The space-time structure of long-period ocean swell fields is investigated, with particular attention given to features in the direction orthogonal to the propagation direction. This study combines space-borne synthetic aperture radar (SAR) data with numerical model hindcasts and time series recorded by in situ instruments. In each data set the swell field is defined by a common storm source. The correlation of swell height time series is very high along a single great circle path with a time shift given by the deep water dispersion relation of the dominant swells. This correlation is also high for locations situated on different great circles in entire ocean basins. Given the Earth radius R, we define the distance from the source Rα and the transversal angle β so that α and β would be equal the colatitude and longitude for a storm centered on the North Pole. Outside of land influence, the swell height field at time t, Hss(α, β,t) is well approximated by a function Hss,0(t - Rα/Cg)/ times another function r2 (β), where Cg is a representative group speed. Here r2 (β) derived from SAR data is very broad, with a width at half the maximum that is larger than 70°, and varies significantly from storm to storm. Land shadows introduce further modifications so that in general r2 is a function of β and α. This separation of variables and the smoothness of the Hss field, allows the estimation of the full field of Hss from sparse measurements, such as wave mode SAR data, combined with one time series, such as that provided by a single buoy. A first crude estimation of a synthetic Hss field based on this principle already shows that swell hindcasts and forecasts can be improved by assimilating such synthetic observations.