Abstract

Differential kinematic flow properties (DKP), such as vertical vorticity, have been estimated from surface drifters. However, previous DKP error estimates were a posteriori and did not include correlated errors across drifters. To accurately estimate submesoscale (≤1 km) DKPs from drifters, errors must be better understood. Here, the a priori vorticity standard error is derived that depends upon the number of drifters in the cluster, the drifter cluster major and minor axes lengths, the instrument velocity error, and the cross-drifter error correlation. Two stationary GPS experiments, with zero vorticity, were performed at separations of O(101–103) m to understand vorticity error and test the derivation using 1 Hz position differences and Doppler shift velocities. Vorticity errors of ±5f (where f is the local Coriolis parameter)were found for ≈40 m separations. The frequency-dependent velocity variances and GPS-to-GPS correlations are quantified. Vorticity estimated with a “blended” velocity has reduced error. The stationary vorticity error can be well predicted given velocity error, correlation, and minor axis length. Vorticity error analysis is applied to submesoscale-sampling in situ GPS drifters near Point Sal, California. The derivation predicts when large high-frequency vorticity fluctuations (indicating noise) occur. Previously, cluster area or ellipticity were used as criteria to distinguish error. We show that the drifter cluster minor axis (narrowness) is a key time-dependent factor affecting vorticity error, and even for velocity errors <0.004 m s−1, the vorticity error exceeds ±5f when cluster minor axis <50 m. These results will aid submesoscale drifter deployment planning.

1. Introduction

Lagrangian drifters play an important role in understanding ocean currents and eddies from large open-ocean scales (hundreds of kilometers; e.g., Lumpkin and Johnson 2013) to small surfzone scales (5 m; e.g., Spydell et al. 2007; Brown et al. 2009). Drifter observations are used to study surface (Lumpkin and Johnson 2013) and subsurface (Ollitrault and Colin de Verdière 2014) currents, estimate absolute and relative diffusivity and Lagrangian time scales, and infer scale-selective diffusivities [for a review, see LaCasce (2008)]. Drifters are tracked with various methods, a brief history of which is found in Lumpkin et al. (2017). Many modern drifters are tracked with GPS due to its affordability and accuracy (e.g., Schmidt et al. 2003; Ohlmann et al. 2017; Novelli et al. 2017).

In addition to mean circulation patterns and diffusivities, surface horizontal divergence (dU/dx + dV/dy) and vertical vorticity (dV/dxdU/dy) have been estimated from drifters. These, and other fluid differential kinematic properties (DKP), are found from the positions and velocities of three or more drifters from which all horizontal velocity gradients (dU/dx, dU/dy, dV/dx, and dV/dy) can be estimated using a least squares (LS) technique, first described in Molinari and Kirwan (1975) and Okubo and Ebbesmeyer (1976). The estimated DKP variance decreases with increasing drifter number N. For three drifters, the LS technique yields an exact fit, hence for three drifters it is not possible to estimate DKP variance.

The LS technique of estimating DKPs was first applied to mesoscale flows, that is, O(10) km length scales and ≥1 day time scales. For clusters of three drifters in the western Caribbean Sea (Molinari and Kirwan 1975), vorticity and divergence estimates were O(10−1f), where f is the local Coriolis parameter. Similar magnitudes of vorticity and divergence were found for mesoscale motions for clusters of five SOFAR floats (sampling at 300 m depth) deployed near the West Spitsbergen Current (Richez 1998). Improved GPS tracking technology has enabled DKP estimation for smaller space- and time-scale flows. For submesoscale eddies and fronts [O(1) km length scales and O(1) h time scales], vorticity and divergence estimates often exceed, and sometimes greatly (10 times), the local f based on clusters of nine GPS tracked near-surface drifters in the Santa Barbara channel (Ohlmann et al. 2017) and clusters of four drifters in the Gulf of Mexico (Ohlmann et al. 2019). Vorticity and divergence magnitudes decrease with increasing space and time scales (e.g., Ohlmann et al. 2017). Although calculating DKP at submesoscales and smaller is now possible, the role of errors in the LS technique at these scales is not completely understood.

The first study that applied the LS technique (Molinari and Kirwan 1975) noted very large “wiggles” in the DKP time series using clusters of three drifters. This is despite low-pass filtering the time series of positions (initial sampled at 15 min), from which the DKP was estimated, to approximately daily positions. These authors found that this error (i.e., wiggles) was inversely related to triangle (cluster) area. Vorticity estimated using clusters of five drifters near Point Reyes (California) was considered erroneous if the cluster area (each drifter is a polygon vertex), became too small (<106 m2) or too large (>4 × 1010 m2) (Paduan and Niiler 1990). For simulated drifters in a mesoscale model of the California Current System, noisy DKP rejection criteria used the longest drifter separation and the cluster ellipticity, defined as the ratio of the major to minor axis of the position covariance matrix (Righi and Strub 2001). The along cluster track Eulerian vorticity and drifter estimated vorticity were similar if the largest drifter separation was <20 km and if the cluster remained fairly circular. However, the grid resolution was approximately 9 km, hence, submesoscale dynamics were not properly resolved. A criterion based only on cluster ellipticity was used in Ohlmann et al. (2017).

DKP error depends on the velocity error, the number of drifters in the cluster, and the drifter cluster geometry (size and shape). Velocity error has two sources (Okubo and Ebbesmeyer 1976; Kirwan 1988): GPS instrument noise or processes noise by assuming spatially uniform velocity gradients in the Taylor series expansion. The velocity error has been a posteriori estimated from the LS misfit (Okubo and Ebbesmeyer 1976; Sanderson et al. 1988). Kirwan and Chang (1979) investigated the role of instrument noise on DKP error. However, explicit dependence of DKP error on cluster geometry was not determined and correlated velocity errors between instruments was not considered. Here, we focus only on the role of instrument noise on the velocity error.

Understanding GPS instrument error is critical as it contributes directly to DKP errors. On smaller scales, GPSs errors could be the main misfit contributor as the velocity Taylor series expansion (upon which the LS technique is based) is increasingly valid for decreasing spatial scales. However, GPS position and velocity errors can vary. For instance, GPS position errors sampling surfzone to shelf flows range from 1 to 10 m (Schmidt et al. 2003; Johnson and Pattiaratchi 2004; Ohlmann et al. 2005; Novelli et al. 2017). GPS position errors can be reduced to <0.01 m (Suara et al. 2015) using real-time kinematic (RTK) positioning. However, RTK systems are uncommon due to their cost, and are not appropriate for inner shelf to open-ocean studies due to needing a nearby base station. Although, GPS position error frequency spectra are red (≈f−2; Johnson and Pattiaratchi 2004; MacMahan et al. 2009; Suara et al. 2015), resulting in white (position differences) velocity error spectra. (MacMahan et al. 2009; Suara et al. 2015), the effect of filtering GPS positions and velocities on DKP errors is not completely understood.

In this article, GPS errors are investigated and quantified, and their effect on DKP, specifically vorticity, is examined. Although the errors associated with only one particular GPS receiver are investigated, the methodology here provides a template for use with any GPS. A simple illustrative formula for the vorticity error is derived and extended for clusters of N GPSs in section 2 and the  appendix. In section 3, the GPSs, the observations, the data processing, and the statistical quantities of interest are described. In section 4, GPS position and velocity errors are presented and scalings for the vorticity error are tested. In the discussion (section 5), the vorticity error for in situ inner-shelf drifters is examined, previous DKP error analyses are contextualized, and the role of GPS satellite coverage investigated. The work is summarized is section 6.

2. Vorticity and vorticity errors from drifters

a. Vorticity error: An illustrative example

Consider velocity gradient error estimated from two still GPSs, one located at X1 = 0, the other at X2 = L. These GPSs measure positions X1 = x1(t) and X2 = L + x2(t) and Doppler velocities u1(t) and u2(t). This analysis is one dimensional for illustration and clarity. The time mean is indicated with an overbar thus, the mean of x1(t) is x¯1, hence, x1(t)=x¯1+x1(t) where x¯1 is the mean error and x1(t) the fluctuating error. For this analysis, we assume that x1, x2, u1, and u2 are correlated Gaussian random variables with nonzero mean and that the error statistics are identical for both GPSs. The position and velocity error variances are (x1)2¯=(x2)2¯=σx2 and (u1)2¯=(u2)2¯=σu2. The position and velocity error correlation on the same GPS is defined as x1u1¯=σxσuρx1u1, where ρab represents correlation between random variables a and b. The position and velocity error correlation across GPSs is x1u2¯=σxσuρx1u2, and the velocity error correlation across different GPSs is u1u2¯=σu2ρu1u2.

