Research On Potential Field Data Processing Based On Low-rank Matrix Decomposition Theory

Potential field exploration uses the density and magnetic difference of rocks to study the structure and distribution of underground space.As the basis of inversion and interpretation,the transformation and processing of potential field data is an important research direction for potential field theory.The observed gravity and magnetic fields are the superposition of gravity and magnetic fields caused by geological bodies of different depths.The potential field separation method is used to extract the gravity and magnetic anomalies caused by shallow or deep geological bodies from the total field,which is the research focus of potential field data processing.At present,potential field separation methods are mainly divided into space domain and frequency domain.The existing spatial domain method has insufficient theoretical basis and low separation accuracy,which makes the frequency domain method the mainstream of potential field separation method.However,the separation accuracy of the frequency domain method is not satisfactory.On the one hand,it is caused by the frequency aliasing,on the other hand,it is caused by the finite effect,discrete effect and superposition effect brought by the data's finite and discrete nature,which makes the spectrum calculated from the data different from the real spectrum.In order to avoid the spectrum estimation error of finite discrete data,the space domain method with clear geophysical significance and high precision separation is the research goal of this paper.Low rank matrix decomposition is one of the most popular research directions in the field of signal and image processing in recent years.It is a space domain method with strong robustness and high calculation accuracy,but it is rarely used in data processing of potential field.This paper mainly studies the applications of the low rank matrix decomposition methods in potential field separation.The idea is from the theoretical basis of the application of the low rank method to the specific method and then to the practical application.The theoretical basis of application is mainly to discuss the singular value characteristics of potential field data delay matrix caused by different depth geological bodies.The specific method is based on the existing low rank decomposition method,and then further improved according to the characteristics of potential field data,focusing on the calculation efficiency.The practical application is to apply the low rank decomposition method to the data processing of gravity and magnetic fields in different regions of China.The results show that the low rank matrix decomposition method can be used to solve the problem of potential field data separation.Compared with the traditional method,the low rank matrix decomposition method has high separation accuracy,strong robustness and simple parameter setting.The proposed fast algorithm can solve the problem of high time complexity to a certain extent and enhance the practical application of the method.The main work for dissertation can be summarized as follows:(1)Taking matched filtering and Wiener filtering as examples,the principles of frequency domain method are introduced.The sampling theorem,finite discrete theorem and error equation of Fourier transform for finite discrete data are summarized.The influence of discrete Fourier transform on spectrum estimation is discussed.(2)The singular value characteristics of one and two dimensional potential data delay matrix are studied.The relationship among model parameters,spectrum,autocorrelation function,autocorrelation matrix,power spectrum,delay matrix,harmonic model and singular value is discussed,and the relationship between model parameters and singular value of delay matrix is obtained.The results show that the delay matrix of potential field data generated by deep geological body has the characteristics of low rank and large non-zero singular value,while the delay matrix of potential field data generated by shallow geological body has the characteristics of high rank and small non-zero singular value.The first several large singular values of total field data delay matrix are the reflection of deep geological body,and the remaining small singular values are mainly the reflection of shallow geological body.(3)The potential field separation method based on singular spectrum analysis is studied.The principle and calculation process of singular spectrum method in one dimension and two dimensions are shown respectively.The parameter selection method is discussed,and the separation accuracy is compared with the traditional model.The results show that the values of K and (?) are related to the anomaly scale,and the truncated position of singular value is related to the decreasing trend of singular value.In the theoretical model experiment,the separation accuracy of singular spectrum analysis is higher than that of the traditional method.(4)The separation method of potential field based on low rank and sparse decomposition is studied.The optimization model and algorithm of potential field separation are presented,and the effect of penalty parameters on the results is discussed.The theoretical model experiment is compared with the traditional method.The results show that the method is robust and the effect of penalty parameters on separation results is limited.The theoretical model experiments show that the method has higher separation accuracy and robustness than the frequency domain method.(5)The fast algorithm of low rank decomposition method is studied.In view of the problem that the block Hankel matrix is too large in scale,which leads to low computational efficiency and large memory consumption,a fast and minimum memory block Hankel matrix singular value decomposition algorithm is proposed.The algorithm can get the singular value and singular vector of the delay matrix without constructing the delay matrix.Furthermore,the fast algorithm is applied to singular spectrum analysis and low rank sum sparse decomposition.The results show that the improved low rank decomposition method has a significant improvement in the computational efficiency,and can calculate the matrix with larger scale.In addition,theoretical model experiments show that the separation accuracy of the fast low rank and sparse decomposition method is further improved.(6)The low rank decomposition method is applied to practical problems.This paper analyzes and processes the gravity and magnetic data of a mining area in Southeast Hubei,a research area in Daye mining area in Hubei,a research area in Beishan Xiangshan mining area in Weining,Ningxia and a research area in Tongling mining area in Anhui Province.Combined with known geological conditions and drilling,the application effect of low rank decomposition method in practical problems is analyzed.The research shows that the low rank matrix decomposition method has a good application in the actual data,and the corresponding relationship between the separated local and regional anomalies and the target geological body is good.The innovative points for dissertation can be summarized as follows:(1)The methods based on low rank theory is applied to the data processing of potential field.(2)Fast computing methods for low rank decomposition methods are proposed,that make the low rank decomposition methods practical.