Statistical Inversion of Surface Ocean Kinematics from Sea Surface Temperature Observations1. Introduction
Sea surface temperature (SST) varies at different time and space scales, and these variations have been linked to the variability in several climate and environmental phenomena (Deser et al. 2010). In addition, the SST data provide valuable information on surface ocean kinematics as it is observed at high spatial and temporal resolution. It is therefore of interest to characterize SST variability and to use the SST data to characterize ocean kinematics. To this end we develop a local linear matrix inversion method (Jeffress and Haine 2014b,a) to quantify the evolution of the SST field with a response function and to derive the associated SST velocity, diffusivity, and decay fields.
In the past, several statistical methods have been used to assess SST variability and related predictability (Hasselmann 1976; Frankignoul 1985; Saravanan and McWilliams 1998; Buckley et al. 2014, 2015; Yamamoto and Palter 2016; Årthun and Eldevik 2016; Årthun et al. 2017; Roberts et al. 2017). In addition, sequential pairs of SST images have been used to infer surface velocities (Emery et al. 1986; Kelly 1989; Vigan et al. 2000a,b; Chen et al. 2008, and references therein). The former studies are usually motivated by climate-related questions, while the latter studies are often more operational in scope. Our methodology is statistical, and therefore similar to the former methods, although some aspects of it, such as usage of spatial covariances, have similarities with the latter approaches. In the following we introduce these approaches, before describing our methodology (see section 5 for more comparison between the different methods).
One of the most widely used statistical approaches to understand SST variability is the lagged correlation analysis, in which the anomaly propagation speed is derived from the time lag of maximum correlation between two locations (e.g., Sutton and Allen 1997; Årthun et al. 2017). These studies often report propagation speeds on the order of a few centimeters per second (cm s−1), which are generally an order of magnitude slower than the mean current speeds in the ocean. Consequently, these studies find climate predictability over lead times of several years and attribute it to the slow SST (or upper-ocean heat content) anomaly propagation (Sutton and Allen 1997; Årthun et al. 2017). Despite its wide usage, the lagged correlation approach has several shortcomings. The lagged correlations tend to be weak unless the underlying data are low-pass filtered in time (Foukal and Lozier 2016). However, such filtering can be problematic, especially if applied over a time scale at which the forcing varies, because the separation between the forcing and the response becomes ambiguous. In fact, Foukal and Lozier (2016) demonstrated that the results of Sutton and Allen (1997) were likely an artifact of the filtering time scale that incorporated the effect of the changing North Atlantic Oscillation on the filtered SST anomaly field. Finally, a more fundamental problem with the lagged correlation approach is that by focusing on the peak correlation, one recovers only one travel time between two points, whereas an advective–diffusive system hosts a distribution of travel times. Such a travel-time distribution can be represented by a response function (Green’s function) that solves the SST anomaly equation for an impulse forcing (Jeffress and Haine 2014a). The ability to derive such a response function is one of the advantages of the matrix inversion presented here.
Another statistical approach is the linear matrix modeling, which is popular in statistical climate prediction (Penland and Magorian 1993; Penland and Sardeshmukh 1995; Penland and Matrosova 1998; Penland and Hartten 2014; Newman 2007; Alexander et al. 2008; Piterbarg and Ostrovskii 1997; Ostrovskii and Piterbarg 2000; Ostrovskii and Font 2003; Deser et al. 2003; Zanna 2012) and in analysis of model predictability (Tziperman and Ioannou 2002; Tziperman et al. 2008; Hawkins and Sutton 2009). Linear matrix modeling assumes that the time evolution of SST (or another parameter of interest) is governed by a linear transport operator and a white noise forcing. As we will show, the linear transport operator can be recovered from the covariance matrices of the SST anomaly vector at different time lags. Because the dimensions of the covariance matrices become very large as the spatial domain size increases, it is a common practice to build the inversion on empirical orthogonal functions (EOF) or some other reduced-size basis set. For example, Frankignoul et al. (1998) invert for the SST anomaly decay rate from autocorrelation by averaging SST over a large region in the North Atlantic. In that case, the transport operator is dominated by air–sea interaction, as the mean current and eddy activity are weak. A related approach moves one step further and focuses on the eigenvectors of the linear transport operator, also known as principal oscillation patterns (POP) (Hasselmann 1988; von Storch et al. 1995, and references therein).
