基于Extrapolation Tikhonov正则化算法的重力数据及梯度多分量数据的3D反演方法研究
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摘要
随着重力及梯度观测数据类型的增加、硬件设施的发展、数学方法的不断进步，重力及梯度数据观测区域日趋广泛、解释技术及方法不断进步，应用领域不断拓宽，重力及梯度张量数据3D反演方法的研究也日益紧迫。反演方法的改进、计算精度的提高、计算时间及计算量的减小、可反演数据类型的丰富成为人们研究的热点。
3D反演方法分为空间域和频率域反演方法。空间域反演方法主要由两类构成，第一类是利用丰富的地质先验信息约束解，在这一分类中具有代表性的方法有种子方法，交互式反演方法等；第二类是尽量少的使用复杂地质信息的约束最小化反演方法，在这类方法中最常用的约束条件包括深度加权约束，平滑约束及上下限约束等。本文重点研究第二类反演方法。除多数学者致力研究的空间域反演方法外，以计算速度快，硬件要求较低为特点的频率域反演在特定的研究领域依然占有重要的地位，本文就这一方面也做了部分研究。
空间域反演方法的计算精度、计算时间、计算量、反演数据类型及反演约束条件等问题是近年来人们研究的焦点，本文就这些问题所做的研究工作如下。
1.深入讨论了Sorokin、Haaz、Nagy及Okabe和Steiner的棱柱体重力模型正演方法，并根据这四个正演方法计算同一棱柱体模型引起的重力异常。计算结果表明，虽然四个计算公式的最后一项不同，但是它们计算的重力异常之间的误差小于10-10，所以，四种方法的计算误差对反演结果没有明显的影响，反演中可忽略这些计算误差，选用其中任意一种正演方法进行反演计算都可满足误差水平，本文选用Nagy的正演方法计算棱柱体正演模型引起的重力值。
2.详细推导计算重力梯度九分量的计算公式，讨论了梯度数据的性质及不同分量之间的对称关系，并建立正演模型，计算此模型的梯度九分量数据，计算结果表明，梯度九分量存在很好的对称性，同时，计算结果也验证了梯度场是无迹的。
3.本文详细分析了反问题病态解的诱因，即解的存在性、唯一性和稳定性三者不能同时满足时，反演解是病态的，并给出了解决病态线性反演的方法。针对线性病态反问题，经典和简单的方法是利用Tikhonov正则化算法解决重力数据反问题中核函数的奇异性问题。为进一步减小因正则化参数的引入带到解中的误差，后人发展的Extrapolation Tikhonov正则化算法成为本文研究反问题的主要工具。
4.深入研究基于Extrapolation Tikhonov正则化算法的重力数据约束反演方法。通过比较基于Tikhonov正则化算法反演结果及基于ExtrapolationTikhonov正则化算法反演结果可知，后者的观测数据与预测数据拟合误差更小。由于正则化算法中参数选择直接关系到反演结果的好坏，文中选用平衡原则、单调误差原则及离散原则三个选择原则综合选择正则化参数n和。
5.在重力及其梯度数据反演中，核函数呈现随深度增加而快速衰减的特性，如果直接进行反演计算，则反演结果存在严重的趋肤效应。为此，前人提出一种可消除这种衰减的方法，即利用与核函数随深度变化具有相反特性的深度加权函数作用与核函数，使核函数沿垂直方向对所有密度块体的作用相等，从而消除重力数据反演中的趋肤效应。本文在深入研究前人深度加权约束函数的基础上，改进了深度加权函数。将前人加权函数、作者改进的加权函数及不加权的核函数应用到不同埋深的单个立方体模型的密度分布反演中。结果表明，不加权核函数反演的密度分布存在严重的趋肤效应，前人加权函数对于立方体的底部分辨率比改进后的加权函数要差，改进后的加权函数对于埋深不同的异常体都有较好的反演效果。
6.密度上下约束函数的引进，对超出先验信息范围的密度差起到约束作用，将反演结果转换到满足先验信息的数值范围内，文中详细分析了上下限约束函数中各参数的选择机制，并将其应用到重力及梯度多分量数据的反演计算中。
7.本文将基于Extrapolation Tikhonov正则化算法及Tikhonov正则化算法的反演方法应用于两组组合模型，就它们的计算精度及计算时间进行对比，结果表明，两个方法都可较准确的反演出异常体密度分布特征。前者能在相同的误差要求下达到更好的拟合误差水平，但是需消耗更多的计算时间。
8.详细讨论了由不同梯度分量及分量组合构成核函数时包含的信息量的计算方法及判断依据。利用奇异值分解方法分解带有深度约束等约束条件的核函数特征值，并将核函数的特征值利用最大特征值规则化，利用规则化之后的特征值谱研究不同的核函数含有的信息量。前人研究结果表明，核函数的特征值越大含有的信息量越多。将不同分量及不同分量组合成的核函数分解后发现单个分量的特征值谱小于分量组合时的特征值谱，因此梯度分量组合反演密度模型时，可获得更多垂向和水平方向的细节。
9.利用基于Extrapolation Tikhonov正则化算法的3D反演方法反演重力梯度多分量数据不同组合方式时的密度模型。结果表明单独反演不同的梯度分量时，Vxy及Vzx在水平及垂向的分辨率较理想，Vxx是各梯度分量数据反演结果中效果最差的，这与数值分析结果相一致。少数梯度分量数据组合反演结果与单个梯度分量作为核函数的反演结果相比，少数梯度分量组合的核函数有更大的特征值谱，因此，可提供更多关于异常源的信息。计算结果表明Vzz|Vzx|Vzy组合的反演结果优于其它的单分量数据及两分量数据反演结果。随着反演中梯度分量的增加，是否分量越多，反演结果越精确成为关注的焦点问题之一，为说明此问题，文中做了五组3个、4个及5个不同分量组合的反演计算，结果表明，这五组反演计算结果相近，由此可见，当参与反演的数据可同时获得较好水平及垂向分辨率时，再继续增加梯度分量数据并不能进一步改善反演结果，只能增加计算量及计算时间，带来时间和空间的耗损。
