Conditional Probability Density Functions of Concentrations for Mixing-Controlled Reactive Transport in Heterogeneous Aquifers

This paper presents an approach conducive to an evaluation of the probability density function (pdf) of spatio-temporal distributions of concentrations of reactive solutes (and associated reaction rates) evolving in a randomly heterogeneous aquifer. Most existing approaches to solute transport in heterogeneous media focus on providing expressions for space–time moments of concentrations. In general, only low order moments (unconditional or conditional mean and covariance) are computed. In some cases, this allows for obtaining a confidence interval associated with predictions of local concentrations. Common applications, such as risk assessment and vulnerability practices, require the assessment of extreme (low or high) concentration values. We start from the well-known approach of deconstructing the reactive transport problem into the analysis of a conservative transport process followed by speciation to (a) provide a partial differential equation (PDE) for the (conditional) pdf of conservative aqueous species, and (b) derive expressions for the pdf of reactive species and the associated reaction rate. When transport at the local scale is described by an Advection Dispersion Equation (ADE), the equation satisfied by the pdf of conservative species is non-local in space and time. It is similar to an ADE and includes an additional source term. The latter involves the contribution of dilution effects that counteract dispersive fluxes. In general, the PDE we provide must be solved numerically, in a Monte Carlo framework. In some cases, an approximation can be obtained through suitable localization of the governing equation. We illustrate the methodology to depict key features of transport in randomly stratified media in the absence of transverse dispersion effects. In this case, all the pdfs can be explicitly obtained, and their evolution with space and time is discussed.     

New in situ techniques for the estimation of the dielectric properties and moisture content of soils

Two original in situ HF techniques of dielectric characterization in two wide bands have been developed in order to estimate the moisture content of soils by complex impedance measurement. These techniques are based on the capacitive effect (1–20 MHz) and on the propagation of electromagnetic waves at high frequencies (0.1–4 GHz). The two measurement techniques use straight conductors that are inserted into the soil. Specific inversion algorithms were developed to estimate the apparent real permittivity of the soil versus frequency from the complex impedance. The validation of both instruments was made in the laboratory in the presence of dry and wet sands. In situ experiments were also made at high frequencies. These complementary devices should enlarge the range of usual soil moisture measurement techniques.     

Sediment deposition from turbidity currents in simulated aquatic vegetation canopies

A laboratory flume experiment was carried out in which the hydrodynamic and sedimentary behaviour of a turbidity current was measured as it passed through an array of vertical rigid cylinders. The cylinders were intended primarily to simulate aquatic vegetation canopies, but could equally be taken to represent other arrays of obstacles, for example forests or offshore wind turbines. The turbidity currents were generated by mixing naturally sourced, poly‐disperse sediment into a reservoir of water at concentrations from 1·0 to 10·0 g l^?1, which was then released into the experimental section of the flume by removing a lock gate. For each initial sediment concentration, runs with obstacle arrays with solid plant fractions of 1·0% and 2·5%, and control cases with no obstacles, were carried out. The progress of the current along the flume was characterized by the array drag term, C_Dax_c (where C_D is the array drag coefficient, at the frontal area of cylinders per unit volume, and x_c is the position of the leading edge of the current along the flume). The downward depositional flux of sediment out of the current as it proceeded was measured at 13 traps along the flume. Analysis of these deposits divided them into fine (2·2 to 6·2 μm) and coarse (6·2 to 104 μm) fractions. At the beginning of their development, the gravity currents proceeded in an inertia‐dominated regime until C_Dax_c = 5. For C_Dax_c > 5, the current transitioned into a drag‐dominated regime. For both fine and coarse sediment fractions, the rate of sediment deposition tended to decrease gradually with distance from the source in the inertial regime, remained approximately constant at the early drag‐dominated regime, and then rose and peaked at the end of the drag‐dominated stage. This implies that, when passing through arrays of obstacles, the turbidity currents were able to retain sufficient sediment in suspension to maintain their flow until they became significantly influenced by the drag exerted by the obstacles.     