The velocity gradient is estimated as a centered difference

 
dU(t)dX=u2(t)u1(t)L+x2(t)x1(t).
(1)

Assuming that L ≫ |x2x1|, the Taylor series expansion of (1) is

 
dUdx=u2u1L[1x2x1L+(x2x1)2L2].
(2)

To first order in (x2x1)/L, the mean velocity gradient is

 
dU¯dx=u¯2u¯1L1L2[x1u1¯+x1u2¯+x2u1¯x2u2¯],
(3)

where the first term on the right-hand side is zero if GPS velocities have no mean error. Because the statistics are the same for all GPSs, x1u2¯=x2u1¯ and x1u1¯=x2u2¯, we have

 
dU¯dx=(u¯2u¯1)L[1+(x¯1x¯2)L]I+2σxσuL2[ρx1u2ρx1u1]II.
(4)

Thus, velocity gradient mean error occurs if velocity mean error is nonzero (term I) or if velocity and position errors are correlated (term II), particularly if x′ and u′ are correlated differently on the same GPS than across GPSs. In practice, term II is much smaller than term I because for oceanographic scales of interest σxL.

The mean square velocity gradient error is

 
(dUdx)2¯=(u2u1L)2[1x2x1L+(x2x1)2L2]2¯
(5)

or

 
(dUdx)2¯=(u12+u222u1u2L2)[12(x2x1)L+3(x2x1)2L2+]¯.
(6)

The expectation of the product of three zero-mean Gaussian random numbers is zero (the Isserlis theorem; Isserlis 1918), that is, u12x1¯=u12x2¯=u1u2x1¯=0, hence, the first two nonzero terms [in (x2x1)/L] are

 
(dUdx)2¯=u12+u222u1u2L2¯+3(u12+u222u1u2)(x12+x222x1x2)¯L4.
(7)

The first term dominates the velocity gradient mean square error; expanding it we have

 
(dUdx)2¯=1L2(u¯12+u¯222u¯1u2¯+2σu22u1u2¯),
(8)

such that the leading term of the squared velocity gradient standard error is

 
σUx2(dUdx)2¯(dU¯dx)2=2σu2L2(1ρu1u2).
(9)

Note that positive GPS to GPS velocity error correlations ρu1u2 decrease the velocity gradient standard error.

The vorticity ζ = dV/dxdU/dy error is found from the gradient error. Assume that two drifters are aligned in the x direction and separated by Lx and two drifters aligned in y direction separated by Ly (a diamond pattern), assume LyLx. For this configuration, the squared vorticity standard error is then

 
σζ2=σVx2+σUy2.
(10)

Assuming u and υ are independent with the same error statistics, we have σVx2=σUx2, σUy2=σVy2, and ρu1u2=ρυ1υ2, the squared vorticity standard error σζ2 is written as

 
σζ2=2σu2Lx2(1+Lx2Ly2)(1ρu1u2).
(11)

b. Estimating vorticity and vorticity errors from drifter cluster observations

Velocity gradients, and thus vorticity, can be estimated from N ≥ 3 drifters (Molinari and Kirwan 1975) using a least squares method. Drifter velocities can be Taylor series expanded about the cluster center,

 
ui=U+dUdxxi+dUdyyi+ui,
(12)

where U is the cluster mean velocity, dU/dx and dU/dy are velocity gradients over the cluster, and (xi, yi) are east–west (E-W) and north–south (N-S) positions relative to cluster center. The velocity residual ui is due to both instrument noise and process noise (the velocity field deviating from a spatially constant velocity gradient). With N drifter velocities and positions, the model parameters

 
β=[UdU/dxdU/dy]T
(13)

are found by least squares fit (minimizing mean square u′) to the observed velocities,

 
β=(RTR)1RTU˜,
(14)

where U˜=[u1u2uN]T is the vector of observed drifter velocities and R is the N × 3 matrix

 
R=(1x1y11x2y21xNyN).
(15)

The alongshore cluster mean velocity V and velocity gradients, dV/dx, and dV/dy, are found similarly. The vorticity ζ is then ζ = dV/dxdU/dy.

For a cluster of N drifters, the vorticity error variance is derived in (A11) in the  appendix and is given by

 
σζ2=1Nσu2la2(1+la2lb2)(1ρu1u2),
(16)

where la and lb (lalb) are the minor and major axis lengths of the drifter cluster, that is, eigenvalues of the position covariance matrix P [see (A10)]. Moreover, la can be considered the width, or narrowness, of the cluster and lb the length of the cluster. The velocity error σu2 is due to both instrument and process error. Here we focus only on the effects of instrument error on DKP error.

The vorticity error variance assumes uncorrelated u′, υ′, but correlated u′ between GPSs [see (16)]. It is analogous to the two drifter illustrative example in (11), but accounts for the drifters number N and cluster size and shape through la and lb. This expression (16) can be used a priori if the velocity error σu is known. Furthermore, (16) indicates that vorticity standard error decreases as N−1/2, indicating that large numbers of drifters are required to reduce vorticity error substantially.

3. GPS instruments, observations, and methods

a. GPS instruments

Off-the-shelf, hand-held, “GT-31” GPSs (henceforth GPSs) manufactured by Locosys Technology Inc. are used here. These GPSs have been used in previous oceanographic studies (Herbers et al. 2012; McCarroll et al. 2014; Pearman et al. 2014; Fiorentino et al. 2014; Slivinski et al. 2017). These GPSs are useful for surface oceanographic drifter applications as they are waterproof to IPX7 standards and, due to their small size (9 cm × 5.8 cm × 2.5 cm), easily fit in a small otter box that can be mounted to various drifter bodies. These GPSs record position and Doppler shift (based on the frequency shift of the GPS carrier frequency) estimated velocity at 1 Hz using the SiRF Star 3 GPS chip. The 1 Hz sampling of these GPS is faster than required to sample submesoscale processes that evolve on tens of minutes to hourly time scales and faster than previous drifter studies of these motions (e.g., D’Asaro et al. 2018). However, as shorter time- and space-scale processes are investigated (e.g., Ohlmann et al. 2017), rapid sampling is needed for 1) understanding GPS errors dependence on sampling frequency, 2) surface wave spectra estimation as submesoscale processes may depend on Stokes drift (Hamlington et al. 2014), and 3) filtering out surface gravity waves. For example, nearshore vorticity can be O(10−2) s−1 (Suanda and Feddersen 2015; Kumar and Feddersen 2017) not far from surface gravity wave frequencies (0.05 s−1).

The SiRF chip GPS position–velocity solution algorithm is proprietary, and thus the relationship between position and Doppler velocity is unknown. The manufacturer states that horizontal positions have 10 m rms absolute accuracy and horizontal Doppler velocities have 0.1 m s−1 rms accuracy. Surface gravity wave spectra at f > 0.05 Hz have been accurately estimated from 1 Hz GT GPS horizontal positions (Herbers et al. 2012). These GPSs return a time series of (latitude, longitude) which is converted to distances using a WGS84 spheroid. The (easting, northing) component of 1 Hz raw position and velocity is Xr(t) = (Xr, Yr) and ur(t) = (ur, υr), respectively. These GPSs also record at 1 Hz the number of satellites in the GPS constellation and a unitless estimate of the horizontal position error (HDOP).

b. Stationary deployments

There were two stationary deployments of multiple GPSs in Monterey, CA. For these stationary deployments, the “exact” position X0 of the GT was obtained by placing a survey grade RTK-GPS (≈1 cm accuracy; Suara et al. 2015) at the same location of the GT. Note that such stationary deployments can be used to quantify the error of any GPS. The deviation from the true position for each GPS is then xr(t) = Xr(t) − X0, where xr(t) is the “raw” 1 Hz position error time series (the subscript r denotes 1 Hz raw quantities).

