## 1. Introduction

The kinematic properties (KPs)—divergence, vorticity, and strain rate—have long been used to characterize fluid flows, due to their prominent role in the linearized representation of the velocities. In ocean flows in particular, however, divergence has often been neglected, since the ocean is close to incompressible. Yet in recent years it has received new attention in its two-dimensional (2D) form. The ocean is certainly 2D compressible, and using the 3D incompressibility (

Unfortunately, observations of surface divergence are also tricky. Velocities derived from radar observations have proven valuable in coastal regions (e.g., Kim 2010; Berta et al. 2020a). In waters further offshore, measurements from ADCPs mounted on ships following parallel tracks (Shcherbina et al. 2013) and ship-mounted X-band radar systems (Berta et al. 2020b) have been successfully used to compute the divergence field, but are expensive to obtain. Supplementing these techniques are Lagrangian methods, which rely on trajectories from groups of drifters. An advantage is that the coverage is less constrained to coastal regions than radar and much cheaper than coordinated ship tracks. Additionally, since the drifters follow water parcels, their trajectories are ideal for observing path-integrated velocity gradient quantities like dilation (Huntley et al. 2015).

Trajectories have been used to estimate KPs for many decades (Molinari and Kirwan 1975; Okubo et al. 1976). The methods from the 1970s have stood the test of time and continue to be used today (e.g., Crane and Wadhams 1996; Ohlmann et al. 2017; Suara et al. 2018; Berta et al. 2020b). The different algorithms fall into two broad categories, with newer developments making up a third: First, the simple, closed curve connecting the positions of a group of drifters can be used, either by considering derivatives of the contained area for divergence and of the “area” in transformed coordinates for the other KPs (method II of Molinari and Kirwan 1975) or by computing line integrals around the perimeter based on Green’s theorem (Kawai 1985). These algorithms generate an estimate of the KPs averaged over the time interval of consecutive measurements and over the region enclosed by the curve. The minimum number of drifters in a group is three. For larger groups, care must be taken that the curve connecting the positions remains simple (i.e., without self-intersections). Even for triads, the methodology breaks down when the orientation of the triangle circumference reverses. Second, velocity gradients—and subsequently the KPs—can be estimated from least squares fits to linearizations around either the centroid position (Okubo et al. 1976) or the centroid velocity (method I of Molinari and Kirwan 1975) of a group of drifters. In either case, the estimate is valid for the centroid location. The minimum number of drifters needed for these methods is also three, whereby the least squares fit collapses into the solution of a linear system of equations for this limiting case. Okubo and Ebbesmeyer (1976) and Essink (2019) have suggested that a minimum of 6 drifters should be used for the estimates to converge. In addition, if more than three drifters are in the group, the residual of the fit can be used as an error estimate. Third, the velocities measured at individual drifter positions can be interpolated before taking derivatives, either analytically from the interpolating equations or numerically on a square grid. These methods trace their roots to data blending and data assimilation techniques (Taillandier et al. 2006) and come in more or less complex flavors, including Lagrangian variational analysis (Berta et al. 2015) and Gaussian regression (Gonçalves et al. 2019; Lodise et al. 2020). They work best when large numbers of drifters are available in a dense cloud, which are not grouped into smaller subsets, in contrast to calculations from the other two categories. Here, we concentrate on the area-based algorithm from Molinari and Kirwan (1975) from the first group.

Generally, it is desirable to maximize the number of KP estimates from a given dataset. This could either produce greater spatial and/or temporal coverage or result in a larger sample for more reliable statistics. It is, therefore, attractive to rely on drifter triads, even though useful nearby information may be ignored. For the area-based algorithm, triads have the additional advantage that the triangle perimeter is uniquely determined by the three vertices; for larger groups choices have to be made regarding convexity and ordering. A comparison of drifter-triplet-based estimates using the area algorithm with KP estimates derived from (i) least squares KP estimates from clusters of at least five drifters and (ii) a completely independent but simultaneous dataset of X-band radar measurements of surface velocities (Berta et al. 2020b) showed that triplets suffice for the area-based algorithm to produce reasonable results. Yet it is also well known that KP estimates suffer when the aspect ratio of the drifter cluster is too small (Berta et al. 2016; Ohlmann et al. 2017; Essink 2019). In practice, this means that triads with small aspect ratios are discarded (e.g., Berta et al. 2020b).

But how small is too small? Here we answer this question for the flow dynamics of a realistic high-resolution regional ocean model. For the model field, the KPs are known and can therefore be compared to the estimates derived from simulated drifter trajectories. This approach sidesteps issues of measurement errors. We recognize, however, that observed positions cannot be perfect, and this clearly also impacts the accuracy of KP estimates (Kirwan and Chang 1979; Spydell et al. 2019). Moreover, the impact of position uncertainty is exacerbated by cluster deformation for least squares KP estimates (Ohlmann et al. 2017). Therefore, we will also consider how position error affects the shape parameter thresholds.

The analysis is based on simulated trajectories. So the question arises whether the threshold values are useful in the context of field observation. To answer this question, we consider an observational dataset of drifter trajectories collected in the general region simulated by the model, the western Mediterranean, during the 2018 CALYPSO field experiment. The kinematic properties along a quasi-permanent density front in this area, derived from a 2019 drifter release using a variational approach, were also studied by Lodise et al. (2020). In this analysis, there is no benchmark to which to compare the divergence estimates, since the “true” divergence cannot be known. However, we demonstrate that the model-derived thresholds effectively and efficiently eliminate the outliers from the distribution of divergence estimates that are most likely associated with errors. For simplicity, we restrict most of our analysis to divergence, but since the algorithms for the other KPs are identical after a coordinate transformation (see Saucier 1955; Molinari and Kirwan 1975), the results are expected to hold for them as well. We present some evidence to this effect from the observational study.

The model, our methods, and the observational data are described in section 2. We present the analysis of divergence estimates for drifter triangles of different aspect ratios, the implied aspect ratio cutoff value, an alternative shape metric, and the impact of position uncertainty on these results in section 3, along with an illustration of how the errors associated with strong deformation arise. This is followed by the application of the thresholds to the observed drifter dataset, in section 4. A summary and conclusions are given in section 5.

## 2. Data and methods

### a. The model

The bulk of the analysis in this report is performed in a model framework, taking the model output as “truth.” The model chosen here is a nested implementation of the Regional Ocean Modeling System (ROMS) (Shchepetkin and McWilliams 2005) for the western Mediterranean. ROMS is a popular and well validated primitive equation ocean model with Boussinesq and hydrostatic assumptions. The outer model at 2-km resolution is the Western Mediterranean Operational Modeling System (WMOP), maintained operationally at Balearic Islands Coastal Observing and Forecasting System (SOCIB). It reaches from the Straits of Gibraltar to the Sardinia Channel (5.8°W–9.2°E) and encompasses the entire Mediterranean Sea latitudinally (34.9°–44.71°N). It uses 32 sigma levels with higher resolution near the surface. Boundary forcing and initialization are obtained from the Mediterranean forecasting model of the Copernicus Marine Service (Clementi et al. 2016). Juza et al. (2016), Mourre et al. (2018), and SOCIB’s website (www.socib.eu) provide additional details, as well as real-time and delayed-mode assessment of the model performance.

The inner nest has a resolution of approximately 425 m and focuses on the eastern Alboran Sea (35.06°–36.84°N, 3.53°–1.02°W), which was the site of the 2018 CALYPSO field experiment (Mahadevan et al. 2020). Velocity fields are saved every 3 h. The simulation analyzed in this study was initialized from WMOP data-assimilative conditions on 23 May 2018 and then evolved freely for a month. Satellite along-track altimetry, sea surface temperature, and Argo temperature and salinity profiles were assimilated prior to that date using ensemble optimal interpolation (Hernandez-Lasheras and Mourre 2018). Thus, the modeled oceanic conditions are expected to be realistic and reasonably close to the actual ones but at the same time the velocity fields are fully governed by the model equations. Drifter and glider data collected during the 2018 CALYPSO experiment confirmed that the model provides a reasonably realistic representation of the salinity gradients and currents during the study period, although the salinity front is offset approximately 30 km to the southwest and the surface velocities are slightly underestimated (Rypina et al. 2021; Garcia-Jove et al. 2022).

The basis for the analysis here is a set of simulated drifter trajectories initialized on the model grid. The launch time was arbitrarily chosen to be at midnight UTC 1 June 2018. Simulated drifters were launched on the model grid points and advected by the surface layer velocities using a fixed time step fourth-order Runge–Kutta scheme with a time step of 10 min and multilinear interpolation of velocities between grid points and archive times. Figure 1 shows the surface salinity field over the model domain with every other grid point indicated with a dot (left). The right panel of the figure displays the positions to which the particles have been advected over 12 h. Clearly, their distribution is no longer uniform, and particularly high densities of particles can be found along fronts of strong salinity gradients. These fronts are also accompanied by a strong strain rate, leading to deformation of the triangles initialized from the grid of drifters. Since our interest is specifically on the effects of such deformation on the divergence estimates, we focus the analysis on a patch across a sharp salinity front that experiences strong deformation over the 12-h period. The primary focus of the analysis will be on a patch consisting of 44 × 47 particles initially on the model grid (shown in red in Fig. 1), for a total of 2068 trajectories and 3956 triplets in a Delaunay triangulation on the initial positions, with average side length of 485 m. (We will refer to this as the 500-m scale.) To investigate scale dependence of our findings, coarser grids of particles are also considered, as indicated by the other brightly colored dots in Fig. 1: a grid of 24 × 22 particles yielding 966 triplets with an average side length of 2.4 km (the “2-km” scale, blue), a grid of 20 × 15 particles yielding 532 triplets with an average side length of 4.9 km (the “5-km” scale, orange), a grid of 13 × 10 particles yielding 216 triplets with an average side length of 7.3 km (the “7-km” scale, green), and a grid of 8 × 6 particles yielding 70 triplets with an average side length of 12.2 km (the “12-km” scale, black).

### b. The local ocean dynamics

The dynamics in the Alboran Sea are dominated by the confluence of warm, salty Mediterranean waters and cold, relatively fresh Atlantic waters entering through the Straits of Gibraltar. The region is therefore rich in strong salinity fronts. In particular, the front targeted by the present analysis is a semipermanent feature of the Alboran Sea, generally known as the Almeria–Oran Front, as it approximately links Almeria, Spain, to Oran, Algeria (Tintoré et al. 1988). The large anticyclonic gyre visible in Fig. 1, centered near 36°N, 2.5°W, is also a common occurrence, known as the Eastern Alboran Gyre (Tintoré et al. 1991; Renault et al. 2012). The fronts and edges of the gyre are associated with strong vertical transport, setting this region up for complex three-dimensional flows (Tintoré et al. 1988; Ruiz et al. 2009; Oguz et al. 2017). Thus, this analysis focuses on a mesoscale feature (the jet) with submesoscale variability resolved by the model.