在空间域3D反演快速发展的同时，频率域重力数据3D反演的研究工作也在不断的推进，其在重磁数据解释方法中占有重要地位，就这一方面本文的研究工作包括以下几点。
1.详细论述了Oldenburg-Parker法计算重磁数据的方法原理，并利用此方法计算余弦球冠模型引起的重力值，为后文的频率域反演提供试算数据。
2.深入研究了重力数据的频率域反演方法，此频率域反演方法的主要思想如下：当等效密度层厚度与待反演的构成异常体的棱柱体水平尺寸满足一定的几何关系时，视密度与异常源深度顶部埋深之间满足拟线性关系，此时可在频率域通过线性反演计算异常源的顶部埋深，然后可计算出均匀异常体的密度值，此反演方法计算快捷，计算量小，可广泛应用于位场数据处理及解释中。
3.本文利用加入了3%高斯噪声的余弦球冠模型数据，反演计算异常源顶部埋深。根据计算的反演数据与理论数据的拟合误差及预测深度与理论模型深度的拟合误差可见，此方法可有效的反演余弦球冠模型的顶部埋深。文中同时给出了不满足条件的反演参数进行反演计算，结果表明，当参数设定范围在视密度及深度不满足线性关系时，反演结果严重失真，观测与预测数据拟合误差及反演与理论模型拟合误差都显著增加。
With the increasing of the types about gravity and gradient data and thedevelopment of hardware facilities and the continuous progress of the mathematicalmethods, the observation areas of the gravity and gradient data are widespread dayby day, and interpretation techniques and methods continue to progress and theapplications of the data continue to expand. The3D inversion of gravity and gradientmulti-component data is urgent increasingly. Researchers put more attention to theimprovement of the inversion method, so they improve the calculated accuracy ofthe inversion, the reduction of the amount of calculation time and the richness of thedata which can be used in the inversion of the gravity and the gradientmulti-component data.
3D inversion method is divided into two parts that are spatial and frequencydomain inversion methods. The spatial domain inversion method consists of twocategories. The first category is constrained inversion which contains rich priorgeological and geophysical information in the inversion progress. Therepresentative methods in this category are the seed inversion method, interactiveinversion method et al. The second category is to minimize the use of the constraintsabout source in the inversion. The constraint conditions which are used mostcommonly in this category are depth weighted constraints, smoothness constraintand upper and lower bounder constraints. This paper focuses on the second inversionmethod. In addition to the spatial domain inversion methods which are studied bymajority of scholars, the frequency-domain3D inversion methodswhich have thefeatures of calculating speed fast, the hardware requirements lower still occupies animportant position in a particular field of study. We also do some work on thissection.
