An Empirical Comparison of Kriging Methods for Nonlinear Spatial Point Prediction

Spatial prediction is a problem common to many disciplines. A simple application is the mapping of an attribute recorded at a set of points. Frequently a nonlinear functional of the observed variable is of interest, and this calls for nonlinear approaches to prediction. Nonlinear kriging methods, developed in recent years, endeavour to do so and additionally provide estimates of the distribution of the target quantity conditional on the observations. There are few empirical studies that validate the various forms of nonlinear kriging. This study compares linear and nonlinear kriging methods with respect to precision and their success in modelling prediction uncertainty. The methods were applied to a data set giving measurements of the topsoil concentrations of cobalt and copper at more than 3000 locations in the Border Region of Scotland. The data stem from a survey undertaken to identify places where these trace elements are deficient for livestock. The comparison was carried out by dividing the data set into calibration and validation sets. No clear differences between the precision of ordinary, lognormal, disjunctive, indicator, and model-based kriging were found, neither for linear nor for nonlinear target quantities. Linear kriging, supplemented with the assumption of normally distributed prediction errors, failed to model the conditional distribution of the marginally skewed data, whereas the nonlinear methods modelled the conditional distributions almost equally well. In our study the plug-in methods did not fare any worse than model-based kriging, which takes parameter uncertainty into account.     

GaRiRO: Gradient and residual integrated rank ordering of stations in rainfall monitoring network

This paper presents a new method that integrates gradient and residual values for rank ordering of stations in a monitoring network (GaRiRO). The innovation is derived from the fact that the parameter (dependent variable) gauged through the monitoring network is modelled using independent variables that influence its measured quantity. And the dependent variable exhibit non-stationary spatial gradient with respect to the independent variables, particularly in complex terrain. GaRiRO technique was developed to prioritize the rain gauge stations for optimizing the existing network and selection of the best locations for relocation or installation of gauges. Although initially aimed to assist hydrologists with a ranking scheme for rain gauge stations, it can be applied to any environmental, meteorological or hydro-meteorological monitoring network. The new procedure is based on deriving gradient and residual value at each station by modeling the spatial relationship of dependent-independent variable using geographically weighted regression (GWR) technique. For the prospective stations with no record, the gradient value is estimated using GWR model and the residual value is derived from the residual map generated by applying kriging technique on the residual derived at all gauged locations. The method combines the decision factor with analytical strength of GIS for prioritizing the stations which results in limited number of trials for installation or relocation of gauges to yield optimized network configuration.     

Tidal rectification on drying estuarine sandbanks

Drying estuarine sandbanks experience only that part of the tidal cycle around high water. In a partially progressive tidal wave, this means that the duration of the flood over the sandbank will be greater than that of the ebb: a process of tidal rectification. In this paper, we propose the hypothesis that this leads to flood-directed tidal residual currents over drying sandbanks. The hypothesis is tested by observation and a 2-D hydrodynamical model in the Conwy estuary, a vertically well-mixed macrotidal estuary in North Wales. The observations include tide gauge data, tidal cycle boat surveys, and fixed current meter data. The data show weak flood-directed residual currents over a drying sandbank and much stronger ebb-directed residuals in the channels along the sides of the sandbank. The model reproduces the observations in the vicinity of the sandbank and shows that the tidal rectification mechanism produces a general pattern of residual circulation in the estuary, with flood-directed flow in the drying areas and ebb-directed flow in the channels. The flood residuals are most marked near the estuary mouth where the tidal wave is most progressive in nature. The main application of this mechanism is believed to be in the transport of bedload sediment. The flood-directed residuals will tend to move the tops of the sandbanks upstream.     

Monthly reservoir inflow forecasting using a new hybrid SARIMA genetic programming approach

Forecasting reservoir inflow is one of the most important components of water resources and hydroelectric systems operation management. Seasonal autoregressive integrated moving average (SARIMA) models have been frequently used for predicting river flow. SARIMA models are linear and do not consider the random component of statistical data. To overcome this shortcoming, monthly inflow is predicted in this study based on a combination of seasonal autoregressive integrated moving average (SARIMA) and gene expression programming (GEP) models, which is a new hybrid method (SARIMA–GEP). To this end, a four-step process is employed. First, the monthly inflow datasets are pre-processed. Second, the datasets are modelled linearly with SARIMA and in the third stage, the non-linearity of residual series caused by linear modelling is evaluated. After confirming the non-linearity, the residuals are modelled in the fourth step using a gene expression programming (GEP) method. The proposed hybrid model is employed to predict the monthly inflow to the Jamishan Dam in west Iran. Thirty years’ worth of site measurements of monthly reservoir dam inflow with extreme seasonal variations are used. The results of this hybrid model (SARIMA–GEP) are compared with SARIMA, GEP, artificial neural network (ANN) and SARIMA–ANN models. The results indicate that the SARIMA–GEP model (R ²=78.8, VAF =78.8, RMSE =0.89, MAPE =43.4, CRM =0.053) outperforms SARIMA and GEP and SARIMA–ANN (R ²=68.3, VAF =66.4, RMSE =1.12, MAPE =56.6, CRM =0.032) displays better performance than the SARIMA and ANN models. A comparison of the two hybrid models indicates the superiority of SARIMA–GEP over the SARIMA–ANN model.     