### c. The divergence algorithms

We will refer to four different estimate of the divergence of the flow field. Ultimately, we are interested in the estimate based solely on the information from the simulated trajectories, analogous to what can be done with actual observations. What results is an estimate of divergence, averaged temporally over the chosen time interval and spatially over the patch of water initially identified. This is not directly comparable to the instantaneous divergence field, which is most commonly reported for models and which constitutes the first estimate. One step closer is the second estimate, the along-trajectory (Lagrangian) temporal average of the divergence, also known as the dilation rate (Huntley et al. 2015). The third estimate computes the spatial average over triangles of the dilation rate, like the drifter-based approach, but relying on the more complete velocity field available from the model. Finally, the fourth estimate uses only the drifter simulations. For linguistic simplicity, we will refer to the Lagrangian temporally averaged divergence as the dilation rate, and to the spatial average thereof as the triangle-mean dilation rate (or simply as the mean dilation rate). Note that all results are reported normalized by the Coriolis parameter for a latitude of 36.5°N, *f* = 8.675 × 10^{−5} s^{−1}.

#### 1) Instantaneous divergence estimates

Instantaneous divergence *d* = **∇** ⋅ **u** is estimated on the native plaid grid of the model, with second-order accuracy, using centered finite differencing. Bilinear interpolation is used for the graphical representation.

#### 2) Dilation rate estimates

*δ*is defined as the product of the singular values of the deformation tensor

*δ*)/(Δ

*t*). It turns out that the dilation rate also equals the along-trajectory average of the divergence (see appendix A and Huntley et al. 2015). Dilation is computed by solving the integration problem for the deformation tensor components along the trajectories (Huntley et al. 2015):

*δ*is the Kronecker delta. For the integration, the velocity gradients are interpolated bilinearly from the gridded finite-difference estimates, and the same fourth-order Runge–Kutta scheme is used as for integrating the trajectories.

_{ij}#### 3) Mean dilation rate estimates

*D*denote the area-averaged divergence for a region Ω(

*t*):

*t*). See appendix A for the derivation.

At time *t _{o}*, all the regions of interest are triangles, but after some time the boundary of Ω(

*t*) will have evolved away from a perfect triangle. For an accurate estimate of

*A*(

*t*+ Δ

_{o}*t*), the initial triangle boundary is discretized with 10-m spacing. This boundary is advected in the flow field. When adjacent boundary points have drifted farther than 10 m apart, a new point is inserted midway between them. Then

*A*(

*t*+ Δ

_{o}*t*) is computed as the area of the resulting polygon. Since the analysis here considers 3-h intervals, new triangle boundaries are initialized from the simulated drifter positions every 3 h, then advected for 3 h This estimate serves as the baseline against which the drifter-based approximation is evaluated.

#### 4) Simulated drifter-based mean dilation rate

After the initially uniform grid of particles is triangulated, the triangles are followed for 12 h. Divergence averaged over each triangle’s area and over 3-h time intervals is computed from the simulated drifter positions every 3 h. This time span coincides with the model output frequency and was chosen as a time span that is long enough to allow significant deformation in the model flow and short enough for trajectories computed using linear interpolation between model output times to be reasonable approximations to those obtained if higher-resolution model output were available (not shown, but tested on a 400-m resolution WMOP run). When the right-hand side of Eq. (4) is evaluated from the drifter trajectories, *A*(*t _{o}* + Δ

*t*) is approximated by the new triangle defined by the positions of the drifter triplet at time

*t*+ Δ

_{o}*t*. It is assumed that this is an adequate approximation to the image under advection by the flow field of the initial triangle. The ramifications of this assumption are discussed in section 3i.

### d. Shape metrics

*A*to the square of the perimeter

*P*(Berta et al. 2016):

*πA*/

*P*

^{2}, which assesses how close a two-dimensional shape is to a circle (e.g., Cox 1927; Blott and Pye 2008), also known as the isoperimetric quotient (Osserman 1978). Intuitively, the velocity gradient information gleaned from the drifter triplet is more robust when the drifters sample both coordinate directions equally, i.e., when Λ is near 1. Moreover, the area method for computing divergence fails completely when the triad collapses, i.e., when Λ = 0.

*θ*and

*γ*, where

*θ*measures the largest interior angle and

*γ*is the ratio of the shortest side length to the middle side length (Merrifield et al. 2010). Unlike several other triangle shape metrics (Pumir et al. 2000; Cressman et al. 2004), these have intuitive geometric interpretations (Merrifield et al. 2010).

*θ*can vary from

*π*/3 for equilateral triangles to

*π*for colinear points;

*γ*can vary from 0 for collapsed triangles with two coincident vertices to 1 for isosceles triangles. Geometry also constrains

*γ*to satisfy

*γ*≥ 2cos

*θ*. Since

*γ*and

*θ*fully describe a triangle, up to similarity, they also determine a unique Λ:

### e. Simulated position uncertainty

To evaluate the impact of position uncertainty inherent to in situ observations on our conclusions, we run a Monte Carlo simulation with 100 samples. For each sample, a random position error is generated for each of the 10 340 simulated drifter positions used in the divergence calculations (2068 trajectories sampled at 5 times). Then the divergence estimates and triangle shape statistics are calculated from the modified positions. The magnitude of the position errors is drawn from a normal distribution with mean 0 and standard deviation 5 m, while the direction is drawn from a uniform distribution on [0°, 360°]. These distributions are consistent with the root-mean-square GPS error of 2 to 50 m reported by Centurioni (2018) for the modern Surface Velocity Program drifters used in the observational example (see next subsection). They also reflect error statistics for other popular modern surface drifters, such as the CARTHE drifters (Novelli et al. 2017).

### f. Drifter observations

The observations used for the application described in section 4 are drawn from a set of Surface Velocity Program (SVP) drifter trajectories collected during the 2018 CALYPSO field experiment in the Alboran Sea in the western Mediterranean Sea (Mahadevan et al. 2020). The cruise took place in May–June 2018. The SVP drifters, part of the Global Drifter Program (Niiler 2001; Centurioni 2018), consist of a spherical surface buoy tethered to a weighted drogue, at a nominal center depth of 15 m. Below the surface buoy, a tether strain gauge measures the tension of the buoy–drogue connection to monitor the drogue presence, and a thermometer measures sea surface temperature. During the 2018 experiment, 52 SVP drifters were deployed, and GPS data were transmitted via Iridium every 10 min. The raw dataset was quality controlled for outliers and interpolated to common times every 15 min.

The analysis here focuses on a subset of 29 SVP drifters that were deployed on 31 May 2018 over 3 h, starting at 1340 UTC, in a symmetric, cross-shaped array in a roughly 12 km × 12 km area. It was chosen to maximize chances of finding nearly collocated and nearly simultaneous triplets sampling the full range of Λ values at a variety of length scales. Drifters forming triangles of average side length up to 10 km were selected every 30 min for a time window of 14.5 h starting at 1430 UTC 31 May 2018. Since the drifters were not released in this formation, we refer to the resulting groups as “chance” groupings. To ensure independent sampling, triplets sharing more than one drifter with another triplet are eliminated. Divergence, averaged over 30-min time intervals, is estimated using the algorithm from Molinari and Kirwan (1975), as described above in section 2c(4). Vorticity, also averaged over 30 min, can be estimated in a similar fashion, after applying a 90° clockwise rotation to the velocity vectors implied by the drifter positions. For details, see Saucier (1955) and Molinari and Kirwan (1975). Note that in contrast to the simulated drifter triads, which are tracked continuously over the entire time interval, the observations are split into new triads for each time interval, in order to maximize the number of samples and minimize the effect of inhomogeneous sampling driven by convergent regions (Pearson et al. 2020; Sun et al. 2020). The shorter averaging time interval also increases the number of available samples. Moreover, the triangles in the real ocean experience greater deformation than the simulated ones, since the model neglects subgrid-scale velocity variability. This suggests that a shorter time interval is appropriate. In total, there are 211 samples in the [0, 2] km bin, 1237 samples in the (2, 4] km bin, 1343 samples in the (4, 6] km bin, and 2517 samples in the (6, 10] km bin.

Like the simulation, the SVP deployment took place near the salinity front associated with the East Alboran Gyre, where the flow is prone to strong deformation processes. The frontal position was captured by an underway conductivity–temperature–depth (UCTD) probe (Rudnick and Klinke 2007) used during the ship survey conducted around the time of the drifter deployment (see Figs. 1 and 3 of Esposito et al. 2021). Details of the UCTD data sampling and processing are provided by Mahadevan et al. (2020) and Dever et al. (2019).

## 3. Model analysis

### a. Simulated drifter triads: Shape statistics

The evolution of the 500-m triangulated grid as it deforms is shown on the salinity field in Fig. 2. Initially, all triangles are approximately identical isosceles right triangles with side lengths 425 m and hypotenuse of length 602 m. This corresponds to a Λ value of 0.89. Over the 12-h period, the middle part of the grid collapses onto the salinity front, while the northwestern portion is rolled up into a cyclonic eddy. The northeastern and southwestern corners meanwhile remain mostly intact. In other words, the 3956 triangles experience a variety of fates.

Triangulated grid (black) on modeled salinity field (colored background) (a) at launch, (b) 3 h after launch, (c) 6 h after launch, (d) 9 h after launch, and (e) 12 h after launch.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Triangulated grid (black) on modeled salinity field (colored background) (a) at launch, (b) 3 h after launch, (c) 6 h after launch, (d) 9 h after launch, and (e) 12 h after launch.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Triangulated grid (black) on modeled salinity field (colored background) (a) at launch, (b) 3 h after launch, (c) 6 h after launch, (d) 9 h after launch, and (e) 12 h after launch.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

This is reflected in the histograms of realized Λ values in Fig. 3. Roughly 2% of triangles retain Λ ≥ 0.89 throughout the analysis time window; including triangles that may have been previously deformed, about 10% satisfy Λ ≥ 0.89 at 6 h postlaunch or later. The number of highly distorted triangles increases significantly over time: While there are no triangles with Λ ≤ 0.2 after 3 h, about a third of the triangles fall into this group after 12 h. If a conservative cutoff value of Λ = 0.7 is used, as by Berta et al. (2020b), almost 30% of triangles after 3 h and more than 75% of triangles after 12 h would have to be discarded. (If the goal is to estimate instantaneous or short-term averaged divergence rather than dilation rate or the evolution of the divergence of a particular patch of water, some of this loss of samples can be remedied by retriangulating at later times.)