The calculation accuracy, the computation time, the amount of calculation, thedata types and the constraint conditions of3D inversion method in the spatialdomain are hot topics in recent years. This paper mainly focuses on the followingparts:
1. The prism forward methods to calculate the gravity data studied by Sorokin,Haaz, Nagy, Okabe and Steiner are discussed in-depth. The data caused by the sameprism model are calculated with the forth forward methods mentioned above and theerror which is less than10-7between them is displayed, so the four methods ofcalculation error had no significant effect on the inversion results. The error can beignored in the progress of the inversion. The level of fitting error can meet no matterwhich forward inversion be chosen. We use the Nagy forward method to calculategravity data caused by the prism.
2. The details about the calculation of the gravity gradient component arediscussed in detail. The geometric symmetry relationships between the differentcomponents of the gradient data are discussed as well. The forward prism model isestablished, and the gradient nine-component data caused by this model is calculated.The results show that the gradient of nine components has a good symmetry. Theresults also verify that the gradient field has null-trace.
3. The inducement that makes the inverse problem ill-conditioned is studied indetail. The existence, uniqueness and stability of the inverse problems are the criticalconditions to make the problem well-conditioned. The classic and simplest way toresolve the linear ill-posed inverse problem of gravity data and gravity and gradientdata is the Tikhonov regularization algorithm. It can solve the singularity in thekernel function by adding the parameter to the eigenvalue of kernel function matrax.To reduce the error in the reverse results caused by the introducing parameters,extrapolation Tikhonov regularization algorithm is introduced by the researcherswhich is used to the inverse problem of gravity data and gradient data in this paper.
4.3D constrained inversion of gravity data based on the extrapolation Tikhonovregularization algorithm is researched further. Comparing the results finverted by theTikhonov regularization algorithm and the extrapolation Tikhonov regularizationalgorithm, it is obvious that the fitting error of the latter one is smaller than the oneof the former one.Whether we can obtain a better result depends on the selection ofthe regularization parameters. The parameter n and are chosen based onbalance principle, monotonous error principle and discrete principle.
5. The kernel function decay rapidly when the depth increases in the3D inversionof gravity and gradient data. The inversion results will concentrate to the surface ifwe inverse directly. The previous reseachers have proposed a depth weightingfunction which can counteract this decay process, and the kernel function will havethe same effect to the block of when the depth increases. The depth weightingfunction proposed by the previous researcher was studied first, and then we makesome improvement in the paper. The initial depth weighting function, the improveddepth weighting function are used in the3D inversion of gravity data caused by theprism with different depth. The results without the depth constraint condition displaythe concentrated surface. The initial depth weighting function get bad resolution inthe bounder of the bottom of the prisms, while the improved depth weightingfunction can make the bounder of the bottom clearer than the initial one.
6. The introduction of the upper and lower bounder constraints can make theinversion density in the meaningful physical range which is from the prioriinformation about the study area. The author analyses the chosen of the parametersin the upper and lower bounder function detailed. This constraint is applied in theinversion of the gravity data and the gradient data.
7. The3D inversion of gravity data based on the Extrapolation Tikhonovregularization algorithm and the Tikhonov regularization algorithm are applied intwo groups of synthetic models and compared the calculation accuracy and thecomputation time of the inverse. The results indicated that these two methods caninverse source accurately and the former one can achieve better fitting error levelunder the same error requirements, but consumes more computation time.
8. The kernel function can be composed with different single component or thecombination of the different single component, so the kernel contains differentinformation when the component different. The calculation method and Base ofjudgment are discussed detailed. The singular value decomposition is used to get theeigenvalue of the kernel function with the depth weighting constraints and otherconstraints. The largest eigenvalue is applied to normalize the other eigenvalue toattain the eigenvalue spectra which can be used to study the information content inthe kernel matrix. The study results of previous indicate that the larger eigenvaluespectra the more information is concluded in the kernel matrix. The eigenvaluespectra of the combination of the components are larger than the one of the singlecomponent after decomposition of the kernel matrix. The vertical and the horizontal details of the resolutions can be improved when the single components combined.