For the first stationary deployment (30 July 2018), denoted the small-scale deployment (SSD), eight GPSs were placed in two squares for 18 h: a small square with 10 m sides (GPSs 1–4) and larger square with ≈40 m sides (GPSs 5–8, Fig. 1a). The SSD relative raw 1 Hz positions xr(t) meander about ±2 m for each GPS (cool colors, Fig. 1b). For the second deployment (12 September 2018, duration of 24 h), denoted the large-scale deployment (LSD), the five GPS separations were larger, O(100–1000) m, than the SSD and GPS placement was not structured (cf. Figs. 1b,a). The LSD relative positions meandered between about ±2 m (GPS 9) to ±5 m (GPS 11) (warm colors, Fig. 1b).

Fig. 1.

Absolute GPS positions (X, Y) for the (a) small-scale stationary deployment (SSD) and (b) absolute GPS positions (X, Y) for the large-scale stationary deployment (LSD). In (b), colored positions are not visible because the scale of the position scatter is too small relative to the GPS separations. In (a) and (b), the deployed location is indicated by “+” and GPS reference numbers are indicated. (c) Relative positions (x, y), to the deployed location (+), for both deployments (with 10 m offsets). Note that colors will be consistent throughout Figs. 2, 4, and 5.

Fig. 1.

Absolute GPS positions (X, Y) for the (a) small-scale stationary deployment (SSD) and (b) absolute GPS positions (X, Y) for the large-scale stationary deployment (LSD). In (b), colored positions are not visible because the scale of the position scatter is too small relative to the GPS separations. In (a) and (b), the deployed location is indicated by “+” and GPS reference numbers are indicated. (c) Relative positions (x, y), to the deployed location (+), for both deployments (with 10 m offsets). Note that colors will be consistent throughout Figs. 2, 4, and 5.

For the SSD, the GPS satellite constellation changed little across GPSs, with 8.9 ± 1.2 satellites in view for each SSD GPS, where 8.9 is the mean (over the 8 GPSs) of the time-mean satellite number and 1.2 is the mean (over the 8 GPSs) of the time-standard deviation. The LSD deployment had slightly worse satellite coverage, with 8.6 ± 1.2 satellites in view (average over GPSs 9, 10, 11, and 13). GPS 12 had the worst satellite coverage, seeing 7.7 satellites in view on average. The GPS estimate of HDOP averaged 1.0 for all GPSs except GPS 12 where HDOP averaged 1.2 over the deployment. The HDOP standard deviation was approximately 0.2 for all GPSs except GPS 12 where it was 0.3.

c. In situ drifter observations

In situ drifter observations from the ONR funded Inner Shelf Experiment conducted near Point Sal, California, during September–October 2017 are used here. Coastal Ocean Dynamics Experiment (CODE) surface drifter bodies (Davis 1985) equipped with 1 Hz Lycocos GT-31 GPSs were deployed for ≈5 h on multiple days on the inner shelf (5–40 m water depth). Drifters followed the mean horizontal flow between approximately 0.3 and 1.2 m below the surface. The water following properties of CODE drifters is well established (Poulain 1999; Novelli et al. 2017).

d. Data processing

The stationary and in situ processing steps are as follows. For the stationary deployments, GPSs had at most one missing velocity or position over the deployment duration (≈1 day) which was filled with linear interpolation. First, outlier Doppler velocities |ur(t)| greater than three standard deviations of ur (but not positions) are removed and filled by linear interpolation. The fraction of outlier velocities was <0.08% for the SSD and between 0.013% and 0.50% for the LSD deployment. Second, velocities based on position differences, denoted x˙r, are obtained by centered difference, x˙r(t)=[Xr(t+δt)Xr(tδt)]/(2δt). These velocities will be called position-derived velocities (PDVs). Thus, x˙r(t) is the PDV error time series for the stationary deployments. Outliers are not removed for the PDV time series because doing so results in PDV time series equal to zero due to the discrete distribution of x˙ (Figs. 2c,d).

Fig. 2.

Time series of (a),(b) E-W relative position (offset in y by 3 m), (c),(d) E-W PDV (offset in y by 0.05 m s−1), and (e),(f) E-W velocity (offset in y by 0.1 m s−1) for (a),(c),(e) small-scale and (b),(d),(f) large-scale stationary deployment.

Fig. 2.

Time series of (a),(b) E-W relative position (offset in y by 3 m), (c),(d) E-W PDV (offset in y by 0.05 m s−1), and (e),(f) E-W velocity (offset in y by 0.1 m s−1) for (a),(c),(e) small-scale and (b),(d),(f) large-scale stationary deployment.

Surface gravity wave motions do not contribute to submesoscale vertical vorticity but can contribute to noise in vorticity estimates. The surface gravity wave influence on raw velocities, positions, and PDVs are removed by low-pass filtering in the frequency domain with a Gaussian filter G(f ) = exp[−(f/fc)2], where fc is the low-pass-filter cutoff frequency. A variety of cutoff frequencies are considered, all at fc ≤ 4 × 10−2 Hz below the sea-swell frequency band. If fc is not specified, the default largest value fc = 4 × 10−2 Hz is used. The resulting low-passed E-W time series are u(t), x(t), and x˙(t). To summarize, raw unfiltered quantities are denoted with a subscript r, low-passed quantities are not subscripted, and if not specified, the default value of fc = 4 × 10−2 Hz is used.

e. Stationary deployment error statistics and spectra

The error statistics are now defined for the E-W components of position x and velocity u. For the jth GPS, the E-W mean position error is estimated as

 
x¯j=1Nji=1Njxj(ti),
(17)

where Nj is the number of 1 Hz samples. For the jth GPS, E-W position standard error σx,j is estimated as

 
σx,j=[1Nji=1Njxj(ti)2x¯j2]1/2.
(18)

Doppler and PDV velocity mean error (u¯ and x˙¯, respectively) and standard error (σu and σx˙, respectively) are defined similarly. The N-S components of these statistics are also defined similarly. The correlation between E-W velocity on GPS j and E-W velocity on GPS k is denoted ρujuk and estimated as

 
ρujuk=1σu,jσu,kNji=1Nj[uj(ti)u¯j][uk(ti)u¯k].
(19)

The correlation between E-W PDV on GPS j and GPS k (ρx˙jx˙k) is defined similarly as are the correlations for N-S velocities. This notation allows statistics to vary between GPSs that are separated by a distance l

 
l=(XkXj)2+(YkYj)2.
(20)

Note, the notation now uses the subscripts j and k, rather than 1 and 2 as in section 2a.

Error frequency spectra for each GPS is calculated from time series of raw GPS position errors xr(t), velocity error ur(t), and PDVs x˙r(t), the respective error spectra are denoted Sxx(f), Suu(f), and Sx˙x˙(f). The spectra are estimated using a multitaper technique (e.g., Prieto et al. 2009) with 6 degrees of freedom (dof) and approximately 3 × 10−4 Hz frequency resolution. Means are removed from each time series prior to spectra calculation.

4. Results: Stationary deployments

a. Time series

GPS E-W relative position x, E-W PDV x˙, and E-W Doppler velocity u, are shown in Fig. 2 (left column: SSD; right column: LSD) for the default low-pass-filter cutoff frequency fc = 4 × 10−2 Hz. N-S position errors y, PDV y˙, and velocity υ are similar to the E-W errors and therefore not shown. E-W positions x meander about ±1.5 m for the SSD (Fig. 2a), the E-W PDV x˙ fluctuates approximately ±0.025 m s−1 (Fig. 2c), and Doppler velocities u fluctuate approximately ±0.05 m s−1 (Fig. 2e). For the LSD, the position error time series x for GPS 9 (red) and GPS 13 (yellow), and PDVs, are similar to SSD GPSs (cf. Figs. 2a,b and 2c,d), whereas positions (especially at higher frequency) and PDVs for GPSs 10–12 are noisier. Velocity errors u for LSD GPSs are similar to the velocity error for SSD GPSs, although more higher-frequency error is evident for GPSs 10–12 (cf. Figs. 2e,f). Position x, and velocity u appear correlated from GPS to GPS (Figs. 2a,b,e,f).

