Discrete stability of the Differential System Method evaluated with geostatistical techniques

The Differential System Method (DSM) permits identification of the physical parameters of finite-difference groundwater flow models in a confined aquifer when piezometric head and source terms are known at each point of the finite-difference lattice for at least two independent flow situations for which the hydraulic gradients are not parallel. Since piezometric head data are usually few and sparse, interpolation of the measured data onto a regular grid can be performed with geostatistical techniques. We apply kriging to the sparse data of a synthetic aquifer to evaluate the stability of the DSM with respect to uncorrelated measurement errors and interpolation errors. The numerical results show that the DSM is stable.     

Effect of Conditioning Randomly Heterogeneous Transmissivity on Temporal Hydraulic Head Measurements in Transient Two-Dimensional Aquifer Flow

We investigated the effect of conditioning transient, two-dimensional groundwater flow simulations, where the transmissivity was a spatial random field, on time dependent head data. The random fields, representing perturbations in log transmissivity, were generated using a known covariance function and then conditioned to match head data by iteratively cokriging and solving the flow model numerically. A new approximation to the cross-covariance of log transmissivity perturbations with time dependent head data and head data at different times, that greatly increased the computational efficiency, was introduced. The most noticeable effect of head data on the estimation of head and log transmissivity perturbations occurred from conditioning only on spatially distributed head measurements during steady flow. The additional improvement in the estimation of the log transmissivity and head perturbations obtained by conditioning on time dependent head data was fairly small. On the other hand, conditioning on temporal head data had a significant effect on particle tracks and reduced the lateral spreading around the center of the paths.     

Effect of Conditioning Randomly Heterogeneous Transmissivity on Temporal Hydraulic Head Measurements in Transient Two-Dimensional Aquifer Flow

We investigated the effect of conditioning transient, two-dimensional groundwater flow simulations, where the transmissivity was a spatial random field, on time dependent head data. The random fields, representing perturbations in log transmissivity, were generated using a known covariance function and then conditioned to match head data by iteratively cokriging and solving the flow model numerically. A new approximation to the cross-covariance of log transmissivity perturbations with time dependent head data and head data at different times, that greatly increased the computational efficiency, was introduced. The most noticeable effect of head data on the estimation of head and log transmissivity perturbations occurred from conditioning only on spatially distributed head measurements during steady flow. The additional improvement in the estimation of the log transmissivity and head perturbations obtained by conditioning on time dependent head data was fairly small. On the other hand, conditioning on temporal head data had a significant effect on particle tracks and reduced the lateral spreading around the center of the paths.     

Thermohaline convection in flowing groundwater

Geothermal activity creates destabilising temperature gradients which are significant in some aquifers. Usually, in such aquifers stabilising salinity gradients also exist. The combination of temperature and salinity distribution in the aquifer may induce various types of hydrodynamic instabilities which were identified in a previous article. The present article concerns the effect of anisotropic characteristics of the hydrodynamic dispersion on the growth of instabilities in the aquifer. Three different mechanisms may lead to instability of the flow field: (a) buoyancy forces may induce convection currents; if the difference between the convection velocity of salt, due to the hydraulic gradients, and that of heat is negligible, then this mechanism is generally most effective in planes parallel to the hydraulic velocity of the fluid (velocity due to the hydraulic gradient); (b) the difference between heat and salt effective diffusivities may lead to overstability; this mechanism is most effective in planes perpendicular to the hydraulic velocity; (c) the difference between the convection velocity of salt and that of heat may induce oscillations which are most effective in planes parallel to the hydraulic velocity. The growth of instabilities in an aquifer of unlimited length is different from their growth in an aquifer of limited length. In the latter thermohaline convection develops in planes perpendicular to the hydraulic velocity, whereas in the former it develops in planes forming an angle θ with the hydraulic gradient. The development of convection cells in the flow field is identified by numerical experiments. These experiments identify the convection cell length and the angle formed between the thermohaline convection plane and the hydraulic gradient.     

