Second-order non-stationary modeling approaches for univariate geostatistical data

A fundamental decision to make during the analysis of geostatistical data is the modeling of the spatial dependence structure as stationary or non-stationary. Although second-order stationary modeling approaches have been successfully applied in geostatistical applications for decades, there is a growing interest in second-order non-stationary modeling approaches. This paper provides a review of modeling approaches allowing to take into account the second-order non-stationarity in univariate geostatistical data. One broad distinction between these modeling approaches relies on the way that the second-order non-stationarity is captured. It seems unlikely to prove that there would be the best second-order non-stationary modeling approach for all geostatistical applications. However, some of them are distinguished by their simplicity, interpretability, and flexibility.     

Performance of geostatistical interpolation methods for modeling sampled data with non-stationary mean

The measured ozone pollution peak in the atmosphere of Mexico City region was considered in order to study the effect of a non-stationary mean of the sampled data in geostatistics interpolation methods. With this objective the local mean value of the sampled data was estimated through a linear regression analysis of their values on the monitoring station’s coordinates. The residuals obtained by removing the data trend are considered as a set of stationary random variables. Several interpolation methods used in geostatistics, such as inverse distance weighted, kriging, and artificial neural networks techniques were considered. In an effort to optimize and evaluate its performance, we fit interpolated values to sampled data, obtaining optimal values for the parameters defining the used model, that means, the values of the parameters that give the lowest mean RMSE between the interpolated value and measured data at 20 stations at 1500 hours for a set of 21 days of December 2001, which was chosen as the training set. The training set is conformed by all the days in December 2001 excepting the days (3,6,9,12,...,27,30) which were considered as the testing set. Once the optimal model is obtained, it is used to interpolate the values at the stations at 1500 hours for the testing days. The RMSE between interpolated and measured values at monitoring stations was also evaluated for these testing values and is shown as a percentage in Table 2. These values and the defined generalization parameter G, can be used to evaluate the performance and the ability of the models to predict and reproduce the peak of ozone concentrations. Scatter plots for testing data are presented for each interpolation method. An interpretation of the ozone pollution levels obtained at 1500 hours at December 21 was given using the wind field that prevailed in the region 1 h before the same day.     

测井、井间地震与地面地震数据联合约束下的地质统计学随机建模方法研究

利用能够整合测井信息与井间地震信息的地质统计学随机模拟方法,结合传统的地质统计学反演思路,得到了一种能够同时整合测井、井间地震与地面地震三种先验信息的地质统计学反演与储层建模方法.由于井间射线信息、测井信息与地面地震数据在随机反演与建模过程当中都得到了尊重,因此与传统地质统计学反演仅利用了测井与地面地震数据相比,本文的地质统计学反演与建模方法更充分地利用了先验信息,有效提高了反演的精度,降低了随机建模中的多解性.基于理论数据的测试证实了上述观点.     

测井、井间地震与地面地震数据联合约束下的地质统计学随机建模方法研究

利用能够整合测井信息与井间地震信息的地质统计学随机模拟方法,结合传统的地质统计学反演思路,得到了一种能够同时整合测井、井间地震与地面地震三种先验信息的地质统计学反演与储层建模方法.由于井间射线信息、测井信息与地面地震数据在随机反演与建模过程当中都得到了尊重,因此与传统地质统计学反演仅利用了测井与地面地震数据相比,本文的地质统计学反演与建模方法更充分地利用了先验信息,有效提高了反演的精度,降低了随机建模中的多解性.基于理论数据的测试证实了上述观点.     

Three-dimensional geostatistical inversion of flowmeter and pumping test data

We jointly invert field data of flowmeter and multiple pumping tests in fully screened wells to estimate hydraulic conductivity using a geostatistical method. We use the steady-state drawdowns of pumping tests and the discharge profiles of flowmeter tests as our data in the inference. The discharge profiles need not be converted to absolute hydraulic conductivities. Consequently, we do not need measurements of depth-averaged hydraulic conductivity at well locations. The flowmeter profiles contain information about relative vertical distributions of hydraulic conductivity, while drawdown measurements of pumping tests provide information about horizontal fluctuation of the depth-averaged hydraulic conductivity. We apply the method to data obtained at the Krauthausen test site of the Forschungszentrum Jülich, Germany. The resulting estimate of our joint three-dimensional (3D) geostatistical inversion shows an improved 3D structure in comparison to the inversion of pumping test data only.     

