Multitaper Spectral Analysis and Wavelet Denoising Applied to Helioseismic Data

Abstract

Estimates of solar normal mode frequencies from helioseismic observations can be improved by using multitaper spectral analysis (MTSA) to estimate spectra from the time series, then using wavelet denoising of the log spectra. MTSA leads to a power spectrum estimate with reduced variance and better leakage properties than the conventional periodogram. Under the assumption of stationarity and mild regularity conditions, the log multitaper spectrum has a statistical distribution that is approximately Gaussian, so wavelet denoising is asymptotically an optimal method to reduce the noise in the estimated spectra. We find that a single m-ν spectrum benefits greatly from MTSA followed by wavelet denoising and that wavelet denoising by itself can be used to improve m-averaged spectra. We compare estimates using two different five-taper estimates (Slepian and sine tapers) and the periodogram estimate for Global Oscillation Network Group (GONG) time series at selected angular degrees ℓ. We compare those three spectra with and without wavelet denoising, both visually and in terms of the mode parameters estimated from the preprocessed spectra using the GONG peak-fitting algorithm. The two multitaper estimates give equivalent results. The number of modes fitted well by the GONG algorithm is 20%-60% larger (depending on ℓ and the temporal frequency) when applied to the multitaper estimates than when applied to the periodogram. The estimated mode parameters (frequency, amplitude, and width) are comparable for the three power spectrum estimates, except for modes with very small mode widths (a few frequency bins), where the multitaper spectra broaden the modes compared with the periodogram. At frequencies below 3 mHz, wavelet denoising of the log multitaper power spectra tends to increase the number of modes for which the GONG peak-fitting algorithm converges well. Close to 3 mHz, where all modes are resolved, wavelet denoising makes little difference. At higher frequencies close to the acoustic cutoff frequency, where modes are blended into ridges, wavelet denoising the multitaper spectra reduces the number of good fits. We tested the influence of the number of tapers used and found that narrow modes at low n-values are broadened to the extent that they can no longer be fitted if the number of tapers is too large. For helioseismic time series of this length and temporal resolution, the optimal number of tapers is less than 10.