An eddy tracking algorithm based on dynamical systems theory

Abstract

This work introduces a new method for ocean eddy detection that applies concepts from stationary dynamical systems theory. The method is composed of three steps: first, the centers of eddies are obtained from fixed points and their linear stability analysis; second, the size of the eddies is estimated from the vorticity between the eddy center and its neighboring fixed points, and, third, a tracking algorithm connects the different time frames. The tracking algorithm has been designed to avoid mismatching connections between eddies at different frames. Eddies are detected for the period between 1992 and 2012 using geostrophic velocities derived from AVISO altimetry and a new database is provided for the global ocean.

Appendix: Tracking algorithm

Description of existing tracking methods

A new tracking method is developed that is intended to increase the overall robustness of the eddy matching process by minimizing the mismatching of eddy pairs. In general, all eddy-tracking methods follow a two-step approach (Penven et al. 2005; Chelton et al. 2011; Mason et al. 2014):

• Preselection. Given an eddy with a determined sense of rotation (C/A) at frame k, the method searches for eddies at frame k + 1 that have the same sense of rotation and within a distance of δ _s = c _max Δt, where c _max is a predefined limit for the propagation velocity and Δt is the time step between two consecutive frames. Some additional preselection criteria may be applied at this point, rejecting eddies at frame k + 1 having extremely different characteristics (i.e., radius, amplitude, etc.).

• Tracking criteria. If an eddy at frame k does not have any preselected eddy at frame k + 1, the eddy is terminated and the search stops. Furthermore, if only a single eddy at frame k + 1 is found in the preselection, it is assumed straightforwardly to be the same structure. If multiple eddies the frame k + 1 are preselected, an additional criterion is applied to identify the correct match. In such cases, Isern-Fontanet et al. (2006a) and Chelton et al. (2011) assume that the successful preselected eddy at frame k + 1 is the nearest one. In order to improve the matching decision, some authors take into account the spatial distances and also the similarity in other eddy properties, such as eddy radii and vertical component of the vorticity (Penven et al. 2005), or the area covered bye the eddy and its amplitude (Mason et al. 2014). Finally, eddies at frame k + 1 that are not linked to any eddy at frame k are labeled as being new eddies.

A new approach to matching of multiple eddy pairs

We implement the preselection as usual: we search for a detected eddy at frame k, eddies at frame k + 1 that are within a prescribed distance (considering distances between eddies as the distance between their centers). Here, we follow the approach of Chelton et al. (2011) where the maximum eddy celerity is 1.75 times the celerity of a long baroclinic Rossby wave. From all eddies satisfying this condition, we discard those satisfying $r_{\max }/r_{\min } > \sqrt {10}$, where r _max (r _min) is the radius of the largest (smallest) eddy of the considered pair. This value eliminates eddies with very different radii, and is flexible enough to allow small differences between consecutive frames. Up to this point, nothing new has been introduced with respect to other methods. However, the main difference with other existing tracking algorithms is that we get the preselecting information forwards and backwards. For each detected eddy i at frame k, we store the preselected eddies at frame k + 1 (hereinafter called descendants, D _k (i) ), and for each eddy j at frame k + 1 we store the list of ascendants, A _{k + 1}(j). By symmetry in the criteria, if an eddy i at frame k is an ascendant of an eddy j at frame k + 1, then eddy j is a descendant of eddy i.

Figure 12 displays some possible combination treated in the tracking criteria. First, if an eddy at frame k does not have any descendants (Fig. 12, first case), the eddy dies. This case includes both the situation of a real eddy vanishing or when the system is not able to detect the descendant due to both inaccuracies in the algorithm or poorly resolved SSH data (sampling errors and noise in the measurements).

Second, if an eddy (i) at k has a single descendant (j) at k + 1, a link is not established directly; instead the algorithm looks through the list ascendants of j. If eddy j has a single ascendant, then A _k (j) = i and the link is made (Fig. 12, third case where eddy #1 at k is linked with eddy 2 at k + 1). If eddy j at k + 1 has several ascendants, the link is not trivial and the method follows the general case described below. This is the situation shown in Fig. 12, third case, where eddies #2 and #3 at k are ascendants of eddy #1 at k + 1. This situation (and others of higher complexity where multiple pairs are found) are not discussed in the above presented tracking methods.