Spectral Simulation of Isotropic Gaussian Random Fields on a Sphere

A spectral algorithm is proposed to simulate an isotropic Gaussian random field on a sphere equipped with a geodesic metric. This algorithm supposes that the angular power spectrum of the covariance function is explicitly known. Direct analytic calculations are performed for exponential and linear covariance functions. In addition, three families of covariance functions are presented where the calculation of the angular power spectrum is simplified (shot-noise random fields, Yadrenko covariance functions and solutions of certain stochastic partial differential equations). Numerous illustrative examples are given.

     

Understanding satellite-derived bathymetry using Sentinel 2 imagery and spatial prediction models

ABSTRACT

Optical satellite data is an efficient and complementary method to hydrographic surveys for deriving bathymetry in shallow coastal waters. Empirical approaches (in particular, the models of Stumpf and Lyzenga) provide a practical methodology to derive bathymetric information from remote sensing. Recent studies, however, have focused on enhancing the performance of such empirical approaches by extending them via spatial information. In this study, the relationship between multibeam depth and Sentinel-2 image bands was analyzed in an optically complex environment using the spatial predictor of kriging with an external drift (KED), where its external drift component was estimated: a) by a ratio of log-transformed bands based on Stumpf’s model (KED_S) and b) by a log-linear transform based on Lyzenga’s model (KED_L). Through the calibration of KED models, the study objectives were: 1) to better understand the empirical relationship between Sentinel-2 multispectral satellite reflectance and depth, 2) to test the robustness of KED to derive bathymetry in a multitemporal series of Sentinel-2 images and multibeam data, and 3) to compare the performance of KED against the existing non-spatial models described by Stumpf et al. and Lyzenga. Results showed that KED could improve prediction accuracy with a decrease in RMSE of 89% and 88%, and an increase in R² of 27% and 14%, over the Stumpf and Lyzenga models, respectively. The decrease in RMSE provides a worthwhile improvement in accuracy, where results showed effective prediction of depth up to 6 m. However, the presence of higher concentrations of suspended materials, especially river plumes, can reduce this threshold to 4 m. As would be expected, prediction accuracy could be improved through the removal of outliers, which were mainly located in the channel of the river, areas influenced by the river plume, abrupt topography, but also very shallow areas close to the shoreline. These areas have been identified as conflictive zones where satellite-derived bathymetry can be compromised.     

Variograms of Order ω: A Tool to Validate a Bivariate Distribution Model

The multigaussian model is used in mining geostatistics to simulate the spatial distribution of grades or to estimate the recoverable reserves of an ore deposit. Checking the suitability of such model to the available data often constitutes a critical step of the geostatistical study. In general, the marginal distribution is not a problem because the data can be transformed to normal scores, so the check is usually restricted to the bivariate distributions. In this work, several tests for diagnosing the two-point normality of a set of Gaussian data are reviewed and commented. An additional criterion is proposed, based on the comparison between the usual variogram and the variograms of lower order: the latter are defined as half the mean absolute increments of the attribute raised to a power between 0 and 2. This criterion is then extended to other bivariate models, namely the bigamma, Hermitian and Laguerrian models. The concepts are illustrated on two real data-sets. Finally, some conditions to ensure the internal consistency of the variogram under a given model are given.     

Apports de la modélisation sur bathymétries historiques dans la compréhension des évolutions des bancs de sable estuariens

Formation and evolution of estuarine sand ridges are some of the less well known among the different classes of sand banks. Simulation of tides and related sand bedload transport on historical bathymetries brings new insights about the secular evolution of two estuarine sand ridges (the Saint-Georges and the ‘Longe de Boyard’ sand banks, Atlantic coast of France). Two different mechanisms are involved in order to explain the evolutions of these banks, suggesting the category of estuarine sand banks to be less homogenous than it is proposed in the last classifications of sand banks. To cite this article: X. Bertin, É. Chaumillon, C. R. Geoscience 337 (2005).