Optimal contour mapping of groundwater levels using universal kriging—a case study

Abstract

Universal kriging is applied to the command area of a set of canal irrigation projects in north-western India to show its applicability for optimal contour mapping of groundwater levels. This command area faces the problem of a rising groundwater table and manifestation of waterlogging over the years at many places. With the use of measured elevations of the water table in September 1990 at 143 observation sites in an area of 4500 km², an omni-directional experimental semivariogram was constructed. The concave upward shape of omni-directional experimental semivariogram indicated the non-stationarity of the data and so the need for universal kriging. Directional semivariograms were calculated to find out the direction along which there is least drift. This directional semivariogram was considered as the underlying semivariogram and various theoretical semivariogram models were fitted to it. The drift order was estimated from the cross-validation procedure. The model and drift order finally selected were used to estimate the groundwater levels and corresponding estimation variances at the nodes of a square grid of 2 km × 2 km, and to develop the corresponding contour map.     

Analytical expressions of conditional mean,covariance, and sample functions in geostatistics

This work presents analytical expressions for the best estimate, conditional covariance function, and conditional realizations of a function from sparse observations. In contrast to the prevalent approach in kriging where the best estimates at every point are determined from the solution of a system of linear equations, the best-estimate function can be represented analytically in terms of basis functions, whose number depends on the observations. This approach is computationally superior when graphing a function estimate and is also valuable in understanding what the solution should look like. For example, one can immediately see that all singularities in the best-estimate function are at observation points.     

一种求解地球物理参数各向异性的新方法一单位步长变异增量法

目前在研究地球物理参数的空间变异特性时常常是基于几何各向异性假设,然而实际中几何各向异性假设常常不成立。本文提出了一种求解地球物理参数各向异性的新方法,首先对各方向上的变异值作归一化处理,然后用拟合的方法得到各方向单位步长上的变异性强度,进而实现对全局各向异性的求解。以上海地区合成孔径雷达干涉测量（InSAR）中差分大气延迟样本数据为例,验证了新方法求解各向异性的优越性：新方法的偏差只有6.4%,而基于几何各向异性假设方法的偏差达到了21.2%,而且新方法的运算效率有非常显著的提高。进一步,通过克里金插值交叉验证实验,证明了基于新算法得到的结构函数的插值效果最好,从而从另外一个角度验证了新算法求解的各向异性更准确,更好地表征了区域化变量的空间结构特征。因此,与几何各向异性假设的方法相比,新方法不仅能更准确的描述地球物理参数的各向异性,而且有更高的运算效率,为更准确地估计出所需的地球物理参数奠定了基础。     

Relationship between point rainfall, average sampled rainfall and ground truth at the event scale in the Sahel

Geostatistical techniques are used to quantify the reference mean areal rainfall (ground truth) from sparse raingaugenetworks. Based on the EPSAT-Niger event cumulative rainfall, a linear relationship between the ground truth considered as the mean area rainfall estimated from the densely available raingauge network and the area rainfall estimated from sparse network are derived. Also, a linear relationship between the ground truth and point rainfall is established. As it was reported experimentally by some authors, the slope of these relationships is less than one. Based on the geostatistical framework, the slope and the ordinate at the origin can be estimated as a function of the spatial structure of the rainfall process. It is shown that the slope is smaller than one. For the special case of one gauge inside a fixed area or a Field Of View (FOV), an areal reduction factor is derived. It has a limit value which depends only on the size of the area and the spatial structure of the rainfall process. The relative variance error of estimating the FOV cumulative rainfall from point rainfall is also given.     

Application of the BME approach to soil texture mapping

In order to derive accurate space/time maps of soil properties, soil scientists need tools that combine the usually scarce hard data sets with the more easily accessible soft data sets. In the field of modern geostatistics, the Bayesian maximum entropy (BME) approach provides new and powerful means for incorporating various forms of physical knowledge (including hard and soft data, soil classification charts, land cover data from satellite pictures, and digital elevation models) into the space/time mapping process. BME produces the complete probability distribution at each estimation point, thus allowing the calculation of elaborate statistics (even when the distribution is not Gaussian). It also offers a more rigorous and systematic method than kriging for integrating uncertain information into space/time mapping. In this work, BME is used to estimate the three textural fractions involved in a texture map. The first case study focuses on the estimation of the clay fraction, whereas the second one considers the three textural fractions (sand, silt and clay) simultaneously. The BME maps obtained are informative (important soil characteristics are identified, natural variations are well reproduced, etc.). Furthermore, in both case studies, the estimates obtained by BME were more accurate than the simple kriging (SK) estimates, thus offering a better picture of soil reality. In the multivariate case, classification error rate analysis in terms of BME performs considerably better than in terms of kriging. Analysis in terms of BME can offer valuable information to be used in sampling design, in optimizing the hard to soft data ratio, etc.     