Fast wavelet-based stochastic simulation using training images

Spatial uncertainty modelling is a complex and challenging job for orebody modelling in mining, reservoir characterization in petroleum, and contamination modelling in air and water. Stochastic simulation algorithms are popular methods for such modelling. In this paper, discrete wavelet transformation (DWT)-based multiple point simulation algorithm for continuous variable is proposed that handles multi-scale spatial characteristics in datasets and training images. The DWT of a training image provides multi-scale high-frequency wavelet images and one low-frequency scaling image at the coarsest scale. The simulation of the proposed approach is performed on the frequency (wavelet) domain where the scaling image and wavelet images across the scale are simulated jointly. The inverse DWT reconstructs simulated realizations of an attribute of interest in the space domain. An automatic scale-selection algorithm using dominant mode difference is applied for the selection of the optimal scale of wavelet decomposition. The proposed algorithm reduces the computational time required for simulating large domain as compared to spatial domain multi-point simulation algorithm. The algorithm is tested with an exhaustive dataset using conditional and unconditional simulation in two- and three-dimensional fluvial reservoir and mining blasted rock data. The realizations generated by the proposed algorithm perform well and reproduce the statistics of the training image. The study conducted comparing the spatial domain filtersim multiple-point simulation algorithm suggests that the proposed algorithm generates equally good realizations at lower computational cost.     

Uncertainty in spatial average undrained shear strength with a site-specific transformation model

ABSTRACT

Transformation models are used to infer geotechnical properties from indirect measurements. A site-specific transformation model can be calibrated with direct and indirect measurements from a site. When such a model is used, then spatial variability, measurement errors and statistical uncertainty propagate into the uncertainty of the spatial average, which is the variable of interest in most geotechnical analyses. This research shows how all components enter the total uncertainty of a transformation model for undrained shear strength from cone resistance. A method is proposed to estimate the uncertainty in the spatial average undrained shear strength, particularly focusing on the role of averaging of all spatially variable error components. The main finding is that if a considerable share of the measurement and transformation errors is random or spatially variable, the uncertainty estimates can be considerably lower compared to methods proposed earlier, and hence, characteristic values can be considerably higher.     

Capturing and modelling metre‐scale spatial facies heterogeneity in a Jurassic ramp setting (Central High Atlas,Morocco)

Each simulation algorithm, including Truncated Gaussian Simulation, Sequential Indicator Simulation and Indicator Kriging is characterized by different operating modes, which variably influence the facies proportion, distribution and association of digital outcrop models, as shown in clastic sediments. A detailed study of carbonate heterogeneity is then crucial to understanding these differences and providing rules for carbonate modelling. Through a continuous exposure of Bajocian carbonate strata, a study window (320 m long, 190 m wide and 30 m thick) was investigated and metre‐scale lithofacies heterogeneity was captured and modelled using closely‐spaced sections. Ten lithofacies, deposited in a shallow‐water carbonate‐dominated ramp, were recognized and their dimensions and associations were documented. Field data, including height sections, were georeferenced and input into the model. Four models were built in the present study. Model A used all sections and Truncated Gaussian Simulation during the stochastic simulation. For the three other models, Model B was generated using Truncated Gaussian Simulation as for Model A, Model C was generated using Sequential Indicator Simulation and Model D was generated using Indicator Kriging. These three additional models were built by removing two out of eight sections from data input. The removal of sections allows direct insights on geological uncertainties at inter‐well spacings by comparing modelled and described sections. Other quantitative and qualitative comparisons were carried out between models to understand the advantages/disadvantages of each algorithm. Model A is used as the base case. Indicator Kriging (Model D) simplifies the facies distribution by assigning continuous geological bodies of the most abundant lithofacies to each zone. Sequential Indicator Simulation (Model C) is confident to conserve facies proportion when geological heterogeneity is complex. The use of trend with Truncated Gaussian Simulation is a powerful tool for modelling well‐defined spatial facies relationships. However, in shallow‐water carbonate, facies can coexist and their association can change through time and space. The present study shows that the scale of modelling (depositional environment or lithofacies) involves specific simulation constraints on shallow‐water carbonate modelling methods.     

