Bayesian maximum entropy approach and its applications: a review

The present paper reviews the conceptual framework and development of the Bayesian Maximum Entropy (BME) approach. BME has been considered as a significant breakthrough and contribution to applied stochastics by introducing an improved, knowledge-based modeling framework for spatial and spatiotemporal information. In this work, one objective is the overview of distinct BME features. By offering a foundation free of restrictive assumptions that limit comparable techniques, an ability to integrate a variety of prior knowledge bases, and rigorous accounting for both exact and uncertain data, the BME approach was coined as introducing modern spatiotemporal geostatistics. A second objective is to illustrate BME applications and adoption within numerous different scientific disciplines. We summarize examples and real-world studies that encompass the perspective of science of the total environment, including atmosphere, lithosphere, hydrosphere, and ecosphere, while also noting applications that extend beyond these fields. The broad-ranging application track suggests BME as an established, valuable tool for predictive spatial and space–time analysis and mapping. This review concludes with the present status of BME, and tentative paths for future methodological research, enhancements, and extensions.     

Uncertainty management of a hydrogeological data set in a greek lignite basin, using BME

The occurrence of chemical pollutants in ground water is an issue of considerable interest. In the case of Ptolemais lignite opencast mining area in Greece, ammonium, nitrites, nitrates, iron total and total manganese concentrations, as well as various other elements have been monitored since the early 2000s through a borehole network. The continuous, though, alteration of the surface topography due to the intensive mining works, limits the life span of the water boreholes and results to irregular spatiotemporal distribution of the samples. Regarding the problem of mapping the water contamination, the coarse and irregular sampling pattern, combined with absence of seasonal variations and temporal trends, does not facilitate spatiotemporal processing of the data. On the other hand, a mere spatial analysis requires the attribution of the whole set of monitored values for each borehole, to a single point in space. The objective of this work is to develop a methodology to cope with the problem of uncertainty caused due to the above assumptions. The proposed solution is based on the consideration of interval data. The Bayesian maximum entropy (BME) theory (Christakos, Modern spatiotemporal geostatistics. Oxford University Press, New York, 2000) is an essential component of this methodology. The main reason for this is that it is the only theory in the framework of Geostatistics that offers powerful tools to merge the uncertainty sources with the rest of conventional measurements. This ability is a result of its generalized view on the problem of estimation that focuses on the knowledge of a natural variable and not the variable itself. The application of the proposed methodology led to sharper posterior distributions, an indication of the increased certainty in estimated values which is induced by the broader utilization of data.     

On the physical geometry concept at the basis of space/time geostatistical hydrology

The objective of this paper is to show that the structure of the spatiotemporal continuum has important implications in practical stochastic hydrology (e.g., geostatistical analysis of hydrologic sites) and is not merely an abstract mathematical concept. We propose that the concept of physical geometry as a spatiotemporal continuum with properties that are empirically defined is important in hydrologic analyses, and that the elements of the spatiotemporal geometry (e.g., coordinate system and space/time metric) should be selected based on the physical properties of the hydrologic processes. We investigate the concept of space/time distance (metric) in various physical spaces, and its implications for hydrologic modeling. More specifically, we demonstrate that physical geometry plays a crucial role in the determination of appropriate spatiotemporal covariance models, and it can affect the results of geostatistical operations involved in spatiotemporal hydrologic mapping.     

Wavelet correlation filter for wide-angle seismic data

A new filtering technique for single‐fold wide‐angle reflection/refraction seismic data is presented. The technique is based on the wavelet decomposition of a set of adjacent traces followed by coherence analysis. The filtering procedure consists of three steps. In the first, a wavelet decomposition of traces into different detail levels is performed. In the second, the coherence attributes for each level are evaluated by calculating cross‐correlation functions of detail portions contained in a space–time moving window. Finally, the filtered traces are obtained as a weighted reconstruction of the trace details. Each weight is obtained from the coherence‐attributes distribution estimated in a proper interval. A sequence of tests is then conducted in order to select possible optimum or unsuitable wavelet bases. The efficiency of the filter proposed was assessed by calculating some properly designed parameters in order to compare it with other standard de‐noising techniques. The proposed method produced a clear signal enhancement in high‐density wide‐angle seismic data, thus proving that it is a useful processing tool for a reliable correlation of seismic phases.     

