HYDRO_GEN: A spatially distributed random field generator for correlated properties

This paper describes a new method for generating spatially-correlated random fields. Such fields are often encountered in hydrology and hydrogeology and in the earth sciences. The method is based on two observations: (i) spatially distributed attributes usually display a stationary correlation structure, and (ii) the screening effect of measurements leads to the sufficiency of a small search neighborhood when it comes to projecting measurements and data in space. The algorithm which was developed based on these principles is called HYDRO_GEN, and its features and properties are discussed in depth. HYDRO_GEN is found to be accurate and extremely fast. It is also versatile: it can simulate fields of different nature, starting from weakly stationary fields with a prescribed covariance and ending with fractal fields. The simulated fields can display statistical isotropy or anisotropy.     

Influence of spatial variations of microtopography and infiltration on surface runoff and field scale hydrological connectivity

Surface runoff on agricultural fields arises when rainfall exceeds infiltration. Excess water ponding in and flowing through local microtopography increases the hydrological connectivity of fields. In turn, an increased level of hydrological connectivity leads to a higher surface runoff flux at the field boundaries. We investigated the functional hydrological connectivity of synthetical elevation fields with varying statistical properties. For this purpose, we developed an object-oriented ponding and redistribution model to which Philip’s infiltration model was coupled. The connectivity behaviour is determined by the presence of depressions with a large area and spatial organization of microtopography in rills or channels. The presence of microdepressions suppresses the effect of the spatial variation of infiltration properties. Connectivity behaviour of a field with a varying spatial distribution of infiltration properties can be predicted by transforming the unique connectivity function that was defined for a designated microtopography.     

Calibration of Parameter Fields Consisting of Multiple Statistical Populations

If a parameter field to be calibrated consists of more than one statistical population, usually not only the parameter values are uncertain, but the spatial distributions of the populations are uncertain as well. In this study, we demonstrate the potential of the multimodal calibration method we proposed recently for the calibration of such fields, as applied to real-world ground water models with several additional stochastic parameter fields. Our method enables the calibration of the spatial distribution of the statistical populations, as well as their spatially correlated parameterization, while honoring the complete prior geostatistical definition of the multimodal parameter field. We illustrate the implications of the method in terms of the reliability of the posterior model by comparing its performance to that of a "conventional" calibration approach in which the positions of the statistical populations are not allowed to change. Information from synthetic calibration runs is used to show how ignoring the uncertainty involved in the positions of the statistical populations not only denies the modeler the opportunity to use the measurement information to improve these positions but also unduly influences the posterior intrapopulation distributions, causes unjustified adjustments to the cocalibrated parameter fields, and results in poorer observation reproduction. The proposed multimodal calibration allows a more complete treatment of the relevant uncertainties, which prevents the abovementioned adverse effects and renders a more trustworthy posterior model.     

STATISTICAL MULTIVARIATE ANALYSIS OF AIRBORNE GEOPHYSICAL DATA ON THE SE BORDER OF THE CENTRAL LAPLAND GREENSTONE COMPLEX*

Statistical multivariate methods for the integrated processing of airborne geophysical data were tested. The data consisted of magnetic, electromagnetic and gamma radiation measurements, to which cluster analysis, principal components analysis and discriminant analysis were applied. Also, auxiliary variables were derived from the original ones and their value was tested. Although the frequency distributions of the data do not favour statistical analysis, the practical results are acceptable. Principal component analyses show geological and technical aspects that are difficult to obtain from the original observations. In cluster analyses, the sources of measured fields control the grouping of variables. Discriminant analysis was applied to the automatic identification of rocks by geophysical data. The rocks investigated are metasediments and metavolcanics, some magnetic and others conductive. When all available geophysical data were included, correct identifications were made in more than 60% of cases. In particular, gamma ray observations were found to improve the discrimination of non-magnetic and non-conductive rocks. The geophysical similarity of rocks studied by cluster analysis depends on electrical and magnetic properties as well as on their origin; the content of radioactive elements in turn is related to the origin.     

Two-dimensional correlation of geophysical data applied to the heat flow and earth's crust thickness patterns in Czechoslovakia

Summary A simple method for the mutual correlation of various geophysical fields is proposed to disclose their prospective inherent correlations. Some problems connected with the statistical and physical confidence of the resulting maps of the correlation coefficient are mentioned. The correlation between the surface heat flow pattern and the crustal thickness on the territory of Czechoslovakia is presented as an example and briefly discussed in terms of crustal structure.     

Statistical estimation of runoff characteristics of watersheds in central Chile

Abstract

Estimation of monthly runoff statistical properties, such as monthly means and variances, is usually needed to design and evaluate water resource systems. If no local recorded data are available, a transfer of information through different alternative procedures can be used. In this paper, the use of linear Transfer Function (TF) models with precipitation series as inputs is proposed to estimate statistical properties of the resulting runoff series. Empirical relationships based on data from watersheds in the mountainous zone of central Chile are suggested to estimate parameters of low-order TF models and some of their properties.     

