Interpolation with Splines in Tension: A Green''s Function Approach

Interpolation and gridding of data are procedures in the physical sciences and are accomplished typically using an averaging or finite difference scheme on an equidistant grid. Cubic splines are popular because of their smooth appearances; however, these functions can have undesirable oscillations between data points. Adding tension to the spline overcomes this deficiency. Here, we derive a technique for interpolation and gridding in one, two, and three dimensions using Green's functions for splines in tension and examine some of the properties of these functions. For moderate amounts of data, the Green's function technique is superior to conventional finite-difference methods because (1) both data values and directional gradients can be used to constrain the model surface, (2) noise can be suppressed easily by seeking a least-squares fit rather than exact interpolation, and (3) the model can be evaluated at arbitrary locations rather than only on a rectangular grid. We also show that the inclusion of tension greatly improves the stability of the method relative to gridding without tension. Moreover, the one-dimensional situation can be extended easily to handle parametric curve fitting in the plane and in space. Finally, we demonstrate the new method on both synthetic and real data and discuss the merits and drawbacks of the Green's function technique.     

逐点激发井深设计网格化方法的选取

随着地震勘探的不断深入,地震资料采集要求主频足够高,频带足够宽,能量足够强,而逐点激发井深设计有助于获得满意的地震信号,要设计合理的井深,网格化方法的选取非常重要。这里首先介绍了Surfer软件中十二种网格化方法原理特征,以及在逐点井深设计上的应用;然后运用Surfer软件中不同的插值方法,对BIN12工区岩性数据进行网格的效果对比与网格的密度探讨,选取了最佳的网格化方法;最后针对几种网格化方法进行井深对比试验,通过资料分析,说明了设计的井深合理,选取网格化方法的效果较好。     

一种实用的等值线型数据网格化方法

数据网格化通常包括三大类:测线型数据网格化、等值线型数据网格化和离散点型数据网格化。文中研究的等值线型数据分块存储、网格点八方位搜索插值的网格化方法较好地解决了平面等值线型数字化数据的网格化计算问题,其计算数据量大,实用性强,精度高,计算速度快。     

Moving Surface Spline Interpolation Based on Green’s Function

Some commonly used interpolation algorithms are analyzed briefly in this paper. Among all of the methods, biharmonic spline interpolation, which is based on Green’s function and proposed by Sandwell, has become the mainstream method for its high precision, simplicity and flexibility. However, the minimum curvature method has two flaws. First, it suffers from undesirable oscillations between data points, which is solved by interpolation with splines in tension. Second, the computation time is approximately proportional to the cube of the number of data constraints, making the method slow for situations with dense data coverage. Focusing on the second problem, this paper introduces the moving surface spline interpolation method based on Green’s function, and the interpolation error equations are deduced. Because the proposed method only chooses the nearest data points by using the merge sort algorithm for interpolating, the computation time is greatly decreased. The optimal number of the nearest points can be determined by using the interpolation error estimation equation. No matter how many data points there are, this method can be implemented without difficulty. Examples show that the proposed method can obtain high interpolation precision and high computation speed at the same time.     

利用MFC和Surfer实现方位加权多重二次曲面插值

受众多因素的影响,我们采集到的地球物理数据是离散的,这也限制了获得更多的地下信息。为了便于后期的解释工作,数据的补插网格化就至关重要。数据网格化是将空间上不均匀分布的数据,按一定方法归算成规则网格节点的代表值。这里依据方位加权二次曲面插值算法,利用MFC对话框界面与Surfer的混合编程,读取实测数据信息后,在可视化界面实现网格化处理过程。最后调用Surfer生成区域等值线图,与其内部插值方法绘制的等值线图进行比较。一方面验证了方位加权二次曲面插值算法的可适性,另一方面通过人机交互的方式简洁快速实现了庞大数据的批处理,便于用户使用。     

Reconstructing the Green's function through iteration of correlations

Correlations of ambient seismic noise are now widely used to retrieve the Earth response between two points. In this study, we reconstruct the surface-wave Green's function by iterating the correlation process over the tail of the noise-based correlation function. It has been demonstrated that the so-called C³ function shows the surface-wave part of the Green's function. Using data from 150 continuously recording stations in Europe, the C³ results help in the extraction of the travel-times from noise-based measurements, especially through the suppression of effects caused by non-isotropic source distributions. We present the results of the next iterative step (i.e. C⁵), which show that some coherent signal is still present in the coda of the C³ function, and we investigate the evolution of the reconstruction of the Green's function throughout the iteration process. Finally, we discuss the interest of combining information from the different correlation functions to improve noise-based tomography analysis.     

