We analyze the pattern of phase and amplitude variations of seismic waves across the NORSAR array on a statistical basis in order to determine the statistical distribution of heterogeneities under NORSAR. Important observables that have been analyzed in the past are the phase (or travel time) and log amplitude variances and the transverse coherence functions (TCFs) of phase and amplitude fluctuations. We propose and develop the theory and methods of using other observables to reduce the degree of nouniqueness and increase the spatial resolution of the analysis. Most important are the angular coherence functions (ACFs), which characterize quantitatively the change in the pattern of fluctuations across the array from one incoming angle (or beam) to another and which have a different sensitivity to the depth distribution of heterogeneities than the TCFs. A combination of the ACFs and TCFs allows estimation of the power spectra of the P wave speed variations under the array as a function of depth. We use data for phase fluctuations from 104 incident beams and amplitude fluctuations from 185 beams with 2-Hz center frequency at NORSAR to calculate the three ACFs and three TCFs (of phase, log amplitude, and their cross coherence). The measured rms travel time fluctuation is 0.135 s, and the rms log amplitude fluctuation is 0.41. The half-coherence widths of the ACFs are 3¿ for log amplitude and 9¿ for phase. The half-coherence widths of the TCFs are 18 km for phase and less than the minimum separation between the elements of the array for log amplitude.

In order to account for these features of the data, we adopt a two-overlapping-layer model for lithospheric and asthenospheric heterogeneities underneath NORSAR, with spectra that are band-limited between the wavelengths of 5.5 and 110 km. Our best model has an upper layer with a flat power spectrum extending from the surface to about 200 km, and a lower with a K-4 power spectrum extending from 15 to 250 km. The latter spectrum corresponds to an exponential correlation function with scale larger than the observation aperture (110 km). The rms P wave speed variations lie in the range 1--4%. The small scale heterogeneities may be attributed to clustered cracks or intrusions; the larger-scale wave-speed heterogeneities are temperature or compositional heterogeneities that may be related to chemical differentiation, or dynamical processes in the boundary layer of mantle convection. ¿ American Geophysical Union 1988