Histograms of Λ for the triangles in the analysis set (a) 3, (b) 6, (c) 9, and (d) 12 h after launch. All triangles are initialized with Λ ≈ 0.89. The red vertical lines indicate the median value of each sample distribution.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Histograms of Λ for the triangles in the analysis set (a) 3, (b) 6, (c) 9, and (d) 12 h after launch. All triangles are initialized with Λ ≈ 0.89. The red vertical lines indicate the median value of each sample distribution.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Histograms of Λ for the triangles in the analysis set (a) 3, (b) 6, (c) 9, and (d) 12 h after launch. All triangles are initialized with Λ ≈ 0.89. The red vertical lines indicate the median value of each sample distribution.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

### b. Early triads: Divergence estimates

We begin our discussion by focusing on the first 3-h time period, when triangles generally remain nearly equilateral, with a median Λ value of 0.79 and less than 15% with Λ < 0.6. Even over this relatively short time window, corresponding to the model archiving time step, the instantaneous divergence field evolves; see Figs. 4a and 4b. Thus, each simulated drifter experiences an evolving divergence field. The Lagrangian along-trajectory temporal average of the divergence is given by the dilation rate in Fig. 4c. Some of the drifters consistently experience positive divergence, for example, in the red swath in the southwest corner. Others experience both positive and negative divergence, as exemplified by those around 36°28′N, 1°48′ W at *t* = 3 h that are convergent at that time (blue in Fig. 4b) but experience, on average, positive divergence (red in Fig. 4c). In fact, the instantaneous divergence changes sign over this short time interval for about 40% of the samples. It is therefore important to remember that the drifter-based estimates from the area-based algorithm represent temporal and spatial averages rather than point estimates, as those obtained from least squares fits applied to drifter trajectories. The spatial mean of the dilation rate over the evolving triangles, shown in Fig. 4d, looks quite similar to the gridded dilation rate, if slightly less smooth, even though it was computed with a completely different algorithm. This quantity will be used as the baseline in the following to assess the drifter-based estimates. The latter are displayed in Fig. 4e.

Divergence in the study area, normalized by *f*. (a) Finite-difference estimates of instantaneous divergence on the grid at launch time, piecewise linearly interpolated for graphing. (b) As in (a), but 3 h after launch. (c) Dilation rate (Lagrangian temporal average divergence) computed on the model grid over the time interval 0–3 h. (d) Triangle-mean dilation rate over the time interval 0–3 h. (e) Drifter-derived mean dilation rate estimate over the time interval 0–3 h.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Divergence in the study area, normalized by *f*. (a) Finite-difference estimates of instantaneous divergence on the grid at launch time, piecewise linearly interpolated for graphing. (b) As in (a), but 3 h after launch. (c) Dilation rate (Lagrangian temporal average divergence) computed on the model grid over the time interval 0–3 h. (d) Triangle-mean dilation rate over the time interval 0–3 h. (e) Drifter-derived mean dilation rate estimate over the time interval 0–3 h.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Divergence in the study area, normalized by *f*. (a) Finite-difference estimates of instantaneous divergence on the grid at launch time, piecewise linearly interpolated for graphing. (b) As in (a), but 3 h after launch. (c) Dilation rate (Lagrangian temporal average divergence) computed on the model grid over the time interval 0–3 h. (d) Triangle-mean dilation rate over the time interval 0–3 h. (e) Drifter-derived mean dilation rate estimate over the time interval 0–3 h.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

The similarity between Figs. 4d and 4e is visually striking and is illustrated more clearly in the scatterplot in Fig. 5a. The correlation between the two estimates is 0.997 with a root-mean-square (RMS) difference of 0.03*f*. The errors roughly follow a Gaussian distribution (Fig. 5b). There is no clear pattern of larger differences being associated with greater (in magnitude) divergence. However, the drifter-based estimates at the ends of the distribution tend to exceed in magnitude the gridded estimates. This is likely a consequence of the drifter-based methodology assuming that the triangle defined by the advected vertices is a good approximation to the evolved boundary of the initial shape. When divergence is high in magnitude, the large changes in area may be accompanied with more significant deviations of the boundary from straight lines. This will be discussed further in section 3i.

(a) Scatterplot of triangle-mean dilation rate for 0 − 3 h from synthetic drifters and the baseline using highly resolved patch boundaries. The least squares linear fit to the data (not shown) is *y* = 0.9969*x* + 0.000 45. Correlation is 0.997, and the RMS difference is 0.03*f*. (b) Empirical probability density function of the differences in the mean dilation rate estimates. The red curve shows the normal distribution with the same standard deviation (0.03).

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

(a) Scatterplot of triangle-mean dilation rate for 0 − 3 h from synthetic drifters and the baseline using highly resolved patch boundaries. The least squares linear fit to the data (not shown) is *y* = 0.9969*x* + 0.000 45. Correlation is 0.997, and the RMS difference is 0.03*f*. (b) Empirical probability density function of the differences in the mean dilation rate estimates. The red curve shows the normal distribution with the same standard deviation (0.03).

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

(a) Scatterplot of triangle-mean dilation rate for 0 − 3 h from synthetic drifters and the baseline using highly resolved patch boundaries. The least squares linear fit to the data (not shown) is *y* = 0.9969*x* + 0.000 45. Correlation is 0.997, and the RMS difference is 0.03*f*. (b) Empirical probability density function of the differences in the mean dilation rate estimates. The red curve shows the normal distribution with the same standard deviation (0.03).

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

The results so far suggest that drifter trajectories can yield a highly accurate estimate of mean dilation rate. However, the first 3-h interval is a gentle test for the methodology, as the triangles start out nearly equilateral, and deformation is minimal over such a short time interval (Fig. 3). Thus, we will next examine a later 3-h time interval, where the starting triangles exhibit more extreme aspect ratios as measured by Λ.

### c. Late triads: Divergence estimates

The time interval from 9 to 12 h after launch offers an opportunity to study the impact of triangle deformation on the divergence estimate: The median Λ at the beginning of this period is 0.51 and drops to 0.38 at the end (Fig. 3). As Fig. 2 illustrates, it is this last time interval when the deformation of the grid along the salinity front results in many highly elongated triangles. At the same time, the triangles around the edges of the eddy also become highly deformed, although those near the center retain a high aspect ratio; see Fig. 6a.

Shape metrics for the triangles 12 h after launch: (a) Λ, (b) *θ*/π, (c) *γ*, and (d) average side length (scale).

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Shape metrics for the triangles 12 h after launch: (a) Λ, (b) *θ*/π, (c) *γ*, and (d) average side length (scale).

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Shape metrics for the triangles 12 h after launch: (a) Λ, (b) *θ*/π, (c) *γ*, and (d) average side length (scale).

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Figure 7 shows the mean dilation rate estimates as computed using highly resolved boundaries (Fig. 7a) and as computed using the triangle areas implied by the synthetic drifter trajectories, as could be achieved with observations (Fig. 7b). The plots look very similar, but close inspection reveals particularly strong disagreement between the two estimates along the front near the southeastern edge of the eddy: In a narrow band in this region, the drifters suggest strong positive divergence, whereas the baseline calculation shows negative divergence (or convergence). In addition, along the western side of the eddy, the drifter-based estimates suggest high spatial variability, with mean dilation rate alternating from strongly positive to negative and back for neighboring triangles. (This is not the case for all triangle sequences, but there are multiple examples.) In contrast, the baseline estimate shows strong but smooth gradients in this area. See the insets in Fig. 7. This observation suggests that the drifter-based estimates have significant errors here.

Mean dilation rate in the study area, normalized by *f*, averaged over triangle areas and over the interval from 9 to 12 h after launch. (a) Baseline estimate based on highly resolved patch boundaries. (b) Estimate derived from the simulated drifters. (c) As in (b), but with data for triangles not meeting the threshold criterion min(Λ) > 0.20 colored in green. The insets show greater detail of the southwestern edge of the eddy, indicated by the orange box.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Mean dilation rate in the study area, normalized by *f*, averaged over triangle areas and over the interval from 9 to 12 h after launch. (a) Baseline estimate based on highly resolved patch boundaries. (b) Estimate derived from the simulated drifters. (c) As in (b), but with data for triangles not meeting the threshold criterion min(Λ) > 0.20 colored in green. The insets show greater detail of the southwestern edge of the eddy, indicated by the orange box.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Mean dilation rate in the study area, normalized by *f*, averaged over triangle areas and over the interval from 9 to 12 h after launch. (a) Baseline estimate based on highly resolved patch boundaries. (b) Estimate derived from the simulated drifters. (c) As in (b), but with data for triangles not meeting the threshold criterion min(Λ) > 0.20 colored in green. The insets show greater detail of the southwestern edge of the eddy, indicated by the orange box.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Indeed, the scatterplot (Fig. 8a) confirms the presence of large outliers, with differences up to 6.22*f*, and the correlation for this time period is much reduced at 0.626. Nonetheless, a correlation of this magnitude indicates that many samples exhibit good agreement, in spite of some with very large errors. This is confirmed when the 10% of data points with the largest errors (Fig. 8c) are excluded: The agreement is still not as tight as for the early time period (Fig. 5), but it is now very good (Fig. 8b), with correlation 0.959 and RMS difference of 0.08*f* (versus 0.37*f* for the whole dataset).