Instead of using EOFs or POPs to reduce the degrees of freedom of a large dataset, it is possible to build the inversion upon covariances that are local in space. In this approach, one inverts a small covariance matrix for each grid cell and its immediate neighbors. For example, Piterbarg and Ostrovskii (1997), Ostrovskii and Piterbarg (2000), and Ostrovskii and Font (2003) perform such a local inversion and invert for velocity, diffusivity, decay rate, and mixed layer entrainment (which form their transport operator) in the North Pacific and in the North Atlantic. In this study we will also use a stencil that is local in space to solve for the transport operator but use a somewhat different approach than Piterbarg and Ostrovskii (1997), Ostrovskii and Piterbarg (2000), and Ostrovskii and Font (2003) in solving for it (see our section 5 for details).
In addition to the statistical approaches, different inversion methods are used to estimate ocean surface velocities from pairs of SST images (or even from single SST images given some additional constraints; e.g., Isern-Fontanet et al. 2017). These studies usually infer the ocean velocities by either using feature tracking (maximum cross correlation) or by solving the heat equation. The focus is also on short time scales (<1 day), and these studies neglect diffusion (or effectively include the effect of diffusion in the velocity estimates) and decay of the anomalies. Finally, it is worth noting that in the fluid dynamics community, there is a wide field of particle image velocimetry (for a recent review, see Westerweel et al. 2013) that focuses on estimating a flow field from a sequence of images through cross correlation.
List of symbols used in this study.
We structure this paper as follows: first we derive an inversion method for solving the transport operator 
2. Theory
The derivation of Eq. (2) is only one possibility to solve for 
3. Implementation
Here we move from the theoretical realm of the previous section to a practical realm. In particular, we always have a finite-length time series and are left with an estimate 
Further, we assume that we can use a local five-point stencil (
Illustration of the five-point stencil, and the notation used in Eqs. (10) and (11).
Citation: Journal of Atmospheric and Oceanic Technology 35, 10; 10.1175/JTECH-D-18-0057.1
a. Preprocessing of the input data
In this study we are interested in characterizing the mechanisms behind the oceanic part of SST variability. However, the SST record is affected not only by oceanic variability but also by external forcing, and atmospheric variability. We aim to remove the latter two sources of variability prior to the inversion.
The largest external signal is seasonality, which we eliminate by removing a smoothed (15-day filter) daily SST climatology at each spatial grid point. Removing only annual or seasonal frequencies instead of the climatology does not affect the results (not shown).
Removing the atmospheric signal is less straightforward. In the previous section, we assumed that we can choose a τ such that the lagged covariance of the atmospheric forcing term vanishes [second term on the right-hand side in Eq. (2)] because 
b. Technical implementation
The inversion is implemented in parallel Python code, which inverts the OISST record in roughly an hour on a standard UNIX cluster using O(10) cores, of which most of the time is spent in data input/output (I/O).
In terms of the matrix operations (matrix division and matrix logarithm), we rely on their Python implementations in the numpy.linalg package. The package provides the logm function for the matrix logarithm, which is similar to, for example, logm in MATLAB. However, Python does not provide an upper-level method for matrix division, so we employ a dot product between the numerator and the pseudoinverted denominator (pseudoinverse with numpy.linalg.pinv). This procedure is similar to, but less error prone than, the linear least squares fit between the numerator and the denominator.
c. Output
The aim of the inversion is to estimate the transport operator 
1) Estimate of the response function 
2) Discretization of 
The transport operator 
The aforementioned discretization implies that both diffusivity and velocity are estimated at a horizontal scale of 
d. Optimizing τ
As mentioned in section 1, a sensible choice for τ is between the forcing decorrelation time scale 
Optimized τ after applying velocity- and diffusivity-based restrictions as discussed in section 3d. Areas that are under sea ice cover (more than 15% ice concentration) more than 50% of the time are masked with light gray.