Using the method described in section 2b, the time series of vorticity ζ (scaled by f at 35°) for the SSD and LSD, using GPSs 5, 6, and 7 and GPSs 9, 12, and 13, respectively, show the influence of GPS separation on vorticity error (Figs. 3a,b). For the small-scale deployment (GPSs separated ≈40 m), using x˙ for the velocities in the best fitting method, that is, (12), results in a vorticity error time series fluctuating between approximately ±5f (red curve, Fig. 3a) whereas for the LSD (GPSs separated ≈1000 m) the error time series is approximately ±0.5f (red curve, Fig. 3b). We note, however, that this vorticity estimate is based on drifter separations smaller (40 and 1000 m), and containing higher-frequency PDVs (up to 25 s motions), than typical for submesoscale vorticity estimation that have separation scales of ≈2000 m and time-scales ≥600 s (e.g., Ohlmann et al. 2017). Using Doppler u for the velocities in (12) results in lower-frequency fluctuations with larger errors than using x˙. For u, for the small-scale deployment −5 < ζ/f < 15 and for the large-scale deployment −1 < ζ/f < 2 (cf. red and black curves in Figs. 3a,b). The difference between vorticity estimated from u and PDV suggests that at lower frequency, vorticity from PDV is more accurate than u.

Fig. 3.

(a) Stationary GPS vorticity (scaled by f at 35°) vs time for the small-scale deployment with separation of ≈40 m. Black line: using ui from GPSs 5, 6, and 7; red line: using x˙i from GPSs 5, 6, and 7. (b) Stationary GPS vorticity (scaled by f at 35°) vs time for the large-scale deployment with separations of ≈1000 m. Black line: using ui from GPSs 9, 12, and 13; red line: using x˙i from GPSs 9, 12, and 13.

Fig. 3.

(a) Stationary GPS vorticity (scaled by f at 35°) vs time for the small-scale deployment with separation of ≈40 m. Black line: using ui from GPSs 5, 6, and 7; red line: using x˙i from GPSs 5, 6, and 7. (b) Stationary GPS vorticity (scaled by f at 35°) vs time for the large-scale deployment with separations of ≈1000 m. Black line: using ui from GPSs 9, 12, and 13; red line: using x˙i from GPSs 9, 12, and 13.

b. Error statistics on an individual GPS

The statistics of stationary GPS positions x(t), velocities u(t), and PDVs x˙(t) for the filter default low-pass-filter cutoff frequency (fc = 4 × 10−2 Hz) time series are now presented. The E-W mean error x¯ is negative for all GPSs except 2 and 7 (circles in Fig. 4a and Table 1) whereas y¯ is positive for all GPSs (squares, Fig. 4a). The mean error x¯ and y¯ is <1 m, except for GPS 12 where |x¯| and |y¯|2.2m (Table 1). The E-W position standard error σx was approximately ≤1 m for the SSD (Fig. 4a, Table 1). E-W position standard errors for the LSD were generally larger than those for the SSD, although position standard errors for GPSs 9 and 13 were similar to SSD values (Table 1). The N-S position standard errors σy were larger in magnitude than σx with GPSs 10, 11, 12, and 13 having noticeably larger σy than the other GPSs (Table 1).

Fig. 4.

(a) Position mean error x¯ (circles) and y¯ (squares) plus and minus position standard error σx vs GPS number. (b) Position differences x˙¯ (circles) and y˙¯ (squares) plus and minus position differences standard deviation σx˙ vs GPS number. (c) Velocity mean error u¯ (circles) and υ¯ (squares) plus and minus velocity standard error σu vs GPS number. GPSs 1–8 (cool colors) are from the small-scale stationary deployment, and GPSs 9–13 (warm colors) are from the large-scale stationary deployment.

Fig. 4.

(a) Position mean error x¯ (circles) and y¯ (squares) plus and minus position standard error σx vs GPS number. (b) Position differences x˙¯ (circles) and y˙¯ (squares) plus and minus position differences standard deviation σx˙ vs GPS number. (c) Velocity mean error u¯ (circles) and υ¯ (squares) plus and minus velocity standard error σu vs GPS number. GPSs 1–8 (cool colors) are from the small-scale stationary deployment, and GPSs 9–13 (warm colors) are from the large-scale stationary deployment.

Table 1.

Statistics for the 13 stationary GPSs. GPSs 1–8 are from the SSD and GPSs 9–13 are from the LSD.

Statistics for the 13 stationary GPSs. GPSs 1–8 are from the SSD and GPSs 9–13 are from the LSD.
Statistics for the 13 stationary GPSs. GPSs 1–8 are from the SSD and GPSs 9–13 are from the LSD.
Table 2.

Terms used to evaluate the vorticity standard error σζ, (16), for the SSD (la = 4 and 16 m) and LSD (la = 177 and 427 m) drifter clusters in Fig. 9. Velocity standard errors (columns 3–5) and correlations (columns 6–8) are low-pass frequency cutoff dependent (column 9).

Terms used to evaluate the vorticity standard error σζ, (16), for the SSD (la = 4 and 16 m) and LSD (la = 177 and 427 m) drifter clusters in Fig. 9. Velocity standard errors (columns 3–5) and correlations (columns 6–8) are low-pass frequency cutoff dependent (column 9).
Terms used to evaluate the vorticity standard error σζ, (16), for the SSD (la = 4 and 16 m) and LSD (la = 177 and 427 m) drifter clusters in Fig. 9. Velocity standard errors (columns 3–5) and correlations (columns 6–8) are low-pass frequency cutoff dependent (column 9).

E-W velocity mean errors u¯0.025ms1 for both SSD and LSD, except GPS 12 which had smaller u¯ (circles, Fig. 4c, Table 1). The N-S velocity mean error is negative with |υ¯|0.01ms1 for both deployments except GPS 12, where υ¯ is positive (squares, Fig. 4c, Table 1). E-W velocity standard errors συ are ≤0.02 m s−1 except for GPS 12 where σu = 0.032 m s−1 (bars, Fig. 4c, Table 1). N-S velocity standard errors συ are larger than σu by approximately 50%. The E-W and N-S PDV mean error x˙¯ and standard error σx˙ are very small relative to the velocity mean and standard error (Fig. 4b). Moreover, the mean errors are x˙¯y˙¯0 whereas the standard error σx˙0.01ms1 except for GPSs 11 and 12. Thus, for velocities containing all frequencies up to f = 4 × 10−2 Hz, PDVs are more accurate than Doppler velocities.

c. GPS-to-GPS correlations

As outlined in section 2a, the velocity correlation from one GPS to another affects vorticity errors [see (11)]. GPS-to-GPS velocity–velocity correlations ρujuk are greater than 0.5 (Fig. 5a), whereas the PDV–PDV correlation ρx˙jx˙k is much smaller (<0.1) (cf. Fig. 5a and Fig. 5b). Both the velocity–velocity and PDV–PDV correlations are smaller for the large-scale deployment (orange and red symbols, l ≥ 100 m) than the small-scale deployment (blue symbols, l < 100 m) suggesting that correlations decrease with increasing GPS separation l. A correlation is red if it includes GPS 12 which had particularly poor satellite coverage relative to the other GPSs. Approximate 95% correlation significant levels are 0.6 for ρujuk and 0.03 for ρx˙jx˙k (gray dashed lines, Fig. 5) with the number of independent data calculated using 83 min and 25 s (e-folding times) as decorrelation time scales (Thomson and Emery 2014). Hence, most ρujuk are significantly nonzero, except for the LSD ρujuk containing GPS 12. For PDVs, only some ρx˙jx˙k for the SSD are significantly nonzero. Note that correlations that include GPS 12, the GPS with the largest errors and worst satellite coverage, are smaller (red symbols, Figs. 5a,b).

Fig. 5.

GPS-to-GPS position correlations vs separation l. (a) Velocity–velocity correlations ρujuk and (b) PDV–PDV correlation ρx˙jx˙k. E-W correlations are circles and N-S correlations are squares. Small-scale deployments are blue and large-scale deployments are orange (or red if the correlation includes GPS 12). All correlations are calculated for raw time series filtered using the default low-pass-filter cutoff frequency fc = 4 × 10−2 Hz. Approximate 95% nonzero correlation values are dashed gray lines.

Fig. 5.