(a) Scatterplot of triangle-mean dilation rate for 9 − 12 h from synthetic drifters and the baseline using highly resolved patch boundaries. The least squares linear fit to the data (not shown) is *y* = 0.37*x* − 0.09. Correlation is 0.626, and the RMS difference is 0.37*f*. (b) As in (a), but only for the best 90% of data points. The least squares linear fit to the data (not shown) is *y* = 0.93*x* − 0.01. Correlation is 0.959, and the RMS difference is 0.08*f*. (c) Differences in the mean dilation rate estimates, sorted in increasing order. The vertical dashed lines indicate the bottom and top 5%. In all panels, each data point is colored by the minimum Λ value of the two triangles entering each drifter-based estimate. Where data points overlap, those with higher Λ values are plotted on top.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

(a) Scatterplot of triangle-mean dilation rate for 9 − 12 h from synthetic drifters and the baseline using highly resolved patch boundaries. The least squares linear fit to the data (not shown) is *y* = 0.37*x* − 0.09. Correlation is 0.626, and the RMS difference is 0.37*f*. (b) As in (a), but only for the best 90% of data points. The least squares linear fit to the data (not shown) is *y* = 0.93*x* − 0.01. Correlation is 0.959, and the RMS difference is 0.08*f*. (c) Differences in the mean dilation rate estimates, sorted in increasing order. The vertical dashed lines indicate the bottom and top 5%. In all panels, each data point is colored by the minimum Λ value of the two triangles entering each drifter-based estimate. Where data points overlap, those with higher Λ values are plotted on top.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

(a) Scatterplot of triangle-mean dilation rate for 9 − 12 h from synthetic drifters and the baseline using highly resolved patch boundaries. The least squares linear fit to the data (not shown) is *y* = 0.37*x* − 0.09. Correlation is 0.626, and the RMS difference is 0.37*f*. (b) As in (a), but only for the best 90% of data points. The least squares linear fit to the data (not shown) is *y* = 0.93*x* − 0.01. Correlation is 0.959, and the RMS difference is 0.08*f*. (c) Differences in the mean dilation rate estimates, sorted in increasing order. The vertical dashed lines indicate the bottom and top 5%. In all panels, each data point is colored by the minimum Λ value of the two triangles entering each drifter-based estimate. Where data points overlap, those with higher Λ values are plotted on top.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Our hypothesis is that the large errors are associated with the distortion of the triangles from a near-equilateral shape. To test this, consider the aspect ratios represented in the group of the 10% of triangles with largest errors, versus those represented in the group of the remaining 90% of triangles, as shown in Fig. 9. Since each mean dilation rate calculation involves the areas at the beginning and at the end of the time interval, we choose as the salient parameter the smaller of the two realized Λ values. Note that none of the samples in the group with large errors has min(Λ) > 0.25. Thus, using a cutoff value of Λ = 0.25 would eliminate all of these instances. However, it would also eliminate 35% of the samples with smaller errors.

Histograms of min(Λ) for the triangles 9 and 12 h after launch, (a) for the 10% of samples with largest errors in the mean dilation rate over the interval 9–12 h and (b) for the 90% with the smallest such errors.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Histograms of min(Λ) for the triangles 9 and 12 h after launch, (a) for the 10% of samples with largest errors in the mean dilation rate over the interval 9–12 h and (b) for the 90% with the smallest such errors.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Histograms of min(Λ) for the triangles 9 and 12 h after launch, (a) for the 10% of samples with largest errors in the mean dilation rate over the interval 9–12 h and (b) for the 90% with the smallest such errors.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

A reasonable compromise is a threshold of Λ = 0.20: This removes 99% of samples with large errors and only 28% of those with small errors, for a total of 35% of all triads. Conversely, with this threshold, more than 99.5% of the retained triangles (all but 11 of the 2561 samples) produce a mean dilation rate estimate with error less than 0.25*f*, while just over 66.7% of the rejected triangles also have such small errors. After thresholding with Λ = 0.20, the synthetic drifter-based and baseline mean dilation rate estimates show a correlation of 0.984. For comparison, when errors are known and the same number of triads is removed by choosing those with the largest errors, the correlation is 0.997.

Retaining only samples with min(Λ) > 0.20 also largely resolves the two problem areas along the southeastern and southwestern edges of the eddy identified above: As shown in Fig. 7c, the area southeast of the eddy and feeding into the northern part of the front that was incorrectly identified as experiencing strong positive divergence has been eliminated as suspect. The same is true for much of the southwestern edge of the eddy, seen in the figure inset, where synthetic drifter-based estimates falsely posited large gradients between strong convergence and strong divergence in neighboring triangles.

### d. Statistics for all 3-h intervals and alternative thresholds

The critical Λ value was identified based on the triads evolving from 9 to 12 h postlaunch, but, of course, to be useful it should hold more generally. Fortunately, the statistics look even better for the larger sample set including mean dilation rate estimates from 0 to 3, 3 to 6, 6 to 9, and 9 to 12 h. Of this larger set, 86% of triangles satisfy min(Λ) > 0.2, and 99.8% of these have errors less than 0.25*f* in magnitude. The correlation for the remaining samples between the synthetic drifter-based and baseline mean dilation rate estimates is 0.991, and the RMS error is 0.04*f*. The price for retaining almost only good estimates by removing 95% of estimates with errors greater than 0.25*f* is that 74% of the removed samples also lead to reasonable estimates with errors less than 0.25*f*. Thus, 11% of decent samples are being discarded. This raises the question whether an alternative threshold can be found that is better at eliminating bad estimates while retaining good ones.

To answer this question, we consider the secondary shape metrics *θ* and *γ*. These are defined in section 2d and illustrated in Fig. 10a with some sample triangles. Low Λ values can be associated either with high *θ* values—triads that are close to colinear (“short and fat” triangles)—or with low *γ* values—triads that have two points very close and the third farther away (“tall and skinny” triangles). Figures 6b and 6c show the geographic distribution of *θ*/*π* and *γ*, respectively, for the triangles 12 h after the launch. Figure 10b illustrates the relationship between the different shape metrics, also defined by Eq. (6), by coloring each sample in the *γ*–*θ* space by the corresponding Λ value. The figure also identifies those samples associated with large mean dilation rate estimate errors (>0.25*f*) by circling them in red. (Note that for this plot the errors are assigned to the shape metrics of the triangle at the end of the averaging time interval.) These large errors are, as expected, predominantly found for low Λ values, colored in dark blue. However, they are also mostly in the region of high *θ*, while covering the full range of *γ* values. The shape metric *γ* thus does not play a role in identifying triangles with potentially large errors in mean dilation rate estimates.

(a) Sample triangle shapes associated with a variety of *θ*–*γ* pairs (after Merrifield et al. 2010). (b) Each triangle at one of the four times (3, 6, 9, or 12 h after launch) is represented by a dot showing its *γ* and *θ* values and colored by its Λ value. Note that triangles are initialized as isosceles right triangles with *γ* = 1, *θ* = *π*/2, and Λ = 0.89. Dots for triangles with large synthetic drifter-based mean dilation rate estimate errors for the preceding 3 h (>0.25*f*) are circled in red. (c) The triangles are located in *θ*–Λ space, where the color corresponds to the mean dilation rate estimate error magnitude. Red lines are drawn at Λ = 0.2 and *θ* = 0.86*π*. In all panels, the gray regions cover portions of the parameter space that cannot be occupied by a triangle; see appendix B.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

(a) Sample triangle shapes associated with a variety of *θ*–*γ* pairs (after Merrifield et al. 2010). (b) Each triangle at one of the four times (3, 6, 9, or 12 h after launch) is represented by a dot showing its *γ* and *θ* values and colored by its Λ value. Note that triangles are initialized as isosceles right triangles with *γ* = 1, *θ* = *π*/2, and Λ = 0.89. Dots for triangles with large synthetic drifter-based mean dilation rate estimate errors for the preceding 3 h (>0.25*f*) are circled in red. (c) The triangles are located in *θ*–Λ space, where the color corresponds to the mean dilation rate estimate error magnitude. Red lines are drawn at Λ = 0.2 and *θ* = 0.86*π*. In all panels, the gray regions cover portions of the parameter space that cannot be occupied by a triangle; see appendix B.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

(a) Sample triangle shapes associated with a variety of *θ*–*γ* pairs (after Merrifield et al. 2010). (b) Each triangle at one of the four times (3, 6, 9, or 12 h after launch) is represented by a dot showing its *γ* and *θ* values and colored by its Λ value. Note that triangles are initialized as isosceles right triangles with *γ* = 1, *θ* = *π*/2, and Λ = 0.89. Dots for triangles with large synthetic drifter-based mean dilation rate estimate errors for the preceding 3 h (>0.25*f*) are circled in red. (c) The triangles are located in *θ*–Λ space, where the color corresponds to the mean dilation rate estimate error magnitude. Red lines are drawn at Λ = 0.2 and *θ* = 0.86*π*. In all panels, the gray regions cover portions of the parameter space that cannot be occupied by a triangle; see appendix B.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Figure 10c shows how triangles with different error magnitudes are distributed in *θ*–Λ space. The group of triangles with Λ ≤ 0.2 (below the horizontal red line in the figure) includes many with low errors. Those with large errors (bright colors) are concentrated in the corner with high *θ* and low Λ values. A better threshold might therefore be imposed on *θ* instead, since all triangles with large *θ* perforce also have low Λ, due to the geometric constraints discussed in appendix B, whereas the converse is clearly not true. The distribution of samples with large errors shown in Fig. 10 suggests a *θ* threshold of 0.86*π*. With this choice, 85% of triangles are retained, of which 99.8% have errors less than 0.25*f*. For this subsample, the correlation between synthetic drifter-based and baseline mean dilation rate estimates is 0.992 and the RMS error is 0.04*f*. This thresholding eliminates 96% of samples with errors greater than 0.25*f* but also 12% of those with smaller errors. In other words, its performance is nearly identical to that of the Λ threshold, as measured by these statistics. About 4% of samples (roughly 690) are only eliminated by one and not the other threshold.

What about other choices for threshold values of these two shape parameters? What value of *θ* or Λ makes the best cutoff is a trade-off between removing a larger percentage of poor estimates and retaining a larger percentage of good estimates. The definition of “poor” and “good” also impacts the choice, and what is considered an acceptable level of error is situation dependent. For our purposes here, we are targeting the removal of errors greater than 0.25*f*, which ultimately results in an RMS error an order of magnitude smaller than that. A variety of different criteria can be applied to choose a threshold quantitatively. The top panels of Figs. 11a and 11b illustrate the consequences of different choices on the fraction of all samples eliminated (black), the fraction of samples with large errors removed (red), and the fraction of samples with small errors kept (green). Λ thresholds are applied to the smaller Λ realized at the time interval endpoints, and similarly for *θ*, where the larger one is considered. The steps in the curves at Λ = 0.89 and *θ* = *π*/2 are a consequence of the oversampling of these values, as they are assumed by all initial triangles. The figure presents results for two different choices for the definition of “large” error, 0.25*f* and 0.1*f*.

