## 2 Materials and Methods
### Optical Flow  
Here, we provide a brief description of the OF method as it is applied to calculate the perceived eddy propagation velocities between the pairs of consecutive SSH anomaly maps (images). Let \(\eta_{1}(x,y,t)\) denote the SSH anomaly signal at a grid point or pixel (\(x,y\)) on the first map at time \(t\). Suppose that during the time interval \(\Delta t\), the water parcel with the signal \(\eta_{1}(x,y,t)\) has moved to a new location (\(x+\Delta x,y+\Delta y\)) on the second map and the SSH anomaly signal has been transformed to \(\eta_{2}(x+\Delta x,y+\Delta y,t+\Delta t)\) (Figure 2a). It is assumed that the SSH change over \(\Delta t\), i.e., between consecutive maps, is small, so that  
\[\eta_{2}(x+\Delta x,y+\Delta y,t+\Delta t)\approx\eta_{1}(x,y,t) \tag{1}\]  
This is known as the brightness-constancy assumption in computer vision. Considering the effective resolution of SSH maps, it is reasonable to assume that the displacements \(\Delta x\) and \(\Delta y\) are relatively small over \(\Delta t=10\) days. Thus, the left-hand side of (1) can be approximated by a Taylor series expansion up to the first-order terms:  
\[\eta_{2}(x+\Delta x,y+\Delta y,t+\Delta t)\approx\eta_{1}(x,y,t)+\eta_{x} \Delta x+\eta_{y}\Delta y+\eta_{t}\Delta t \tag{2}\]  
where (\(\eta_{\times}\), \(\eta_{y}\), \(\eta_{t}\)) is a vector of the zonal, meridional, and temporal derivatives of \(\eta\). Substituting (2) into (1) and dividing by \(\Delta t\) yields the OF constraint equation [24]:  
\[\eta_{x}u+\eta_{y}v+\eta_{t}=0 \tag{3}\]  
where \(u=\Delta x/\Delta t\) and \(v=\Delta y/\Delta t\) are the zonal and meridional propagation velocities (optical flow) of the SSH signal, respectively. While the vector derivative (\(\eta_{\times}\), \(\eta_{y}\), \(\eta_{t}\)) can be estimated by spatio-temporal finite differencing at each pixel, the two unknowns (\(u\) and \(v\)) cannot be uniquely determined from (3) alone. Known as the aperture problem, this ambiguity translates to the unique computation of motion along the direction of the gradient (\(\eta_{\times}\), \(\eta_{y}\)), but not perpendicular to it. Additional constraints are required to arrive  
Figure 2: (**a**) An example of an anticyclonic eddy propagating westward over the 10-day time interval (\(\Delta t\)). The black and blue rectangles show \(5\times 5\) pixel patches, over which the flow is assumed to be constant. During \(\Delta t\), the patch shown by the black rectangle has moved in the direction shown by the black and blue arrows to the position outlined by the blue rectangle. The red rectangles outline individual pixels \(\eta_{1}\) and \(\eta_{2}\). (**b**) An illustration of the pyramid method with three layers. The optical flow (OF) is computed at each pyramid layer starting from the top layer at the lowest resolution down to the original image. at a unique solution. Among many approaches to address this problem, the Lucas-Kanade method assumes that the OF within a small neighborhood \(W\) (e.g., an \(n\times m\) rectangular patch) of each pixel (\(x,y\)) is constant [43]. This yields a linear system of equations for the \(n\times m\) pixels within \(W\) with only two unknowns:  
\[ \begin{bmatrix}\eta_{x}(1)&\eta_{y}(1)\\ \eta_{x}(2)&\eta_{y}(2)\\ \vdots&\vdots\\ \eta_{x}(nm)&\eta_{y}(nm)\end{bmatrix}\begin{bmatrix}u\\ v\end{bmatrix} = \begin{bmatrix}-\eta_{t}(1)\\ -\eta_{t}(2)\\ \vdots\\ -\eta_{t}(nm)\end{bmatrix}  \tag{4}\]  
The least-squares optimization applied to the above over-constrained linear system of the form \(A_{nm\times 2}\mathbf{u}=B_{nm}\), where \(\mathbf{u}(u,v)\) is the optical flow vector, yields the following solution:  
\[\mathbf{u}=\left(A^{T}A\right)^{-1}A^{T}B \tag{5}\]  
Mathematically, we require a near-unity condition number of \(A^{T}A\) for a robust solution. In our application, overcoming the aperture problem, this translates to having sufficient variation of \(\eta\) within \(W\).