GPS-to-GPS position correlations vs separation l. (a) Velocity–velocity correlations ρujuk and (b) PDV–PDV correlation ρx˙jx˙k. E-W correlations are circles and N-S correlations are squares. Small-scale deployments are blue and large-scale deployments are orange (or red if the correlation includes GPS 12). All correlations are calculated for raw time series filtered using the default low-pass-filter cutoff frequency fc = 4 × 10−2 Hz. Approximate 95% nonzero correlation values are dashed gray lines.

d. Error spectra

The spectra of raw positions Sxx, PDVs Sx˙x˙, and velocities Suu are now presented (Fig. 6). The spectra of N-S and E-W position for each individual GPS is averaged (light colored thin lines in Fig. 6a) resulting in 12 dof for each spectra (thin lines). The N-S and E-W spectra are also averaged for PDV and velocity (light colored thin lines in Figs. 6b,c). The spectra for each GPS are averaged over the SSD and LSD (thick blue and red curves, respectively, Figs. 6a–c) resulting in 96 (2 × 6 × 8) dof for average SSD spectra and 60 (2 × 6 × 5) dof for average LSD spectra. Note that spectra are shown only for frequencies less than the default low-pass-filter cutoff frequency fc = 4 × 10−2 Hz.

Fig. 6.

(a) Position error spectra Sxx, (b) velocity error spectra Suu, and (c) PDV error spectra Sx˙x˙ vs frequency f for the small-scale deployment (blue) and large-scale deployment (red). E-W and N-S components are averaged. Thin light-colored curves are from each GPS and the thick lines are the mean over the GPSs. In (a) and (b) f−2 is shown as the dashed black line. In (c), the gray lines are Suu(f) from (b) for reference.

Fig. 6.

(a) Position error spectra Sxx, (b) velocity error spectra Suu, and (c) PDV error spectra Sx˙x˙ vs frequency f for the small-scale deployment (blue) and large-scale deployment (red). E-W and N-S components are averaged. Thin light-colored curves are from each GPS and the thick lines are the mean over the GPSs. In (a) and (b) f−2 is shown as the dashed black line. In (c), the gray lines are Suu(f) from (b) for reference.

For the small-scale deployment, the position error spectra Sxx(f) follows an approximate f−2 scaling for f > 10−4 Hz (cf. blue curve and dashed line, Fig. 6a). A similar spectral slope for GPS position errors has been previously reported (Johnson and Pattiaratchi 2004; MacMahan et al. 2009; Suara et al. 2015). The spectra Sxx(f) flattens for f ≤ 10−4 Hz because this is the approximate frequency resolution.

The velocity error spectra Suu(f) is also red but falls of more slowly, than the position error spectra Sxx(f) (Fig. 6b). The SSD and LSD PDV error spectra is nearly white (blue and red curves, Fig. 6c) because the PDV is the time derivative of position. Comparing Suu (reproduced in gray in Fig. 6c) and Sx˙x˙(f) indicates that Doppler velocity u errors are smaller than PDV x˙ errors for approximately f > 10−3 Hz whereas for lower frequencies PDVs have smaller error than Doppler velocities.

e. Blended trajectory

The velocity Suu and PDV spectra Sx˙x˙ indicate that for low frequencies (f ≲ 10−3 Hz), x˙ velocities have less error than u velocities, whereas for higher frequencies (f ≳ 10−3 Hz), u velocities have less error than x˙ velocities. Thus, Sx˙x˙(f)<Suu(f) for f < fT whereas Sx˙x˙(f)>Suu(f) for f > fT. The transition frequency is approximately fT ≈ 10−3 Hz because Sx˙x˙ and Suu cross at approximately this frequency (Fig. 6c). Note, that 04×102HzSx˙x˙(f)df<04×102HzSuu(f)df so that σx˙<σu (see Figs. 4b,c and Table 1). A reduced error velocity time series, however, would use x˙ for the low frequency and u for the high frequency. This blended velocity uB is then

 
uB(t)=u(t)u˜(t)+x˙˜(t),
(21)

where u˜(t) and x˙˜(t) are low passed at low-pass-filter cutoff frequency fc = 10−3 Hz. Recall that u(t) is filtered at the default low-pass-filter cutoff frequency fc1=25s, so that blended velocities contain motions with periods ≥25 s. For all 13 GPSs, blended raw velocity time series uB(t) are calculated. Note, that the blended time series are shorter than u(t) and x˙(t) due to the low-pass filtering at fc = 10−3 Hz which necessitates removing some of the beginning and end of u(t) and x˙(t). The difference in trajectory lengths results in small blended velocity and PDV spectra differences and small PDV–PDV correlation and blended velocity–velocity correlation differences for frequencies < 10−3 Hz. These differences, however, are not statistically significant. Blended trajectories xB,

 
xB(t)=0tuB(t)dt,
(22)

are also be constructed and follow the low-frequency drifter trajectory but have more accurate higher-frequency excursions. Blended trajectories are used for the in situ observations (section 5a). We note that typical studies of submesoscale motions (where the sampling f < 10−3 Hz) do not require using blended velocities or trajectories because PDVs have smaller errors at these frequencies. However, blended velocities should be used, if possible, when sampling motions with time scales >10−3 Hz.

f. Dependence of velocity errors and correlation on the low-pass-filter cutoff frequency

The effect of the low-pass-filter cutoff frequency fc on the velocity standard error and velocity–velocity correlation is now explored for SSD and LSD. The velocity standard error dependence on fc [σu(fc)] is calculated as the square root of the integral of the velocity error spectra (Fig. 6, thick curves)

 
σu(fc)=[0fcSuu(f)df]1/2.
(23)

The fc dependence of the PDV standard error σx˙(fc) and blended velocity error σuB(fc) are similarly estimated. Recall that spectra for the SSD and LSD are averaged over all GPSs in the deployment and across E-W and N-S components. Frequency-dependent velocity–velocity correlations are calculated using velocities filtered at various low-pass-filter cutoff frequencies fc > 2 × 10−4 Hz. At lower frequencies, too few independent data exist for reliable correlations. Low-pass-filter cutoff-frequency–dependent SSD and LSD correlations are averaged over all GPS pairs (i.e., all j and k for jk in ρujxk) and averaged over the E-W and N-S components.

The PDV error σx˙(fc) monotonically increases with the low-pass-filter cutoff frequency fc (see blue and red dotted curves in Fig. 7a). Because the PDV error spectra is approximately white, the error generally follows σx˙(fc)~fc1/2 for fc < 5 × 10−3 Hz. For the lowest frequencies, the velocity error σu starts large, relative to σx˙, but quickly asymptotically approaches a constant value (dashed curves in Fig. 7a) owing to the redness of Suu(f). Only near the highest frequency (fc = 4 × 10−2 Hz), are the velocity error σu and the PDV error σx˙ similar (cf. dashed and dotted curves in Fig. 7a). Due to their construction, blended velocity errors follow PDV errors for low frequencies and asymptotically approach a constant value for approximately fc > 10−3 Hz, less than either the u or x˙ error. For fc < 10−3 Hz, σuB is not statistically different from σx˙(fc). Consistent with previous error statistics, the velocity standard error dependence on the low-pass-filter cutoff frequency fc is larger for the LSD deployment than the SSD (red curves are above blue in Fig. 7a).

Fig. 7.

The frequency dependence of quantities related to vorticity standard error. Small-scale (SSD) and large-scale (LSD) deployments are blue and red curves, respectively. (a) Velocity error σu and (b) the velocity–velocity correlation between different GPSs ρujuk (jk) vs low-pass-filter cutoff frequency fc. Statistics for x˙ are dotted, u are dashed, and blended velocity uB are solid. Correlations for f < 2 × 10−4 are not shown because there are not enough independent samples for these frequencies.

Fig. 7.

The frequency dependence of quantities related to vorticity standard error. Small-scale (SSD) and large-scale (LSD) deployments are blue and red curves, respectively. (a) Velocity error σu and (b) the velocity–velocity correlation between different GPSs ρujuk (jk) vs low-pass-filter cutoff frequency fc. Statistics for x˙ are dotted, u are dashed, and blended velocity uB are solid. Correlations for f < 2 × 10−4 are not shown because there are not enough independent samples for these frequencies.