Building on crossvalidation for increasing the quality of geostatistical modeling

The random function is a mathematical model commonly used in the assessment of uncertainty associated with a spatially correlated attribute that has been partially sampled. There are multiple algorithms for modeling such random functions, all sharing the requirement of specifying various parameters that have critical influence on the results. The importance of finding ways to compare the methods and setting parameters to obtain results that better model uncertainty has increased as these algorithms have grown in number and complexity. Crossvalidation has been used in spatial statistics, mostly in kriging, for the analysis of mean square errors. An appeal of this approach is its ability to work with the same empirical sample available for running the algorithms. This paper goes beyond checking estimates by formulating a function sensitive to conditional bias. Under ideal conditions, such function turns into a straight line, which can be used as a reference for preparing measures of performance. Applied to kriging, deviations from the ideal line provide sensitivity to the semivariogram lacking in crossvalidation of kriging errors and are more sensitive to conditional bias than analyses of errors. In terms of stochastic simulation, in addition to finding better parameters, the deviations allow comparison of the realizations resulting from the applications of different methods. Examples show improvements of about 30% in the deviations and approximately 10% in the square root of mean square errors between reasonable starting modelling and the solutions according to the new criteria.     

A discrete space continuous time modeling approach to nonsteady flow in a leaky aquifer system of finite configuration

Existing analytical procedures for nonsteady flow in a leaky confined aquifer assume that the aquifer system is areally infinite. A technique is presented that treats a leaky confined aquifer system of finite configuration. By means of a discrete space continuous time (DSCT) modeling approach, the partial differential equation governing the flow system is transformed into a set of ordinary differential equations that can be easily integrated numerically on a high speed digital computer using available scientific subroutines. The finite difference formulation is in effect an explicit scheme. A criterion is developed for which the scheme is computationally stable. A numerical example is presented.     

A practical review of geostatistical processing applied to geophysical data: methods and applications

Nowadays, geostatistics is commonly applied for numerous gridding or modelling tasks. However, it is still under used and unknown for classical geophysical applications. This paper highlights the main geostatistical methods relevant for geophysical issues, for instance to improve the quality of seismic data such as velocity cubes or interpreted horizons. These methods are then illustrated through four examples. The first example, based on a gravity survey presents how a geostatistical interpolation can be used to filter out a global trend, in order to better define real anomalies. In the second case study, dedicated to refraction surveying, geostatistical filtering is used to filter out acquisition artefacts and identify the main geological structures. The third one is an example of porosity being integrated geostatistically with a seismic acoustic impedance map. The last example deals with geostatistical time to depth conversion; the interest of performing geostatistical simulations is finally discussed.     

On some aspects of peaks-over-threshold modeling of floods under nonstationarity using climate covariates

This paper discusses some aspects of flood frequency analysis using the peaks-over-threshold model with Poisson arrivals and generalized Pareto (GP) distributed peak magnitudes under nonstationarity, using climate covariates. The discussion topics were motivated by a case study on the influence of El Niño–Southern Oscillation on the flood regime in the Itajaí river basin, in Southern Brazil. The Niño3.4 (DJF) index is used as a covariate in nonstationary estimates of the Poisson and GP distributions scale parameters. Prior to the positing of parametric dependence functions, a preliminary data-driven analysis was carried out using nonparametric regression models to estimate the dependence of the parameters on the covariate. Model fits were evaluated using asymptotic likelihood ratio tests, AIC, and Q–Q plots. Results show statistically significant and complex dependence relationships with the covariate on both nonstationary parameters. The nonstationary flood hazard measure design life level (DLL) was used to compare the relative performances of stationary and nonstationary models in quantifying flood hazard over the period of records. Uncertainty analyses were carried out in every step of the application using the delta method.     