Citation: Journal of Atmospheric and Oceanic Technology 35, 10; 10.1175/JTECH-D-18-0057.1
While the resulting 
SST autocorrelation e-folding time scale with changing time resolution 
Citation: Journal of Atmospheric and Oceanic Technology 35, 10; 10.1175/JTECH-D-18-0057.1
In this context note that the usefulness of the inversion for inferring oceanic properties is limited by the processes affecting the SST variability. Specifically, the transport operator 
4. Results
First we provide estimates of the inverted velocity, diffusivity, and decay fields, after which we evaluate the sensitivity of the inversion to τ, 
a. General description
1) Velocity
We expect the inverted velocity field at 0.25° resolution to capture the velocity at which mesoscale eddies (in the extratropics) and linear Rossby waves (in the tropics) propagate. Both features move at the Rossby wave speed Doppler shifted by the depth-averaged background flow (Klocker and Marshall 2014).
Indeed, both the inverted velocity field and the eddy velocity field [derived from the sea surface height–based eddy atlas of Chelton et al. (2011); similar to Fig. 1a in Klocker and Marshall (2014)] show a comparable structure dominated by ubiquitous westward propagation (Figs. 4a,b). Outside the tropics (i.e., poleward of 20°S–20°N), the root-mean-square error (RMSE) between the two velocity fields is 3 cm s−1 for both velocity components. In the tropics the inversion velocities are notably smaller than the eddy velocities (Figs. 4a,b) with an RMSE of 9 and 5 cm s−1 for zonal and meridional velocities, respectively. A more detailed analysis, and an inversion of sea level anomaly data (not shown), reveals that in the extratropics the SST-based velocities closely follow the velocity estimates from the eddy atlas and sea level anomaly–based inversion (i.e., record the Rossby wave speed) but that this relationship breaks in the tropics. There, the SST-based velocities are close to the underlying current velocities, and we suggest that the 0.25° resolution is enough in the tropics to see the deformation of the SST field by the strong zonal jets (and their shear).
Comparison of different velocity estimates. (a) Inverted velocity field, (b) velocity field estimated from the daily atlas of sea surface height–based eddy tracks (covering 1993–2016; Chelton et al. 2011), (c) satellite altimetry–based OSCAR velocity climatology (mean over 1993–2016; ESR 2009; Bonjean and Lagerloef 2002), and (d) drifter velocity climatology (based on data from surface drifters between 1979 and 2015; Laurindo et al. 2017). Color illustrates speed, and the streak lines are plotted with the Python matplotlib streamplot function. Gray shading in (a) indicates areas where the inversion becomes unreliable as the SST autocorrelation drops below 1 day, and gray shading in (b) indicates regions where the velocity estimate is based on fewer than 10 eddies.
Citation: Journal of Atmospheric and Oceanic Technology 35, 10; 10.1175/JTECH-D-18-0057.1
As expected for the surface flow, the sea surface height–based geostrophic estimates (OSCAR; ESR 2009; Bonjean and Lagerloef 2002), and drifter data (Laurindo et al. 2017), render much higher velocities and different circulation features than the inversion- or eddy-atlas-based velocity estimates (Figs. 4b–d). Globally, a comparison to inverted velocities yields RMSE of (
In appendix B we show that the velocities are unreliable when the decay term becomes large compared to the advection (specifically if 
2) Diffusivity
Diffusivity peaks in the most energetic regions of the ocean in boundary currents and in the tropics (Fig. 5b) as one would expect based on the eddy kinetic energy distribution (e.g., Klocker and Abernathey 2014). We note that the inversion produces negative diffusivities in regions where the SST variability is reduced to white noise after the data is high-pass filtered, that is, 
(a) Inverted speed, (b) mean diffusivity, and (c) decay time scale. Gray shading indicates areas where the inversion becomes unreliable as the SST autocorrelation drops below 1 day. In (b) the cyan contour indicates regions where 
Citation: Journal of Atmospheric and Oceanic Technology 35, 10; 10.1175/JTECH-D-18-0057.1
The largest diffusivities occur in the tropical eastern Pacific and tropical Atlantic. Note that Klocker and Abernathey (2014) argued that these regions are dominated by linear Rossby waves and therefore an eddy-diffusivity closure (see appendix A) is not valid. However, we suggest that while linear waves themselves do not mix diffusively, mixing caused by wave breaking can be represented by a diffusive closure. Therefore, we relate the tropical maximum in diffusivity to mixing by breaking Rossby waves.
