Orthonormal wavelet expansion method is applied to an analysis of atmospheric turbulence data which shows more than two decades of the inertial subrange spectrum. The result of the orthonor mal wavelet analysis of the turbulence data is discussed in comparison with those of an artificial random noise. The local wavelet spectra of turbulence show a characteristic structure. which is absent in the artificial random noise and is identified with the trace of the energy cascade process. The higher· order structure function of velocity. obtained by the wavelet analysis. shows the intermit· tent structure of the flow field. In 1941 Kolmogorov proposed a universal theory of fluid turbulence/) in which every statistical quantity concerning the inertial subrange of flow field is assumed to be determined only by the energy dissiation rate. According to this theory, the average of the n-th order of velocity difference between two points separated by spatial distance r is proportional to r13. This prediction has been repeatedly examined in accurate experiments in high Reynolds number flows, and it is now widely accepted that as far as lower order of velocity difference is concerned, the r-dependence agrees well with the Kolmogorov theory. In particular, experimental forms of the energy spectrum of the velocity field, which corresponds to n=2, coincide with the Kolmogorov form of k- 5/3 • However, it has been repeatedly confirmed that higher order of the velocity difference has a statistical property different from Kolmogorov's prediction, so that the n-th order structure function of the velocity field, when normalized by the second order of the velocity difference, shows a clear r-dependence. This fact implies that the energy cascade process in the inertial subrange has an unnegligible deviation from the Kolmogorov picture. The deviation is reflected in the shape of the probability distribution function of the velocity difference, which has longer tail for smaller distance r. This deviation is often called intermittency, and its characterization is regarded as one of the central problems of fluid turbulence. The fact that intermittency becomes more prominent at smaller scales indicates that the intermittency is an essential part of the energy cascade process. However, this cascade process itself, sometimes called Richardson cascade, has been only a matter of theoretical consideration, and its characteristic structure has not yet been clearly captured in experiments or in numerical simula tions.