(a),(b) (top) The fraction of triads removed from the sample (black), the fraction of triads with “large” errors removed (red), and the fraction of triads with “small” errors retained (green) as a function of the chosen threshold in either (left) Λ or (right) *θ* at the 500-m scale. For (a), “large” errors are defined as those greater than 0.25*f*, while (b) uses 0.1*f*. (bottom) The gain defined in the text as a function of threshold. The vertical gray lines highlight threshold values where the gain reaches 0. (c) (top) The RMS error (RMSE) in the normalized mean dilation rate estimates of the retained samples and (bottom) the related cost function *C* defined by Eq. (7) as a function of (left) Λ and (right) *θ* thresholds for the five sampled scales. The vertical lines correspond to Λ = 0.2 and *θ* = 0.86*π*.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

(a),(b) (top) The fraction of triads removed from the sample (black), the fraction of triads with “large” errors removed (red), and the fraction of triads with “small” errors retained (green) as a function of the chosen threshold in either (left) Λ or (right) *θ* at the 500-m scale. For (a), “large” errors are defined as those greater than 0.25*f*, while (b) uses 0.1*f*. (bottom) The gain defined in the text as a function of threshold. The vertical gray lines highlight threshold values where the gain reaches 0. (c) (top) The RMS error (RMSE) in the normalized mean dilation rate estimates of the retained samples and (bottom) the related cost function *C* defined by Eq. (7) as a function of (left) Λ and (right) *θ* thresholds for the five sampled scales. The vertical lines correspond to Λ = 0.2 and *θ* = 0.86*π*.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

(a),(b) (top) The fraction of triads removed from the sample (black), the fraction of triads with “large” errors removed (red), and the fraction of triads with “small” errors retained (green) as a function of the chosen threshold in either (left) Λ or (right) *θ* at the 500-m scale. For (a), “large” errors are defined as those greater than 0.25*f*, while (b) uses 0.1*f*. (bottom) The gain defined in the text as a function of threshold. The vertical gray lines highlight threshold values where the gain reaches 0. (c) (top) The RMS error (RMSE) in the normalized mean dilation rate estimates of the retained samples and (bottom) the related cost function *C* defined by Eq. (7) as a function of (left) Λ and (right) *θ* thresholds for the five sampled scales. The vertical lines correspond to Λ = 0.2 and *θ* = 0.86*π*.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

The suggested thresholds of Λ = 0.2 and *θ* = 0.86*π* are marked by gray lines in Fig. 11a. As discussed above, these two thresholds perform comparably. If Λ is increased or *θ* decreased, more samples are removed, both with large and with small errors. This trade-off may be desirable. However, at some point, very little is gained in terms of removing more poor samples while incurring substantial losses of good samples. One way to measure this quantitatively is to consider the gain, defined as the sum of the gain in poor sample removal and that in good sample retention; the gain in each of these quantities is taken to be the difference in the fraction of samples between consecutive sample points (a scaled discrete forward-difference derivative). A positive gain indicates that an increase in the threshold leads to a greater increase in the fraction of poor samples removed than the decrease in the fraction of good samples retained (or, for the *θ* threshold, a greater increase in the retained fraction than the decrease in the discarded fraction). If both criteria are valued equally, one would target a zero gain. The two suggested thresholds satisfy this condition (to three decimal places). Nearby values of Λ and *θ* also show minimal gain, but when the threshold for Λ is decreased below 0.2 or that for *θ* is increased beyond 0.86*π*, the figures show steep losses in the fraction of poor samples being eliminated, which do not justify the small gain in good samples being kept.

These windows of optimal thresholds in this sense move somewhat when smaller errors are targeted. Thus, Fig. 11b shows the gain reaches 0 for Λ = 0.3 and for *θ* = 0.8*π* when “large” errors are defined as those exceeding 0.1*f*. Again, slightly larger values of Λ—up to about 0.4—and slightly smaller values of *θ*—down to about 0.72*π*—also correspond to very small gains.

*f*. As shown by the red line in the top panels of Fig. 11c, the gains in RMSE reduction for the 500-m scale level off right around Λ = 0.2 and

*θ*= 0.86

*π*. Minima, however, are only reached when practically all samples are removed, which is not a useful threshold. To take into account the need to retain as many samples as possible while reducing the RMSE as much as possible, a cost function

*C*is defined as

*θ*= 0.88

*π*.

### e. Scale dependence

All the results discussed so far are obtained using the basic set of simulated drifter trajectories launched on the original model grid of 425 m. Here we investigate the scale dependence of the results, considering increasingly larger initial triangles for trajectory launches up to 12 km apart (Fig. 1). As a result, a total of 5 triad scales are considered: 500 m, 2 km, 5 km, 7 km, and 12 km. To maintain the sampling within the same frontal feature, the number of available trajectories decreases with increasing grid size going from 3956 sampled triads for 500 m to 70 triads for 12 km (see section 2a).

The results are summarized in Fig. 11c in terms of RMSE in the mean dilation rate estimates and the cost function *C* defined in Eq. (7) as functions of the thresholds for parameters Λ and *θ*, for each of the five scales. Generally, the RMSE tends to increase with scale. This is due to the fact that the three points used to track the patch boundary become less and less effective as the scales increase and the deformation of the boundary segments into filaments becomes more prominent. For the 500-m triads this effect is likely underrepresented, given that this is the model resolution and subgrid-scale variability is limited. In the real ocean, of course, this is not the case so that a somewhat greater degree of error related to the triad approximation is expected to occur even at small scales, while the increase in error at increasing scales might be less pronounced than in the model.

The dependence on the parameter thresholds shows a qualitatively similar trend across scales, with a slight decrease in RMSE as the Λ threshold increases (and *θ* decreases) and a most significant decrease at the lower end for Λ and the upper end for *θ*. For very large Λ and small *θ* thresholds the curves get noisy, because the statistics become less reliable (fewer samples). Overall, the thresholds Λ = 0.2 and *θ* = 0.86 (gray vertical lines in Fig. 11c) still appear to be reasonable choices for the larger scales, since they identify the range of reduced gradient in the error. This is confirmed also by the scatterplots of mean dilation rate estimates versus the baselines at various scales (not shown), which indicate that the thresholds are effective in eliminating the worst estimates. For the 2- and 5-km scales, the thresholds are close to the values minimizing *C*: 0.22 and 0.23, respectively, for Λ and 0.84*π* and 0.87*π* for *θ*. For the largest scales, the minimizing threshold values are Λ = 0.44 and 0.37 and *θ* = 0.78*π* and 0.79*π* for the 7- and 12-km scales, respectively. However, the cost function is very flat for the 7-km scale. At the largest scale, the cost function shows a true minimum for most stringent thresholds. This is partly a consequence of there being fewer highly distorted samples at the larger scales: By increasing the threshold from Λ = 0.2 to Λ = 0.37, e.g., few additional samples are being discarded, while the RMSE decreases somewhat.

It should be noted that drifter triads at different scales measure slightly different things, since the resulting estimate is a spatial average of the dilation rate rather than a point estimate. Observational studies have suggested that the RMS mean dilation rate decreases with increasing scale in general (Esposito et al. 2021; Berta et al. 2016). This is to be expected, as the spatial averaging constitutes a smoothing over larger and larger areas. It is also reflected in the model, where the RMS baseline mean dilation rate computed over the sample triangles decreases from 0.31*f* for the 500-m scale to 0.22*f* for the 2-km scale, 0.17*f* for the 5-km scale, 0.15*f* for the 7-km scale, and 0.12*f* for the 12-km scale.

### f. Subsampling bias

One concern with using such a shape-metric threshold for weeding out potentially bad mean dilation rate estimates is that it may bias the sampling. For example, strong convergence is sometimes accompanied by strong deformation, so that removing strongly deformed triads could preferentially remove triads sampling such strong convergence regions. That, in turn, could impact the statistics derived from the drifter data. Deriving domainwide statistics from drifter trajectories is known to include a Lagrangian bias (Berta et al. 2020b), as drifters tend to spend more time in convergence areas (Jacobs et al. 2016). Here we explore whether the shape-based subsampling introduces an additional bias.

We check the subsampling bias by considering the distributions of the mean dilation rate estimates based on the highly resolved boundary calculation (our baseline) before and after subsampling. The RMS mean dilation rate over the sample triangles at all scales is not impacted by the proposed subsampling for Λ > 0.2 or *θ* < 0.86*π*. For a more detailed look at the smallest scale, Fig. 12 shows the histograms, in Fig. 12a for triangles from all time periods and in Figs. 12b and 12c for triangles from an early and a late interval only, respectively. In each panel, the blue histogram shows the distribution for all the data from the given time period, the red histogram shows the distribution for the samples satisfying Λ > 0.2, and the yellow histogram shows the distribution for the samples satisfying *θ* < 0.86*π*. These mean dilation rate estimates are not influenced by the triangle shape; so the blue histogram can be thought of as the desired distribution, while the other histograms are those achievable with subsampling.

(a) Histograms for the normalized triangle-mean dilation rate, for all triangles (blue), for triangles with Λ > 0.2 (red), and for triangles with *θ* < 0.86*π* (yellow). (b) As in (a), but considering only triangles from the time interval from 3 to 6 h after launch. (c) As in (a), but considering only triangles from the time interval from 9 to 12 h after launch. (d) Spatial averages over the Eulerian domain shown in Fig. 2 of instantaneous divergence (purple with crosses) and gridded 3-h dilation rate (green with stars). Also shown is the average over all triangles of the triangle-mean dilation rate estimates for each 3-h time period (blue with circles), whereby red and yellow circles show means computed over the Λ and *θ* subsamples only, respectively. (For the first time interval, no triads are discarded due to shape distortion.) Dilation rates are shown at the midpoint of the computational time interval.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

(a) Histograms for the normalized triangle-mean dilation rate, for all triangles (blue), for triangles with Λ > 0.2 (red), and for triangles with *θ* < 0.86*π* (yellow). (b) As in (a), but considering only triangles from the time interval from 3 to 6 h after launch. (c) As in (a), but considering only triangles from the time interval from 9 to 12 h after launch. (d) Spatial averages over the Eulerian domain shown in Fig. 2 of instantaneous divergence (purple with crosses) and gridded 3-h dilation rate (green with stars). Also shown is the average over all triangles of the triangle-mean dilation rate estimates for each 3-h time period (blue with circles), whereby red and yellow circles show means computed over the Λ and *θ* subsamples only, respectively. (For the first time interval, no triads are discarded due to shape distortion.) Dilation rates are shown at the midpoint of the computational time interval.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

(a) Histograms for the normalized triangle-mean dilation rate, for all triangles (blue), for triangles with Λ > 0.2 (red), and for triangles with *θ* < 0.86*π* (yellow). (b) As in (a), but considering only triangles from the time interval from 3 to 6 h after launch. (c) As in (a), but considering only triangles from the time interval from 9 to 12 h after launch. (d) Spatial averages over the Eulerian domain shown in Fig. 2 of instantaneous divergence (purple with crosses) and gridded 3-h dilation rate (green with stars). Also shown is the average over all triangles of the triangle-mean dilation rate estimates for each 3-h time period (blue with circles), whereby red and yellow circles show means computed over the Λ and *θ* subsamples only, respectively. (For the first time interval, no triads are discarded due to shape distortion.) Dilation rates are shown at the midpoint of the computational time interval.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

In absolute terms, the subsampling over all times leads to fewer samples in 16 of the 21 histogram bins, excluding the smallest and the four largest bins (not shown). But the reduction is not quite uniform: Fig. 12a demonstrates that the subsampling leads to a slight relative oversampling of positive mean dilation rates (higher probabilities in the subsample histograms) and a slight relative undersampling of negative mean dilation rates, except in the bin of smallest magnitude (higher probabilities in the complete histogram). The impact on the computed mean and standard deviation of the normalized mean dilation rate is relatively small: For the full sample, they are −0.046 ± 0.303, for the Λ-based subsample −0.018 ± 0.301, and for the *θ*-based subsample −0.019 ± 0.302.