In appendix B we show that because of the central differencing scheme, the inversion overestimates diffusivities at large Peclet numbers, particularly when 
3) Decay time scale
Patterns of the local decay time scale reflect the patterns of the SST autocorrelation (cf. Figs. 3, 5c) and one can indeed derive a close relation between the two (not shown). The decay time scale peaks at both frontal regions, suggesting that frontal perturbations are long lived, and in eddy-rich locations it is similar to eddy lifetime (cf. Fig. 5c to Fig. 5 in Chelton et al. 2011). In regions with strong atmospheric coupling (e.g., tropical Indian Ocean and tropical western Pacific), the decay time is short even when the velocities are small, as the atmosphere effectively controls the upper-ocean temperature anomalies.
In appendix B we show that the decay time scale is overestimated (r is underestimated) when 
b. Sensitivity
1) Sensitivity to the length of the time series
The 
Figure 6 shows the inverted velocity, diffusivity, and decay fields based on a 25-yearlong SST anomaly time series (1985–2016; Figs. 6a–c), and the error when these fields are the means of shorter segments (21–24 yr) over the same time period (Figs. 6d–o).
Effect of the length of the time series. (a)–(c) As in Fig. 5, but calculated for 1986–2016. (d)–(o) Difference between the estimate from the full 32-yr SST time series and the mean of the estimates based on the shorter segments of the time series (full time series minus mean of the segments).
Citation: Journal of Atmospheric and Oceanic Technology 35, 10; 10.1175/JTECH-D-18-0057.1
The error is smallest in regions with the strongest oceanic variability, and it decreases throughout the ocean as the length of the time series increases. In particular, the area-averaged root-mean-square error (Fig. 7) decreases roughly as the inverse of the square root of the length of the time series (similar to Jeffress and Haine 2014b). For example, moving from a 2-yr-long time series to a 8-yr-long time series would halve the error.
Area-weighted RMSE in ice-free regions where 
Citation: Journal of Atmospheric and Oceanic Technology 35, 10; 10.1175/JTECH-D-18-0057.1
Apart from the error caused by the length of the time series, the shorter segments of the time series reveal interannual- to decadal-scale variability that is larger than the error compared to the long time series (not shown). Assessing this variability is left for a future study.
2) Effect of
In section 3d we created an objective set of parameters to optimize τ. Here we show that the optimization is indeed well justified and demonstrate the need for the spatial high-pass filtering of the SST data.
Diffusivity estimates in Figs. 8a and 8b show that at long τ, the original (green) and the high-pass-filtered (orange) data approach a background global average around 500 m2 s−1. However, the two sets of data approach the background value from different directions. Diffusivity decreases toward the background value when the inversion is based on the original data, while it increases toward the background value when diffusivity is inverted from the high-pass-filtered data. This difference follows from the imprint of atmospheric variability in the original data; the oceanic variability starts to dominate only when one moves beyond the atmospheric autocorrelation time scale.
Effect of the forcing decorrelation time scale τ given the original data and spatially high-pass-filtered data. We illustrate the area-weighted global distribution with the median shown by the solid line, and the area between 25% and 75% quartiles shown by the shading.
Citation: Journal of Atmospheric and Oceanic Technology 35, 10; 10.1175/JTECH-D-18-0057.1
The decay time scale increases with τ because anomalies that have a lifetime shorter than τ are not accounted for by the inversion (Fig. 8c). High-pass filtering reduces the mean decay time scale, because it removes some of the large and persistent SST anomalies, such as those related to El Niño–Southern Oscillation in the tropical Pacific.
Velocity distributions are largely independent of τ, except zonal velocities, which show a peak around τ = 8 days (Figs. 8d–f). This reflects the global mean time that it takes for an anomaly to propagate from a central cell to the surrounding cells in the five-point stencil. In addition, high-pass filtering increases westward velocities, because it removes any eastward-propagating atmospheric anomalies present in the original data, leaving behind mostly westward-propagating eddies [see also section 4b(4)].
3) Effect of time resolution
Here we assess the sensitivity of the inversion to the time resolution and low-pass filtering (in time) of the input data (Fig. 9). This assessment is relevant because many of the observed surface parameters are archived at resolutions 
Effect of averaging and low-pass filtering in time. We illustrate the area-weighted global distribution with the median shown by the solid line, and the area between 25% and 75% quartiles shown by the shading.
