基于空间散乱点插值的曲面重构

采用最近邻点法、距离反比加权法和克里格法等空间插值方法,基于IDL语言实现插值算法,对我国西部某金属矿山坑道内,利用大功率充电法测得的163个电位数据进行曲面重构,结果表明,克里格插值方法更能够真实地反映空间数据变量(场)的分布特性,具有广泛的实用价值。     

地球物理勘探中几种二维插值方法的误差分析

为了研究采样和网格化方法对地球物理数据成图精度的影响,为野外数据采集布设提供一定的依据,采用数值模拟确定重力异常场场值,通过不同采样间距和不同插值方法计算重力异常绝对误差均方根值和节点处的绝对误差值,对比不同插值方法的误差,得到了如下认识：1）对于同一插值方法而言,存在小间距绝对误差均方根值小于大间距绝对误差均方根值的关系。2）对不同的插值方法而言:当采样间距小于最小异常地质体尺度时,绝对误差均方根值由小到大的顺序是径向基函数法、改进的谢别德法、克里金插值法、自然邻点法、反距离加权插值法、最近邻点法、最小曲率法,并且线性插值三角网法与自然邻点法具有几乎相同的数值;当采样间距大于最小异常地质体尺度时,绝对误差均方根值由小到大的顺序是径向基函数法、改进的谢别德法、克里金插值法、自然邻点法、最小曲率法、最近邻点法、反距离加权插值法,并且线性插值三角网法和自然邻点法具有几乎相同的数值。3）从绝对误差均方值看,径向基函数方法、改进的谢别德方法和克里金方法数值较小,其中径向基函数值绝对误差均方根值最小。4）从节点处绝对误差值来看,径向基函数方法、克里金方法、改进的谢别德方法相对其他插值方法具有更小的误差,不存在局部误差较小或较大的情况,是相对较好的插值方法,并且径向基函数方法是最好的。     

自然邻点插值算法及其在二维不规则数据网格化中的应用

由于观测数据的不足,对不规则散点数据的二维插值在地球物理中应用非常广泛。这里研究并实现了自然邻点二维插值算法,并以一个大地电磁测深反演数据为例,验证了该方法的可行性与效果。实际资料试算与成图结果表明,该方法具有精度高、插值效果好、速度快以及便于模拟地形数据等优点。     

关于地球化学数据处理方法的讨论——以大兴安岭喜桂图地区1/20万区域地球化学数据为例

针对地球化学数据中存在的离群数据点在地球化学图面上出现的等量线畸变或"牛眼点"现象,本文使用多种数据网格化插值方法和参数对比以消除"公牛眼"现象,结果表明,克里格、径向基本函数、改进的谢别德、最小曲率、移动平均等方法均不能较好地消除该现象。各种网格化算法编制的地球化学等量线图和交叉验证结果证明,以改进的指数距离幂倒数加权网格化方法效果最好,并且网格参数选择中以取样间距为网格间距,以2.5倍网格间距为搜索半径和幂指数因子的网格化参数既能消除等量线畸变现象,又能对原始数据进行较接近的表达。上述各种地球化学数据网格化插值方法的对比及结果可为地球化学数据处理中遇到同类问题提供信息。     

元素空间分布插值方法的对比研究:以铜陵地区土壤中的重金属元素为例

文中以铜陵地区As、Cd、Cu、Pb、Tl、Zn等6种土壤污染元素为例,选取常用且具有代表性的反距离加权法、径向基函数法、普通克里格法、多维分形法4种空间插值方法,进行土壤元素空间插值,并对其结果进行验证分析和评价。各方法均选取最优参数进行插值对比,土壤样本数共372个,其中337个用于插值计算,35个不参与插值计算而用于验证插值结果。对比研究显示,普通克里格法对刻画区域土壤元素的空间分布趋势效果最佳,但其半变异函数模型及参数的优选仍有待进一步研究;多维分形法对刻画土壤元素局部异常和污染效果最佳,但其对土壤元素分布普遍特征的反映仍需深入研究;反距离加权法和径向基函数法对土壤元素分布的空间插值精度一般,但其简单易用、插值最优参数易于选择。     

