The Inversion of Poisson’s Integral in the Wavelet Domain (original) (raw)
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Application of wavelet theory to the analysis of gravity data
Various Green's functions occurring in Poisson potential field theory can be used to construct non-orthogonal, non-compact, continuous wavelets. Such a construction leads to relations between the horizontal derivatives of geophysical field measurements at all heights, and the wavelet transform of the zero height field. The resulting theory lends itself to a number of different applications in the processing of potential field data. Some simple, synthetic examples in 2-D illustrate one inversion approach based upon the maxima of the wavelet transform (multiscale edges). These examples are presented to illustrate, by way of explicit demonstration, the information content of the multiscale edges. We do not suggest the methods used in these examples be taken literally as a practical algorithm or inversion technique. Rather, we feel that the real thrust of the method is toward physically based, spatially local filtering of geophysical data images using Green's function wavelets, or compact approximations thereto. To illustrate our first steps in this direction, we present some preliminary results of a 3-D analysis of an aeromagnetic survey.
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2000
Wavelets can be used in the decomposition and analysis of airborne gravity data. In this paper, multiresolution analysis is applied to de-noise gravity disturbance and different de-noising techniques are studied. The first objective is testing the usefulness of wavelets for analyzing and filtering airborne gravity data. The second one is a comparison between the usage of the wavelet transform and
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Regularization methods are used to recover a unique and stable solution in ill-posed geophysical inverse problems. Due to the connection of homogeneous operators that arise in many geophysical inverse problems to the Fourier basis, for these operators classical regularization methods possess some limitations that one may try to circumvent by wavelet techniques.
Interpretation of gravity data using 2-D continuous wavelet transformation and 3-D inverse modeling
Journal of Applied Geophysics, 2015
Recently the continuous wavelet transform has been proposed for interpretation of potential field anomalies. In this paper, we introduced a 2D wavelet based method that uses a new mother wavelet for determination of the location and the depth to the top and base of gravity anomaly. The new wavelet is the first horizontal derivatives of gravity anomaly of a buried cube with unit dimensions. The effectiveness of the proposed method is compared with Li and Oldenburg inversion algorithm and is demonstrated with synthetics and real gravity data. The real gravity data is taken over the Mobrun massive sulfide ore body in Noranda, Quebec, Canada. The obtained results of the 2D wavelet based algorithm and Li and Oldenburg inversion on the Mobrun ore body had desired similarities to the drill-holes depth information. In the all of inversion algorithms the model non-uniqueness is the challenging problem. Proposed method is based on a simple theory and there is no the model non-uniqueness on it.
Wavelet Evaluation of Inverse Geodetic Problems
International Association of Geodesy Symposia, 2008
A computational scheme using the wavelet transform is employed for the numerical evaluation of the integrals involved in geodetic inverse problems. The integrals are approximated in finite multiresolution analysis subspaces. The wavelet algorithm is built on using an orthogonal wavelet base function. A set of equations is formed and solved using preconditioned conjugate gradient method. The full solution with all equations requires large computer memory, therefore, multiresolution properties of the wavelet transform are used to divide the full solution into parts. Each part represents a level of wavelet detail coefficients or the approximation coefficients. Wavelet hard thresholding technique is used for the compression of the kernel. Numerical examples are given to illustrate the use of this procedure for the numerical evaluation of inverse Stokes and Vening Meinesz integrals. Conclusions and recommendations are given with respect to the suitability, accuracy and efficiency of this method.
Iidentification of Suitable Discrete Wavelet for Gravity Data Decomposition
— So far, various edge detection methods have been proposed for potential field interpretation. Recognition of the anomaly source boundary can accelerate and facilitate the gravity field analysis. Wavelet transform (WT) is one of these suggested approaches. Several discrete and continuous mother wavelets have been defined. In this study, has been used of 2D discrete wavelet transform (DWT) as a method for determination of gravity anomaly source boundary. The DWT leads to a decomposition of the approximation coefficients in four distinct components: the approximation, horizontal, vertical and diagonal. For comparing the efficiency of wavelets, the synthetic gravity anomalies, with and without added random noise, have been decomposed at 1 level with six discrete, two-dimensional wavelets: Haar, Biorthogonal, Coiflets, Symlets, Discrete Meyer and Daubechies. In this study, for anomaly edge enhancement has been proposed a new formula namely HVC that is computed from the square root of the sum of the squares of the horizontal and vertical components. The results indicate the acceptable performance of the Haar and Biorthogonal wavelets in delineating the edges of the gravity anomaly sources.
