3D Inversion Of Gravity Data And Gravity Gradiometry Data Based On The Extrapolation Tikhonov Regularization Algorithm

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.