地震道的非线性混沌反演——Ⅰ.理论和数值试验

全文以地震道反演的数值实验为例,根据混沌理论讨论了非线性地震反演的以下特性:1.对于带误差的地震数据和极其平滑的初始模型,逐次线性化的迭代过程产生的输出序列最终走向无序,这是非线性反演系统本身的特性决定的;2.迭代（非线性反演）系统是由Poin-care映的或系统方程描述的;3.可以根据地震反问题的特点,用多种不同的方法定义相应的Lyapunov指数,它们的数值和组合是非线性反演系统状态的有效指示;4.取决于系统参数的选取,发现在二维相空间有奇异吸引子的对应物.总之,从混沌理论的角度来研究纯数学的非线性地震反问题,可以揭示某些过去鲜为人知的内在规律性.本文为全文的上篇,主要讨论前两个问题,即理论和地震道反演数值试验结果.     

Improving kriging of groundwater level data using nonlinear normalizing transformations—a field application

Abstract

The present research study investigates the application of nonlinear normalizing data transformations in conjunction with ordinary kriging (OK) for the accurate prediction of groundwater level spatial variability in a sparsely-gauged basin. We investigate three established normalizing methods, Gaussian anamorphosis, trans-Gaussian kriging and the Box-Cox method to improve the estimation accuracy. The first two are applied for the first time to groundwater level data. All three methods improve the mean absolute prediction error compared to the application of OK to the non-transformed data. In addition, a modified Box-Cox transformation is proposed and applied to normalize the hydraulic heads. The modified Box-Cox transformation in conjunction with OK is found to be the optimal spatial model based on leave-one-out cross-validation. The recently established Spartan semivariogram family provides the optimal model fit to the transformed data. Finally, we present maps of the groundwater level and the kriging variance based on the optimal spatial model.

Editor D. Koutsoyiannis; Associate editor A. Montanari

Citation Varouchakis, E.A., Hristopoulos, D.T., and Karatzas, G.P., 2012. Improving kriging of groundwater level data using nonlinear normalizing transformations—a field application. Hydrological Sciences Journal, 57 (7), 1404–1419.     

求解线性方程组的一种迭代方法——ART算法的一种改进

本文用Ｓｃｈｍｉｄｔ正交化方法对阻尼ＡＲＴ算法进行了改进，并以此为基础，通过与一种投影迭代法的结合给出一种求解线性方程组的迭代方法，同时指出即使线方程组不相容时，由此法所构造的向量序列将收敛于其Ｍｏｏｒｅ－Ｐｅｎｒｏｓｅ广义逆解。     

Kalman filtering in groundwater flow modelling: problems and prospects

The popularity of applying filtering theory in the environmental and hydrological sciences passed its first climax in the 1970s. Like so many other new mathematical methods it was simply the fashion at the time. The study of groundwater systems was not immune to this fashion, but neither was it by any means a prominent area of application. The spatial-temporal characteristics of groundwater flow are customarily described by analytical or, more frequently, numerical, physics-based models. Consequently, the state-space representations associated with filtering must be of a high order, with an immediately apparent computational over-burden. And therein lies part of the reason for the but modest interest there has been in applying Kalman filtering to groundwater systems, as reviewed critically in this paper. Filtering theory may be used to address a variety of problems, such as: state estimation and reconstruction, parameter estimation (including the study of uncertainty and its propagation), combined state-parameter estimation, input estimation, estimation of the variance-covariance properties of stochastic disturbances, the design of observation networks, and the analysis of parameter identifiability. A large proportion of previous studies has dealt with the problem of parameter estimation in one form or another. This may well not remain the focus of attention in the future. Instead, filtering theory may find wider application in the context of data assimilation, that is, in reconstructing fields of flow and the migration of sub-surface contaminant plumes from relatively sparse observations. Received: October 27, 1997