9. The3D inverse method of multi-component data based on extrapolationTikhonov regularization inversion in the spatial domain is used in the data composedwith different component to inverse the density model. The inverse results indicatedthat the Vxy component and Vzx component can get better resolution in horizontaland vertical direction than other component. The inverse results about Vxx is theworst one when inverse the single component which is consistent with the results ofnumerical analysis phase. The combination of a small number of the componentsdata which is regarded as kernel matrix has larger eigenvalue spectra than the singleone, so it can provide more information about the source. The calculated resultsindicate that the combination of Vzz|Vzx|Vzy contain more information othersingle-component data because it has the larger eigenvalue than the others. With theincrease of the component which can be combined whether the information willincrease at the same time is the focus to the researchers. To illustrate this problem,five sets of three, four and five different component combinations inversion aretested. the results show that these groups are similar to each other. Thus, when thedata inversion can get a better horizontal and vertical resolution, and then continue toincrease the gradient components does not further improve the anti-speech results,but only increase the amount of computation and calculation time and bring thedepletion of time and space.
The3D inversion method of gravity data in frequency domain are progressingconstantly when the method in spatial domain develop fast. This category plas animportant role in the interpretation of gravity and magnetic data, The research workof in this regard include the following point in this paper.
1. The theory of Oldenburg-Parker forward method about gravity and magneticdata is discussed in detail. The gravity caused by cosine spherical cap modelcalculated by this method which can be used in the inversion in the frequencydomain.
2. The3D inversion of gravity data in frequency domain is researched in depth.The main idea of the inversion method in frequency domain is as follows. When theequivalent density layer thickness to be inversion has the certain numerical relationsto the horizontal size of the prism, it meets the quasi-linear relationship between theapparent densities with the top of the source depth. We can inverse the depth of thesource by linear relationship in frequency domain, and then calculate the uniform density contrast value. This inversion method calculated fast, and has small amountof calculation, so it can be widely used in the potential field data processing andinterpretation.
3.3%Gaussian noise is added to the gravity data caused by cosine spherical capmodel. The depth of the source top is calculated by the method mentioned above.The fitting error between the theoretical data and the predicted data and the fittingerror between the theoretical model and the inverse depth indicate that this methodcan inverse the depth of the source top of cosine spherical cap model. The parameterthat does not meet the conditions is set to inversion test. The results show that whenthe parameter range does not satisfy the linear relationship between the apparentdensity and the depth of the source top, the fitting errors on both data and models areincreased significantly.