Physically based modelling of shallow landslide sediment yield at a catchment scale

 A shallow landslide erosion and sediment yield component, applicable at the basin scale, has been incorporated into the physically based, spatially distributed, hydrological and sediment transport modelling system, SHETRAN. The component determines when and where landslides occur in a basin in response to time-varying rainfall and snowmelt, the volume of material eroded and released for onward transport, and the impact on basin sediment yield. Derived relationships are used to link the SHETRAN grid resolution (up to 1 km), at which the basin hydrology and final sediment yield is modelled, to a subgrid resolution (typically around 10–100 m) at which landslide occurrence and erosion is modelled. The subgrid discretization, landslide susceptibility and potential landslide impact are determined in advance using a geographic information system (GIS), with SHETRAN then providing information on temporal variation in the factors controlling landsliding. The ability to simulate landslide sediment yield is demonstrated by a hypothetical application based on a catchment in Scotland. Received: 30 October 1996 · Accepted: 25 June 1997     

Trend-surface analysis of structure and thickness of Bulli Seam, Sydney Basin, New South Wales

Methods are presented which permit the analysis of structural and thickness variations of a stratigraphic interval into a number of components and testing of the components for relationship between structure and thickness. The methods are tested first in the axial region of the South Bulli Syncline area where the Bulli Seam is known to be thicker. Maps of the residual values from the structural and thickness first-degree trends are similar and there is a significant inverse relationship between the residuals. Data from the area where the Bulli Seam occurs also give seam structural and thickness residuals which have a significant inverse relationship. Local exceptions to this relationship are present but in most instances are due to laccolithic doming or changes in lithology of the Bulli Seam. A number of the trend components of structural and thickness variation also show an inverse relationship. It is postulated that most of the present structures defined at the Bulli Seam horizon were active during deposition of the Bulli Seam, and within the limit of the conditions suitable for peat accumulation, controlled thickness of the Bulli Seam. Thickness changes in overlying units suggest that a major part of the warping of the Bulli Seam horizon had occurred by the end of the Triassic Period.     

Effect of Multipoint Heterogeneity on Nonlinear Transformations for Geological Modeling： Porosity-Permeability Relations Revisited

An analysis of statistical expected values for transformations is performed in this study to quantify the effect of heterogeneity on spatial geological modeling and evaluations. Algebraic transformations are frequently applied to data from logging to allow for the modeling of geological properties. Transformations may be powers, products, and exponential operations which are commonly used in well-known relations (e.g., porosity-permeability transforms). The results of this study show that correct computations must account for residual transformation terms which arise due to lack of independence among heterogeneous geological properties. In the case of an exponential porosity-permeability transform, the values may be positive. This proves that a simple exponential model back-transformed from linear regression underestimates permeability. In the case of transformations involving two or more properties, residual terms may represent the contribution of heterogeneous components which occur when properties vary together, regardless of a pair-wise linear independence. A consequence of power- and product-transform models is that regression equationswithin those transformations need corrections via residual cumulants. A generalization of this result isthat transformations of multivariate spatial attributes require multiple-point random variable relations. This analysis provides practical solutions leading to a methodology for nonlinear modeling using correct back transformations in geology.     

Axisymmetric finite element analysis of pile loading tests

In this paper a typical soil–structure interaction problem is considered, the case of a vertical pile installed in sand and submitted to an axial compression loading. Results from two full scale pile tests are analysed and the tests are reproduced by numerical simulations via finite elements method (FEM). The choice of the mechanical parameters for the soil and the sand–pile interface and the modelling approach are first described. A new numerical strategy is outlined to account for pile installation effects due to jacking and driving via FEM. The proposed approach is based on the application of existing empirical correlations available for the quantification of residual radial and shear stresses along the pile shaft as well as residual pressures around the pile base after the installation. This approach is proposed as an alternative to more complex methods based on the numerical modelling of the pile penetration problem. The role of the constitutive modelling of the interface is also discussed. Finally, comparative analyses of pile loading tests using FEM are provided and the comparisons between numerical and experimental results are presented and discussed.     