The velocity–velocity correlation ρujuk decreases with increasing low-pass-filter cutoff frequency fc for u, x˙, and uB (dashed curves, Fig. 7b). For u, ρujuk(fc) starts large (≥0.7) and decreases by about 0.1 with increasing fc. Both ρx˙x˙ and ρuBuB, are relatively large for fc < 10−3 Hz and decreases quickly with values <0.1 for larger fc (dotted and solid curves, Fig. 7b). Like the standard error, blended velocity correlation is not statistically different from the PDV correlation for fc < 10−3 Hz. SSD (blue curves Fig. 7b) velocity, PDV, and blended velocity GPS-to-GPS correlations are greater than their corresponding LSD correlations (red curves Fig. 7b) consistent with the individual GPS-to-GPS correlations in Fig. 5.

g. Scaling the vorticity error

Here, scalings for the mean vorticity and vorticity error for the stationary GPSs are tested. The effect of the velocity type (u, x˙, or uB), the effect of low-pass-filter cutoff frequency fc, and the effect of the minor axis length la are examined. Clusters of three drifters are defined from triplets of SSD and LSD GPSs. The SSD has eight clusters (Fig. 1a) made of GPS numbers (1, 2, 3), (2, 3, 4), (3, 4, 1), (4, 1, 2), (5, 6, 7), (6, 7, 8), (7, 8, 5), and (8, 5, 6). From the eigenvalues of the cluster position covariance matrix, clusters 1–4 have minor and major axes of la ≈ 4 m and lb ≈ 7 m and clusters 5–8 have la ≈ 16 m and lb ≈ 7 m. The LSD has three clusters (Fig. 1b) made up of GPS numbers (9, 11, 12), (9, 12, 13) and (10, 12, 13). For the first cluster, (la, lb) = (180, 530) m and for the remaining clusters (la, lb) ≈ (430, 800) m. For each cluster and velocity type, least squares vorticity is calculated for three different low-pass-filter cutoff frequencies fc1=(25,300,1800)s resulting in 99 (11 clusters of three GPSs, 3 velocity products, and 3 fc) vorticity time series from which (as GPSs are stationary) vorticity mean error ζ¯ and standard error σζ are calculated. Additionally, σζ is estimated a priori from (16) using σu (and σx˙, σuB) and ρujuk (and ρx˙jx˙k, ρuBjuBk) associated with the appropriate fc (Fig. 7) and respective SSD or LSD la (Table 1). Recall that velocity variances and correlations are averages of E-W and N-S values.

The vorticity mean error magnitude |ζ¯|/f is inversely related to the cluster minor axis la (dashed line, Fig. 8). For the Doppler velocities u, |ζ¯|/f>0.1 regardless of minor axis la (black triangles, Fig. 8), whereas for x˙ and uB, |ζ¯|/f<0.1f (red and blue triangles, Fig. 8), as the x˙¯ and u¯B are much smaller than u¯ (Fig. 4). Note that |ζ¯|/f is independent of the low-pass-filter cutoff frequency fc because it derives from the time mean of the velocity.

Fig. 8.

Absolute value of the mean vorticity (scaled by f at 35°) for clusters of three stationary GPSs (triangles) vs the minor axis length la of the GPS cluster. Small- and large-scale deployments are separated by la = 100 m. Vorticity derived from Doppler velocities u are black, PDVs x˙ are red, and blended velocities uB are blue. All velocities are filtered at the low-pass-filter cutoff frequency fc = 4 × 10−2 Hz. The dashed line is la1. There is no frequency dependence since the standard deviations are of the mean velocity, which does not change with fc. Blue and red triangles are not exactly the same since the length of the time series are different due to the low passing required in the construction of uB.

Fig. 8.

Absolute value of the mean vorticity (scaled by f at 35°) for clusters of three stationary GPSs (triangles) vs the minor axis length la of the GPS cluster. Small- and large-scale deployments are separated by la = 100 m. Vorticity derived from Doppler velocities u are black, PDVs x˙ are red, and blended velocities uB are blue. All velocities are filtered at the low-pass-filter cutoff frequency fc = 4 × 10−2 Hz. The dashed line is la1. There is no frequency dependence since the standard deviations are of the mean velocity, which does not change with fc. Blue and red triangles are not exactly the same since the length of the time series are different due to the low passing required in the construction of uB.

For all low-pass-filter cutoff frequencies fc and all velocity products, the vorticity standard error σζ/f decreases as la1 from la of 5–400 m (Fig. 9). For the highest fc1=25s, the vorticity standard error σζ/f varies from 30 to 0.2 over the la = 5 m to la = 400 m and the different velocity products (Fig. 9a). The σζ/f is 2–4 times larger using u relative to uB, with x˙ in between (cf. colored triangles in Fig. 9a). Using the Doppler velocity u, σζ/f does not decrease substantially with decreasing fc (cf. black triangles across Fig. 9), as much of the u error is at low frequencies and filtering has little effect (Figs. 6 and 7). In contrast, for the x˙ and uB, σζ/f decreases an order of magnitude from fc1=25s to fc1=1800s at all la, consistent with the reduced error (Fig. 7). At fc1=25s and fc1=300s, the uB-based σζ/f is smallest as it combines the optimal frequency-dependent noise properties of u and x˙ (Fig. 7). At fc1=1800s, uB and x˙ σζ/f are nearly identical. The small difference is because the time series uB is shorter than x˙.

Fig. 9.

Vorticity standard error for clusters of three stationary GPSs for the small- and large-scale stationary deployment. Standard deviation of cluster vorticity (triangles, scaled by f at 35°) vs la for velocities filtered at (a) fc1=25s, (b) fc1=300s, and (c) fc1=1800s. The small- and large-scale deployments are separated by la = 100 m. Vorticity is derived from velocities u (black), PDVs x˙ (red), or blended velocities uB (blue). Dashed black lines in all panels are la1. Large circles are found using (16): values of each term are listed in Table 2.

Fig. 9.

Vorticity standard error for clusters of three stationary GPSs for the small- and large-scale stationary deployment. Standard deviation of cluster vorticity (triangles, scaled by f at 35°) vs la for velocities filtered at (a) fc1=25s, (b) fc1=300s, and (c) fc1=1800s. The small- and large-scale deployments are separated by la = 100 m. Vorticity is derived from velocities u (black), PDVs x˙ (red), or blended velocities uB (blue). Dashed black lines in all panels are la1. Large circles are found using (16): values of each term are listed in Table 2.

For each velocity product, the vorticity standard error σζ/f dependence on the length scales la and lb is well reproduced by the error scaling (16) with N = 3 (cf. circles and triangles in Fig. 9). The small differences between stationary GPS σζ/f and (16) are due to using SSD or LSD values of σu and ρujuk in (16) and not specific values for the GPS triplet. In contrast to previous, a posteriori error analysis (e.g., Okubo and Ebbesmeyer 1976), the similarity between the stationary GPS σζ/f and the scaling (16) indicates that vorticity errors can be anticipated a priori with knowledge of the velocity error (and potential correlation) and the scale of drifter cluster. This will be useful in drifter experiment planning where vorticity and divergence are being estimated. Note that vorticity errors for drifters instrumented with other GPSs may differ due to different GPS instrument noise σu. However, the methods outlined here (sections 4b, 4c, 4d, and 4f) can be applied to any GPS in order to accurately estimate σu and therefore σζ.

5. Discussion

a. Vorticity error for in situ data

The scaling (16) is now used to assess the influence of GPS error for in situ derived vorticity. Surface drifters were deployed in clusters on 10 October 2016 near Point Sal, California (34.9°N, −120.67°E). Here, an example of three clusters of three drifters over 5 h reveals a complex surface flow (Fig. 10). The two northern clusters (red and blue in Fig. 10) initially heads offshore, before advecting onshore and northward about 2 km. In contrast, the southernmost cluster (black in Fig. 10) has weak advection. For all clusters, drifters are entrained in a frontal feature and end up aligned alongfront. The frontal nature of this feature was identified by temperatures recorded by nearby moorings: the surface temperature increased by ≈1°C from the shoreward to seaward side of the front (not shown).

Fig. 10.

Tracks of three clusters of three drifters for the 10 October drifter release (colored curves). Initial positions are indicated by dots. Bathymetry is contoured and thick white contours are at 10 m intervals while thin white contours are at 5 m intervals. N-S (y) and E-W (x) distances are relative to the tip of Point Sal, CA (34.9°N, −120.67°E).

Fig. 10.

