New anisotropic covariance models and estimation of anisotropic parameters based on the covariance tensor identity

 Many heterogeneous media and environmental processes are statistically anisotropic. In this paper we focus on range anisotropy, that is, stochastic processes with variograms that have direction dependent correlation lengths and direction independent sill. We distinguish between two classes of anisotropic covariance models: Class (A) models are reducible to isotropic after rotation and rescaling operations. Class (B) models can be separated into a product of one-dimensional functions oriented along the principal axes. We propose a new Class (A) model with multiscale properties that has applications in subsurface hydrology. We also present a family of Class (B) models based on non-Euclidean distance metrics that are generated by superellipsoidal functions. Next, we propose a new method for determining the orientation of the principal axes and the degree of anisotropy, i.e., the ratio(s) of the correlation lengths. This information reduces the degrees of freedom of anisotropic variograms and thus simplifies the estimation procedure. In particular, Class (A) models are reduced to isotropic and Class (B) models to one-dimensional functions. Our method is based on an explicit relation between the second-rank slope tensor (SRST), which can be estimated from the data, and the covariance tensor. The procedure is conceptually simple and numerically efficient. It is more accurate for regular (on-grid) data distributions, but it can also be used for sparse (off-grid) spatial distributions. In the case of non-differentiable random fields the method can be extended using generalized derivatives. We illustrate its implementation with numerical simulations.     

Testing linear models of sea-floor topography

Stochastic models can generate profiles that resemble topography by taking uncorrelated, zero-average noise as input, introducing some correlation in the time series of noise, and integrating the resulting correlated noise. The output profile will depict a nonstationary, randomly rough surface. Two models have been chosen for comparison: a fractal model, in which the noise is correlated even at large distances, and an autoregressive model of order 1, in which the correlation of the noise decays rapidly. Both models have as an end-member a random walk, which is the integration of uncorrelated noise. The models have been fitted to profiles of submarine topography, and the sample autocorrelation, power spectrum and variogram have been compared to the theoretical predictions. The results suggest that a linear system approach is a viable method to model and classify sea-floor topography. The comparison does not show substantial disagreement of the data with either the autoregressive or the fractal model, although a fractal model seems to give a better fit. However, the amplitudes predicted by a nonstationary fractal model for long wavelengths (of the order of 1000 km) are unreasonably large. When viewed through a large window, ocean floor topography is likely to have an expected value determined by isostasy, and to be stationary. Nonstationary models are best applied to wavelengths of the order of 100 km or less.     

Relationships between correlation lengths and integral scales for covariance models with more than two parameters

In geostatistical applications, the terms correlation length and range are often used interchangeably and refer to a characteristic covariance length ξ that normalizes the lag distance in the variogram or the covariance model. We present equations that strictly define the correlation length (r _c ) and integral range (ℓ _c ). We derive analytical expressions for r _c and ℓ _c of the Whittle–Matérn, fluctuation gradient curvature and rational quadratic covariances. For these covariances, we show that the correlation length and integral range for a given model are not fully determined by ξ. We define non-trivial covariance functions, and we formulate an ergodicity index based on ℓ _c . We propose using the ergodicity index to compare coarse-grained measures corresponding to non-trivial covariance functions with different parameters. Finally, we discuss potential applications of the proposed covariance models in stochastic subsurface hydrology.     

Field evaluation of some sand transport models

Fluorescent tracer procedures, originally developed for research on fluvial sediment movement, were used to monitor the movement of three commercially purchased sands in a natural dune environment. Results were compared with estimates of transport rates from three theoretical models. Estimates from models by Bagnold (1941) and Hsu (1971) were larger by as much as one order of magnitude than the rates observed in the tracer study. The model of A. A. Kadib (1965) provided closer correspondence to observed transport rates for medium sand (mean diam. 0.653 mm) but underestimated rates for coarse sand (mean diam. 0.992 mm).     

Correlation and covariance of runoff

The application of objective methods for interpolation of stochastic fields is based on the assumption of homogeneity with respect to the correlation function, i.e. only the relative distance between two points is of importance. This is not the case for runoff data which is demonstrated in this paper. Taking into consideration the structure of the river network and the related drainage basin supporting areas theoretical expressions are derived for the correlation function for flow along a river from its outlet and upstream. The results are exact for a rectangular drainage basin. For more complex basin geometry a grid approximation is suggested. The found relations are demonstrated on a real world example with a good agreement between the theoretically calculated correlation functions and empirical data.     

