Study on Mercer condition extension of support vector regression based on Ricker wavelet kernel

Support vector regression (SVR) based on the Ricker wavelet kernel is a new filtering method for suppressing strong random noise in seismic records. The Mercer condition, which is the rule to determine a support vector admissible kernel, is used to discuss the validity of the Ricker wavelet kernel. By computing the minimum eigenvalues of kernel matrixes, we find that there exist some small negative values in a wider region which have orders of magnitude 10~(-13- 10~(-16, and also exist some small positive values which have orders of magnitude 10~(-13)-10~(-15). Considering the same mechanism resulting in the positive and negative computational errors, we conclude that the Mercer condition can be moderately relaxed, that is, the kernel matrix can be not exactly positive semi-definite and close to positive semi-definite. In order to apply the SVR based on the Ricker wavelet kernel to practical applications, we compare the performances of our method, the wavelet transform-based method and adaptive Wiener filtering in detail, including the waveforms in the time domain, the amplitude spectrums in frequency domain, the SNRs before and after filtering and the MSEs. The results show that our method works better than the two other methods, which lays a foundation for practical applications of the SVR based on the Ricker wavelet kernel.