Figures 12b and 12c show that the differences between the histograms grow when a larger proportion of triads needs to be discarded, as is the case in the late time interval. The entire sample—deformed and nondeformed triads together—shifts toward negative mean dilation rate, but more so for the full sample than for either subsample. For the early time interval, the mean and standard deviation for the computed normalized mean dilation rate are 0.024 ± 0.279 for the full sample, 0.031 ± 0.281 for the Λ-based subsample, and 0.034 ± 0.282 for the *θ*-based subsample. For the late time interval, the mean and standard deviation for the computed normalized mean dilation rate are −0.159 ± 0.279 for the full sample, −0.121 ± 0.273 for the Λ-based subsample, and −0.129 ± 0.265 for the *θ*-based subsample.

The differences between full samples and subsamples are small compared to the differences between samples from different time intervals. These differences are only partially explained by the evolving flow field, as can be seen in Fig. 12d, which shows averages over the geographic domain in Fig. 2 of the instantaneous divergence (purple) and of the gridded 3-h dilation rate (green) as a function of time, in comparison to the triangle-mean dilation rate over all triads (blue). As the flow evolves, domainwide the divergence becomes weaker and ultimately slightly negative on average. The dilation rates are consistently lower for all time intervals, reflecting the propensity of drifters to spend more time in the converging regions of the ocean. The grid of trajectories entering the triad calculations covers only a portion of the domain; see Fig. 2. Initially, its mean dilation rate is greater than that over the entire domain, but as the grid evolves over several hours, without reseeding (unlike for the gridded dilation rate), it concentrates more and more in the converging regions. The subsampling actually mitigates this to a small degree, as shown by the red and yellow circles in Fig. 12d. The greater concern with deriving divergence averages from groups of drifters, thus, is the limited spatial sampling by a finite number of trajectories and the bias introduced by drifters preferentially staying in convergence areas. Addressing this issue is beyond the scope of this paper, but it has been touched on by others, e.g., Pearson et al. (2020) and Sun et al. (2020).

### g. Identification of regions of extreme convergence or divergence

Beyond the divergence statistics, for some applications, it is desirable specifically to identify locations of strong convergence. This is the case, for example, when regions of persistent subduction are sought based on accessible surface information alone (Aravind et al. 2022). This raises the question whether strong triangle deformation can be used as an indicator of large negative mean dilation rate, even if the mean dilation rate itself cannot be reliably quantified. Unfortunately, the triangle shape metrics by themselves are not a sufficient proxy for convergence. They are, in fact, a reasonable proxy for the stretch rate, the Lagrangian average strain rate (not shown), but strong strain rate and divergence are not generally coincident, even when restricting consideration to areas of extreme values.

Nonetheless, the subsampling helps in identifying such target regions by eliminating triads with large errors that would fall in the top or bottom percentiles of mean dilation rate estimates. Figure 13 shows an example of this. In each panel, all triads are plotted in their initial orientation but colored by whether they would fall into the top or bottom 1% of estimates from the drifter-based method or the baseline method using the highly resolved boundaries. In the top row, no subsampling has been performed; for the bottom row, triads with min(Λ) ≤ 0.2 have been marked as eliminated and were not included in identifying the percentiles for the drifter-based method. Dark blue indicates that both methods place a triad’s mean dilation rate into the bottom 1% (strongest convergence), bright blue indicates that only the drifter-based method does so, while turquoise indicates that only the baseline method does so. The dark turquoise in the bottom row is used for triads where the baseline method finds strong convergence but that are eliminated due to their shape from the drifter-based estimation. Similarly, dark red indicates that both methods place a triad’s mean dilation rate into the top 1% (strongest divergence), bright red indicates that only the drifter-based method does so, while orange indicates that only the baseline method does so. The dirty orange in the bottom row is used for triads where the baseline method finds strong divergence but that are eliminated due to their shape from the drifter-based estimation. Triads that are eliminated from the drifter-based estimation that are not in the top or bottom 1% according to the baseline calculation are colored gray.

Highest and lowest 1% target regions for mean dilation rate estimates from the baseline (highly resolved boundary) and the synthetic drifters for (top) all triangles and (bottom) triangles satisfying min(Λ) > 0.2. Each column corresponds to one of the four 3-h intervals. In each panel, the triangles are shown in their orientation at launch for easier visualization. Triangles with mean dilation rate in the top 1% are colored in shades of red, differentiating whether this classification holds for both algorithms (dark red), only for the drifter algorithm (bright red), or only for the baseline algorithm (orange). The dirty orange indicates those triangles in the top 1% for the baseline algorithm and eliminated in the subsampling. Triangles with mean dilation rate in the bottom 1% (strongest convergence) are colored in shades of blue, with dark blue indicating that the classification holds for both algorithms, bright blue that it holds only for the drifter algorithm, cyan that it holds only for the baseline algorithm, and the dark turquoise that it holds for the baseline algorithm and was eliminated in subsampling. Eliminated triangles not in a target region according to the baseline algorithm are colored gray.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Highest and lowest 1% target regions for mean dilation rate estimates from the baseline (highly resolved boundary) and the synthetic drifters for (top) all triangles and (bottom) triangles satisfying min(Λ) > 0.2. Each column corresponds to one of the four 3-h intervals. In each panel, the triangles are shown in their orientation at launch for easier visualization. Triangles with mean dilation rate in the top 1% are colored in shades of red, differentiating whether this classification holds for both algorithms (dark red), only for the drifter algorithm (bright red), or only for the baseline algorithm (orange). The dirty orange indicates those triangles in the top 1% for the baseline algorithm and eliminated in the subsampling. Triangles with mean dilation rate in the bottom 1% (strongest convergence) are colored in shades of blue, with dark blue indicating that the classification holds for both algorithms, bright blue that it holds only for the drifter algorithm, cyan that it holds only for the baseline algorithm, and the dark turquoise that it holds for the baseline algorithm and was eliminated in subsampling. Eliminated triangles not in a target region according to the baseline algorithm are colored gray.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Highest and lowest 1% target regions for mean dilation rate estimates from the baseline (highly resolved boundary) and the synthetic drifters for (top) all triangles and (bottom) triangles satisfying min(Λ) > 0.2. Each column corresponds to one of the four 3-h intervals. In each panel, the triangles are shown in their orientation at launch for easier visualization. Triangles with mean dilation rate in the top 1% are colored in shades of red, differentiating whether this classification holds for both algorithms (dark red), only for the drifter algorithm (bright red), or only for the baseline algorithm (orange). The dirty orange indicates those triangles in the top 1% for the baseline algorithm and eliminated in the subsampling. Triangles with mean dilation rate in the bottom 1% (strongest convergence) are colored in shades of blue, with dark blue indicating that the classification holds for both algorithms, bright blue that it holds only for the drifter algorithm, cyan that it holds only for the baseline algorithm, and the dark turquoise that it holds for the baseline algorithm and was eliminated in subsampling. Eliminated triangles not in a target region according to the baseline algorithm are colored gray.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

For the first time period, where triangle distortion is minimal, there are only a few isolated triangles that are misidentified from the drifter data as belonging to the target regions (bright blue or red). In the second time period, the drifter-based estimates miss one region of strong convergence (cyan patch on the left border) and misidentify another (bright blue patch near the top left). This problem is eliminated by the subsampling, in which case only a single misplaced triangle, identified as strongly convergent when it should not have been (bright blue), remains. As the deformation gets stronger in the last two time periods, the story is similar: Without subsampling, the drifter estimates show strong convergence and strong divergence in the wrong places (bright blue and red), missing one or two of the strong convergence zones altogether (cyan). With the subsampling, all target zones consisting of more than a single triangle are at least partially identified, and the incorrectly identified target zones are mostly (though not entirely) removed.

### h. Impact of position uncertainty

Positions of synthetic drifters in a model ocean can easily be accurately identified, but in the real ocean drifter data are subject to GPS errors. Ohlmann et al. (2017) showed that errors in divergence estimates from drifter clusters due to position error increase rapidly as the cluster deforms. Consequently, we seek to address here whether errors due to position uncertainty require a tightening of the shape metric thresholds derived above assuming perfectly known positions. For this purpose, the analysis is repeated on a 100-member Monte Carlo simulation, as described in section 2e. The RMS error and correlation values for the drifter-based mean dilation rate estimates in comparison to the baseline estimates are given in Table 1 with and without position errors for the whole dataset and for the Λ- and *θ*-based subsamples.

Correlations and RMS error for the synthetic drifter-based estimates of mean dilation rate, relative to the estimates using highly resolved patch boundaries. Statistics are reported for all samples and for subsamples, satisfying either Λ > 0.2 or *θ* < 0.86. For each category, the first column shows the result when positions are known exactly, and the second column shows the result from the Monte Carlo simulation including position error.

The results demonstrate that the position error has a negligible impact on these metrics, although the subsampling is slightly less effective in the presence of position error. Thus, the RMS error over all triads remains the same with or without position error, but the reduction in the error achieved by subsampling is 80% without position error and only 75% with position error. Note that by itself, the error due to position uncertainty in the subsample is not negligible. Since *f* (and up to more than 14.5*f*) occur only for triads with Λ < 0.16 and *θ* ≥ 0.92*π*. However, differences down to 0.25*f* can be found in triads with Λ < 0.50 and *θ* > 0.49*π*. These cause the skill reduction of the subsampling. That the impact of position uncertainty depends on the triangle shape is consistent with previous results by Spydell et al. (2019), who found that position error impact on a point vorticity estimate from a drifter cluster depends on the cluster’s shape.

