Multiple taper spectral analysis was introduced 20 years ago. The algorithm by Thomson (1982) has been widely used and works extremely well for short time series and high dynamic range signals, but also has some limitations. As explained by Riedel and Sidorenko (1995) the Slepian tapers used in this algorithm can have considerable bias due to curvature of the spectrum in the frequency bandwidth of interest. The best example of this feature is the box-like response of the multiple-taper spectrum when the signal has a periodic line component. In these study we go further and introduce a new adaptive multitaper algorithm that is unbiased to curvature of the spectrum (d2S/df) and thus outperforms Thomson’s original algorithm in some situations. We present several examples comparing methods of spectrum estimation for both synthetic and geophysical signals. As an example of the application we present applications of this method to marine topography data obtained from ship tracks in the central Pacific ocean.