Combined discrete and continuous gravity observations at Mount Etna

Systematic investigation of discrete gravity measurements has continued at Mount Etna since 1986. The network now covers an area of 400 km² with about 70 stations 0.5–3 km apart. Mass redistributions occurring at depths ranging between about 8 km below sea level and a few hundred metres below the surface (magma level changes within the shallower parts of the feeding conduits) have been identified from these data. Conventional (discrete) microgravity monitoring on a network of stations furnishes only instantaneous states of the mass distribution at continuously active systems. In order to obtain information on the rate at which the volcanic processes (and thus mass transfers) occur, three stations for continuously recording gravity where installed on Mount Etna in 1998. A 16-month long sequence from one of the continuously running stations (PDN, located 2 km from the active northeast crater at the summit of Etna volcano) is presented. After removing the effects of Earth Tide and tilt, the correlation of the residual gravity sequence with simultaneous recordings of meteorological parameters acquired at the same station was analysed. Once the meteorological effects have also been removed, continuous gravity changes are within 10 μGal of gravity changes measured using conventional microgravity observations at sites very close to the continuous station. This example shows how discrete and continuous gravity observations can be used together at active volcanoes to get a fuller and more accurate picture of the spatial and temporal characteristics of volcanic processes.     

起伏地表复杂介质波动方程有限元数值模拟方法

波动方程数值模拟是深入研究地震波传播规律的有效方法.有限差分法因其方法简单、精度高而得到了广泛的应用.但其缺点是不能准确模拟具有复杂几何形态的物性界面.因而当遇到起伏地表或复杂构造时,求解精度低.为了准确模拟起伏地形、复杂构造和复杂介质条件下的地震波场,本文采用有限元法模拟二维声波方程.用三角形单元模拟地形和速度界面;把单元内的场和波速均看作单元上的线性函数,以适应复杂介质压制边角散射;采用吸收边界条件去除来自截断边界上的反射;采用集中质量矩阵和集中阻尼矩阵使得显式时间递推无需对矩阵求逆,提高了计算效率.对模型的计算表明该方法正确有效.     

Approximate copula-based estimation and prediction of discrete spatial data

The present paper reports on the use of copula functions to describe the distribution of discrete spatial data, e.g. count data from environmental mapping or areal data analysis. In particular, we consider approaches to parameter point estimation and propose a fast method to perform approximate spatial prediction in copula-based spatial models with discrete marginal distributions. We assess the goodness of the resulting parameter estimates and predictors under different spatial settings and guide the analyst on which approach to apply for the data at hand. Finally, we illustrate the methodology by analyzing the well-known Lansing Woods data set. Software that implements the methods proposed in this paper is freely available in Matlab language on the author’s website.     

Compiling and verifying 3D models of 2D induced polarization and resistivity data by geostatistical methods

A 3D model of collected time-domain induced polarization (IP) and electrical resistivity tomography (ERT) data is compiled by geostatistical methods as well as studying spatial correlation among the database. Mesgaran copper deposit, located in Birjand eastern Iran, was chosen to compile and verify the model, leading to five parallel surveyed IP and ERT profiles with dipole–dipole arrays. The collected data were inverted, and then 2D models of IP and ER were prepared; also 3D inversion was done. Afterward, the 3D model has been built by geostatistical methods. Correspondingly, the anomalies threshold was detected by fractal methods and the estimation variance and Kriging efficiency were calculated to validate the modeling. The mineralization zones were determined according to the classified anomalies and those with the lowest error. Results indicated a high correlation between anomalies identified from the model and mineralization. The results made it possible to construct 3D models from surveyed 2D data with acceptable error level.     

离散介质波场正演模拟与弱化性影响分析

从流变力学角度建立的离散介质本构关系同时考虑了介质的弹性、粘性与弱化性,与损伤力学建立的本构关系比较表明弱化因子具有损伤因子的力学意义。本文采用交错网格伪谱法正演模拟了离散介质中波的传播,并以平面波解分析了体波的频散和介质的吸收系数。高频和低频情况下波场数值模拟的结果是:饱和多孔介质中体波速度随弱化因子的增加而降低,介质吸收系数随弱化因子的增加而增加,弱化性对不同体波频散的影响程度不同。     

Geologic heterogeneity and a comparison of two geostatistical models: Sequential Gaussian and transition probability-based geostatistical simulation

