Satellite Synthetic Aperture Radar (SAR) observations can provide a global view of ocean swell fields when using a specific "wave mode" sampling. A methodology is presented to routinely derive integral properties of the longer wavelength (swell) portion of the wave spectrum from SAR Level 2 products, and both monitor and predict their evolution across ocean basins. SAR-derived estimates of swell height, and energy-weighted peak period and direction, are validated against buoy observations, and the peak directions are used to project the peak periods in one dimension along the corresponding great circle route, both forward and back in time, using the peak period group velocity. The resulting real time dataset of great circle-projected peak periods produces two-dimensional maps that can be used to monitor and predict the spatial extent, and temporal evolution, of individual ocean swell fields as they propagate from their source region to distant coastlines. The methodology is found to be consistent with the dispersive arrival of peak swell periods at a mid-ocean buoy. The simple great circle propagation method cannot project the swell heights in space like the peak periods, because energy evolution along a great circle is a function of the source storm characteristics and the unknown swell dissipation rate. A more general geometric optics model is thus proposed for the far field of the storms. This model is applied here to determine the attenuation over long distances. For one of the largest recorded storms, observations of 15 s period swells are consistent with a constant dissipation rate that corresponds to a 3300 km e-folding scale for the energy. In this case, swell dissipation is a significant term in the wave energy balance at global scales.