空间插值方法模拟地下水空间分布适宜性探讨——以敦煌盆地东湖保护区为例

选取敦煌盆地东湖保护区稀少地下水控制点区域为研究区,以潜水埋深数据为例,利用克里金、反距离加权法和径向基函数法等3种插值方法获取地下水潜水埋深栅格图像,并通过交叉验证和拟合分析对插值结果进行比较,分析总结3种插值方法在缺少足够控制点区域获取插值图像的适用特点及差异性,结果认为:在干旱半干旱地区艰苦环境下,利用取之不易的少量地下水控制点通过克里格插值方法模拟区域地下水位埋深结果是可靠的,同时对于空间相关性较强的类似空间变量,利用克里格插值可以较好地获取区域变量空间分布趋势。     

土壤碳储量计算中不同插值方法对比研究——以吉林省大安市为例

土壤碳储量问题是大气温室效应和全球变化研究的热点问题。本文采用“单位土壤碳量”方法,计算了吉林省大安市表层土壤有机碳总储量和平均土壤有机碳储量,以为全球碳循环研究提供精确的基础数据。分析和比较了反距离加权、全局多项式、局部多项式、径向基函数和普通克里格等不同插值模型对表层单位土壤有机碳储量空间插值结果的影响,结果表明全局多项式插值的均差最小,径向基函数中张力样条法的平均绝对误差最小,普通克里格法的标准差最小。综合分析来看,普通克里格的有理二次方程式模型插值精度最高。      

合肥义城地区土壤重金属污染评价中典型插值方法的对比

空间插值对于土壤中重金属元素的空间分布及污染评价具有重要意义。对合肥义城地区土壤中的Cu、Pb、Zn、Cd、As、Hg等污染重金属元素,以常用且具有代表性的反距离加权法、径向基函数法、普通克里格法,进行了空间插值的对比验证分析和评价。通过对各种元素的空间插值各种误差进行综合比较的结果表明：Cu、Pb、As元素采用普通克里格法进行插值结果最优,而Zn元素采用反距离加权法最优,对于Cd、Hg元素则径向基函数插值法最优。     

降雨量分布的空间插值方法研究——以美国爱达荷州为例

以美国爱达荷州为例,根据105个气象站点及其30年年平均降水量数据,采用反距离加权平均法、普通克立金法、规则样条函数法及趋势面法等方法进行插值,分析了不同插值方法中站点数量变化、像元尺度变化对降雨数据空间插值结果的影响.通过对数据和精度指标的对比分析,得出每一种插值方法在某一标度范围插值的精度及适用范围.研究表明:[1]插值精度并不与样本点数量成正比例关系;[2]像元尺度的变化对空间插值精度的影响很小;[3]不同插值方法对插值结果有很大的影响,规则样条函数具有优秀的插值精度.要得到最理想的插值结果,须对不同研究区,进行实测样本数据进行分析,反复试验比较,以选择最佳插值方法.     

地学数据的分形插值与成图

本文将分形几何学的思想和方法引进对地学数据的插值处理与计算机成图的研究中，建立了适合于具有随机形特征的地学数据的“局部分数布朗场”数学模型，在讨论和分析了Ｖｏｓｓ提出的“逐次随机添加”与Ｂａｒｎｓｌｅｙ提出的“分形插值函数”这两种可用于分形插值法，并将它们应用到对地貌，化探，物探等地学数据进行插值，网格化和计算机成图上。     

Neighborhood discontinuities in bivariate interpolation of scattered observations

Because of the need for computational efficiency, bivariate interpolation methods applied to scattered observations often involve two stages. Initially the variable is estimated at regular grid nodes using a running subset of data (usually of fixed number). This, however, will produce discontinuities in the interpolated surface. Thus a second stage, curvilinear interpolation technique, is applied to estimated values to smooth out the effect of discontinuities. Such problems can be overcome efficiently in processing large data sets by interpolating over natural neighbor subsets. Interpolation procedures that generate discontinuities in the interpolated surface are inappropriate for geological applications, where dislocations due to structural complications may be present.     

