Power spectral estimates using two-dimensional Morlet-fan wavelets with emphasis on the long wavelengths: jackknife errors, bandwidth resolution and orthogonality properties

Abstract

We present a method for estimating the errors on local and global wavelet power spectra using the jackknife approach to error estimation, and compare results with jackknifed multitaper (MT) spectrum estimates. We test the methods on both synthetic and real data, the latter being free air gravity over the Congo basin. To satisfy the independence requirement of the jackknife we investigate the orthogonality properties of the two-dimensional Morlet wavelet. Although Morlet wavelets are non-orthogonal, we show that careful selection of parameters can yield approximate orthogonality in space and azimuth. We also find that, when computed via the Fourier transform, the continuous wavelet transform (CWT) contains errors at very long wavelengths due to the discretisation of large-scale wavelets in the Fourier domain. We hence recommend the use of convolution in the space-domain at these scales, even though this is computationally more expensive. Finally, in providing an investigation into the bandwidth resolution of CWT and MT spectra and errors at long wavelengths, we show that the Morlet wavelet is superior in this regard to Slepian tapers. Wavelets with higher bandwidth-resolution deliver smaller spectral error estimates, in contrast to the MT method, where tapers with higher bandwidth-resolution deliver larger errors. This results in the fan-WT having better spectral estimation properties at long wavelengths than Slepian multitapers.