A traveling epidemic model of space–time disease spread

This work presents a random field model of disease attribute (incidence, mortality etc.) that transfers the study of the attribute distribution from the original spatiotemporal domain onto a lower-dimensionality traveling domain that moves along the direction of disease velocity. The partial differential equations connecting the disease attribute covariances in the original and the traveling domain are derived with coefficients that are functions of the disease velocity. These equations offer epidemiologic insight concerning the strength of the space–time dependence between the disease attribute values in the two domains. The traveling disease model has certain theoretical and computational advantages in the study and prediction of space–time disease attribute distributions in conditions of uncertainty. Estimates of the disease attribute are derived in the traveling domain and then used to generate maps of space–time disease attribute distribution in the original domain. The theoretical model is illustrated and additional insight is gained by means of a numerical mortality simulation study, which shows that the proposed model is at least as accurate but computationally more efficient than mainstream mapping techniques of higher dimensionality. These findings concerning the very good predictability of the proposed model also strongly support its adequacy to represent the space–time mortality distribution.     

Fast hyperbolic deconvolutive Radon transform using generalized Fourier slice theorem

The hyperbolic Radon transform has a long history of applications in seismic data processing because of its ability to focus/sparsify the data in the transform domain. Recently, deconvolutive Radon transform has also been proposed with an improved time resolution which provides improved processing results. The basis functions of the (deconvolutive) Radon transform, however, are time-variant, making the classical Fourier based algorithms ineffective to carry out the required computations. A direct implementation of the associated summations in the time–space domain is also computationally expensive, thus limiting the application of the transform on large data sets. In this paper, we present a new method for fast computation of the hyperbolic (deconvolutive) Radon transform. The method is based on the recently proposed generalized Fourier slice theorem which establishes an analytic expression between the Fourier transforms associated with the data and Radon plane. This allows very fast computations of the forward and inverse transforms simply using fast Fourier transform and interpolation procedures. These canonical transforms are used within an efficient iterative method for sparse solution of (deconvolutive) Radon transform. Numerical examples from synthetic and field seismic data confirm high performance of the proposed fast algorithm for filling in the large gaps in seismic data, separating primaries from multiple reflections, and performing high-quality stretch-free stacking.     

Parameter estimation of subsurface flow models using iterative regularized ensemble Kalman filter

A new parameter estimation algorithm based on ensemble Kalman filter (EnKF) is developed. The developed algorithm combined with the proposed problem parametrization offers an efficient parameter estimation method that converges using very small ensembles. The inverse problem is formulated as a sequential data integration problem. Gaussian process regression is used to integrate the prior knowledge (static data). The search space is further parameterized using Karhunen–Loève expansion to build a set of basis functions that spans the search space. Optimal weights of the reduced basis functions are estimated by an iterative regularized EnKF algorithm. The filter is converted to an optimization algorithm by using a pseudo time-stepping technique such that the model output matches the time dependent data. The EnKF Kalman gain matrix is regularized using truncated SVD to filter out noisy correlations. Numerical results show that the proposed algorithm is a promising approach for parameter estimation of subsurface flow models.     

Space–time forecasting of groundwater level using a hybrid soft computing model

Forecasting of space–time groundwater level is important for sparsely monitored regions. Time series analysis using soft computing tools is powerful in temporal data analysis. Classical geostatistical methods provide the best estimates of spatial data. In the present work a hybrid framework for space–time groundwater level forecasting is proposed by combining a soft computing tool and a geostatistical model. Three time series forecasting models: artificial neural network, least square support vector machine and genetic programming (GP), are individually combined with the geostatistical ordinary kriging model. The experimental variogram thus obtained fits a linear combination of a nugget effect model and a power model. The efficacy of the space–time models was decided on both visual interpretation (spatial maps) and calculated error statistics. It was found that the GP–kriging space–time model gave the most satisfactory results in terms of average absolute relative error, root mean square error, normalized mean bias error and normalized root mean square error.     

