Research On Applications Of Wavelet Multiscale Analysis In The Earth's Gravity Field

The earth's gravity field, one of the important physical features of the earth, is the overall reaction of the mass distribution inside the earth and the rotation of the earth. The knowledge of the earth's gravity field is the information indispensable to geosciences, especially to geodesy, solid geophysics and oceanography. So comprehension and investigation of the earth's gravity field is one of the most active areas in modern geodesy and other relative sciences.The rapid progress of gravimetry technique as well as spatial technique and satellite positioning technique have radically changed the situation that the theory of geophisical geodesy is behind its technique. Multiform gravimetry techniques with high precision provide a plentiful supply of information about the earth's gravity field, which makes it possible to determine the refined structure of the earth's gravity field with unprecedented precision and resolution. On the other side, the breakthrough of the technique makes the earth's gravity field plays a more important role in geosciences. Facing the new situation, breakthroughs in theories and methods for the determination of the earth's gravity field are expected urgently.Wavelet analysis is a new branch of mathematics which has been developed rapidly in recent decades. It has extensive applications in many areas of science because of its ability in localization and multiscale analysis. There are also lots of applications of wavelet analysis in the research of the earth's gravity field. In this thesis, we will apply the theory and methods of wavelet multiscale analysis to study the earth's gravity field, and expect to get some beneficial results for the processing and interpretation of gravity field data. There are some advantages to study the earth's gravity field using multiscale analysis. Firstly, the earth's gravity field is a multiscale field, which behaves different characteristics on different space or time scales. So multiscale analysis is helpful to understand the characteristics of the earth's gravity field. Secondly, spatio-temporal multiscale is an impersonal property of processes and phenomena in geosciences. To study these processes and phenomena needs information of the gravity field on various scales. Therefore, multiscale analysis of the earth's gravity field is beneficial to recognize geodynamic phenomena with different scales. Finally, besides surface gravity data, there are also airborne gravity data,satellite-to-satellite tracking data, altimeter data and satellite gravity gradient data available to determine the earth's gravity field. These data have different resolving power to the earth's gravity field, because they are observed at different altitudes using various sensors. In order to approximate the earth's gravity field optimally, all these observations should be combined together. Multiscale analysis has a good property for multiscale data fusion. In a word, multiscale analysis can be used to improve some traditional theories and methods of the earth's gravity field.Taking "multiscale" as the theme, applications of wavelet multiscale analysis in downward continuation of gravity signal, gravity anomaly inversion, multiresolution gravity data fusion, and so on, are studied in this dissertation. The detailed contents and main results are as following:1. The general concept of scale and its notions in geosciences are introduced. The importance of multiscale analysis to understanding is discussed. And the multiscale characteristic of the earth's gravity field is analyzed. It is pointed out that applying multiscale analysis to investigate the earth's gravity field has a great importance and practical value. Whereafter applications of wavelet analysis in geodesy and its progress are reviewed, especially focused on approximation of gravity field, filtering and estimation, improvement of numerical algorithms, and geophysical interpretation. Plenty of successful applications show that wavelet analysis has been a powerful tool for theoretical research and data processing in geodesy.2. As the research basis for this thesis, wavelet analysis theory is introduced, including concept of wavelet, continuous wavelet transform and its properties, dyadic wavelet transform, 2D orthonormal wavelet transform, multiscale analysis and its fast algorithm, etc. By contrasting wavelet transform with Fourier transform in detail, the advantage of wavelet analysis in multiscale analysis and time-frequency localization analysis is illustrated. More than that, the basic principles and methods for singularity detection, signal filtering and denoising by wavelet are also discussed in this thesis.3. The theory of wavelet multiscale edge analysis is introduced into the earth's gravity field, and applied to downward continuation of gravity signals. The downward continuation is a seriously ill-posed problem, which is always a focus and difficulty in the research of the earth's gravity field. Summary remarks on present methods of downward continuation are made, and a new downward continuation method based on multiscale edges constraint is proposed. Using the potential wavelet, the physicalsignification and the characteristic of the wavelet transform and the multiscale edge of gravity signals are analyzed, and a relation between the scale parameter and the altitude is founded. It is proved that, the wavelet transform on scale .s of the gravity signal fz at altitude z is the horizontal gradient of upward continued signal/z+bo multiplying by a factor s, and the multiscale edges are composed of pointswhere the gravity signals at different heights change sharply. Multiscale edges of gravity signals at different heights are relative to the distribution of anomalous mass inside the earth, so they should have the same shape except for a translation of the scale. Based on this fact, multiscale edges can be used as a constraint condition. Simulated examples show that the multiscale edges constraint method can efficiently restrain noise and improve the downward continued results, and has practical value for the downward continuation of airborne gravimetry data.4. Applications of multiscale analysis to the gravity anomaly inversion are explored. Firstly, a simple introduction to the concept of the gravity anomaly inversion and some of existing inversion methods is given. And some of applications of wavelet analysis to the potential field data processing are reviewed. Secondly, the method for the separation of gravity anomalies based on wavelet transform is modified using multiscale edge analysis. In contrast to the traditional approaches, the modified method can be used to perform simultaneously radial multiple decomposition and transverse separation of gravity anomalies to isolate an individual anomaly. Finally, to some simple geological units, the forward models of multiscale edges are deduced, and their properties are summarized. Sequently, an inversion method based on multiscale edges is put forward, which models are very simple and have a good antinoise performance. The outstanding ability of this new method is that geometry parameters and density can be inverted at the same time. The inversion method based on multiscale edges provides a new way to the gravity anomaly inversion, which has a fruitful prospect for geophysical inversion.5. At present, Least-Squares Collocation (LSC) is the unique method which can combine heterogenous types of gravity field information to determine the earth's gravity field. There are different viewpoints on collocation depicted by different mathematical concepts, such as stochastic collocation, deterministic collocation and statistical collocation. A descriptive review about the advantages and drawbacks of these LSC methods is given in the thesis. Based on the Fourier transform form ofminimum mean square error principle;the connection between the collocation results and data resolution is discussed. The collocation formulas depending on data resolution are deduced;and the estimation formula for the aliasing error due to the finite sampling resolution is also given. Following on;the theory of multiresolution least square collocation(MLSC) is expatiated and its formulas are derived combining the stepwise collocation with wavelet multiscale analysis. Results of MLSC are compared with those of traditional collocation. MLSC provides a new possible technique to fuse gravity data with different precision and resolution;however;more work should be done to verify whether the MLSC method is superior to the traditional collocation.6. When processing a huge amount of data;collocation needs to solve large systems of linear equations. In the stationary case collocation equations can be solved in the frequency domain using Fourier transform. But Fourier transform can not be used in the non-stationary case because of its disability to deal with the non-stationary signals. Wavelet transform can be used to solve the non-stationary collocation equations;however;it can not diagonalize the system matrix like Fourier transform. Based on the wavelet method;Wavelet-Vaguelette method for the solution of non-stationary collocation is presented. The new method integrates the good performances of Fourier transform and wavelet transform;and may improve the computational efficiency significantly for some non-stationary collocation problems.