Modelling in-situ upgrading of heavy oil using operator splitting method

The in-situ upgrading (ISU) of bitumen and oil shale is a very challenging process to model numerically because of the large number of components that need to be modelled using a system of equations that are both highly non-linear and strongly coupled. Operator splitting methods are one way of potentially improving computational performance. Each numerical operator in a process is modelled separately, allowing the best solution method to be used for the given numerical operator. A significant drawback to the approach is that decoupling the governing equations introduces an additional source of numerical error, known as the splitting error. The best splitting method for modelling a given process minimises the splitting error whilst improving computational performance compared to a fully implicit approach. Although operator splitting has been widely used for the modelling of reactive-transport problems, it has not yet been applied to the modelling of ISU. One reason is that it is not clear which operator splitting technique to use. Numerous such techniques are described in the literature and each leads to a different splitting error. While this error has been extensively analysed for linear operators for a wide range of methods, the results cannot be extended to general non-linear systems. It is therefore not clear which of these techniques is most appropriate for the modelling of ISU. In this paper, we investigate the application of various operator splitting techniques to the modelling of the ISU of bitumen and oil shale. The techniques were tested on a simplified model of the physical system in which a solid or heavy liquid component is decomposed by pyrolysis into lighter liquid and gas components. The operator splitting techniques examined include the sequential split operator (SSO), the Strang-Marchuk split operator (SMSO) and the iterative split operator (ISO). They were evaluated on various test cases by considering the evolution of the discretization error as a function of the time-step size compared with the results obtained from a fully implicit simulation. We observed that the error was least for a splitting scheme where the thermal conduction was performed first, followed by the chemical reaction step and finally the heat and mass convection operator (SSO-CKA). This method was then applied to a more realistic model of the ISU of bitumen with multiple components, and we were able to obtain a speed-up of between 3 and 5.     

Stepwise Conditional Transformation for Simulation of Multiple Variables

Most geostatistical studies consider multiple-related variables. These relationships often show complex features such as nonlinearity, heteroscedasticity, and mineralogical or other constraints. These features are not handled by the well-established Gaussian simulation techniques. Earth science variables are rarely Gaussian. Transformation or anamorphosis techniques make each variable univariate Gaussian, but do not enforce bivariate or higher order Gaussianity. The stepwise conditional transformation technique is proposed to transform multiple variables to be univariate Gaussian and multivariate Gaussian with no cross correlation. This makes it remarkably easy to simulate multiple variables with arbitrarily complex relationships: (1) transform the multiple variables, (2) perform independent Gaussian simulation on the transformed variables, and (3) back transform to the original variables. The back transformation enforces reproduction of the original complex features. The methodology and underlying assumptions are explained. Several petroleum and mining examples are used to show features of the transformation and implementation details.     

Long-term changes in teleseismic P residuals at new zealand stations

Teleseismic P residuals in New Zealand have been studied to examine the hypothesis that they change prior to a local earthquake. The period 1965 to 1977 has been used, and to provide a statistically significant sample a six-monthly sliding mean has been calculated for all data from earthquakes

1. (a) more than 25 degrees away, and

2. (b) with a residual less than 3 seconds.

Results are discussed for all New Zealand seismograph stations where sufficient teleseismic arrivals have been reported, and are compared with the results from Scott Base, Antarctica, which is situated in an area devoid of local earthquakes.The study reveals a complex situation with certain events being well correlated with changes in residuals, and others showing no correlation at all. Moreover, some changes in residual occur at times of quiet seismic activity.A dominant feature of plots of mean residual versus time for each station is a gradual increase in P residual from about 1968 to 1975, followed by a slow decrease. No satisfactory explanation has been found for this long-term change in residual.The data detailed in this note was presented at the Earthquake Prediction Workshop, Canberra 1979, in the hope that a cause of the long-term drift might be suggested. No attempt has been made to analyse the short-term variations whilst the long term one remains unexplained.     

趋势面分析在高原地区音频大地电磁测深中的应用

常规音频大地电磁法,如EH-4电磁成像系统,在资料处理时通常不能有效分辨部分异常区域。以青海海北某公路隧道勘探为例,研究趋势面分析法在音频大地电磁测深中的应用。结果表明：通过拟合趋势面方程,可获取研究区内任一位置的视电阻率值,在拟合度较高的情况下可对区外的视电阻率值进行预测与估计;野外观测数据通常受随机因素的影响,可通过趋势面分析法对随机噪声进行平滑滤波处理;剩余值反映了视电阻率异常特征,绘制剩余值等值线图可以在去除区域背景视电阻率值后对异常区域进行凸显。音频大地电磁探测成果与趋势面分析法结合能更充分地利用地质信息。     

Stochastic model for settlement: footings on cohesionless soil

In this paper, given an estimate of the bearing capacity of the soil, by treating settlement at a given load as a random variable and the evolution of settlement of footing on cohesionless soil with the increasing load as a stochastic process, a tri-level homogeneous Markov chain (TLHMC) model is proposed for prediction of settlement. Comparison of the predicted mean and bounds on settlements, obtained using TLHMC, with the respective field values obtained from literature shows that the stochastic evolution can be modelled using TLHMC with a correlation coefficient of 0.90. A methodology for reliability-based design of footings is also presented and its use is demonstrated through a numerical example.     