Citation: Journal of Atmospheric and Oceanic Technology 35, 10; 10.1175/JTECH-D-18-0057.1
Both diffusivity and velocity estimates decrease with increasing 
Altogether these results suggest that with the caveats discussed above, one can base the inversion on data with a lower time resolution, and, for example, carry out the inversion based on satellite-derived sea surface salinity data or Argo float–based products (both are usually available on a 5–7-day time resolution).
4) Effect of spatial scales
Here we assess the effect of the spatial resolution of the anomalies by applying both high-pass and low-pass spatial filters to the original SST data, which have a 0.25° horizontal resolution (Fig. 10). This assessment is motivated by the varying spatial resolution between different tracer datasets. Physically, one might expect the spatial scales of the anomalies to matter for the anomaly propagation; for example, one would expect larger-scale SST anomalies to decay slower because of their larger heat content and ability to influence the atmosphere above. Alternatively, large-scale anomalies would be expected to be relatively slow because of a lack of coherent flow at large spatial scales.
Effect of spatial high- and low-pass filtering, where x axis is the filter size in latitude and the filter size in longitude is twice that. As in Fig. 9, we illustrate the area-weighted global distribution with the median shown by the solid line, and the area between 25% and 75% quartiles shown by the shading. Note the logarithmic y scale for diffusivity.
Citation: Journal of Atmospheric and Oceanic Technology 35, 10; 10.1175/JTECH-D-18-0057.1
Overall, the inversion appears insensitive to the size of the high-pass filter (green; Fig. 10), which supports our assumption of the scale separation between atmospheric and oceanic influence on SST anomalies. At the global scale, one achieves similar results with any filter that is below the atmospheric scales but still above the local stencil size. In the rest of this study, we used a 4° × 8° filter (latitude × longitude) because it preserves the equatorial waves that are O(100) km in scale but still removes the atmospheric signals that are generally an order of magnitude larger.
In contrast, low-pass filtering reveals strong filter size dependence of the inversion (orange; Fig. 10). We interpret this dependence as a shift from a mixed ocean–atmosphere signal when no filter is used (0.25° scale) to one dominated by the atmosphere at larger filter sizes. The low-pass-filtered diffusivities (orange; Figs. 10a,b) approach an asymptotic value O(105) m2 s−1 that is relatively close to effective diffusivities reported in the upper troposphere and lower stratosphere [O(105)–O(106) m2 s−1; Haynes and Shuckburgh 2000a,b]. The decay time scale increases with the low-pass filter size (orange; Fig. 10c), because of the increase in the spatial autocorrelation length scale, which translates to a larger autocorrelation time scale. The speed of the anomalies increases with the size of the low-pass filter (orange; Fig. 10f) primarily because of increasing eastward velocities in the midlatitudes, and increasing westward velocities in the tropics (note the increasing spread of the zonal velocity in Fig. 10d). Increasing eastward propagation in the midlatitudes is unlikely to have an oceanic origin, because Rossby waves propagate westward. However, the midlatitude atmosphere has eastward-moving anomalies (weather systems), albeit they move with speeds much faster than the inversion suggests. Therefore, we argue that the eastward-moving anomalies in the atmosphere, together with a lagged response from the ocean mixed layer, produce midlatitude SST anomalies that seem to propagate eastward at speeds faster than the underlying ocean current speeds [see Nilsson (2000, 2001), who suggests that the SST anomaly speed depends on the speed of the atmospheric temperature/heat flux anomaly and the ratio between the heat capacities of the atmosphere and the ocean]. The largest low-latitude increase in the westward anomaly propagation coincides with the trade wind–related surface wind stress maxima in the tropical North Atlantic and in the tropical Pacific (Risien and Chelton 2008). Similar to the midlatitudes, we suggest that the lagged ocean mixed layer response to wind stress anomalies produces relatively fast, westward-propagating SST anomalies.
5. Discussion
As mentioned in section 1, a widely used approach to linear matrix modeling is to estimate the linear matrix operator from a reduced number of EOFs instead of the full SST field. Another related method is to analyze the leading eigenvectors, POPs, of the transport operator 
Previously, Piterbarg and Ostrovskii (1997), Ostrovskii and Piterbarg (2000), and Ostrovskii and Font (2003) applied a linear matrix modeling approach very similar to ours to the SST field. It appears that the main differences between their approach and ours are in the implementation of the methodology. For example, they used a nine-point stencil, and instead of our logarithmic relation between the SST covariance matrices [Eq. (2)], their derivation ends with a set of nine discretized equations that depend linearly on the covariance matrices (Piterbarg and Ostrovskii 1997). Their results are qualitatively similar to ours in the North Pacific and in the North Atlantic (which are the domains they cover). However, quantitatively, the results differ, which is partly explained by their coarser and shorter SST dataset, and partly because they do not spatially high-pass filter the data.