Accuracy Analysis of Digital Elevation Model Relating to Spatial Resolution and Terrain Slope by Bilinear Interpolation

This study investigates the accuracy analysis of the digital elevation model (DEM) with respect to the following two major factors that strongly affect the interpolated accuracy: (1) spatial resolution of a DEM and (2) terrain slope. Unlike existing studies based mainly on a simulation approach, this research first provides an analytical approach in order to build the relationship between the interpolated DEM accuracy and its influencing factors. The bi-linear interpolation model was adopted to produce this analytic model formalized as inequalities. Then, our analytic models were verified and further rectified by means of experimental studies in order to derive a practical formula for estimating the DEM accuracy together with an optimization model for calculating the required resolution when a prescribed upper bound to the DEM accuracy is given. Moreover, this analytic approach can cope with either a grid-based DEM or a randomly scattered scenario whose efficacies have been validated by the experiments using both synthetic and realistic data sets. In particular, these findings first establish the rules for directly correlating the horizontal resolution of DEM data with vertical accuracy.     

断层的三维可视化建模研究

断层的三维可视化建模是实现断层三维动态模拟的前提和基础。本文在探讨三维可视化建模的数据模型及剖面插值方法的基础上,提出将混合插值方法与B-Rep模型结合起来用于断层建模。基于GeoView三维可视化地学信息系统,将其应用于临清坳陷东部的三维构造-地层格架建模中,效果良好。     

Hybrid Gaussian-cubic radial basis functions for scattered data interpolation

Scattered data interpolation schemes using kriging and radial basis functions (RBFs) have the advantage of being meshless and dimensional independent; however, for the datasets having insufficient observations, RBFs have the advantage over geostatistical methods as the latter requires variogram study and statistical expertise. Moreover, RBFs can be used for scattered data interpolation with very good convergence, which makes them desirable for shape function interpolation in meshless methods for numerical solution of partial differential equations. For interpolation of large datasets, however, RBFs in their usual form, lead to solving an ill-conditioned system of equations, for which, a small error in the data can cause a significantly large error in the interpolated solution. In order to reduce this limitation, we propose a hybrid kernel by using the conventional Gaussian and a shape parameter independent cubic kernel. Global particle swarm optimization method has been used to analyze the optimal values of the shape parameter as well as the weight coefficients controlling the Gaussian and the cubic part in the hybridization. Through a series of numerical tests, we demonstrate that such hybridization stabilizes the interpolation scheme by yielding a far superior implementation compared to those obtained by using only the Gaussian or cubic kernels. The proposed kernel maintains the accuracy and stability at small shape parameter as well as relatively large degrees of freedom, which exhibit its potential for scattered data interpolation and intrigues its application in global as well as local meshless methods for numerical solution of PDEs.     

Scattered Data Approximation on the Bisphere and Application to Texture Analysis

The paper deals with the approximation and optimal interpolation of functions defined on the bisphere \mathbb S²×\mathbb S²\mathbb {S}^{2}\times \mathbb {S}^{2} from scattered data. We demonstrate how the least square approximation to the function can be computed in a stable and efficient manner. The analysis of this problem is based on Marcinkiewicz–Zygmund inequalities for scattered data which we present here for the bisphere. The complementary problem of optimal interpolation is also solved by using well-localized kernels for our setting. Finally, we discuss the application of the developed methods to problems of texture analysis in material science.     