Mathematical models for long-term prediction of mountainous river runoff: methods,information and results

Abstract

Unlike traditional statistical methods, a mathematical model is used to solve the problem of the long-term prediction of mountainous river runoff. The paper describes the structure of a basic model and ways of numerical presentation of macroscale rainfall fields in mountains as well as the computation of their parameters in conditions with complex relief and lack of information. The encouraging results and estimates of forecasts are discussed.     

Two-dimensional stochastic analysis of flow in leaky confined aquifers subject to spatial and periodic leakage

This paper addresses the issue of flow in heterogeneous leaky confined aquifers subject to leakage. The leakage into the confined aquifer is driven by spatial and periodic fluctuations of water table in an overlying phreatic aquifer. The introduction of leakage leads to non-uniformity in the mean head gradient and results in nonstationarity in hydraulic head and velocity fields. Therefore, a nonstationary spectral approach based on Fourier–Stieltjes representations for the perturbed quantities is adopted to account for the spatial variability of nonstationary head fields. Closed-form expressions for the variances of hydraulic head and specific discharge are developed in terms of statistical properties of hydraulic parameters. The results indicate that the spatiotemporal variations in leakage leads to enhanced variability of the hydraulic head and of the specific discharge, which increase with distance from any arbitrary reference point. The coefficient of leakage and the spatial structure of log transmissivity field and of the amplitude of water table fluctuation are critical in quantifying the variability of the hydraulic head and of the specific discharge.     

Role of Wavelets in the Physical and Statistical Modelling of Complex Geological Processes

— Today, wavelets are recognized to have a wide range of useful properties that allow them to treat effectively multifacet problems, such as data compression, scale-localization analysis, feature extraction, statistics, numerical simulation, visualization, and communication. Second-generation wavelets represent a recent improvement of the wavelet algorithm, allowing for greater flexibility in the spatial domain and other computational advantages. We will show how these wavelets can be employed to extract large-scale coherent structures from (1) three-dimensional turbulent flows and (2) high Rayleigh number thermal convection. We will discuss the concept of modelling via decomposition into coherent and incoherent fields, taking into account the effect of the incoherent field via statistical modelling. We will discuss wavelet properties and how they can be utilized and integrated in handling large-scale problems in earthquake physics and other nonlinear phenomena in the geosciences.     

Identification of the environmentally safe fairway in the South-Western Baltic Sea and Kattegat

Application of the preventive techniques for the optimisation of fairways in the south-western Baltic Sea and the Kattegat in terms of protection of the coastal regions against current-driven surface transport of adverse impacts released from vessels is considered. The techniques rely on the quantification of the offshore domains (the points of release of adverse impacts) in terms of their ability to serve as a source of remote, current-driven danger to the nearshore. An approximate solution to this inverse problem of current-driven transport is obtained using statistical analysis of a large pool of Lagrangian trajectories of water particles calculated based on velocity fields from the Denmark’s Meteorological Institute (DMI)/BSH cmod circulation model forced by the DMI-HIRHAM wind fields for 1990–1994. The optimum fairways are identified from the spatial distributions of the probability of hitting the coast and for the time (particle age) it takes for the pollution to reach the coast. In general, the northern side of the Darss Sill area and the western domains of the Kattegat are safer to travel. The largest variations in the patterns of safe areas and the properties of pollution beaching occur owing to the interplay of water inflow and outflow. The gain from the use of the optimum fairways is in the range of 10–30?% in terms of the decrease in the probability of coastal hit within 10?days after pollution release or an increase by about 1–2?days of the time it takes for the hit to occur.     

Statistical characteristics of geophysical fields in Hilbert's space

Summary A review of the terms of functional analysis, which can be exploited for statistical treatment of geophysical fields, is given. The procedure for determining the statistical characteristics of the fields, defined by a single realization, which does not require the limiting assumption of homogeneity to be introduced, is outlined.     

A heuristic model of three-dimensional spectra of temperature inhomogeneities in the stably stratified atmosphere

Many measurements have shown that the random temperature fields in the stably stratified atmosphere are not locally isotropic (LI). The local axial symmetry (LAS) hypothesis looks more appropriate under these conditions. The object of this paper consists in the development of a flexible tool for spectral studies of LAS scalar fields independently of their origin in stably stratified geophysical flows. A heuristic model of a 3D spatial spectrum is proposed in order to describe and study statistical properties of LAS temperature inhomogeneities from LI structures up to quasi-layered ones. To simplify the solution of this problem, a main assumption was accepted: the consideration is restricted to 3D spectra which may be given on a one-parametric family of surfaces of rotation. Such 3D spectra may be represented by a single function of one variable which is the parameter of the family. This approach allows one to introduce the generalized energy spectrum which describes an energy distribution according to inhomogeneity sizes. The family of surfaces determines the shape of inhomogeneities. The family of ellipsoids of rotation and power-law generalized energy spectrum is used as the simplest example of the model application in order to study the general properties of LAS-structure spectra. The behavior of vertical, horizontal, and oblique 1D spectra and coherency spectra is studied. The relationship between the suggested model and some existing models of temperature spectra is considered. The application of the model for the analysis of experimental data is shown for two sets of measurements. It is shown that the suggested model may quantitatively describe experimental 1D spectra and coherency spectra from a unique point of view. It is noted that the model may be used for both the planning of measurements and data processing. Possible generalizations of the model are considered for random fields with more degenerate symmetry and for space-temporal spectra.     