A covariance-based model-fitting approach is often considered valid to represent field spatial variability of hydraulic properties. This study examines the representation of geologic heterogeneity in two types of geostatistical models under the same mean and spatial covariance structure, and subsequently its effect on the hydraulic response to a pumping test based on 3D high-resolution numerical simulation and field data. Two geostatistical simulation methods, sequential Gaussian simulation (SGS) and transition probability indicator simulation (TPROGS) were applied to create conditional realizations of alluvial fan aquifer systems in the Lawrence Livermore National Laboratory (LLNL) area. The simulated K fields were then used in a numerical groundwater flow model to simulate a pumping test performed at the LLNL site. Spatial connectivity measures of high-K materials (channel facies) captured connectivity characteristics of each geostatistical model and revealed that the TPROGS model created an aquifer (channel) network having greater lateral connectivity. SGS realizations neglected important geologic structures associated with channel and overbank (levee) facies, even though the covariance model used to create these realizations provided excellent fits to sample covariances computed from exhaustive samplings of TPROGS realizations. Observed drawdown response in monitoring wells during a pumping test and its numerical simulation shows that in an aquifer system with strongly connected network of high-K materials, the Gaussian approach could not reproduce a similar behavior in simulated drawdown response found in TPROGS case. Overall, the simulated drawdown responses demonstrate significant disagreement between TPROGS and SGS realizations. This study showed that important geologic characteristics may not be captured by a spatial covariance model, even if that model is exhaustively determined and closely fits the exponential function.     

最优化辛格式广义离散奇异核褶积微分算子地震波场模拟

将波动方程变换至Hamilton体系,构造了一种新的保结构算法,即最优化辛格式广义褶积微分算子(OSGCD). 在时间离散上,首先引入了Lie算子设计二级二阶辛格式,基于最小误差原理得到了优化的辛格式. 在空间离散上,引入广义离散奇异核褶积微分算子计算空间微分,提出了一种有效方法优化GCD并得到了稳定的算子系数. 针对本文发展的新方法,给出了OSGCD稳定性条件. 在数值实验中,将OSGCD与多种方法比较,从精度和计算效率两方面分析了OSGCD的计算优势,计算结果也表明OSGCD长时程以及非均匀介质中地震波模拟亦具有较强能力.     

Numerical electromagnetic modeling including studies of characteristic dimensions: A review

Recent activity in important approximate methods used in numerical electromagnetic (EM) modeling is reviewed. Comparisons between the results obtained by different numerical methods and between analytical and numerical solutions are presented. The importance of 3D modeling and thin sheet approximations are pointed out.This review also considers and summarizes studies of characteristic dimensions in three topics: source fields, numerical modeling and physical phenomena in the earth and interpretation. The skin depth (i.e., generally the attenuation) of the EM energy is considered to be the most important and fundamental characteristic dimension.     

Integrated spatial sampling modeling of geospatial data

Sampling is to, by efficient selection of samples, acquire the accurate information about the population (the research object) at less cost. Spatial sampling is a kind of sampling toward geospatial objects or features with spatial correlation. The differences between effi-cient sampling and completely universal survey lie in quality, time and cost. Sampling provides a kind of economical, prompt and accurate survey[13]. Efficient spatial sampling can be regarded as the optimization of the sampl…     

Rainfall data simulation by hidden Markov model and discrete wavelet transformation

In many regions, monthly (or bimonthly) rainfall data can be considered as deterministic while daily rainfall data may be treated as random. As a result, deterministic models may not sufficiently fit the daily data because of the strong stochastic nature, while stochastic models may also not reliably fit into daily rainfall time series because of the deterministic nature at the large scale (i.e. coarse scale). Although there are different approaches for simulating daily rainfall, mixing of deterministic and stochastic models (towards possible representation of both deterministic and stochastic properties) has not hitherto been proposed. An attempt is made in this study to simulate daily rainfall data by utilizing discrete wavelet transformation and hidden Markov model. We use a deterministic model to obtain large-scale data, and a stochastic model to simulate the wavelet tree coefficients. The simulated daily rainfall is obtained by inverse transformation. We then compare the accumulated simulated and accumulated observed data from the Chao Phraya Basin in Thailand. Because of the stochastic nature at the small scale, the simulated daily rainfall on a point to point comparison show deviations with the observed data. However the accumulated simulated data do show some level of agreement with the observed data.