Tracks of three clusters of three drifters for the 10 October drifter release (colored curves). Initial positions are indicated by dots. Bathymetry is contoured and thick white contours are at 10 m intervals while thin white contours are at 5 m intervals. N-S (y) and E-W (x) distances are relative to the tip of Point Sal, CA (34.9°N, −120.67°E).

For these three clusters and 5 h deployment, the minor la and major lb axis lengths (Figs. 11a–c) are calculated from the drifter-position covariance matrix P, (A10), using blended positions xB, (22), at every time step. The LS vorticity is estimated with the blended positions xB and blended velocities uB. Blended velocities uB use a low-pass cutoff frequency of fc1=300s. For each cluster, vorticity standard error σζ is estimated using the scaling (16) with parameters N = 3, (la, lb), the LSD derived σuB=0.0036ms1, and with ρuBjuBk=0. Note, assuming ρuBjuBk=0 is a worst-case assumption and makes this error estimate an upper bound. The σζ time dependence is due only to the changing cluster geometry (la, lb).

Fig. 11.

(a)–(c) The minor la (dashed curves) and major la (solid curves) axis lengths vs time for the (top to bottom) red, blue, and black drifter clusters in Fig. 10. (d)–(f) The LS vorticity normalized by the local f (ζ/f) vs time for the three clusters. Vorticity is calculated using a 5-min low-pass frequency cutoff (fc1=300s) blended velocities. In (d)–(f), thin dashed curves are plus and minus the vorticity standard error σζ/f calculated using (16). Gray shading in all panels indicates times for which la < 50 m, corresponding to the time when σζ/f increases rapidly.

Fig. 11.

(a)–(c) The minor la (dashed curves) and major la (solid curves) axis lengths vs time for the (top to bottom) red, blue, and black drifter clusters in Fig. 10. (d)–(f) The LS vorticity normalized by the local f (ζ/f) vs time for the three clusters. Vorticity is calculated using a 5-min low-pass frequency cutoff (fc1=300s) blended velocities. In (d)–(f), thin dashed curves are plus and minus the vorticity standard error σζ/f calculated using (16). Gray shading in all panels indicates times for which la < 50 m, corresponding to the time when σζ/f increases rapidly.

For all clusters except the southern most (black), the cluster major axis lb(t) grows with time (solid curves, Figs. 11a–c). For all clusters, the minor axis la(t) is at first largely constant (between 100 and 300 m depending on cluster) before decreasing rapidly at approximately 1100, 1100, and 1200 Pacific daylight time (PDT) for the red, blue, and black clusters, respectively (dashed curves, Figs. 11a–c). This rapid decrease is due to drifters being entrained into the frontal feature (Fig. 10). Times of frontal entrainment, that is, when la < 50 m, are shaded gray in Fig. 11. This alongfront drifter alignment with very small la/lb ratio is typical of for surface drifter clusters in submesoscale features (Ohlmann et al. 2017). Prior to frontal entrainment, LS vorticity for each cluster is generally between ±2f and can change by f on 1 h time scales with additional higher-frequency (0.3–0.5 h time scales) variability (Figs. 11d–f). During this time, the a priori vorticity standard error σζ/f is relatively small, nearly always smaller than the σζ/f (dashed curves in Fig. 11). The σζ/f is largest for cluster “red” due to the smaller la and lb (top panels of Fig. 11). Some of the higher-frequency σζ/f variability in Figs. 10d–f is attributable to error as this variability has approximate amplitude of σζ/f. Coincident with the la rapid decrease, σζ/f begins oscillating widely beyond ±5f at very high frequencies at approximately 1130, 1200, and 1300 PDT for the red, blue, and black cluster, respectively. This suggests that the vorticity is dominated by noise. Commensurate with these oscillations, the vorticity standard error σζ/f increases rapidly precisely when la becomes very small (<50 m, shaded gray in Fig. 11). Hence, even for velocity errors of σu ≈ 0.004 m s−1 , vorticity cannot be accurately estimated on scales of la < 50 m with N = 3 m if that vorticity is O(f).

b. Previous vorticity estimates and errors

Large errors in estimated DKP (i.e., vorticity, divergence) were previously evident (e.g., Molinari and Kirwan 1975; Paduan and Niiler 1990; Ohlmann et al. 2017) and a method to estimate DKP standard errors had been developed (Okubo and Ebbesmeyer 1976; Kirwan and Chang 1979) establishing the role of the velocity misfit σu, due to instrument error and process noise by assuming uniform velocity gradients. However, σu had to be estimated a posteriori from the LS fit, with some fraction of contribution from instrument noise. The relative contribution of instrument and process noise to σu will depend on the flow scales, drifter cluster size, and instrument noise. Here, the instrument velocity error σu and GPS-to-GPS error correlations ρuj,uk are estimated a priori for different low-pass-filter cutoff frequencies fc. From this, DKP errors such as vorticity standard error σζ can be known a priori rather a posteriori estimated. This method, (16), can be used to guide drifter experiment planning for accurately estimating vorticity and divergence.

The previous DKP error estimation method, (A4), embedded cluster geometry within (RTR)−1, and thus did not explicitly reveal the importance of the geometry. Because the precise role of cluster geometry on the errors was unknown, various ad hoc methods for determining when the DKP error was too large were developed. Previous authors have used area, effectively lalb, (Paduan and Niiler 1990), ellipticity la/lb (Ohlmann et al. 2017), or a combination of maximum drifter separation and ellipticity (Righi and Strub 2001), to determine when DKP errors become too large. Here, the precise role of the drifter cluster minor la and major lb axes were established and

 
σζ2=N1σu2la2[1+(la/lb)2](1ρujuk).

Thus, for highly elliptic clusters (lalb), the vorticity error is largely due to la, while for more circular clusters (lalb), the squared error depends on both la and lb and for lbla the squared error is ≈2 times the squared error for highly elliptic clusters. As such, large vorticity errors should be identified by la2+lb2, rather than the area or ellipticity.

For the Point Sal estimated vorticity, comparing our direct predictions of the vorticity error (dashed lines in Figs. 11d–f) to the previous criterion is instructive. Recall that the time dependence of these errors arise only from the changing geometry of the drifter cluster, that is, la(t) and lb(t), and that large vorticity errors resulted from la < 50 m. Previous authors suggested that vorticity errors depend on cluster area (Molinari and Kirwan 1975; Paduan and Niiler 1990) setting lower and upper limits to the area of the drifter cluster beyond which DKP errors are too large. Setting an upper limit is appropriate for large drifter separations where the process noise becomes large as the Taylor series approximation of the velocity, (12), from which the LS technique is built on, is no longer valid. A minimum ellipticity la/lb criterion for detecting DKP noise has also been suggested (Ohlmann et al. 2017). Both a minimum area criterion and a minimum ellipticity criterion could be applied to the Point Sal clusters because for these clusters, area and ellipticity criterion are similar to the minimum la criterion (<50 m) used here. The time dependence of la and lb in Figs. 11a–c indicate that until approximately 1100 PDT la(t) remains fairly constant whereas lb(t) increases. For times greater than 1100, 1100, and 1200 PDT (red, blue, and black clusters, respectively), however, la rapidly decreases whereas lb is relatively constant. Thus, the time dependence of la, lalb, and la/lb are all similar when la(t) is rapidly decreasing. However, cluster minimum area or ellipticity criteria to distinguish high DKP noise will not in general give accurate results due to direct dependence of the vorticity standard error, (16), on la.

c. Effect of satellite coverage on errors

To get the most accurate estimate of vorticity errors from GPS tracked drifters, the underlying GPS velocity error must be known. In addition to depending on the particular GPS receiver, this error will depend on the quality of the GPS satellite constellation. The quality of the constellation depends on the number of satellites ns in view and the position of these satellites in the sky. Although satellite position is not recorded by the GPSs used here, ns and a nondimensional estimate of the absolute position error (HDOP), are recorded by these GPSs at 1 Hz. Because ns only varied by 1.5 satellites over the SSD and LSD, effect of satellite number on GPS velocity errors cannot be thoroughly examined. Here, we briefly explore how satellite coverage affects GPS position and velocity standard errors for the stationary GPS dataset.

