位场向下延拓的波数域迭代法及其收敛性

提出了位场向下延拓的波数域迭代法. 对水平面上的位场观测值进行Fourier变换,得到其波谱. 根据第一类Fredholm积分方程的空间域迭代解法,推导出计算向下延拓水平面上位场波谱的波数域迭代公式. 在波数域中进行迭代,一直进行到相继两次迭代近似解的差值最大绝对值小于给定的精度,或迭代达到给定的最大迭代次数. 对这种迭代近似解进行Fourier逆变换,得到向下延拓的位场. 数值计算结果表明：与空间域迭代法比较,这种波数域迭代法简单、快速,并有同样好的向下延拓效果. 本文还证明了这种迭代法是收敛的,并给出了它的收敛特性和滤波特性.     

重力场向下延拓Milne法

重力场向下延拓能够突出局部和浅部的异常信息,分离叠加的异常特征.但是向下延拓通常具有过程不稳定、下延深度小、结果不准确等问题.针对向下延拓所存在的不足,本文利用重力场及其垂向一阶导数,基于辛普森(Simpson)求积公式,推导出重力场向下延拓米尔尼(Milne)公式.将本文向下延拓方法应用于模型数据,向下延拓模型结果及误差曲线表明,相对于向下延拓快速傅里叶变换(FFT)法和积分迭代法,向下延拓Milne法的深度更大,相对误差更小;相对模型值,向下延拓Milne法能够获得稳定且准确的结果.对加拿大乃查科(Nechako)盆地地区实测航空重力数据进行本文方法向下延拓验证,处理结果表明,相对于实测异常,本文方法向下延拓结果能够很好还原实测数据,并且在进一步向下延拓中反映原始异常的趋势,增强局部和细小异常信息.     

位场向下延拓的波数域广义逆算法

位场向下延拓是位场数据处理和反演中的重要运算,但是它的不稳定性影响了它在许多处理和反演方法技术中的应用.本文通过把位场向下延拓视为向上延拓的反问题,得到向下延拓的褶积型线性积分方程,再利用Fourier变换矩阵的正交对称特性,并结合矩阵的奇异值分解和广义逆原理,提出了一种稳定的不需要进行求逆运算的位场向下延拓广义逆方法——波数域广义逆算法,解决了位场大深度向下延拓的不稳定性问题.把这种方法用于三维理论模型数据和实际磁场数据的向下延拓获得了理想的结果.     

补偿向下延拓方法研究及应用

位场向下延拓是重、磁处理和解释的常用方法,但其不稳定性限制了其在资料处理及反演中的应用.本文基于补偿圆滑滤波思想以及空间域向下延拓迭代法,通过逐次补偿的办法实现位场的稳定向下延拓.同时,在频率域空间给出了该下延方法的频率域响应因子,并讨论了其低通滤波特性,理论模型和实际位场资料试验表明该方法向下延拓稳定性具有较高的延拓精度.将其应用于重力密度界面反演中,改进反演的稳定性,实际莫霍界面反演表明下延因子具备实用性.     

积分迭代法的正则性分析及其最优步长的选择

位场积分迭代法是一种解决位场大数据量、大深度向下延拓的有效方法.本文基于Kirsch正则化子理论,推导了积分迭代法对应的正则化滤子函数,并证明积分迭代法为一种求解位场向下延拓不适定反问题的正则化方法.针对积分迭代法迭代步长固定、迭代次数较多影响收敛速度的问题,提出该迭代法最优迭代步长的选择原理.理论模型和实测数据对比分...     

位场向下延拓迭代法收敛性分析及稳健向下延拓方法研究

地磁导航作为一种新的无源导航方式,具有重要的国防意义.构建空间地磁数据库是实现地磁导航的基础,位场延拓是解决地磁数据库构建的有效方法.积分-迭代法是一种解决位场大深度向下延拓的实用方法.本文着重对积分-迭代法的收敛性进行了分析,从数学角度证明积分-迭代法能够收敛到直接下延法理论解.同时对积分-迭代法的抗干扰性进行了初步分析,当观测数据含有噪声时,积分-迭代过程中使得噪声得到累加,影响延拓数据的精度.本文利用正则化方法和递增型维纳滤波方法,提出了波数域位场向下延拓新算法.模型检验表明,新算法稳定、抗干扰能力强、计算速度快.     

