Data from synthetic aperture radars (SAR's) can be used to measure the surface height spectrum of the ocean. Traditionally, only the real SAR image is used; we show, both theoretically and with SAR data, that the complex image contains more information about the ocean scene. We derive an expression for the ambiguity function of the complex image, which is an extension of the concept of a power spectrum, but generalized to phenomena with time-dependent properties. The ambiguity function of the complex image is derived as an integral of the surface height spectrum; this allows us to calculate the statistics of the complex SAR image directly, without having to generate individual realizations of the surface. In the derivation of the ambiguity function, we find that a simple assumption about the complex reflectivity of the surface leads to a formulation of the SAR imaging process that circumvents many traditional problems. By using ambiguity functions, we can remove some of the smearing that is otherwise seen in the images of time-dependent scenes. One result of this analysis is that we can explain SAR focusing in a simple way, and extend it to range-traveling waves as well as azimuth-traveling waves. Also, we can calculate the correlation length and time of the surface directly from the surface height spectrum. While the correlation length seems to play only a minor role in this theory, the correlation time determines the minimum wavelength for azimuth waves that can be imaged. This paper shows how the ambiguity function of the image is related to the surface height spectrum and presents an example using SAR data. ¿ American Geophysical Union 1993