Testing alternative ground water models using cross-validation and other methods

Many methods can be used to test alternative ground water models. Of concern in this work are methods able to (1) rank alternative models (also called model discrimination) and (2) identify observations important to parameter estimates and predictions (equivalent to the purpose served by some types of sensitivity analysis). Some of the measures investigated are computationally efficient; others are computationally demanding. The latter are generally needed to account for model nonlinearity. The efficient model discrimination methods investigated include the information criteria: the corrected Akaike information criterion, Bayesian information criterion, and generalized cross-validation. The efficient sensitivity analysis measures used are dimensionless scaled sensitivity (DSS), composite scaled sensitivity, and parameter correlation coefficient (PCC); the other statistics are DFBETAS, Cook's D, and observation-prediction statistic. Acronyms are explained in the introduction. Cross-validation (CV) is a computationally intensive nonlinear method that is used for both model discrimination and sensitivity analysis. The methods are tested using up to five alternative parsimoniously constructed models of the ground water system of the Maggia Valley in southern Switzerland. The alternative models differ in their representation of hydraulic conductivity. A new method for graphically representing CV and sensitivity analysis results for complex models is presented and used to evaluate the utility of the efficient statistics. The results indicate that for model selection, the information criteria produce similar results at much smaller computational cost than CV. For identifying important observations, the only obviously inferior linear measure is DSS; the poor performance was expected because DSS does not include the effects of parameter correlation and PCC reveals large parameter correlations.     

关于地质材料内部构造层次的若干模型

材料的内部结构对于材料的宏观行为有显著的影响。同时外部作用也会对于材料的内部结构产生影响，在岩石及岩体的破坏过程中也会形成新的结构，并进而对于材料的宏观行为产生影响。因此在研究地质材料的变形与破坏时必须考虑其内部构造。本文对这方面的某些模型进行了回顾，并对需要研究的问题作一些的展望。     

Dispersion and amplitudes of love waves for some models of the earth's crust and mantle

Summary The properties of the partial derivatives of dispersion curves, amplitudes and theoretical seismograms, in particular the occurrence of short-period phases, are studied on four crustmantle models. The individual models differ in the number of low-velocity channels. The fundamental and first higher mode of Love waves are studied in detail.     

Fatigue failure characteristics of some models of jointed rock

The time-dependent failure of rock subjected to repeated loading can be explained in terms of the residual strength locus: catastrophic failure occurs when the accumulation of deformation is such that the residual strength locus is intercepted. In a preliminary experimental study of the fatigue characteristics of rock masses, highly idealized models of jointed rock were subjected to a series of cyclic uniaxial compression tests. The results of these experiments show that intact and failed models of jointed rock are extremely susceptible to cyclic fatigue failure, and that in these cases the residual strength locus interpretation of time-dependent failure is correct.     

A novel geostatistical approach combining Euclidean and gradual-flow covariance models to estimate fecal coliform along the Haw and Deep rivers in North Carolina

Incorporating flow in the covariance function is important for geostatistical water quality estimation that accounts for hydrological transport. Very few studies have successfully incorporated flow due to various reasons including implementation difficulties. To address this critical issue, we introduce here the first implementation of a flow weighted covariance model that uses gradual flow, and we use this model in a novel hybrid Euclidean/Gradual-flow covariance model to estimate fecal coliform along the Haw and Deep rivers from 2006 to 2010. The hybrid Euclidean/Gradual-flow model results in a 12.4% reduction in estimation mean square error compared to the Euclidean model, indicating that this is the first study to successfully incorporate gradual flow and demonstrate an improvement in estimation accuracy over the purely Euclidean approach. Furthermore, results show that the Euclidean/Gradual-flow model is more accurate and easier to implement than the Euclidean/Pipe-flow model. Our assessment found that 96 river miles were detected as being impaired according to the Euclidean/Gradual-flow method, which is more than twice the 39 river miles found according to the Euclidean estimate. These results demonstrate that the Euclidean/Gradual-flow model substantially increase the sensitivity in detecting fecal impairment, which provide critical new information for watershed management and public health protection measures.     

Smoothing and high risk areas detection in space-time disease mapping: a comparison of P-splines,autoregressive, and moving average models

Recently, several models have been proposed for smoothing risks in disease mapping. These models consider different ways of introducing both spatial and temporal dependence as well as spatio-temporal interactions. In this work, a comparison among some autoregressive, moving average, and P-spline models is performed. Firstly, brain cancer mortality data are used to analyze the degree of smoothness introduced by these models. Secondly, two separate simulation studies (one model-based and the other model-free) are carried out to evaluate the model performance in terms of bias, variability, sensitivity, and specificity. We conclude that P-spline models seem to be a good alternative to autoregressive and moving average models when analyzing highly sparse disease mapping data.     