HydRun: A MATLAB toolbox for rainfall–runoff analysis

Understanding the nature of streamflow response to precipitation inputs is at the core of hydrological applications and water resource management. Indices such as the base flow index, recession constant, and response lag of a watershed retain an important place in hydrology as metrics to compare watersheds and understand the impact of human activity, geology, geomorphology, soils, and climate on precipitation–runoff relations. Extracting characteristics of the hyetograph–hydrograph relationship is often done manually, which is time consuming and may result in subjective and potentially inconsistent outcomes. Here, we present a MATLAB‐based toolbox, called HydRun, for rapid and flexible rainfall–runoff analysis. HydRun uses a series of flexible routines to extract base flow from the hydrograph and then computes commonly used time instants of the rainfall–runoff relationship. HydRun provides users the flexibility to decide thresholds and limits of analysis, but objectively computes hydrometric indices. The toolkit includes a graphical user interface and example files. In this paper, we apply HydRun to 4 watersheds, 3 in Scotland and 1 in Canada, to demonstrate the software functions and highlight important decisions the user must make in its application.     

Improving the resolution of seismic traces based on the secondary time–frequency spectrum

The resolution of seismic data is critical to seismic data processing and the subsequent interpretation of fine structures. In conventional resolution improvement methods, the seismic data is assumed stationary and the noise level not changes with space, whereas the actual situation does not satisfy this assumption, so that results after resolution improvement processing is not up to the expected effect. To solve these problems, we propose a seismic resolution improvement method based on the secondary time–frequency spectrum. First, we propose the secondary time-frequency spectrum based on S transform (ST) and discuss the reflection coefficient sequence and time-dependent wavelet in the secondary time–frequency spectrum. Second, using the secondary time–frequency spectrum, we design a twodimensional filter to extract the amplitude spectrum of the time-dependent wavelet. Then, we discuss the improvement of the resolution operator in noisy environments and propose a novel approach for determining the broad frequency range of the resolution operator in the time–frequency–space domain. Finally, we apply the proposed method to synthetic and real data and compare the results of the traditional spectrum-modeling deconvolution and Q compensation method. The results suggest that the proposed method does not need to estimate the Q value and the resolution is not limited by the bandwidth of the source. Thus, the resolution of the seismic data is improved sufficiently based on the signal-to-noise ratio (SNR).     

A stochastic framework to assess the performance of flood warning systems based on rainfall‐runoff modeling

Nowadays, Flood Forecasting and Warning Systems (FFWSs) are known as the most inexpensive and efficient non‐structural measures for flood damage mitigation in the world. Benefit to cost of the FFWSs has been reported to be several times of other flood mitigation measures. Beside these advantages, uncertainty in flood predictions is a subject that may affect FFWS's reliability and the benefits of these systems. Determining the reliability of advanced flood warning systems based on the rainfall–runoff models is a challenge in assessment of the FFWS performance which is the subject of this study. In this paper, a stochastic methodology is proposed to provide the uncertainty band of the rainfall–runoff model and to calculate the probability of acceptable forecasts. The proposed method is based on Monte Carlo simulation and multivariate analysis of the predicted time and discharge error data sets. For this purpose, after the calibration of the rainfall–runoff model, the probability distributions of input calibration parameters and uncertainty band of the model are estimated through the Bayesian inference. Then, data sets of the time and discharge errors are calculated using the Monte Carlo simulation, and the probability of acceptable model forecasts is calculated by multivariate analysis of data using copula functions. The proposed approach was applied for a small watershed in Iran as a case study. The results showed using rainfall–runoff modeling based on real‐time precipitation is not enough to attain high performance for FFWSs in small watersheds, and it seems using weather forecasts as the inputs of rainfall–runoff models is essential to increase lead times and the reliability of FFWSs in small watersheds. Copyright © 2013 John Wiley & Sons, Ltd.     