The choice of time step will influence this conclusion. Here, all estimates are based on 3-h intervals. Over smaller time intervals, in general, the drifters will move shorter distances, the triangle areas will change less, and position errors of the same magnitude will impact the estimates more. More samples may have to be discarded to eliminate estimates that are poor due to the position uncertainty. These could be highly deformed triangles; they could also be small triangles, since a fixed position error has a relatively larger impact on smaller triangles. [Some of these issues have been discussed in the supplemental materials by Berta et al. (2016) and Esposito et al. (2021).] Position uncertainty can be reduced through filtering, using adjacent positions. This is less effective when drifters move quickly, as is the case along the edge of the East Alboran Gyre. Ultimately, these additional conditions will relax as technology improves and the uncertainty in measured positions decreases. The same cannot be said about the errors in the mean dilation rate estimates arising from the shape deformation. These can only be mitigated by increasing the number of drifters that resolve the perimeter of the polygon over which divergence is being averaged, as will be discussed in the next subsection.

### i. Why deformation matters

Equation (4) is an exact statement that holds for any region Ω(*t*). In principle, Ω(*t*) can be a highly deformed triangle, and the equation still holds. The problem with the synthetic-drifter-based estimates arises when the area of Ω(*t _{o}* + Δ

*t*) is not—and cannot be—computed accurately. For drifter triplets, only three points are known along the boundary of Ω. At

*t*=

*t*, this is not a problem, because one can freely decide to define Ω to be a triangle, but these straight line segments generally do not remain such under a flow map. Over short time intervals, the deviations tend to stay small. For triangles close to equilateral, with high Λ and low

_{o}*θ*, these small deviations only have a small impact on the area, as they add or subtract only a small percentage of the total area. For highly deformed triangles, with low Λ and high

*θ*, however, adding or subtracting a thin sliver to a highly elongated area may be substantial.

Figure 14 illustrates these concepts. The example on the left is close to equilateral, with large Λ and *γ* values and low *θ*. The evolved boundaries shown in the bottom row do deviate slightly from the straight lines connecting the vertices, but the impact on the final area is minimal (less than 100 m^{2} out of 85.7 × 10^{3} m^{2}), and the resulting mean dilation rate calculations are accurate to two decimal places. In the middle is an example of a triangle with low Λ and large *θ*. While the distance between the evolved boundary and the triangle approximation is small, the deviation extends over a side of the triangle that is more than 2 km long, and the final areas are vastly different (38 × 10^{3} versus 0.9 × 10^{3} m^{2}). Correspondingly, the mean dilation rate computed from the triangle approximation shows a large error. The last example, on the right, shows a triangle that has low Λ but also low *θ*, with a small *γ* value. Here, the area from the triangle approximation is a reasonable estimate of the true evolved area, and the average divergence estimate has an error of 0.04*f*, illustrating that a small *γ* is not predictive of error.

Three triangle examples. (top) Initial definition of the triangle based on three corner points. (bottom) Position 3 h later of the triangle defined by the three advected vertices (black) and of the evolved triangle, discretized with maximum gap of 10 m (red). The shape metrics for each triangle are given in each panel on the left (black), as well as the area of the evolved shape for the bottom row (red). On the right side of each panel in the bottom row are the two mean dilation rate (or average divergence) estimates, from the synthetic drifter triangles (black) and from the highly resolved boundaries (red).

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Three triangle examples. (top) Initial definition of the triangle based on three corner points. (bottom) Position 3 h later of the triangle defined by the three advected vertices (black) and of the evolved triangle, discretized with maximum gap of 10 m (red). The shape metrics for each triangle are given in each panel on the left (black), as well as the area of the evolved shape for the bottom row (red). On the right side of each panel in the bottom row are the two mean dilation rate (or average divergence) estimates, from the synthetic drifter triangles (black) and from the highly resolved boundaries (red).

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Three triangle examples. (top) Initial definition of the triangle based on three corner points. (bottom) Position 3 h later of the triangle defined by the three advected vertices (black) and of the evolved triangle, discretized with maximum gap of 10 m (red). The shape metrics for each triangle are given in each panel on the left (black), as well as the area of the evolved shape for the bottom row (red). On the right side of each panel in the bottom row are the two mean dilation rate (or average divergence) estimates, from the synthetic drifter triangles (black) and from the highly resolved boundaries (red).

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

## 4. Application: Divergence and vorticity estimates in the Alboran Sea

The real ocean, of course, is more complex than the modeled one, raising the question whether the model results will hold in an application to actual observed drifter trajectories. Here we test the proposed subsampling strategy, retaining only triangles with Λ > 0.2, on a set of SVP drifter trajectories observed in the Alboran Sea. See section 2f. As mentioned in section 1, for real ocean drifters the errors on divergence estimates cannot be directly computed, since baseline results are obviously not available. As a consequence, the effectiveness of the parameter threshold is evaluated here with statistical analyses, i.e., in terms of outliers eliminated by the threshold and stability of the resulting statistical values.

The triangle-mean dilation rate is estimated from drifter observations using the same methodology as was used for the synthetic drifters [section 2c(4); Molinari and Kirwan 1975], although here it is applied to 30-min time averages. The shorter time span increases the sample size and enables closer collocation of different samples. As mentioned above (section 3h), position errors tend to have a greater impact on estimates from shorter averaging times. So this is a more stringent test for the methodology. In addition to divergence, we also estimate vorticity, using the coordinate transformation described in section 2f, following Molinari and Kirwan (1975).

To account for scale dependence of the estimated KPs (Berta et al. 2016, 2020b), chance groupings of drifters into triplets are sorted into several scale bins, where a triangle’s scale is determined by its average side length. The bins considered here are (0, 2], (2, 4], (4, 6], and (6, 10] km, similar to the scales investigated in the simulation. As for the model analysis, we start the analysis of the observations with the smallest scale. Within the smallest size bin, (0, 2] km, 74 of the 211 triplets (35%) have Λ ≤ 0.2, while over all size ranges (0–10 km), 1383 of the 5308 triads (26%) have Λ ≤ 0.2. With the idealized launch positions and the smoother flows in the simulation, only 11% of the modeled triads are that distorted. In other words, the thresholding eliminates a larger portion of the observed triads.

The left panels of Fig. 15 show the geographic distribution of the mean dilation rate estimates for drifter triads in the smallest size bin, the highly distorted ones (Λ ≤ 0.2) on the top and the rest on the bottom. All samples are located within a strong northward flow along the edge of a large anticyclone. The right panels show the corresponding probability distributions. Extreme values of mean dilation rate, up to ±8*f*, only appear for triplets with Λ ≤ 0.2 (top panels). The maximum mean dilation rate estimated from the other triplets is about ±2*f*. Given that all samples are nearly collocated in both space and time, it is likely that the outliers are, in fact, a result of large estimation errors. Moreover, the independent analysis of divergence along a density front in this geographic region, though in another year, by Lodise et al. (2020) found that divergence did not exceed ±7.4*f*, with very few samples stronger than ±2*f*. Note that mean and mean square values of the mean dilation rate, respectively, are one and two orders of magnitude larger for the distorted samples than for the rest. The small negative skewness for the samples with Λ > 0.2 is consistent with previously reported values from Eulerian velocity measurements of a submesoscale field (Shcherbina et al. 2013).

(a),(b) Maps of normalized mean dilation rate, where each colored dot represents the estimate from an SVP chance triplet in the size range (0,2] km from 1430 UTC 31 May to 0450 UTC 1 Jun 2018. The gray dots indicate launch positions, and the lines show the drifter trajectories in the Alboran Sea. (c),(d) Probability distributions of the normalized mean dilation rate shown in the maps, with values for the mean, standard deviation (STD), mean square (MS), and skewness (skew). Rows show results for triplets with (top) Λ ≤ 0.2 and (bottom) Λ > 0.2.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

(a),(b) Maps of normalized mean dilation rate, where each colored dot represents the estimate from an SVP chance triplet in the size range (0,2] km from 1430 UTC 31 May to 0450 UTC 1 Jun 2018. The gray dots indicate launch positions, and the lines show the drifter trajectories in the Alboran Sea. (c),(d) Probability distributions of the normalized mean dilation rate shown in the maps, with values for the mean, standard deviation (STD), mean square (MS), and skewness (skew). Rows show results for triplets with (top) Λ ≤ 0.2 and (bottom) Λ > 0.2.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

(a),(b) Maps of normalized mean dilation rate, where each colored dot represents the estimate from an SVP chance triplet in the size range (0,2] km from 1430 UTC 31 May to 0450 UTC 1 Jun 2018. The gray dots indicate launch positions, and the lines show the drifter trajectories in the Alboran Sea. (c),(d) Probability distributions of the normalized mean dilation rate shown in the maps, with values for the mean, standard deviation (STD), mean square (MS), and skewness (skew). Rows show results for triplets with (top) Λ ≤ 0.2 and (bottom) Λ > 0.2.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

The probability distributions for larger scales restricted to samples with Λ ≤ 0.2 also exhibit large outliers. Moreover, for vorticity as well, large outliers are found only among the highly deformed samples, as is shown in Fig. 16, suggesting that they are unrealistic. Discarding samples with Λ ≤ 0.2 efficiently eliminates these outliers in both KPs. For the remaining data, the distributions of divergence and vorticity estimates from collocated samples across different size bins are in good agreement. We also tested thresholds of Λ = 0.1 and 0.15; neither of these eliminated all the outliers (not shown), suggesting that Λ = 0.2 is a good compromise between retaining valuable data and discarding error-prone samples. We note that many of the discarded samples also produced estimates consistent with those in the retained set, similar to what we found in the simulation.

