actions) of the European Union's Seventh Framework Program FP7/2007-2013/ under REA Grant Agreement No. 290201 MEDGATE’. We are very grateful to REPSeOL, TGS–NOPEC, and the CSIC-Institut Jaume Almera (http://geodb.ictja.csic.es) for allowing us to use an unpublished seismic data from the Gulf of Cadiz. We thank J. Aguire (UGR, Spain) for comments and suggestions concerning the Pliocene and Quaternary outcrops, B. van den Berg (USAL) for organizing a thought-provoking field-trip to Cadiz, Spain in November, 2014, M. Ángel Caja, L. García Diego, and J. Tritlla (REPSOL) for provenance and diagenetic analysis of early Pliocene sandstones and debrites, and L.J. Lourens (Utrecht University) for providing us the eccentricity and 200-Kys glacio-eustatic sea-level curves included in the Figure 16. Both Prof. D.A.V. Stow (Heriot-Watt Univ., UK) and F.J. Hernández-Molina (RHUL, UK), as the main co-proponents of the IODP Proposal 644 and the co-chiefs of the IODP Exp. 339, thanks to IODP, Exp. IODP 339 Scientists; JR crew and technicians, as well as all people, institutions and companies involved in making IODP a success since 2003. Finally, we also thank the editor, Gert J. De Lange and the reviewers T. Mulder (Bourdeaux Univ.); D. Van Rooij (Ghent Univ) and J. Duarte (Monash Univ.) for their very positive and helpful feedback and discussions in publishing this research.Peer reviewedPublisher PD

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

Pattern Recognition in a Bimodal Aquifer Using the Normal-Score Ensemble Kalman Filter

The ensemble Kalman filter (EnKF) is now widely used in diverse disciplines to estimate model parameters and update model states by integrating observed data. The EnKF is known to perform optimally only for multi-Gaussian distributed states and parameters. A new approach, the normal-score EnKF (NS-EnKF), has been recently proposed to handle complex aquifers with non-Gaussian distributed parameters. In this work, we aim at investigating the capacity of the NS-EnKF to identify patterns in the spatial distribution of the model parameters (hydraulic conductivities) by assimilating dynamic observations in the absence of direct measurements of the parameters themselves. In some situations, hydraulic conductivity measurements (hard data) may not be available, which requires the estimation of conductivities from indirect observations, such as piezometric heads. We show how the NS-EnKF is capable of retrieving the bimodal nature of a synthetic aquifer solely from piezometric head data. 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