Simplified modelling of mass transport in a fissured rock-clay buffer system

A simplifying concept of modelling of transport of dissolved species through a clay backfill from the disposal canister to a fissure in rock, or vice versa, is presented to describe the transport fairly well with a wide range of parameter values. The modelling is an improvement to the previously used simplifying expressions.

The new approach is very illustrative and conserves the volume of the clay section under consideration. Conservation of the volume gives the right time dependency in the case of a well mixed volume. The nonstationary phase of transport can thus be modelled using in principle the presented simplifying concept as a starting point.

Comparisons of stationary mass transport calculations with previous and present simplifying expressions to more accurate calculations are presented.     

Direct Sequential Simulation and Cosimulation

Sequential simulation of a continuous variable usually requires its transformation into a binary or a Gaussian variable, giving rise to the classical algorithms of sequential indicator simulation or sequential Gaussian simulation. Journel (1994) showed that the sequential simulation of a continuous variable, without any prior transformation, succeeded in reproducing the covariance model, provided that the simulated values are drawn from local distributions centered at the simple kriging estimates with a variance corresponding to the simple kriging estimation variance. Unfortunately, it does not reproduce the histogram of the original variable, which is one of the basic requirements of any simulation method. This has been the most serious limitation to the practical application of the direct simulation approach. In this paper, a new approach for the direct sequential simulation is proposed. The idea is to use the local sk estimates of the mean and variance, not to define the local cdf but to sample from the global cdf. Simulated values of original variable are drawn from intervals of the global cdf, which are calculated with the local estimates of the mean and variance. One of the main advantages of the direct sequential simulation method is that it allows joint simulation of N _v variables without any transformation. A set of examples of direct simulation and cosimulation are presented.     

Sveconorwegian igneous complexes beneath the Norwegian–Danish Basin

Gravity and magnetic anomalies have previously been interpreted to indicate strongly magnetic Permian or even Tertiary intrusive bodies beneath the Skagerrak waterway (such as the ‘Skagerrak volcano’) and beneath Silkeborg in Denmark. Our combined modelling of the magnetic and gravity anomalies over these rock bodies indicates that a steep upward magnetisation is required to explain the magnetic anomalies at the surface, reminiscent of the magnetic direction in the Sveconorwegian rocks of the Rogaland Igneous Province in southern Norway. The younger rocks of the Permian Oslo Rift region have intermediate and flat magnetisation that is inadequate to explain the observed magnetic field. The positive part of the Skagerrak aeromagnetic anomaly is continuous with the induced anomalies associated with the eastward extension of the Rogaland Igneous Province. This relation also suggests that rocks of the Rogaland Igneous Province and its offshore extension are responsible for the Skagerrak anomalies. Both the negative, remanence-dominated aeromagnetic anomaly and the positive gravity anomaly can be modelled using constraints from seismic reflection lines and available density data and rock-magnetic properties. A 7 km thick complex of ultramafic/mafic intrusions is located below a southward dipping 1–4 km thick section of Mesozoic sediments and 1–2 km of Palaeozoic sediments. The enormous body of dense, ultramafic/mafic rocks implied by the modelling could be the residue of the parental magma that produced the voluminous Rogaland anorthosites. The application of similar petrophysical properties in the forward modelling of the Silkeborg source body provides an improved explanation of the observed gravity and magnetic anomalies compared with earlier studies. The new model is constrained by magnetic depth estimates (from the Located Euler method) ranging between 6 and 8 km. Forward modelling shows that a model with a reverse magnetic body (anorthosite?) situated above a dense, mafic/ultramafic body may account for the Silkeborg anomalies. The anorthosites may have formed by differentiation of the underlying mafic intrusion, similar to the intrusive relations in the Rogaland Igneous Province. We conclude that there is strong evidence for a Sveconorwegian age for both the Skagerrak and the Silkeborg anomalous rock bodies.