Theoretical seismogram constraints on some crustal velocity models in the central United States

Summary Earthquake investigations in the central United States require a realistic crustal model for accurate location of seismic events, for the determination of earthquake source parameters, and for theoretical ground motion prediction for an assumed earthquake source. A number of crustal models have been proposed which were based on first arrival refraction data obtained from artificial and natural seismic sources.The reflectivity method for generating realistic theoretical seismograms is used together with recently acquired refraction survey data to distinguish among several of the proposed crustal models. The data used in this study preclude a definitive statement about the nature of the lower crust in the region. However, a crust with two or more layers in the upper 20 km immediately below the basement is preferred. The absence of prominent near yertical reflections suggests that the layers may be separated by rapid velocity gradients rather than by sharp discontinuities.     

A comparative study of nonlinear Markov chain models for conditional simulation of multinomial classes from regular samples

Simulating fields of categorical geospatial variables from samples is crucial for many purposes, such as spatial uncertainty assessment of natural resources distributions. However, effectively simulating complex categorical variables (i.e., multinomial classes) is difficult because of their nonlinearity and complex interclass relationships. The existing pure Markov chain approach for simulating multinomial classes has an apparent deficiency—underestimation of small classes, which largely impacts the usefulness of the approach. The Markov chain random field (MCRF) theory recently proposed supports theoretically sound multi-dimensional Markov chain models. This paper conducts a comparative study between a MCRF model and the previous Markov chain model for simulating multinomial classes to demonstrate that the MCRF model effectively solves the small-class underestimation problem. Simulated results show that the MCRF model fairly produces all classes, generates simulated patterns imitative of the original, and effectively reproduces input transiograms in realizations. Occurrence probability maps are estimated to visualize the spatial uncertainty associated with each class and the optimal prediction map. It is concluded that the MCRF model provides a practically efficient estimator for simulating multinomial classes from grid samples.     

Impact of time-scale of the calibration objective function on the performance of watershed models

Many of the continuous watershed models perform all their computations on a daily time step, yet they are often calibrated at an annual or monthly time-scale that may not guarantee good simulation performance on a daily time step. The major objective of this paper is to evaluate the impact of the calibration time-scale on model predictive ability. This study considered the Soil and Water Assessment Tool for the analyses, and it has been calibrated at two time-scales, viz. monthly and daily for the War Eagle Creek watershed in the USA. The results demonstrate that the model's performance at the smaller time-scale (such as daily) cannot be ensured by calibrating them at a larger time-scale (such as monthly). It is observed that, even though the calibrated model possesses satisfactory ‘goodness of fit’ statistics, the simulation residuals failed to confirm the assumption of their homoscedasticity and independence. The results imply that evaluation of models should be conducted considering their behavior in various aspects of simulation, such as predictive uncertainty, hydrograph characteristics, ability to preserve statistical properties of the historic flow series, etc. The study enlightens the scope for improving/developing effective autocalibration procedures at the daily time step for watershed models. Copyright © 2007 John Wiley & Sons, Ltd.     

Investigation of the scale-dependent variability of radar-rainfall and rain gauge error covariance

There is a significant spatial sampling mismatch between radar and rain gauge data. The use of rain gauge data to estimate radar-rainfall error variance requires partitioning of the variance of the radar and rain gauge difference to account for the sampling mismatch. A key assumption in the literature pertaining to the error variance separation method used to partition the variance is that the covariance between radar-rainfall error and the error of rain gauges in representing radar sampling domain is negligible. Our study presents the results of an extensive test of this assumption. The test is based on empirical data and covers temporal scales ranging from 0.25 to 24 h and spatial scales ranging from 1 to 32 km. We used a two-year data set from two high quality and high density rain gauge networks in Oklahoma and excluded the winter months. The results obtained using a resampling procedure show that covariance can be considerable at large scales due to the significant variability. As the variability of the covariance rapidly increases with larger spatial and shorter temporal scales, applications of the error variance separation method at those scales require more caution. The variability of the covariance and one of its constituting variables, the variance ratio of radar and gauge errors, shows simple scaling behavior well characterized by a power-law.     

Linear vertical vibrations of suspension bridges: A review of continuum models and some new results

The classical continuum model for the linear vertical vibrations of a suspension bridge (Bleich et al., 1950 [1]) is re-examined. The primary objective of the study is to extend the definitive analytical and numerical results of Irvine and Caughey (1974) [2], Irvine and Griffin (1976) [3] and Irvine (1980, 1981) [4], [5] for the natural frequencies, mode shapes, and modal participation factors for an extensible suspension cable, which depend on one dimensionless parameter related to the elasticity of the cable, to the case of a stiffened suspension bridge in which the response depends also on a second dimensionless parameter related to the stiffness of the girder. The continuum suspension bridge model is also used to understand the pattern of variation of mode shapes as a function of cable elasticity and girder stiffness, which has been shown by West et al. (1984) [6] to be considerably more complex than that for a suspension cable. Finally, the threshold amplitudes of free vibrations that would result in the incipient slackening of the hangers are determined.