Green's functions for a fully coupled thermoporoelastic material

The Green's functions for a set of fully coupled linear equations governing thermal consolidation of a homogeneous isotropic material are presented. The Green's functions are evaluated for one, two and three spatial dimensions, and in the Laplace and real-time domains. The Green's functions are employed in a boundary integral analysis of a simple problem to demonstrate the influence of full coupling on the field quantities.     

交叉验证在离散数据网格化时的应用

根据数据的特点,选择不同的算法和参数对离散数据进行网格化,所得网格化数据对原始数据的反映程度不同。因此,在网格化时,可以利用交叉验证(Cross Validation)对不同的网格化方法进行定量的评估和比较,以选择最能尊重原始数据的网格化算法和参数。     

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。     

等值线绘图软件SURFER7.O中九种插值法介绍

SURFER软件是一个功能强大的绘制等值线图及三维立体图软件包，能迅速地将离散的测量数据通过插值转换为连续的数据曲面。SURFER7．0提供的内插方法多达九种，其中每一种内插方法都有其意义及相关的参数设置。作者在本文中主要介绍了该软件中的九种内插方法，并进行了应用示例。     

多目标区域地球化学编图数据整理方法—以海河流域编图为例

多目标区域地球化学编图是科学表达区域土壤地球化学分布和特征的图示方法,大区域的编图一般可能涉及到对不同的工作区、不同实验室和不同时间的测试数据,以及不同景观条件和网度的表层、深层土壤元素数据或不同分度带的网格数据分别进行拼接合并,因此区域编图对原始数据的整理和网格化处理不可避免.而不同的数据整理方法及其计算参数所形成的结果是非常不同的,数据本身的特点也对编图具有很大的影响.为了真实客观地反映区域内元素的地球化学分布特征,数据整理和处理的方法及技术参数的选择是关键技术问题.作者就化探数据处理中常用的几种插值方法进行了对比评价,并分别从网格间距及搜索半径大小等方面做比较,提出了自己在编图中采用的数据处理方法及选用的技术参数.     

Elastostatic response of a pile embedded in a transversely isotropic half‐space under transverse loading

In the framework of elastostatics, a mathematical treatment is presented for the boundary value problem of the interaction of a flexible cylindrical pile embedded in a transversely isotropic half‐space under transverse loadings. Taking the pile region as a stiffened subdomain of an extended half‐space, the formulation of the interaction problem is reduced to a Fredholm integral equation of the second kind. The necessary set of Green's functions for the transversely isotropic half‐space is obtained by means of a method of potentials. The resulting Green's functions are incorporated into a numerical procedure for the solution of the integral equation. The theoretical response of the pile is presented in terms of bending moment, displacement and slope profiles for a variety of transversely isotropic materials so that the effect of different anisotropy parameters can be meaningfully discussed. Copyright © 2013 John Wiley & Sons, Ltd.     

Characterizing a fractured aquifer in Mexico using geological attributes related to open-pit groundwater

A multivariable analysis of the Rock Quality Designation (RQD) and its relation with the hydraulic conductivity of 17 dewatering wells in an open-pit mine (central Mexico) is presented as a tool for groundwater exploration in fractured aquifers. A solid model was constructed with the RQD data using three sizes for each grid cell and four interpolation methods. The inverse-distance method with a small grid gives the best results. The resulting RQD solid model was used to locate 22 pilot holes, on which an air-lift test was performed as a qualitative assessment of their usefulness. The results showed a lower water production (1.8 l/s) in shale that has low alteration, whereas in highly altered shale, breccias, and intrusive rock, the flow rate was 3.9 l/s. This implies an important relationship between the pilot-hole performance, the lithology, and the rock alteration, but it was also found that some fractures or faults, which cannot be detected clearly by the RQD, play an important role in the hydrodynamics of the aquifer. In conclusion, it is necessary to consider all available factors that can help to identify the hydrodynamic behavior of the aquifer because using only RQD data can lead to errors in prospecting for groundwater.     

Nested gridding and streamline-based simulation for fast reservoir performance prediction

Detailed reservoir models routinely contain 10⁶–10⁸ grid blocks. These models often cannot be used directly in a reservoir simulation because of the time and memory required for solving the pressure grid on the fine grid. We propose a nested gridding technique that efficiently obtains an approximate solution for the pressure field. The domain is divided into a series of coarse blocks, each containing several fine cells. Effective mobilities are computed for each coarse grid block and the pressure is then found on the coarse scale. The pressure field within each coarse block is computed using flux boundary conditions obtained from the coarse pressure solution. Streamline-based simulation is used to move saturations forward in time. We test the method for a series of example waterflood problems and demonstrate that the method can give accurate estimates of oil production for large 3D models significantly faster than direct simulation using streamlines on the fine grid, making the method overall approximately up to 1,000 times faster than direct conventional simulation.     