This multi-pair scenario embraces several possibilities: an eddy i at k having multiple descendants; an eddy having a single descendant but the former with multiple ascendants, etc. In the general case, the problem can be expressed as an eddy having multiple descendants, each of them having single or multiple ascendants and, further, each of them may have single or multiple descendants and so on. In these situations, the tracking algorithm considers all involved eddies at frames k and k + 1 in a coupled configuration since each single link might influence the other ones. Starting from an eddy at frame k, the configuration must be constructed iteratively, first listing its descendants (at k + 1) and, then, for each descendant adding its non repeated ascendants (at k). In a third iteration, for every new (non-repeated) ascendant, we add its (non-listed) descendants to the whole configuration. The iteration ends when no new eddies are found. A simple example is shown in Fig. 12, third case, where eddies #3 and #2 at frame k are connected sharing the same descendant (#1 at k + 1). A more complex configuration is shown in Fig. 12, last case, where eddies #1, #2, and #3 at k are linked with eddies #2, #1, and #4 at k + 1. For this example, Table 1 lists the ascendants and descendants and Table 2 shows the iteration process where black lines indicate previous connections and blue lines new connections. In the first iteration, D _k (#1) = #2(new); in the second iteration A _{k + 1}(#2)={#1(old),#2(new)}; the third iteration is D _k (#2)={#1(new),#2(old)}; the fourth iteration is A _{k + 1}(#1)={#2(old),#3(new)}; the fifth iteration is D _k (#3)={#1(old),#4(new)}. The last iteration, A _{k + 1}(#4) = #3(old) does not provide any new eddy at k and therefore the iterative process finishes, returning connected eddies #1, #2, and #3 at k and eddies #1, #2, and #4 at k + 1 (Table 3).

Table 1 Preselected descendants and ascendants of eddies at frames k and k + 1, respectively, derived from the configuration shown in Fig. 12, last panel

Table 2 Iteration scheme for the linking of eddies between frames k and k + 1 in the case of multiple ascendant/descendant possibilities derived from the configuration shown in Fig. 4, right

Table 3 Statistical properties from time series of active eddies over the global domain for the period January 1994 - December 2010 shown in Fig. 9

Once connections are established, all permutations of allowed pairs at k and k + 1 are computed including permutations with unmatched eddies. If there is a single permutation with the maximum number of links, it is the one chosen. Otherwise, the criteria is to choose the permutation which minimizes the collective distance that is defined as the sum of distances between the centers of eddy-pairs. If there are unmatched eddies at k in the selected permutation, they are considered to disappear, while unmatched eddies at k + 1 are considered to be newborns. In the example shown in Fig. 12, third case, two permutations are allowed, i.e., #3 at k with #1 at k + 1 leaving eddy #2 at k unmatched, and eddy #2 at k linked with eddy #1 at k + 1 leaving now #3 at k unmatched. Both permutations have the same number of links (one), and the collective distance criteria is reduced to link #1 at k + 1 with its closest eddy at k. It is worth noting that this very simple multi-pair case cannot be solved properly with methods that just perform the pre-selection forwards (and not backwards). In the example shown in Fig. 12, fourth case, there is a single permutation that allows all eddies to be linked, [#1(k)−#2(k + 1)],[#2(k),#1(k + 1)],[#3(k),#4(k + 1)], so this is the one the is chosen. Again, using a tracking system that is based only on minimization distances between individual pairs would match together [#2(k),#2(k + 1)],[#3(k), #1(k + 1)], leaving unpaired eddies #1 at frame k and #4 at frame k + 1.

The above methodology is applied iteratively to all frames to obtain the evolution of the eddies. Figure 13 shows for illustration purposes the trajectory of an eddy detected in the North Atlantic. The panels correspond to snapshots at 0, 10, 20, and 30 weeks from its birth. The detected SNs and CTs are plotted as circles and crossed circles, respectively, while the red circle represents the eddy traveling to the west in a path represented by the dashed red line.

Fig. 13

Example of eddy tracking in the North Atlantic ocean. Snapshots correspond to, from top to bottom, the 1^st, 11^th, 21^st, and 31^st weeks since the birth of the encircled eddy (red contour). Red dashed line shows the eddy trajectory at each given week. Black arrows display the velocity field direction (not magnitude), which have been subsampled every three mesh nodes (in each direction) and the background color is the vorticity (units in 10⁻⁵ s ⁻¹). Otimes symbol are the CT-FPs and ∘ are the SN-FPs