2006
Continuous gravity recordings in volcanic area could play a fundamental role in the monitoring of active volcanoes and in the prediction of eruptive events too. This geophysical methodology is utilized, on active volcanoes, in order to detect mass changes linked to magma transfer processes and, thus, to recognize forerunners to paroxysmal volcanic events. Spring gravimeters are still the most utilized instruments for microgravity studies because of their relatively low cost and small size, which make them easy to transport and install. Continuous gravity measurements are now increasingly performed at sites very close to active craters, where there is the greatest opportunity to detect significant gravity changes due to a volcanic activity. Unfortunately, spring gravity meters show a strong influence of meteorological parameters (i.e. pressure, temperature and humidity), especially in the adverse environmental conditions usually encountered at such places. As the gravity changes due ...
Wavelet decomposition in the Earth's gravity field investigation
Acta Geodynamica et Geomaterialia, 2013
This paper presents the results of the application of wavelet decomposition to processing data from the GGP sites (The Global Geodynamics Project). The GGP is an international project within which the Earth's gravity field changes are recorded with high accuracy at a number of stations worldwide using superconducting gravimeters. Data with a 5-second sampling interval from Wettzell and Bad Homburg were used for the research. The wavelet transform enables the investigation of the temporal changes of the oscillation amplitudes or the decomposition of the time series for the analysis of the required frequencies. The wavelet decomposition was performed using the regular orthogonal symmetric Meyer wavelet. The research concerned data from an earthquake period recorded at various locations and a quiet period when the gravimeters worked without any disturbances. The decomposition was followed by the Fast Fourier Transform for signal frequency components and then by correlation analyses of corresponding frequency components (for periods from 10 to 60 000 seconds) for all sensor combinations, for the quiet and the earthquake periods separately. Frequency components defining long term changes for all sensor combinations, as well as combinations between two sensors at the same site for the quiet days are characterised by high correlation coefficients. For the time of the earthquake, the Wettzell site data proved strong correlation for all frequency components, while the Bad Homburg site data showed an unexpected decrease of correlation for the majority of frequency components. The authors also showed that wavelet decomposition can be a good method of data interpolation, especially from the time of earthquakes. Moreover, it is a very useful tool for filtering the data and removing the noises.
Geophysical Journal International, 2013
We solve the 3-D gravity inverse problem using a massively parallel voxel (or finite element) implementation on a hybrid multi-CPU/multi-GPU (graphics processing units/GPUs) cluster. This allows us to obtain information on density distributions in heterogeneous media with an efficient computational time. In a new software package called TOMOFAST3D, the inversion is solved with an iterative least-square or a gradient technique, which minimizes a hybrid L 1-/L 2-norm-based misfit function. It is drastically accelerated using either Haar or fourthorder Daubechies wavelet compression operators, which are applied to the sensitivity matrix kernels involved in the misfit minimization. The compression process behaves like a preconditioning of the huge linear system to be solved and a reduction of two or three orders of magnitude of the computational time can be obtained for a given number of CPU processor cores. The memory storage required is also significantly reduced by a similar factor. Finally, we show how this CPU parallel inversion code can be accelerated further by a factor between 3.5 and 10 using GPU computing. Performance levels are given for an application to Ghana, and physical information obtained after 3-D inversion using a sensitivity matrix with around 5.37 trillion elements is discussed. Using compression the whole inversion process can last from a few minutes to less than an hour for a given number of processor cores instead of tens of hours for a similar number of processor cores when compression is not used.
Gravity Gradient Data Filtering Using Translation Invariant Wavelet
ASEG Extended Abstracts
Full tensor gradient (FTG) data is highly useful in hydrocarbon exploration and the detection of some geological targets with small size as its higher detailed information abundance and finer resolution. At the same time, there are some high-frequency Gaussian white noise mixed in the target signal and which has closer frequency range than the conventional gravity data. Thus, one key step before inversion is to remove as much Gaussian white noise as possible and reserve the subtle details. For this pre-processing step, several effective methods have been used, including low-pass filters, least square fitting methods based on Laplace equation and wavelet filtering methods. In this paper, we would utilize the translation invariant wavelet for the reason that it can suppress Gaussian white noise through multi-resolution analysis and at the same time can avoid pseudo-Gibbs phenomenon. The other point different from wavelet method used before is that we applied a mixed threshold constructed according to the curve of both soft threshold and hard threshold. Compared to soft and hard threshold, mixed threshold can keep more details and remove more noise respectively in terms of the energy distribution of signal and noise. Then we process wavelet coefficients with mixed threshold and do inverse transform to recover the data. The results demonstrate that translation invariant wavelet can not only remove the major part of Gaussian white noise, but also reserves high-frequency detailed information of FTG data. Obviously, translation invariant wavelet with mixed thresholding has preferable application effect in filtering FTG data.