A modification of Burger's phi-normal approximation for the moments of grain-size data

This paper gives a modification of Burger's method of approximation for the moments of grain-size data. It is shown that formal integration can be carried out in conjunction with Burger's phi-normal interpolation scheme. This approach obviates Burger's recourse to a discrete numerical integration procedure because exact integration yields a computationally more efficient approximation formula for the moments.     

SURFER软件图形数据的进一步处理和利用

ＳＵＲＦＥＲ软件可将原始数据网格化，自动生成等值线图形，使用方便，但它的图形编辑功能较弱，无中文标注功能，多图层操作功能方面也不能令人满意，为了弥补这些不足，需对其数据做进一步处理。本文讨论了ＳＵＲＦＥＲ软件数据的进一步处理和利用中的一些问题，作为例子，着重探讨ＳＵＲＦＥＲ软件与其它系统的接口     

High-resolution regional gravity field model of Pakistan based on best residual terrain model (RTM) of topography and interpolation techniques

This study focuses on the development of absolute gravity model for Pakistan based on best possible residual terrain model of gravity using residual terrain modeling technique. The datasets used for the development of model are observed gravity, global gravity models, and Shuttle Radar Topographic Mission (SRTM30) elevation data. The residual terrain modeling technique has been used in the remove-restore procedure for smoothing the observed gravity field. Different topographic elevation models were tested in the model selection and one best possible model with minimum mean and standard deviation was selected for residual terrain effects. Least square collocation technique has been used for quality control and error estimates. The best possible covariance model was established from residual gravity for onward prediction of gravity anomalies at the earth surface for error and prediction analysis. The residual terrain effect of gravity, value of free air anomaly from EGM96, and observed free air anomaly are added to normal gravity to compute the absolute gravity at earth surface. The prediction of these parameters is made by employing Lagrange interpolation with least square adjustment. The results are compared with ～5% randomly selected data points not utilized for the development of covariance function and/or model development. Spline interpolation technique has also been used for the prediction of gravity field-related parameters. Lagrange interpolation exhibits relatively superior results over spline-based interpolation. This is as per expectation due to the reason that additional gridding for spline interpolation filters the signal part as well. This fact is evident from the results of spline interpolation of Grid-I and Grid-II with relatively better prediction results in Grid-I. This version of the model is capable of prediction having limiting error of 30 mGal. The predicted results show that 96.16% of prediction data falls within above-mentioned limit with Lagrange interpolation technique with least square adjustment for whole Pakistan area. The adverse effect of gridding is absent in case of Grid-I due to relatively flat areas and predicted data matches totally with control values for both spline as well as Lagrange interpolations. However, in case of Grid-II which includes high mountains of Himalaya, gridding effect is present and the accuracy of the predicted results falls to ～92%. The computed results have been compared with absolute values predicted using EGM96 and EGM2008 models as well. The gravity field recovered with PAKGM model is much better, i.e., ～ 96.16%, than both with EGM96 and EGM2008 which is about 85% only.     

Wave scattering and diffraction of subsurface cavities in layered half-space for incident SV-P and SH waves

A novel approach is proposed to deal with the problem of wave scattering and diffraction of subsurface cavities embedded in stratified half-space. The subsurface cavity with complex surroundings is treated as a substructure. The continuity condition at the interface between the substructure and the far field of stratified half-space is maintained by applying a free-field approach. As the boundary of the free-field ground is regular, the construction of the dynamic matrices and the evaluation of the wave input on the interface become considerably easier. Based on the previous work with some improvement, a novel approach for evaluation of Green's functions in stratified half-space is presented. The wave equation is decoupled into the one for SV-P wave components and the other one for SH wave component. The precise integration technique ensures high accuracy of the solution of wave equations. The layer merging technique and the dual form equation make it possible to obtain Green's function in closed-form solution of matrix equations. Numerical examples validate accuracy and efficiency of the proposed approach.     

油藏数值模拟中地质模型的建模流程与方法

围绕油藏数值模拟过程中三维地质模型的建模技术进行了分析与探讨,详细介绍了基于角点网格模型的建模方法,给出了相应的实现步骤。其主要流程是:首先根据断层数据构造断层模型,在断层模型的基础上构建骨架模型;然后在骨架模型约束下采用地层恢复技术实现含断层的地层模型;最后基于结构模型插值物性参数完成属性模型。以塔河油田缝洞型油藏为例,对建模流程和技术的可行性进行验证,结果表明,其建模结果与专业地质建模软件Petrel相符。     

地球物理不规则分布数据的空间网格化法

不规则分布数据的网格化处理是地球物理数据处理和解释的基础问题,是保证许多地球物理数据处理方法得以成功实施的前提.文中介绍了线性插值法、多元二次函数法、普通克里格法、反插值法等4种空间网格化法的原理,并利用这些网格化方法进行了理论模型和实际航磁数据的试验分析.