## 1 Introduction  
The ocean is always in turbulent, highly variable motion. Ocean circulation redistributes mass, heat, salt, nutrients, carbon, and other properties, and plays a crucial role in regulating the global and regional climate, weather, and ecosystems. The advent of satellite observations of sea surface temperature (SST) in the 1970s and sea surface height (SSH) in the 1980s has revolutionized our view on ocean variability. The nearly global coverage of satellite measurements has revealed the distribution and spectral characteristics of ocean variability at various time scales. It has been discovered that ocean variability is dominated by processes with spatial scales ranging from a few tens to a few hundreds of kilometers and a time scale on the order of 100 days [1, 2, 3]. This type of ocean variability is usually termed as transient mesoscale eddies, including individual coherent eddies (vortices characterized by a clear boundary), meanders, fronts, filaments, and planetary Rossby waves [4]. Transient mesoscale eddies can be found virtually everywhere in the ocean, but the largest and most energetic eddies emerge from the instabilities of the strongest currents, in particular the Antarctic Circumpolar Current (ACC) and western boundary currents, such as the Gulf Stream, Kurosho, Agulhas Current, etc. Away from strong currents that can significantly influence their motion, eddies tend to propagate westward as Rossby waves. They carry properties horizontally and redistribute them vertically, contribute to the climatically meaningful meridional heat transport, drive regional sea level changes, influence shipping routes and fisheries, etc., e.g., refs. [5, 6, 7, 8]. The importance of mesoscale eddies in ocean dynamics has stimulated numerous efforts to track the movement of individual eddies and to estimate their propagation velocities. The tracking of individual eddies consists of following their trajectories in a Lagrangian manner. Existing techniques include the Okubo-Weiss method [9; 10; 11; 12], algorithms based on the geometry of velocity vectors and streamlines [13; 14; 15; 16; 17], a hybrid geometric Okubo-Weiss method [18], a wavelet-based approach [19; 20], and an objective method based on geodesic transport theory [21]. Most procedures of eddy detection rely on using either surface geostrophic velocities derived from SSH or closed SSH contours. In altimetry applications, however, the use of geostrophic velocities and moreover the relative vorticity is not optimal, because it requires computation of the first and the second derivatives of SSH, respectively, which amplifies any noise present in satellite measurements and interpolation errors [15]. Different from individual eddy detection and tracking, the regional and global patterns of transient eddy propagation velocities in the Eulerian frame of reference have also been obtained by carrying out a space-time-lagged correlation analysis of SSH fields [22; 23]. The objective of this study is to demonstrate the use of the optical flow (OF) technique to derive the time-dependent propagation velocities of transient mesoscale eddies in the Eulerian framework. The OF is a computer vision algorithm that estimates the apparent motion of brightness patterns in a sequence of images, e.g., [24]. Some examples of the earlier implementation of the OF in geosciences include the assessment of the quality of forecasts of meteorological fields [25; 26; 27], and the estimates of sea-ice motion [28; 29] and surface currents from SST images [30; 31]. The study reported in this paper is, to our knowledge, the first implementation of the OF method to satellite altimetry SSH data. Like the space-time-lagged correlation analysis of [22; 23], the OF method cannot distinguish various forms of eddy variability, e.g., isolated cyclones and anticyclones, planetary waves, meanders, fronts, etc.