The ability to invert for the diffusivity globally is intriguing, and it would be interesting to relate our results to other approaches in estimating ocean mesoscale diffusivity. While we leave a detailed comparison for further studies, we want to point out some correspondence with prior works that use higher moments of tracer time series to infer oceanic mixing. For example, Hughes et al. (2010) and David et al. (2017) use skewness and kurtosis of potential vorticity to identify jets and mixing barriers, and based on idealized simulations Hughes et al. (2010) further suggest a linear relation between diffusivity and kurtosis of potential vorticity (in the sense that higher kurtosis equals higher diffusivity). We suggest that future studies should attempt to link these higher moments of temporal variability to properties of spatial variability. For example, it is clear that the local spatial covariance drops in the presence of mixing barriers, but it would be fruitful to understand how spatial covariance relates to the local time variability in general.
In this study the velocity estimates are primarily linked to the mesoscale eddy propagation, because the OISST product does not resolve submesoscale features. However, note that any velocity estimation that is based on deformation of the tracer field is fundamentally limited. A deforming tracer field holds information only about the velocity component perpendicular to the tracer contours, which causes the deformation, while no information exists about the velocity component that is parallel to the tracer contours. This is of particular importance in temperature-based estimates, because temperature is an active tracer affecting the density field, which is why the velocity field itself tends to align with the temperature field (because of geostrophy).
Rhines and Young (1983) estimated that a time scale for the tracer alignment with the velocity field is 
On the other hand, our estimates are a function of the spatial and temporal resolution of the SST dataset. For example, given hourly time resolution and submesoscale [O(1) km] spatial resolution, one could detect structures that are directly advected by the mean current. In such a case, MicroInverse would return the underlying current velocities. In our analysis the tropical regions seem to be at the border of this limit, as the retrieved velocities are closer to the current velocities than the local wave speeds. We believe that at finer temporal and spatial scales, our statistical estimate would converge to estimates from feature tracking. The advantage of MicroInverse over the feature tracking (especially maximum covariance) methods is the lack of region-specific tuning needed for the method to work (Heuzé et al. 2017), and the ability to estimate also diffusivity and decay rate. The advantage of feature tracking is the ability to work on image pairs rather than time series.
In this study we have used OISST data that are preprocessed (interpolated and filtered) to yield a complete dataset on a regular latitude–longitude grid. In principle MicroInverse could be used with less-processed data products, for example, pure satellite images. Control on the filtering and interpolation routines is desirable, and one could use existing methods to estimate covariance matrices with missing data (Dempster et al. 1977; Schneider 2001), instead of calculating them directly with interpolated data [as in Eq. (2)]. An added benefit of satellite images over processed data products is the higher spatiotemporal resolution.
6. Conclusions
We developed a local linear matrix inversion method to assess ocean kinematics. Given a gridded time series of tracer data, one can use this inversion method to estimate a mean transport operator that governs the evolution of tracer anomalies. One can then use the transport operator to derive a global response function (Green’s function) and to estimate physical fields such as velocity, diffusivity, and decay time scale.
In this paper we applied the inversion to the 35-yearlong daily SST record (OISST, 1982–2016; Reynolds et al. 2007) at 0.25° horizontal resolution. We show that the inversion is in general feasible at 0.25° resolution as long as the time resolution is below the SST decorrelation time scale. The decorrelation time scale of the SST field is O(10) days, which suggests that it could be feasible to invert other data products that have approximately weekly time resolution (such as Argo data). The inversion is also relatively robust to the length of the time series (error decreases as an inverse square root of the length of the time series), enabling assessments of subdecadal variability.
However, care should be taken with the combination of spatial and time resolution of the input data. The inversion of high-spatial-resolution data but low-time-resolution data should be avoided, as one would expect anomalies to leave a local stencil within a time step, which would cause underestimation of diffusivity and velocity. In such a case it would make more sense to regrid the data to a coarser spatial resolution if a higher time resolution is not available. The opposite scenario—high time resolution but low spatial resolution—is not as problematic, although the assumption of separating atmospheric and oceanic SST imprints by spatial scales becomes invalid as the spatial resolution decreases.