3D反演方法分为空间域和频率域反演方法。空间域反演方法主要由两类构成，第一类是利用丰富的地质先验信息约束解，在这一分类中具有代表性的方法有种子方法，交互式反演方法等；第二类是尽量少的使用复杂地质信息的约束最小化反演方法，在这类方法中最常用的约束条件包括深度加权约束，平滑约束及上下限约束等。本文重点研究第二类反演方法。除多数学者致力研究的空间域反演方法外，以计算速度快，硬件要求较低为特点的频率域反演在特定的研究领域依然占有重要的地位，本文就这一方面也做了部分研究。
空间域反演方法的计算精度、计算时间、计算量、反演数据类型及反演约束条件等问题是近年来人们研究的焦点，本文就这些问题所做的研究工作如下。
1.深入讨论了Sorokin、Haaz、Nagy及Okabe和Steiner的棱柱体重力模型正演方法，并根据这四个正演方法计算同一棱柱体模型引起的重力异常。计算结果表明，虽然四个计算公式的最后一项不同，但是它们计算的重力异常之间的误差小于10-10，所以，四种方法的计算误差对反演结果没有明显的影响，反演中可忽略这些计算误差，选用其中任意一种正演方法进行反演计算都可满足误差水平，本文选用Nagy的正演方法计算棱柱体正演模型引起的重力值。
2.详细推导计算重力梯度九分量的计算公式，讨论了梯度数据的性质及不同分量之间的对称关系，并建立正演模型，计算此模型的梯度九分量数据，计算结果表明，梯度九分量存在很好的对称性，同时，计算结果也验证了梯度场是无迹的。
3.本文详细分析了反问题病态解的诱因，即解的存在性、唯一性和稳定性三者不能同时满足时，反演解是病态的，并给出了解决病态线性反演的方法。针对线性病态反问题，经典和简单的方法是利用Tikhonov正则化算法解决重力数据反问题中核函数的奇异性问题。为进一步减小因正则化参数的引入带到解中的误差，后人发展的Extrapolation Tikhonov正则化算法成为本文研究反问题的主要工具。
4.深入研究基于Extrapolation Tikhonov正则化算法的重力数据约束反演方法。通过比较基于Tikhonov正则化算法反演结果及基于ExtrapolationTikhonov正则化算法反演结果可知，后者的观测数据与预测数据拟合误差更小。由于正则化算法中参数选择直接关系到反演结果的好坏，文中选用平衡原则、单调误差原则及离散原则三个选择原则综合选择正则化参数n和。
5.在重力及其梯度数据反演中，核函数呈现随深度增加而快速衰减的特性，如果直接进行反演计算，则反演结果存在严重的趋肤效应。为此，前人提出一种可消除这种衰减的方法，即利用与核函数随深度变化具有相反特性的深度加权函数作用与核函数，使核函数沿垂直方向对所有密度块体的作用相等，从而消除重力数据反演中的趋肤效应。本文在深入研究前人深度加权约束函数的基础上，改进了深度加权函数。将前人加权函数、作者改进的加权函数及不加权的核函数应用到不同埋深的单个立方体模型的密度分布反演中。结果表明，不加权核函数反演的密度分布存在严重的趋肤效应，前人加权函数对于立方体的底部分辨率比改进后的加权函数要差，改进后的加权函数对于埋深不同的异常体都有较好的反演效果。
6.密度上下约束函数的引进，对超出先验信息范围的密度差起到约束作用，将反演结果转换到满足先验信息的数值范围内，文中详细分析了上下限约束函数中各参数的选择机制，并将其应用到重力及梯度多分量数据的反演计算中。
7.本文将基于Extrapolation Tikhonov正则化算法及Tikhonov正则化算法的反演方法应用于两组组合模型，就它们的计算精度及计算时间进行对比，结果表明，两个方法都可较准确的反演出异常体密度分布特征。前者能在相同的误差要求下达到更好的拟合误差水平，但是需消耗更多的计算时间。
8.详细讨论了由不同梯度分量及分量组合构成核函数时包含的信息量的计算方法及判断依据。利用奇异值分解方法分解带有深度约束等约束条件的核函数特征值，并将核函数的特征值利用最大特征值规则化，利用规则化之后的特征值谱研究不同的核函数含有的信息量。前人研究结果表明，核函数的特征值越大含有的信息量越多。将不同分量及不同分量组合成的核函数分解后发现单个分量的特征值谱小于分量组合时的特征值谱，因此梯度分量组合反演密度模型时，可获得更多垂向和水平方向的细节。
9.利用基于Extrapolation Tikhonov正则化算法的3D反演方法反演重力梯度多分量数据不同组合方式时的密度模型。结果表明单独反演不同的梯度分量时，Vxy及Vzx在水平及垂向的分辨率较理想，Vxx是各梯度分量数据反演结果中效果最差的，这与数值分析结果相一致。少数梯度分量数据组合反演结果与单个梯度分量作为核函数的反演结果相比，少数梯度分量组合的核函数有更大的特征值谱，因此，可提供更多关于异常源的信息。计算结果表明Vzz|Vzx|Vzy组合的反演结果优于其它的单分量数据及两分量数据反演结果。随着反演中梯度分量的增加，是否分量越多，反演结果越精确成为关注的焦点问题之一，为说明此问题，文中做了五组3个、4个及5个不同分量组合的反演计算，结果表明，这五组反演计算结果相近，由此可见，当参与反演的数据可同时获得较好水平及垂向分辨率时，再继续增加梯度分量数据并不能进一步改善反演结果，只能增加计算量及计算时间，带来时间和空间的耗损。
在空间域3D反演快速发展的同时，频率域重力数据3D反演的研究工作也在不断的推进，其在重磁数据解释方法中占有重要地位，就这一方面本文的研究工作包括以下几点。
1.详细论述了Oldenburg-Parker法计算重磁数据的方法原理，并利用此方法计算余弦球冠模型引起的重力值，为后文的频率域反演提供试算数据。
2.深入研究了重力数据的频率域反演方法，此频率域反演方法的主要思想如下：当等效密度层厚度与待反演的构成异常体的棱柱体水平尺寸满足一定的几何关系时，视密度与异常源深度顶部埋深之间满足拟线性关系，此时可在频率域通过线性反演计算异常源的顶部埋深，然后可计算出均匀异常体的密度值，此反演方法计算快捷，计算量小，可广泛应用于位场数据处理及解释中。
3.本文利用加入了3%高斯噪声的余弦球冠模型数据，反演计算异常源顶部埋深。根据计算的反演数据与理论数据的拟合误差及预测深度与理论模型深度的拟合误差可见，此方法可有效的反演余弦球冠模型的顶部埋深。文中同时给出了不满足条件的反演参数进行反演计算，结果表明，当参数设定范围在视密度及深度不满足线性关系时，反演结果严重失真，观测与预测数据拟合误差及反演与理论模型拟合误差都显著增加。
With the increasing of the types about gravity and gradient data and thedevelopment of hardware facilities and the continuous progress of the mathematicalmethods, the observation areas of the gravity and gradient data are widespread dayby day, and interpretation techniques and methods continue to progress and theapplications of the data continue to expand. The3D inversion of gravity and gradientmulti-component data is urgent increasingly. Researchers put more attention to theimprovement of the inversion method, so they improve the calculated accuracy ofthe inversion, the reduction of the amount of calculation time and the richness of thedata which can be used in the inversion of the gravity and the gradientmulti-component data.