From registration to visualization of geological data with MO2GEO: applications of the FieldModule,interpolation methods,and computer visualization

The registration and screen visualization of geological data are basic steps in modelling of 3D structures and geoscientific processes. Therefore, a standardized coding of geological field observations is reasonable to enable an independent modelling approach. The field module of MO2GEO provides diverse standards for data coding and was tested in the frame of a mapping project. The visualization of profiles and the construction of cross-sections improve the database for a variety of interpolation methods. The subsequently developed 3D models, constructed with diverse modelling tools, can be visualized and serve as the base for diverse applied scientific projects.     

Kriging插值方法在地层模型生成中的应用

为了建立三维数字地层,采用了一种适合城市工程地质和岩土工程特点的地层数据模型-基于钻孔信息的3棱柱模型。由于钻孔之间的距离稀疏程度、方向、数据值存在差异,钻孔以外未知的地质特性需要插值和推断,传统的数理统计方法无法很好地解决空间样本点的选取、空间估值和2组以上空间数据的关系等问题。借鉴地质统计学的Kriging方法给出一种距离加权插值算法,即先根据空间数据得到统计特征,再根据统计特征进行插值。通过对地层模型插值结果的观察,得出该算法可以获得良好的插值效果。     

Regional dome evolution and its control on ore-grade distribution: Insights from 3D implicit modelling of the Navachab gold deposit,Namibia

We introduce a novel approach to analyse and assess the structural framework of ore deposits that fully integrates 3D implicit modelling in data-rich environments with field observations. We apply this approach to the early Palaeozoic Navachab gold deposit which is located in the Damara orogenic belt, Namibia. Compared to traditional modelling methods, 3D implicit modelling reduces user-based modelling bias by generating open or closed surfaces from geochemical, lithological or structural data without manual digitisation and linkage of sections or level plans. Instead, a mathematically defined spatial interpolation is used to generate 3D models that show trends and patterns that are embedded in large drillhole datasets. In our 3D implicit model of the Navachab gold deposit, distinctive high-grade mineralisation trends were identified and directly related to structures observed in the field. The 3D implicit model and field data suggest that auriferous semi-massive sulphide ore shoots formed near the inflection line of the steep limb of a regional scale dome, where shear strain reached peak values during fold amplification. This setting generated efficient conduits and traps for hydrothermal fluids and associated mineralisation that led to the formation of the main ore shoots in the deposit. Both bedding-parallel and highly discordant sets of auriferous quartz-sulphide veins are interpreted to have formed during the later lock-up stage of the regional scale dome. Additionally, pegmatite dykes crosscut and remobilise gold mineralisation at the deposit scale and appear to be related to a younger joint set. We propose that kilometre-scale active folding is an important deformation mechanism that influences the spatial distribution and orientation of mineralisation in ore deposits by forming structures (traps and pathways for fluids) at different preferred sites and orientations. We also propose that areas that experience high shear strain, located along the inflection lines of folds can act as preferred sites for syn-deformational hydrothermal mineralisation and should be targeted for regional scale exploration in fold and thrust belts. Our research also suggests that examination of existing drillhole datasets using 3D implicit modelling is a powerful tool for spatial analysis of mineralisation patterns. When combined with fieldwork, this approach has the potential to improve structural understanding of a variety of ore deposits.     

三维地层建模钻孔数据预处理研究

钻孔数据是三维地层建模的主要数据来源之一,在三维地层建模中,地层界面一般基于钻孔数据插值而成,正确处理钻孔数据是构建理想三维地层模型的前提和基础。三维地层拓扑关系极其复杂,地层界面插值数据在数学上选择也不是惟一的,因此,常导致构建地层模型出现与事实相悖的情形。针对这一点,根据地层地质演变历史,建立地层层序序列,此序列记录地层地质年代顺序及其拓扑关系。根据地层地质特征,可将地层拓扑关系概化为两类：“覆盖”和“侵蚀”。据此地层层序序列可惟一确定地层界面插值数据,从而构建正确地层模型。采用python语言实现了该钻孔数据预处理算法,实例表明,该预处理方案能正确地预测实际地层。