WebGIS基本原理及其在地学研究中的应用前景

万维网地理信息系统 (WebGIS)是建立在Internet上、具有浏览器 /服务器体系结构 (B/S)的网络GIS ,它改变了传统GIS的运行模式 ,使用户可以远程使用GIS ,共享地学空间信息资源。首先对WebGIS的概念、基本原理、实现技术、存在问题及发展现状与趋势作了较详细的介绍 ;然后对WebGIS在地学研究中的应用前景进行了探讨 ,认为WebGIS已成为地学研究中新的用户需求和发展的必然趋势 ,必将促进地学研究的信息化进程     

基于GIS的地质数据库系统:研究现状和发展趋势

有效地存储、管理、交流、进而充分利用正日益增多的地质资料和数据，离不开功能强大的数据库管理系统，然而，地学数据显著的空间特征和复杂的结构属性又不能简单运用常规的数据库管理系统进行表述、处理，地理信息系统（GIS）技术，以其对空间数据强大的储存查询和分析处理功能而鲜明地区 地普通管理信息系统，正适合于对复杂的地球空间数据进行采集、储存、分类、检索查询、刻划表达、以及分析建模，因此，先进的GIS技术与强大的地质数据库系统相结合，亦即是，基于GIS的地质数据库系统的开发和应用，是计算机技术应用于地学研究的发展方向和应用趋势，是当今地学发展所必需的基础技术之一，本文结合我们的近期工作概述了这一新兴领域的研究现状和发展趋势。     

Strict positive definiteness in geostatistics

Geostatistical modeling is often based on the use of covariance functions, i.e., positive definite functions. However, when interpolation problems have to be solved, it is advisable to consider the subset of strictly positive definite functions. Indeed, it will be argued that ensuring strict positive definiteness for a covariance function is convenient from a theoretical and practical point of view. In this paper, an extensive analysis on strictly positive definite covariance functions has been given. The closure of the set of strictly positive definite functions with respect to the sum and the product of covariance functions defined on the same Euclidean dimensional space or on factor spaces, as well as on partially overlapped lower dimensional spaces, has been analyzed. These results are particularly useful (a) to extend strict positive definiteness in higher dimensional spaces starting from covariance functions which are only defined on lower dimensional spaces and/or are only strictly positive definite in lower dimensional spaces, (b) to construct strictly positive definite covariance functions in space–time as well as (c) to obtain new asymmetric and strictly positive definite covariance functions.     

Modern geostatistics: computational BME analysis in the light of uncertain physical knowledge – the Equus Beds study

This paper is concerned with a computational formulation of the Bayesian maximum entropy (BME) mapping method, which can handle rigorously and efficiently spatiotemporal applications of considerable practical importance. BME is a method of modern geostatistics that can integrate and process physical knowledge that belongs to two major bases: general knowledge (i.e., obtained from general principles and laws, summary statistics and background information), and specificatory knowledge (i.e., obtained through experience with the specific situation). BME allows considerable flexibility regarding the choice of an appropriate spatiotemporal map, offers a complete assessment of the mapping uncertainty and contributes to the scientific understanding of the underlying natural phenomenon. Valuable insight is gained by studying a spatiotemporal data set representing water-level elevations at the Equus Beds aquifer (Kansas). Numerical results show that, as was expected in theory, classical geostatistics analysis is obtained as a special case of the considerably more general BME approach. Moreover, modern geostatistical analysis in terms of BME offers more accurate and informative results in practice, by incorporating various sources of physical knowledge that cannot be processed by the classical methods.     