3D inversion method is divided into two parts that are spatial and frequencydomain inversion methods. The spatial domain inversion method consists of twocategories. The first category is constrained inversion which contains rich priorgeological and geophysical information in the inversion progress. Therepresentative methods in this category are the seed inversion method, interactiveinversion method et al. The second category is to minimize the use of the constraintsabout source in the inversion. The constraint conditions which are used mostcommonly in this category are depth weighted constraints, smoothness constraintand upper and lower bounder constraints. This paper focuses on the second inversionmethod. In addition to the spatial domain inversion methods which are studied bymajority of scholars, the frequency-domain3D inversion methodswhich have thefeatures of calculating speed fast, the hardware requirements lower still occupies animportant position in a particular field of study. We also do some work on thissection.
The calculation accuracy, the computation time, the amount of calculation, thedata types and the constraint conditions of3D inversion method in the spatialdomain are hot topics in recent years. This paper mainly focuses on the followingparts:
1. The prism forward methods to calculate the gravity data studied by Sorokin,Haaz, Nagy, Okabe and Steiner are discussed in-depth. The data caused by the sameprism model are calculated with the forth forward methods mentioned above and theerror which is less than10-7between them is displayed, so the four methods ofcalculation error had no significant effect on the inversion results. The error can beignored in the progress of the inversion. The level of fitting error can meet no matterwhich forward inversion be chosen. We use the Nagy forward method to calculategravity data caused by the prism.
2. The details about the calculation of the gravity gradient component arediscussed in detail. The geometric symmetry relationships between the differentcomponents of the gradient data are discussed as well. The forward prism model isestablished, and the gradient nine-component data caused by this model is calculated.The results show that the gradient of nine components has a good symmetry. Theresults also verify that the gradient field has null-trace.
3. The inducement that makes the inverse problem ill-conditioned is studied indetail. The existence, uniqueness and stability of the inverse problems are the criticalconditions to make the problem well-conditioned. The classic and simplest way toresolve the linear ill-posed inverse problem of gravity data and gravity and gradientdata is the Tikhonov regularization algorithm. It can solve the singularity in thekernel function by adding the parameter to the eigenvalue of kernel function matrax.To reduce the error in the reverse results caused by the introducing parameters,extrapolation Tikhonov regularization algorithm is introduced by the researcherswhich is used to the inverse problem of gravity data and gradient data in this paper.
4.3D constrained inversion of gravity data based on the extrapolation Tikhonovregularization algorithm is researched further. Comparing the results finverted by theTikhonov regularization algorithm and the extrapolation Tikhonov regularizationalgorithm, it is obvious that the fitting error of the latter one is smaller than the oneof the former one.Whether we can obtain a better result depends on the selection ofthe regularization parameters. The parameter n and are chosen based onbalance principle, monotonous error principle and discrete principle.