Finally, the inversion method provides several avenues for future work. Propagation of temperature anomalies, and their lifetimes, has been a long-standing research question with direct implications for predictability. The global response function can be used to assess the lifetime of anomalies, and to find regions where there might be potential predictability linked to temperature anomalies alone (similar to, e.g., Wang and Chang (2004), but using a local stencil rather than EOFs to derive the transport operator]. Another research question is the estimation of subgrid-scale diffusivity; here the ability to estimate horizontal diffusivity from different observational, as well as model products, can provide a new understanding of the scale dependency of diffusivity. One could also investigate changes in the velocity and diffusivity at depth, with, for example, Argo-based products, or perform the inversion on isopycnal surfaces to estimate diffusivities along isopycnals.
We thank Peter Cornillon and an anonymous reviewer for their useful suggestions, which helped to improve the paper. We also thank M. Almansi, A. Gnanadesikan, R. Gelderloos, M.-A. Pradal, and A. Saberi for the fruitful discussions and feedback. A.N. and T.H. were supported by a grant from the NSF Directorate for Geosciences (1536554). NOAA Optimum Interpolation SST data (NOAA OISST) are provided by NOAA/OAR/ESRL PSD, Boulder, Colorado, from its web site (at https://www.esrl.noaa.gov/psd/). The Mesoscale Eddy Trajectory Atlas products were produced by SSALTO/DUACS and distributed by AVISO+ (http://www.aviso.altimetry.fr/) with support from CNES, in collaboration with Oregon State University with support from NASA. The data and further details are available online (at https://www.aviso.altimetry.fr/en/data/products/value-added-products/global-mesoscale-eddy-trajectory-product.html). The OSCAR data were obtained from JPL Physical Oceanography DAAC (https://podaac.jpl.nasa.gov/dataset/OSCAR_L4_OC_third-deg) and developed by ESR. The drifter-based climatology of surface currents can be found online (http://www.aoml.noaa.gov/phod/dac/dac_meanvel.php). The inversion code with examples is distributed under the name MicroInverse (https://github.com/AleksiNummelin/MicroInverse), and the code to generate the data and figures in this paper is distributed through Github (https://github.com/AleksiNummelin/MicroInverse_paper). Please contact the lead author for help with MicroInverse implementation.
APPENDIX A
Derivation of Perturbation Advection–Diffusion Equation
APPENDIX B
Idealized Sensitivity Analysis
Here we provide two simple illustrations of the inversion method by inverting results from a numerically solved advection–diffusion–relaxation equation. We use the FiPy (Guyer et al. 2009) Python package to integrate the forward equation [Eq. (A6)] into a 50 × 50 gridcell domain. In the first test case, we carry out multiple simulations, varying the constant velocity, diffusivity, and decay fields. In the second case, we use a similar approach but advect the temperature field with the observed OSCAR velocity field. Detailed results are presented below for both cases.
a. Constant velocity field
In the first test case, we use constant velocity, diffusivity, and decay fields, and start from an initial condition of 25 Gaussian SST anomalies within the domain. We apply no forcing and as the velocity field is constant in time, the right-hand side of Eq. (A6) vanishes. Each anomaly initially has a peak value of 1 K. We integrate forward with a time step of 900 s for a total of 
After generating this idealized dataset, we invert it choosing 
The median error between the known (constant) and estimated diffusivity, velocity, and decay fields in a square domain with an initial decaying anomaly but without a forcing.