利用微分进化法确定海洋磁场向下延拓中的最优参数

位场的解析延拓是实现不同高度海洋地磁场相互转换的主要途径,是构建海洋三维磁空间背景场模型的关键技术.针对位场向下延拓迭代法中最优正则化参数及最佳迭代次数难以确定问题,尝试引入微分进化法,以正则化参数及迭代次数为种群变量,以延拓结果的熵值为目标函数,以目标函数最小化为搜索准则,实现两种参数的并行全局寻优.采用实测数据对微分进化法在几种常用的迭代法中最优正则化参数及最佳迭代次数的确定进行了分析,与传统L-曲线准则确定的最优正则化参数及多次试验确定的最佳迭代次数进行对比,结果表明：微分进化法确定的最优参数能使三种迭代法取得最佳迭代效果,延拓结果与真实地磁场最为接近,并且该法计算稳定、自适应强,建议在海洋磁场数据向下延拓中应用.     

噪声对磁场向下延拓迭代法的计算误差影响分析

磁场向下延拓可以增强数据的分辨率.在现有的向下延拓方法中,迭代法具有较好的计算效果和较大的向下延拓高度.首先介绍了迭代法的基本原理,分析迭代法的特点;然后利用迭代法对不同噪声水平的模型数据和实测数据进行向下延拓计算,研究噪声对迭代法的计算误差影响.通过延拓结果和实际数据比较分析,客观评价了迭代法的计算精度、可靠性及适应性,对推广应用该方法具有实质性意义.     

位场向下延拓的最小曲率方法

针对位场向下延拓的不适定性,我们将位场向下延拓视为向上延拓的反问题,提出以位场最小曲率作为约束条件来求解稳定的下延位场.我们将剖面位场向上延拓表达式用傅里叶矩阵的形式表示,以矩阵乘法形式给出延拓的表达式,同时向待反演的下延位场引入最小曲率约束,得到向下延拓的最小曲率解,并利用正交变换给出了更为简洁的频率域解.随后,利用Kronecker积将上述全部结果拓展至三维位场,给出了三维位场向下延拓的最小曲率解.此外,我们将位场数据的填充、扩充问题与向下延拓问题统筹考虑,提出一种新的向下延拓迭代格式,该算法面向实际资料处理需求、无须预扩充或填补数据.下延迭代时,对原始数据直接向下延拓,而空白部分利用上一次下延位场估计的上延值替代其空白值并对其向下延拓,直至获得最小曲率约束下稳定的向下延拓结果.同时,我们也讨论了利用改进L曲线和广义交叉验证(GCV)计算正则参数最优估计的问题.对理论模型和实际航空重力资料进行了向下延拓检验,处理结果表明位场向下延拓的最小曲率方法解能满足实际位场资料对向下延拓处理的需求,具有较高的下延精度.     

2D and 3D potential-field upward continuation using splines

The dominant upward‐continuation technique used in the potential‐field geophysics industry is the fast Fourier transform (FFT) technique. However, the spline‐based upward‐continuation technique presented in this paper has some advantages over the FFT technique. The spline technique can be used to carry out level‐to‐uneven surface 2D and 3D potential‐field upward continuation. An example of level‐to‐uneven surface upward continuation of 3D magnetic data using the spline technique is shown, and it is evident that the continued anomalies are very close to the theoretical values. The spacing can be irregular. Synthetic examples using the spline technique to continue noise‐contaminated gravity and magnetic data upward to an altitude of 15 km on irregular grids are shown. Gaussian noise with a zero mean and a standard deviation of 1% does not cause much error and can readily be tolerated. Through comparison with the FFT technique, it is found that for low‐altitude gravity and magnetic upward continuation, both the FFT technique and the spline technique are suitable; for high‐altitude upward continuation, the FFT technique is inaccurate, whereas the spline technique works very well. Also, upward continuation by the spline technique has a smaller edge effect than upward continuation by the FFT technique. The spline‐based upward continuation technique works fairly well even when the periphery of a grid is not quiet: it is rather robust in general. A real example shows that the spline technique can be employed to perform upward continuation of total‐field magnetic data and to de‐emphasize near‐surface noise.     