For this dataset, increasing satellite number n¯s decreases the position standard error σx (Fig. 12a). Overall, LSD had smaller n¯s (red dots, especially GPS 12) and larger σx than the SSD (blue dots). The PDV standard error σx˙ and Doppler velocity standard error σu are similarly related to satellite number n¯s (Figs. 12b,c). The GPS estimate of horizontal position error HDOP decreases approximately linearly with n¯s (Fig. 12d), hence, HDOP provides a useful measure of position error. The relationship between error and n¯s indicates that more precise estimates of GPS velocity errors are possible given satellite coverage information. This could be useful in order to distinguish signal and noise for vorticity estimates where the noise and signal magnitudes are similar, that is, for vorticity estimates at smaller space and time scales.

Fig. 12.

(a) Position standard error σx, (b) PDV standard error σx˙, (c) Doppler velocity standard error σu, and (d) HDOP vs the mean number of satellites in view n¯s. Colors refer to SSD (blue) and LSD (red). The standard errors are the square root of averaged E-W and N-S squared standard errors in Table 1. Values are found using the time series filtered at the default low-pass-filter frequency fc = 4 × 10−2 Hz.

Fig. 12.

(a) Position standard error σx, (b) PDV standard error σx˙, (c) Doppler velocity standard error σu, and (d) HDOP vs the mean number of satellites in view n¯s. Colors refer to SSD (blue) and LSD (red). The standard errors are the square root of averaged E-W and N-S squared standard errors in Table 1. Values are found using the time series filtered at the default low-pass-filter frequency fc = 4 × 10−2 Hz.

6. Summary

The a priori vorticity standard error σζ is derived (16) based on the least squares method of estimating vorticity from drifters. The σζ depends upon the velocity error (from instrument noise or process noise due to assuming uniform velocity gradients), cross-drifter correlation, drifter number, and drifter cluster shape. This derivation extended previous vorticity standard error estimates by including the effect of the correlated velocity errors and showing how the drifter cluster minor and major axes la, lb affect the error.

Two stationary GPS experiments, with identically zero vorticity, were performed at separations of 10–700 m to understand drifter derived vorticity error and test the derivation using 1 Hz position differences (PDV) and Doppler shift velocities. Standard vorticity estimation reveals error of ±5f at separations of 40 m. For low frequencies (<10−3 Hz), PDVs velocities are more accurate than Doppler velocities, whereas at higher frequencies (>10−3 Hz), the opposite occurs. A “blended” velocity is derived which has the low-frequency characteristics of PDV and the higher-frequency characteristics of the Doppler velocities, resulting in the smallest velocity error. The frequency-dependent velocity variances and GPS-to-GPS correlations were quantified as a function of low-pass-filter cutoff frequency. For the two stationary GPS experiments, the vorticity standard error as a function of cluster minor axis la is well predicted given velocity error and GPS-to-GPS correlation.

Vorticity error analysis is applied to three clusters of three GPS drifters released on the inner shelf off of Point Sal, California, that sampled submesoscale flow features. The value σζ due to GPS noise was estimated a priori using (16). For these clusters, the vorticity was O(f) but began to oscillate widely as the drifters were entrained in a frontal feature. The a priori estimated σζ increases dramatically coincident with the large vorticity oscillations. This σζ increases is due to small la (<50 m), and the drifter cluster minor axis (narrowness) is the key time-dependent factor affecting vorticity error. Even for velocity errors of 0.004 m s−1, the vorticity error exceeds ±5f when cluster minor axis <50 m. Large vorticity standard error cannot be anticipated based on cluster area (lalb) or ellipticity (la/lb). This a priori method for estimating vorticity standard error can be used in planning submesoscale drifter deployments where vorticity or divergence are being estimated.

Acknowledgments

The Office of Naval Research supported this research through grants N00014-5-1-2631 (SIO) and N0001418WX00229 (NPS). Casey Gon assisted in the stationary field deployments. For the Point Sal fieldwork, Bill Boyd, Greg Boyd, Tucker Freismuth, Casey Gon, Matt Gough, Rob Grenzeback, Derek Grimes, Ami Hansen, Paul Jessen, Michael Kovatch, Paul Lenz, Aaron Morrone, Andy O’Neill, Lucian Parry, Brett Pickering, Greg Sinnett, Kent Smith, Marla Stone, Ata Suanda, Brian Woodward, and Keith Wyckoff are acknowledged for their help in deployment and recovery. The authors thank Derek Grimes, Michael Kovatch, Jen Mackinnon, Sean Celona, Andre Paloczy, Nirnimesh Kumar, Ata Suanda, and Amy Waterhouse for providing useful feedback. The data and files necessary to reproduce the results herein, can be obtained by contacting the corresponding author (mspydell@ucsd.edu). We thank two reviewers for helping to improve the manuscript.

APPENDIX

Derivation of Vorticity Error Variance

Here, the vorticity standard error σζ is derived explicitly including both cluster geometry (size and shape), and velocity correlations across drifters, extending previous work (Okubo and Ebbesmeyer 1976). For N drifters, the least squares inversion for β=[UdU/dxdU/dy]T, (13), is given in (14) and the position matrix R is given in (15). The covariance of least squares fit model parameters β is given by (e.g., Wunsch 1996)

 
cov(β)=(RTR)1RTU˜U˜TR(RTR)1.
(A1)

Because positions (xi and yi) are relative to the cluster center, ⟨x⟩ = 0 and ⟨y⟩ = 0, where ⟨⋅⟩ represent an average over all drifters. Thus, the matrix RTR takes the form

 
RTR=N(1000sxx2sxy20sxy2sy2),
(A2)

where the position covariances are defined as sab2=ab, again averaged over all drifters. The (RTR)−1 matrix is given by

 
(RTR)1=1N(sxx2syy2sxy4)(sxx2syy2sxy4000syy2sxy20sxy2sxx2).
(A3)

The velocity error covariance is U˜U˜T. Previously the velocity error covariance matrix was assumed diagonal (e.g., Okubo and Ebbesmeyer 1976), that is, errors are uncorrelated and homogeneous, and the covariance of the LS fit parameters is given by

 
cov(β)=σu2(RTR)1forU˜U˜T=σu2I,
(A4)

where I is the identity matrix and σu is the velocity error due to both instrument noise and process error (incorrectly assuming that velocity gradients are constant). Here, however, the velocities errors are assumed to be equally correlated across all GPSs with coefficient ρujuk, thus

 
U˜U˜T=σu2(1ρujukρujukρujuk1ρujuk)=σu2ρujuk(111111)A+σu2(1ρujuk)IB.
(A5)

Correlations of velocity error are approximately constant if drifter separations do not vary much, for example, correlations for the small- and large-scale (if GPS 12 is omitted) GPS deployments are approximately constant for each deployment (see Fig. 5a). The variance of the estimated mean velocity U, that is, cov(β)1,1, is

 
σU2=ρujukσu2+1N(1ρujuk)σu2,
(A6)

where the first term is from term A and the second from term B in (A5). The (3, 3) term of (A1) is the variance of the velocity gradient σdU/dy2, and it is σu2(1ρujuk)a33 where a33 is the (3, 3) term of (A3), thus,

 
σdU/dy2=σu2N(1ρujuk)sxx2sxx2syy2sxy4.
(A7)

The vorticity error variance is

 
σζ2=σdV/dx2+σdU/dy2
(A8)

and if the E-W and N-S velocity errors are the same (συ2=σu2), and assuming u and υ errors are uncorrelated, the squared vorticity standard error is

 
σζ2=σu2N(1ρujuk)sxx2+syy2sxx2syy2sxy4.
(A9)

If these assumptions are relaxed, a less elegant expression results.

The position covariances sxx2, syy2, and sxy2 can be expressed as the eigenvalues la2 and lb2 (lalb) of the position covariance matrix P:

 
P=(sxx2sxy2sxy2syy2)
(A10)

in such a way that the vorticity error, (A9), in terms of la and lb is

 
σζ2=σu2N(1ρuu)(1la2+1lb2)
(A11)

which is (16). The eigenvalues la and lb can be considered the width and length of the drifter cluster. This concise formula for the vorticity error variance differs from previous analysis (e.g., Okubo and Ebbesmeyer 1976; Kirwan and Chang 1979) in that explicitly accounts for the cluster geometry and accounts for potentially correlated velocity errors.

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Footnotes

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