A ring-diagram analysis of solar oscillations has been applied to a 3 day time series of full-disk Doppler images obtained at Mount Wilson. A model of the three-dimensional power spectrum assuming a Lorentzian profile in frequency and the advection of the wave front by horizontal flows has been fitted to the observed spectrum. The model provides estimates of the two horizontal components of subsurface flows averaged over depth. These estimates are then input to a least-squares smoothness-constrained inversion procedure to infer the depth dependence of the horizontal velocity. The method has been applied at nine different heliographic positions. The results indicate the presence of three east-west shear layers at all longitudes and latitudes, associated in depth with the ionization zones of hydrogen and helium in the outermost 20 Mm. The direction of the shear with respect to the average surface rotation rate alternates with depth, with organized north-south flows apparent between the east-west flows. Thus, the resultant vector velocity field appears to execute a spiral as a function of depth. Below ˜30 Mm, the organized character of the flow disappears, and a chaotic spatial distribution dominates. There is also an indication of a convergent flow at ˜10° north latitude, suggesting the presence of a toroidal convective roll. The magnitudes of the flows are on the order of 100-200 ms-1 in the outer 20 Mm, rising to several hundred ms-1 at 50 Mm.