5. The kernel function decay rapidly when the depth increases in the3D inversionof gravity and gradient data. The inversion results will concentrate to the surface ifwe inverse directly. The previous reseachers have proposed a depth weightingfunction which can counteract this decay process, and the kernel function will havethe same effect to the block of when the depth increases. The depth weightingfunction proposed by the previous researcher was studied first, and then we makesome improvement in the paper. The initial depth weighting function, the improveddepth weighting function are used in the3D inversion of gravity data caused by theprism with different depth. The results without the depth constraint condition displaythe concentrated surface. The initial depth weighting function get bad resolution inthe bounder of the bottom of the prisms, while the improved depth weightingfunction can make the bounder of the bottom clearer than the initial one.
6. The introduction of the upper and lower bounder constraints can make theinversion density in the meaningful physical range which is from the prioriinformation about the study area. The author analyses the chosen of the parametersin the upper and lower bounder function detailed. This constraint is applied in theinversion of the gravity data and the gradient data.
7. The3D inversion of gravity data based on the Extrapolation Tikhonovregularization algorithm and the Tikhonov regularization algorithm are applied intwo groups of synthetic models and compared the calculation accuracy and thecomputation time of the inverse. The results indicated that these two methods caninverse source accurately and the former one can achieve better fitting error levelunder the same error requirements, but consumes more computation time.
8. The kernel function can be composed with different single component or thecombination of the different single component, so the kernel contains differentinformation when the component different. The calculation method and Base ofjudgment are discussed detailed. The singular value decomposition is used to get theeigenvalue of the kernel function with the depth weighting constraints and otherconstraints. The largest eigenvalue is applied to normalize the other eigenvalue toattain the eigenvalue spectra which can be used to study the information content inthe kernel matrix. The study results of previous indicate that the larger eigenvaluespectra the more information is concluded in the kernel matrix. The eigenvaluespectra of the combination of the components are larger than the one of the singlecomponent after decomposition of the kernel matrix. The vertical and the horizontal details of the resolutions can be improved when the single components combined.
9. The3D inverse method of multi-component data based on extrapolationTikhonov regularization inversion in the spatial domain is used in the data composedwith different component to inverse the density model. The inverse results indicatedthat the Vxy component and Vzx component can get better resolution in horizontaland vertical direction than other component. The inverse results about Vxx is theworst one when inverse the single component which is consistent with the results ofnumerical analysis phase. The combination of a small number of the componentsdata which is regarded as kernel matrix has larger eigenvalue spectra than the singleone, so it can provide more information about the source. The calculated resultsindicate that the combination of Vzz|Vzx|Vzy contain more information othersingle-component data because it has the larger eigenvalue than the others. With theincrease of the component which can be combined whether the information willincrease at the same time is the focus to the researchers. To illustrate this problem,five sets of three, four and five different component combinations inversion aretested. the results show that these groups are similar to each other. Thus, when thedata inversion can get a better horizontal and vertical resolution, and then continue toincrease the gradient components does not further improve the anti-speech results,but only increase the amount of computation and calculation time and bring thedepletion of time and space.
The3D inversion method of gravity data in frequency domain are progressingconstantly when the method in spatial domain develop fast. This category plas animportant role in the interpretation of gravity and magnetic data, The research workof in this regard include the following point in this paper.
1. The theory of Oldenburg-Parker forward method about gravity and magneticdata is discussed in detail. The gravity caused by cosine spherical cap modelcalculated by this method which can be used in the inversion in the frequencydomain.
2. The3D inversion of gravity data in frequency domain is researched in depth.The main idea of the inversion method in frequency domain is as follows. When theequivalent density layer thickness to be inversion has the certain numerical relationsto the horizontal size of the prism, it meets the quasi-linear relationship between theapparent densities with the top of the source depth. We can inverse the depth of thesource by linear relationship in frequency domain, and then calculate the uniform density contrast value. This inversion method calculated fast, and has small amountof calculation, so it can be widely used in the potential field data processing andinterpretation.
3.3%Gaussian noise is added to the gravity data caused by cosine spherical capmodel. The depth of the source top is calculated by the method mentioned above.The fitting error between the theoretical data and the predicted data and the fittingerror between the theoretical model and the inverse depth indicate that this methodcan inverse the depth of the source top of cosine spherical cap model. The parameterthat does not meet the conditions is set to inversion test. The results show that whenthe parameter range does not satisfy the linear relationship between the apparentdensity and the depth of the source top, the fitting errors on both data and models areincreased significantly.
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