The geomagnetic activity of the Dst index is analyzed using wavelet transforms and it is shown that the Dst index possesses properties associated with self-affine fractals. For example, the power spectral density obeys a power-law dependence on frequency, and therefore the Dst index can be viewed as a self-affine fractal dynamic process. In fact, the behaviour of the Dst index, with a Hurst exponent H≈0.5 (power-law exponent β≈2) at high frequency, is similar to that of Brownian motion. Therefore, the dynamical invariants of the Dst index may be described by a potential Brownian motion model. Characterization of the geomagnetic activity has been studied by analysing the geomagnetic field using a wavelet covariance technique. The wavelet covariance exponent provides a direct effective measure of the strength of persistence of the Dst index. One of the advantages of wavelet analysis is that many inherent problems encountered in Fourier transform methods, such as windowing and detrending, are not necessary.