基于ANSYS的新型乡村抗震防灾适宜性规划模型分析

当前新型乡村抗震防灾适宜性规划分析中通常采用地质分区方法对勘测点进行分析,在分析过程中忽略了GIS空间的复杂性,且未对评价指标加权分析,导致抗震适宜性评价指标量化过程过于主观,存在计算结果与实际结果拟合度低的问题。据此,提出基于ANSYS的新型乡村抗震防灾适宜性规划模型分析。考虑到GIS的空间复杂性,采用ANSYS在GIS空间进行有限元结构场修正操作,结合Logistic非线性回归模型,对乡村土地抗震防灾适宜性规划中的二分类变量数据进行非线性回归分析。为了防止计算数值过于主观,采用组合熵系数模型对Logistic方程计算得来的评价指标加权,由此完成基于ANSYS的新型乡村抗震防灾适宜性规划模型分析。经过实例分析证明,所提方法求出的计算结果与实际结果拟合度较高,能成功完成评价指标的量化,对乡村抗震防灾适宜性规划分析更加客观。     

Topography-adjusted solar radiation indices and their importance in hydrology

Abstract

Solar radiation, direct and diffuse, is affected by surface characteristics, such as slope, aspect, altitude and shading. The paper examines the effects of topography on radiation, at multiple spatiotemporal scales, using suitable geometric methods for the direct and diffuse components. Two indices are introduced for comparing the direct radiation received by areas at the same and different latitudes. To investigate the profile of direct radiation across the whole of Greece, these are evaluated from an hourly to annual basis, via GIS techniques. Moreover, different approaches are examined for estimating the actual global radiation at operational spatial scales (sub-basin and terrain), according to the available meteorological data. The study indicates that the errors of typical hydrometeorological modelling formulas, which ignore the topographic effects and the seasonal allocation of direct and diffuse radiation, depend on the spatial scale and are non-uniformly distributed in time. In all cases, the estimations are improved by applying the proposed adjustment approaches. In particular, the adjustment of the measured global radiation ensures up to 10% increase of efficiency, while the modified Angström formula achieves slight (i.e. 2–4%) increase of efficiency and notable reduction of bias.

Editor Z.W. Kundzewicz

Citation Mamassis, N., Efstratiadis, A. and Apostolidou, I.-G., 2012. Topography-adjusted solar radiation indices and their importance in hydrology. Hydrological Sciences Journal, 57 (4), 756–775.     

Multi‐scale relief model (MSRM): a new algorithm for the visualization of subtle topographic change of variable size in digital elevation models

Morphological analysis of landforms has traditionally relied on the interpretation of imagery. Although imagery provides a natural view of an area of interest (AOI) images are largely hindered by the environmental conditions at the time of image acquisition, the quality of the image and, mainly, the lack of topographical information, which is an essential factor for a correct understanding of the AOI's geomorphology. More recently digital surface models (DSMs) have been incorporated into the analytical toolbox of geomorphologists. These are usually high‐resolution models derived from digital photogrammetric processes or LiDAR data. However, these are restricted to relatively small areas and are expensive or complex to acquire, which limits widespread implementation. In this paper, we present the multi‐scale relief model (MSRM), which is a new algorithm for the visual interpretation of landforms using DSMs. The significance of this new method lies in its capacity to extract landform morphology from both high‐ and low‐resolution DSMs independently of the shape or scale of the landform under study. This method thus provides important advantages compared to previous approaches as it: (1) allows the use of worldwide medium resolution models, such as SRTM, ASTER GDEM, ALOS, and TanDEM‐X; (2) offers an alternative to traditional photograph interpretation that does not rely on the quality of the imagery employed nor on the environmental conditions and time of its acquisition; and (3) can be easily implemented for large areas using traditional GIS/RS software. The algorithm is tested in the Sutlej‐Yamuna interfluve, which is a very large low‐relief alluvial plain in northwest India where 10 000 km of palaeoriver channels have been mapped using MSRM. The code, written in Google Earth Engine's implementation of JavaScript, is provided as Supporting Information for its use in any other AOI without particular technical knowledge or access to topographical data. © 2017 The Authors. Earth Surface Processes and Landforms published by John Wiley & Sons Ltd.