On the problem of the existence and uniqueness of a discrete,surface, optimum wiener filter,applied to the filtering of geophysical fields

Summary Proof is given of the existence and uniqueness of a discrete, surface, optimum (in the Wiener sense) filter for filtering geophysical fields considering a spectral approach to the construction of the said filter. It is also shown that the filtration coefficients and the filtration errors of the filter, constructed with the help of statistical estimates of the required spectral densities, converge towards their theoretical values. From the mathematical point of view, this paper concludes the building-up of the mathematical model of the discrete Wiener optimum surface filtration, suitable for geophysical fields.     

Covariance functions and models for complex-valued random fields

In Geostatistics, primary interest often lies in the study of the spatial, or spatial-temporal, correlation of real-valued random fields, anyway complex-valued random field theory is surely a natural extension of the real domain. In such a case, it is useful to consider complex covariance functions which are composed of an even real part and an odd imaginary part. Generating complex covariance functions is not simple at all, but the procedure, developed in this paper, allows generating permissible covariance functions for complex-valued random fields in a straightforward way. In particular, by recalling the spectral representation of the covariance and translating the spectral density function by using a shifting factor, complex covariances are obtained. Some general aspects and properties of complex-valued random fields and their moments are pointed out and some examples are given.     

Interpolation of Steady-State Concentration Data by Inverse Modeling

In most groundwater applications, measurements of concentration are limited in number and sparsely distributed within the domain of interest. Therefore, interpolation techniques are needed to obtain most likely values of concentration at locations where no measurements are available. For further processing, for example, in environmental risk analysis, interpolated values should be given with uncertainty bounds, so that a geostatistical framework is preferable. Linear interpolation of steady-state concentration measurements is problematic because the dependence of concentration on the primary uncertain material property, the hydraulic conductivity field, is highly nonlinear, suggesting that the statistical interrelationship between concentration values at different points is also nonlinear. We suggest interpolating steady-state concentration measurements by conditioning an ensemble of the underlying log-conductivity field on the available hydrological data in a conditional Monte Carlo approach. Flow and transport simulations for each conditional conductivity field must meet the measurements within their given uncertainty. The ensemble of transport simulations based on the conditional log-conductivity fields yields conditional statistical distributions of concentration at points between observation points. This method implicitly meets physical bounds of concentration values and non-Gaussianity of their statistical distributions and obeys the nonlinearity of the underlying processes. We validate our method by artificial test cases and compare the results to kriging estimates assuming different conditional statistical distributions of concentration. Assuming a beta distribution in kriging leads to estimates of concentration with zero probability of concentrations below zero or above the maximal possible value; however, the concentrations are not forced to meet the advection-dispersion equation.     

German tanks and historical records: the estimation of the time coverage of ungauged extreme events

The use of historical data can significantly reduce the uncertainty around estimates of the magnitude of rare events obtained with extreme value statistical models. For historical data to be included in the statistical analysis a number of their properties, e.g. their number and magnitude, need to be known with a reasonable level of confidence. Another key aspect of the historical data which needs to be known is the coverage period of the historical information, i.e. the period of time over which it is assumed that all large events above a certain threshold are known. It might be the case though, that it is not possible to easily retrieve with sufficient confidence information on the coverage period, which therefore needs to be estimated. In this paper methods to perform such estimation are introduced and evaluated. The statistical definition of the problem corresponds to estimating the size of a population for which only few data points are available. This problem is generally refereed to as the German tanks problem, which arose during the second world war, when statistical estimates of the number of tanks available to the German army were obtained. Different estimators can be derived using different statistical estimation approaches, with the maximum spacing estimator being the minimum-variance unbiased estimator. The properties of three estimators are investigated by means of a simulation study, both for the simple estimation of the historical coverage and for the estimation of the extreme value statistical model. The maximum spacing estimator is confirmed to be a good approach to the estimation of the historical period coverage for practical use and its application for a case study in Britain is presented.     

Intermittency of the solar wind density near the interplanetary shock

The properties of turbulent fluctuations of the solar wind plasma near the interplanetary shock observed at September 12, 2014 by the BMSW instrument are considered. The spectra of the density fluctuations in the solar wind and their statistical characteristics up-and downstream of the shock front are analyzed. They are compared with each other and with characteristics corresponding to different turbulence models. It is shown that the spectral and statistical characteristics of the density fluctuations in the solar wind conserve their basic properties after the arrival of an interplanetary shock. Intermittency is observed both before and after the front, but its level increases on average in the second case. In both regions, the scaling of the structure functions of the density fluctuations in the solar wind differ from the scaling of the classical Kolmogorov model and can be described by the log-Poisson turbulence model. Parameterization of the scaling of the structure functions revealed the presence of filamentary structures in the solar wind plasma, which provide the density intermittency in the studied space regions.