The Mathematical Principle And Application Of Two Dimensional Filtering Methods

It is an important task of the seismic signal processing to improve thesignal-noise ratio（SNR） of the seismic data. So as to improve the SNR ofseismic data, we need to eliminate the random noise and coherent noise ofseismic data. The seismic data that studied in this dissertation is atwo-dimensional signal of the time and the space. So called two-dimensionalfiltering is two-dimensional signal in the seismic data to reach the signal-noiseseparation or improve the SNR according to some aspects of the signal andnoise.According to the different characteristics of random noise and coherentnoise, this dissertation introduces in detail the mathematical principle andapplication of four methods. KL transformation method and singular valuedecomposition filtering method are applied to remove the random noise; τ-ptransformation filtering method and the F-K filtering method of frequency wavenumber domain are applied to remove the coherent noise.(1)KL transformationMethod is used for calculating characteristic value and quantities of the matrix covariance by the Jacobi method. Strong relative effective signal is rebuilt byfirst principal component, and the poorly correlated random noise isreconstructed through those behind principal components.(2)Singular valuedecomposition filtering method is used for real matrix singular valuedecomposition by the Householder transformation and deformation of QRalgorithm, then it design filter to achieve filtering according to the singularvalue curve.(3) τ-ptransformation filtering method contains three processes:firstly, the signal of the time space domain is converted into slope interceptdomain by transformation. Then the signal is separated according to slopedifference of effective signal and noise. Finally, the signal of slope interceptdomain is converted into the time space domain by inverse transformation.(4)F-K filtering method also contains three processes: firstly, the signal of thetime space domain is converted into frequency wave number domain throughFourier transform. Then the filter is designed to separate the signal in thedomain. Finally, the signal is converted into the time space domain by inverseFourier transform. At last, we use FORTRAN programming to realize andcalculate through the model, it is proved that the feasibility of the four methods to remove the noise.Model data shows that: the KL transformation method and singular valuedecomposition filtering method can remove well the random noise, especiallythe bigger energy difference between effective signal and the random noise, thebetter filtering effect. Singular value decomposition filtering is higher accuratethan KL transformation. τ-ptransformation filtering method and F-K filteringmethod are both good ways to remove the coherent noise, various signal byτ-ptransformation filtering method can be clearly separated in the (τ-p)domain, which is very helpful for removing the coherent noise, while F-Kfiltering method does not have this effect in the (f,k)domain.