The search for weak oscillations in noisy records requires the development of analysis tools more sophisticated than the Fourier spectrum analysis. We present in this paper a new non-linear damped harmonic analysis (NLDHA) method that enables us to detect any damped harmonic signal hidden in noise and to simultaneously estimate the amplitude, frequency, phase and damping factor of the harmonic signal. We also compare this newly developed NLDHA method with a conventional auto-regressive method called ESPRIT. The application of the NLDHA method to the seismic normal modes 0S2 and 0S3 excited after the 2004 Sumatra-Andaman earthquake highlights its ability to precisely recover the frequencies and quality factors of the singlets that constitute these two seismic modes. The application of the NLDHA in the sub-seismic frequency band reveals the presence of some unmodelled waves but no detection of the translational oscillation of the inner core has been claimed.