位场数据曲化平的迭代法

位场数据曲化平是位场数据处理解释中的重要运算,但是它的计算量和计算的复杂性影响了它在许多处理和解释方法技术中的应用.本文提出一种位场数据曲化平的迭代方法,即通过把位场数据曲化平视为平面位场数据向上延拓的反问题,得到曲化平的线性积分方程,再把曲面上位场数据视为曲面平均高程面上的位场数据,利用向下延拓的波数域广义逆算法把平均高程面上的位场数据向下延拓到设定平面上,再根据曲面和其平均高程面的相对起伏对设定平面上的向下延拓数据进行起伏校正,最后再把所得平面上的位场数据向上延拓得到曲面上的位场数据,并进行迭代.把这种方法用于三维理论模型数据和实际磁场数据的曲化平处理均获得了理想的结果.     

High‐accuracy practical spline‐based 3D and 2D integral transformations in potential‐field geophysics

Potential, potential field and potential‐field gradient data are supplemental to each other for resolving sources of interest in both exploration and solid Earth studies. We propose flexible high‐accuracy practical techniques to perform 3D and 2D integral transformations from potential field components to potential and from potential‐field gradient components to potential field components in the space domain using cubic B‐splines. The spline techniques are applicable to either uniform or non‐uniform rectangular grids for the 3D case, and applicable to either regular or irregular grids for the 2D case. The spline‐based indefinite integrations can be computed at any point in the computational domain. In our synthetic 3D gravity and magnetic transformation examples, we show that the spline techniques are substantially more accurate than the Fourier transform techniques, and demonstrate that harmonicity is confirmed substantially better for the spline method than the Fourier transform method and that spline‐based integration and differentiation are invertible. The cost of the increase in accuracy is an increase in computing time. Our real data examples of 3D transformations show that the spline‐based results agree substantially better or better with the observed data than do the Fourier‐based results. The spline techniques would therefore be very useful for data quality control through comparisons of the computed and observed components. If certain desired components of the potential field or gradient data are not measured, they can be obtained using the spline‐based transformations as alternatives to the Fourier transform techniques.     

位场曲化平积分方程的迭代解

提出了位场曲化平的新方法. 给定观测曲面S上的位场、S对下方水平面P的相对高程,确定P上的位场. 利用由P向上延拓到S的积分式,建立这两个面上位场及相对高程三者所满足的方程,它是第一类Fredholm积分方程. 用Fourier逆变换式把这一空间域积分式化为波数域积分式,再由指数函数的Taylor展开进一步化为级数式. 积分方程的解采用逐次逼近法迭代计算,即用S上的位场观测值作为P上位场的初始迭代值,用导出的级数式求得S上的位场计算值、由S上的位场观测值与计算值之差校正P上的位场,多次迭代,直到满足迭代终止准则. 我们还给出该积分方程的波数域迭代计算方法. 模型算例表明,重力异常曲化平的均方差和磁异常曲化平的均方差分别为0.0008 mGal和0.0019 nT,在主频为2.26 GHz的笔记本电脑运行,2048×2048数据量,计算时间是975 s. 野外磁场实际资料处理也证实这种方法的有效性.     

位场向下延拓的改进迭代维纳滤波法

根据维纳滤波理论导出的位场向下延拓滤波器为最佳下延滤波器,但因其实现需要已知待求位场和噪声的功率谱而在实际应用中受到限制.针对该问题,本文首先提出一种基于位场径向平均功率谱的位场噪声水平估计方法,进而利用偏差准则求取正则化参数,实现位场正则化向下延拓;然后将位场正则化下延结果的功率谱作为待求位场功率谱的估计初值,采用带修正项的迭代维纳滤波方法来更新对待求位场功率谱的估计,最后提出本文的位场向下延拓改进迭代维纳滤波方法.基于理论重力模型数据及航磁实测数据进行了向下延拓对比试验,结果表明,改进迭代法具有较好的收敛性,且下延精度优于Tikhonov正则化法和递增型维纳滤波法.     

Interpretation of Gravity Anomalies with the Normalized Full Gradient (NFG) Method and an Example

The Normalized Full Gradient (NFG) method which was put forward about 50 years ago has been used for downward continuation of gravity potential data, especially in the former Union of Soviet Socialist Republics. This method nullifies perturbations due to the passage of mass depth during downward continuation. The method depends on the downwards analytical continuation of normalized full gradient values of gravity data. Analytical continuation discriminates certain structural anomalies which cannot be distinguished in the observed gravity field. This method has been used in various petroleum and tectonic studies. The Trapeze method was used for the determination of Fourier coefficients during the application of this method. No other techniques for calculating these coefficients have been used. However, the Filon method was used for the determination of Fourier coefficients during the application of the NFG method in this work. This method, rather than the Trapeze method, should be preferred for indicating abnormal mass resources at the lower harmonics. In this study, the NFG method using the Filon method has been applied the first time to theoretical models of gravity profiles as example field at the Hasankale-Horasan petroleum exploration province where successful results were achieved. Hydrocarbon presence was shown on the NFG sections by the application of NFG downward continuation operations on theoretical models. Important signs of hydrocarbon structure on the NFG section for field and model data at low harmonics are obtained more effectively using this method.     