Scatterplots of normalized mean dilation rate as a function of normalized spatiotemporally averaged vorticity derived from SVP chance triplets from 1430 UTC 31 May to 0450 UTC 1 Jun 2018 for (a) highly distorted triangles with Λ ≤ 0.2 and (b) less distorted triangles with Λ > 0.2. Different colors correspond to different triplet size ranges: [0, 2] km in blue, (2, 4] km in yellow, (4,6] km in green, and (6,10] km in gray.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Scatterplots of normalized mean dilation rate as a function of normalized spatiotemporally averaged vorticity derived from SVP chance triplets from 1430 UTC 31 May to 0450 UTC 1 Jun 2018 for (a) highly distorted triangles with Λ ≤ 0.2 and (b) less distorted triangles with Λ > 0.2. Different colors correspond to different triplet size ranges: [0, 2] km in blue, (2, 4] km in yellow, (4,6] km in green, and (6,10] km in gray.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

Scatterplots of normalized mean dilation rate as a function of normalized spatiotemporally averaged vorticity derived from SVP chance triplets from 1430 UTC 31 May to 0450 UTC 1 Jun 2018 for (a) highly distorted triangles with Λ ≤ 0.2 and (b) less distorted triangles with Λ > 0.2. Different colors correspond to different triplet size ranges: [0, 2] km in blue, (2, 4] km in yellow, (4,6] km in green, and (6,10] km in gray.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

The bottom panel of Fig. 16 demonstrates the robustness and consistency of divergence and vorticity estimates from collocated triads with Λ between 0.2 and 1. In particular, estimates from triplets with 0.2 < Λ ≤ 0.7 are consistent with those from triplets with Λ ≥ 0.7. For the latter, more stringent threshold, although applied to a different drifter dataset, Berta et al. (2020b) showed the reliability of drifter-based KP estimates by comparing them directly to independent KP estimates based on collocated Eulerian X-band radar current measurements.

To further investigate how different choices for the subsampling threshold impact the outcome, we consider the RMS value of the normalized mean dilation rate estimates from subsets obtained from a variety of thresholds, for all the scale bins (Fig. 17). To ensure statistical significance, only values from bins with more than 85 samples are shown, along with 95% confidence intervals. For all thresholds, the RMS mean dilation rate decreases as the scale increases, in agreement with other analyses (Esposito et al. 2021; Berta et al. 2016). The dependence on the Λ threshold is shown in the top panel, while the *θ* thresholds are tested in the bottom panel. Generally, the RMS of the normalized mean dilation rate decreases as more and more samples are discarded, i.e., as the threshold Λ increases or the threshold *θ* decreases, especially for smaller scales. However, this decrease is minimal for Λ > 0.2 and *θ* = 0.86*π*, whereas it is exponential for smaller Λ and larger *θ* thresholds. Thus, KP estimates derived from triplets that are nearly collocated in space and time are consistent with each other when Λ > 0.2 or *θ* = 0.86*π*, but inconsistent outliers are found among the samples with Λ < 0.2 or *θ* = 0.86*π* that contaminate the RMS estimates. In other words, the observations suggest that the shape parameter thresholds found to be optimal in the simulation are also applicable to real observations for all the tested scales.

RMS normalized mean dilation rate after subsampling, with 95% confidence intervals, as a function of the shape parameter thresholds, (a) Λ and (b) *θ*. Values are estimated from SVP chance triplets in the different size bins from 1430 UTC 31 May to 0450 UTC 1 Jun 2018. Vertical lines indicate the thresholds proposed by the model study: Λ = 0.2 and *θ* = 0.86*π*.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

RMS normalized mean dilation rate after subsampling, with 95% confidence intervals, as a function of the shape parameter thresholds, (a) Λ and (b) *θ*. Values are estimated from SVP chance triplets in the different size bins from 1430 UTC 31 May to 0450 UTC 1 Jun 2018. Vertical lines indicate the thresholds proposed by the model study: Λ = 0.2 and *θ* = 0.86*π*.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

RMS normalized mean dilation rate after subsampling, with 95% confidence intervals, as a function of the shape parameter thresholds, (a) Λ and (b) *θ*. Values are estimated from SVP chance triplets in the different size bins from 1430 UTC 31 May to 0450 UTC 1 Jun 2018. Vertical lines indicate the thresholds proposed by the model study: Λ = 0.2 and *θ* = 0.86*π*.

Citation: Journal of Atmospheric and Oceanic Technology 39, 10; 10.1175/JTECH-D-21-0161.1

## 5. Summary and conclusions

Recent advances in drifter technology (Lumpkin et al. 2017) have permitted the near-simultaneous deployment of large numbers of drifters. Such deployments provide the data necessary to derive estimates of the KPs to characterize the fluid flow and identify surface signatures of subduction and upwelling. Yet not all the resulting estimates are reliable. Here we have derived conditions on the shape of drifter triads that will optimize the balance between retaining reliable samples and discarding unreliable and error-prone ones. Across all studied scales, the results indicate that at least 85% of all available triplets can be retained. However, at larger scales, the subsampling is less effective at removing the error that is introduced by insufficient resolution of the evolving triangle boundary.

In the model, where we have access to a baseline calculation of the spatiotemporal average divergence (the triangle-mean dilation rate) as computed by the area-based KP algorithm to be applied to drifter trajectories (Molinari and Kirwan 1975), we found that a good balance can be found using the threshold values Λ = 0.2 for the aspect ratio or *θ* = 0.86*π* for the largest interior angle. In either case, samples with very large errors are sifted out, without also losing too many samples with small errors. These two shape parameter thresholds perform roughly equally well. On the other hand, the shape parameter *γ*, measuring the ratio of the smallest to the middle side length, is not indicative of reliability, with large errors being found in triangles from the entire range of *γ*. The threshold can be adjusted to modify the balance between removing large errors and retaining good estimates, giving each a different weight. However, the band of reasonable choices is fairly narrow, before a steep drop in one of the objectives makes further gains in the other too costly. The optimal threshold also changes when the level of error in an individual sample targeted for removal is reduced. When the fraction of samples retained is balanced against minimizing the total RMS error of the remaining samples, the threshold values of Λ = 0.2 and *θ* = 0.88*π* are found to be optimal at the 500-m scale. These are also nearly optimal for all the other tested scales, except for at 7 km, where the optimal thresholds were found to be stricter (Λ = 0.44 and *θ* = 0.78*π*).

Subsampling drifter triads by removing highly deformed triangles does introduce some sampling bias into the distribution and average of the mean dilation rate estimates. Any KP distribution derived from Lagrangian data will show some bias, simply because drifters tend to spend more time in convergence zones than in other areas of the ocean (Jacobs et al. 2016). This bias worsens over time, reducing the value of chance pairings over time for this application. The bias introduced by removing highly deformed samples is small compared to this Lagrangian observation bias; it also slightly corrects for the latter.

Applying a shape parameter threshold eliminates large outliers, resulting from large errors in the model study. In this way, it actually improves the identification of regions in the ocean that are prone to very high divergence or very high convergence. Spurious outliers can otherwise contaminate the result. This is particularly encouraging when divergence from drifter observations is to be used as a proxy for subduction regions, which are much harder to observe directly (Aravind et al. 2022).

Position uncertainty, to which all observations are subject, only has a small impact on the reliability of the KP estimates, provided subsampling is implemented using one of the shape parameter thresholds. Position errors have a particularly large effect on KP estimates from highly deformed triangles. Thus, the subsampling can assist in dealing with position uncertainty.

An application of the proposed shape parameter thresholds to an observational dataset showed that Λ > 0.2 or *θ* < 0.86*π* are useful criteria for identifying reliable samples even in the complex real ocean, experiencing processes at a full range of scales. The analysis of the observed SVP trajectories showed that the thresholding eliminates outliers from a set of samples that are nearly collocated in space and time. In addition, the thresholding is similarly effective for vorticity estimates as for divergence.

In conclusion, we recommend that estimates of KPs from drifter triplets are subsampled using one of our proposed shape parameter thresholds to optimize the reliability of the results.

## Acknowledgments.

This work was supported by generous financial support from the U.S. Office of Naval Research (ONR) through Grants N00014-18-1-2461, N00014-18-1-2782, and N00014-16-1-3130 under the CALYPSO DRI, as well as by support from the European Union’s JERICO-S3 project through Grant 871153 and from the National Academies of Sciences, Engineering and Medicine’s Gulf Research Program through Grant 200011071. The drifter equipment and LC were supported by ONR Grants N00014-17-1-2517 and N00014-19-1-2692. The authors thank Eugenio Cutolo for the initial technical support in the implementation of the nested WMOP simulation, and the Spanish National Meteorological Agency (AEMET) for providing the hourly and 5 km spatial resolution HIRLAM atmospheric model outputs. The authors also acknowledge the considerable effort involved in the planning, coordination, and execution of the CALYPSO field experiment that yielded the observational drifter data and wish to thank chief scientists Amala Mahadevan and Eric D’Asaro, as well as the many other scientists who participated, for making it possible. Three anonymous reviewers provided excellent comments leading to a much improved manuscript.

## Data availability statement.

The model data used here are archived by the Data Centre at SOCIB, Spain (http://www.socib.es/), and can be obtained by request to Baptiste Mourre (bmourre@socib.es). The observational data are part of the Global Drifter Program, and hourly trajectories can be accessed at NOAA’s website at https://www.aoml.noaa.gov/phod/gdp/index.php. The higher temporal resolution is archived by the Istituto Nazionale di Oceanografia e di Geofisica Sperimentale (OGS) and can be obtained from http://nettuno.ogs.trieste.it/sire/drifter/project.php?country_www=CALYPSO.

## APPENDIX A

### Equivalence of Lagrangian Averaged Divergence, Area Ratios, and Dilation Rate

#### a. Average divergence and area ratios

*t*) and a time interval [

*t*,

_{o}*t*+ Δ

_{o}*t*] on one hand and the logarithm of the area ratio for Ω at the final and initial times on the other. This fact follows from the relationship between the time derivative of the area of an evolving patch and divergence: Let

*A*(

*t*) denote the area of Ω(

*t*), and observe that the determinant of the deformation tensor

**F**defined in Eq. (1) is the Jacobian for the coordinate transformation from instantaneous positions

**x**to initial positions

**x**

*:*

_{o}*D*the area-averaged divergence, defined as in Eq. (3), this becomes

#### b. Average divergence and dilation rate

*δ*is the product of the singular values of

**F**and therefore equal to |det(

*F*)|. Thus,

## APPENDIX B

### Relationships between Triangle Shape Metrics

The three triangle shape metrics Λ, *γ*, and *θ* can generally vary independently, but they must satisfy several constraints detailed here.

Since the triangle shape metrics are independent of the triangle size, we rescale the triangle for all the derivations, so that its middle side has length 1 and denote the longest side length by *s*. Thus, the three sides have lengths

*γ*and

*θ*as follows:

*γ*is bounded from below by

*θ*, Λ increases with

*γ*. Note that

*γ*.

*γ*≥ 0.

*γ*ranges from max(0, 2 cos

*θ*) to 1 (by constraint 2) and

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