Citation: Journal of Atmospheric and Oceanic Technology 35, 10; 10.1175/JTECH-D-18-0057.1
b. OSCAR velocity field
In the second test case, we also keep the diffusivity and decay fields constant but use the OSCAR velocity field (ESR 2009). We focus on a 50 × 50 gridcell subdomain (similar to the more idealized case) in the Southern Pacific (30°–46°S, 157°–173°W) away from the boundary currents. We modify the original OSCAR data by first making the velocity fields divergence free [following Marshall et al. 2006; Abernathey and Marshall (2013), with no flux boundary conditions]. We then randomly sample the velocities in time to form a new dataset with 1-day time resolution (which is then linearly interpolated to the time step scale). The resampling of the velocity field was done because of the constraints discussed in section 3d: The original OSCAR velocities are 5-day averages with decorrelation time scale 
We relax the SST field toward an atmospheric temperature field that is constant in time, but has a meridional gradient of 10 K. Because there are no time fluctuations in this relaxation term, the atmospheric part of the forcing term in Eq. (A6) is zero. However, since the OSCAR velocities vary in time, the 
After generating the data for the second test case, we carry out the inversion and optimize τ following the procedure laid out in section 3d. Figure B2 shows the results for subgrid-scale diffusivity D ranging from 250 to 1500 m2 s−1.
Histograms of relative error between the known and estimated diffusivity (D is subgrid-scale diffusivity), velocity, and decay fields. The variable 
Citation: Journal of Atmospheric and Oceanic Technology 35, 10; 10.1175/JTECH-D-18-0057.1
Accuracy and precision of the diffusivity estimates (Figs. B2a–d) increases as a logarithmic function of D. Inversion overestimates small diffusivities, because the Peclet number is close to 1 (see previous test case) but also because small D allows for large velocity–temperature fluctuation covariance that contributes to the eddy diffusivity (note that inversion of the OISST field, Fig. 5b, suggest that that the eddy diffusivity is around 500 m2 s−1 in this region). As the subgrid-scale diffusivity increases, the temperature field becomes relatively smooth and the velocity–temperature fluctuation covariance small, and the inverted diffusivity is mostly due to the subgrid-scale diffusivity alone. As expected from the previous test case, estimates of the decay time scale are relatively accurate and precise compared to the diffusivity and velocity fields, and appear robust to changes in the subgrid-scale diffusivity (Figs. B2e–h). Finally, velocity estimates are rather accurate, but not very precise (Figs. B2i–l). This is because the velocity field itself is highly variable (as we randomly sampled the field) and consequently there is also large variability in the alignment of the temperature and velocity fields both in time and space (see section 5). Note that the velocity estimation also grows less accurate as the subgrid-scale diffusivity begins to dominate the deformation of the SST field over the influence of the highly variable velocity field.
In all cases the precision is in large part controlled by the length of the time series (not shown; similar to what was found with the OISST data), but also by τ. Choosing a larger τ improves especially the accuracy of diffusivity estimates, as the solution asymptotes to a true value [similar to section 4b(2)], but at the same time precision decreases because the covariance matrix becomes noisier with increasing lag.
APPENDIX C
Comparison between NOAA OISST AVHRR Only and OISST AVHRR + AMSR-E Products
Here we compare the two NOAA OISST products available, using the period 2003–11 when there are additional data from the AMSR-E instrument. AVHRR is an infrared instrument, but there are data gaps in cloudy regions where it cannot see the surface. AMSR-E is a microwave instrument that can penetrate the clouds, but the data are available for only 2003–11 (full years). Including the AMSR-E data is expected to yield better results, as the data coverage is more complete. Both datasets went through the same pre- and postprocessing independently; that is, the data were spatially high-pass filtered and the climatologies were removed, and afterward τ was optimized as described in section 3d.
The comparison appears in Fig. C1, which demonstrates that while the underlying patterns are practically the same, the inclusion of AMSR-E data leads to larger velocities and diffusivities, and to a longer decay time scale. Note the differences in the boundary currents where the inclusion of AMSR-E data yields stronger velocities and in the interior where they yield fewer negative diffusivities. The inclusion of AMSR-E data also diminishes regions with decay time scales below the time resolution of the input data (1 day), shrinking the regions with unreliable results. In general, the anomalies also have longer lifetimes, which is likely due to fewer gaps in the data.
Comparison between the (left) NOAA OISST AVHRR only and (middle) NOAA OISST AVHRR + AMSR-E data products over the period 2003–11. (right) The difference between the products, shown as a ratio. (a)–(c) U (m s−1), (d)–(e) V (m s−1), (g)–(i) κx (m2 s−1), (j)–(l) κy (m2 s−1), and (m)–(o) r (days). Note that land areas are masked white, and negative diffusivities are masked in (g), (h), (j), and (k).
Citation: Journal of Atmospheric and Oceanic Technology 35, 10; 10.1175/JTECH-D-18-0057.1
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