位场延拓的积分-迭代法

本文介绍一种新的位场延拓方法——积分-迭代法.将起伏面上的实测位场值,垂直投影至起伏面下部的一个水平面上,作为该水平面上的位场初始值.根据该水平面上的初始值,用积分方法计算起伏面上的位场值.用起伏面上的实测值与计算值的差值,对水平面上的位场值进行校正.如此反复迭代,直至起伏面上的实测值与计算值的差值小到可以忽略.有了水平面上的位场值后,就可以用积分的方法或其他方法计算水平面以上的任意曲面或水平面的位场值.该方法原理简单,不用解线性代数方程组,有较高的计算速度.它特别适用于位场向下延拓,有良好的延拓效果.本文还介绍了积分迭代法的应用实例.     

A refined method of recovering potential coefficients from surface gravity data

Summary A new method for computing the potential coefficients of the Earth's external gravity field is presented. The gravimetric boundary-value problem with a free boundary is reduced to the problem with a fixed known telluroid. The main idea of the derivation consists in a continuation of the quantities from the physical surface to the telluroid by means of Taylor's series expansion in such a way that the terms whose magnitudes are comparable with the accuracy of today's gravity measurements are retained. Thus not only linear, but also non-linear terms are taken into account. Explicitly, the terms up to the order of the third power of the Earth's flattening are retained. The non-linear boundary-value problem on the telluroid is solved by an iteration procedure with successive approximations. In each iteration step the solution of the non-linear problem is estimated by the solutions of two linear problems utilizing the fact that the non-linear boundary condition may be split into two parts; the linear spherical approximation of the gravity anomaly whose magnitude is significantly greater than the others and the non-linear ellipsoidal corrections. Finally, in order to solve the problem in terms of spherical harmonics, the transform method composed of the fast Fourier transform and Gauss Legendre quadrature is theoretically outlined. Immediate data processing of gravity data measured on the physical Earth's surface without any continuation of gravity measurements to a reference level surface belongs to the main advantage of the presented method. This implies that no preliminary data handling is needed and that the error data propagation is, consequently, maximally suppressed.     

重力位场谱分析方法研究综述

利用Fourier变换将空间域位场数据变换到频率域进行谱分析是重力位场数据处理中的一种重要方法．这种方法在位场的延拓及转换、场分离、物性界面的正反演和位场的曲化平处理等方面已得到广泛的应用．本文详细介绍了位场谱分析的基本理论和方法,总结了国内外最近几十年来在这方面取得的主要成果,并提出了未来值得深入研究的方向．     

A stable iterative downward continuation of potential field data

Downward continuation is a useful tool in the processing of potential field data, which can effectively enhance weak anomalies and identify overlap anomalies, but we all know that the computation of downward continuation is unstable, and easily distorts the true feature of potential field data. Because the computation of upward continuation and horizontal derivatives is stable, we proposed using the combination of upward continuation and horizontal derivative to accomplish the downward continuation of potential field data. The proposed method is demonstrated on synthetic potential field data, and the results show that the proposed method can finish the downward continuation of the data stably and precisely, and the precision of the proposed method is higher than the traditional method. We also apply it to real potential field data, and the results show that the proposed method accomplishes the downward continuation of the real data stably.     

迭代法与FFT法位场向下延拓效果的比较

将水平观测面上的实测位场值,垂直投影至下部的延拓水平面上,作为该水平面上的位场初始值. 根据该水平面上的初始值,用快速傅里叶变换(FFT)的方法向上延拓计算观测面上的位场值. 用观测面上的实测值与计算值的差值,对延拓面上的位场值进行校正. 如此反复迭代,直至观测面上的实测值与计算值的差值小到可以忽略. 这种空间域的迭代法原理简单,不用解线性代数方程组,有较高的计算速度和良好的延拓效果. 本文用迭代法对模型数据和实际数据进行向下延拓,对比了迭代法与常规的FFT法在位场向下延拓中的效果,迭代法显著优于FFT法.