A space-time interpolator for creating average geophysical fields from satellite measurements is presented and tested. It is designed for optimal spatiotemporal averaging of heterogeneous data. While it is illustrated with satellite surface wind measurements in the tropics, the methodology can be useful for interpolating, analyzing, and merging a wide variety of heterogeneous and satellite data in the atmosphere and ocean over the entire globe. The spatial and temporal ranges of the interpolator are determined by averaging satellite and in situ measurements over increasingly larger space and time windows and matching the corresponding variability at each scale. This matching provides a relationship between temporal and spatial ranges, but does not provide a unique pair of ranges as a solution to all averaging problems. The pair of ranges most appropriate for a given application can be determined by performing a spectral analysis of the interpolated fields and choosing the smallest values that remove any or most of the aliasing due to the uneven sampling by the satellite. The methodology is illustrated with the computation of average divergence fields over the equatorial Pacific Ocean from SeaWinds-on-QuikSCAT surface wind measurements, for which 72 h and 510 km are suggested as optimal interpolation windows. It is found that the wind variability is reduced over the cold tongue and enhanced over the Pacific warm pool, consistent with the notion that the unstably stratified boundary layer has generally more variable winds and more gustiness than the stably stratified boundary layer. It is suggested that the spectral analysis optimization can be used for any process where time-space correspondence can be assumed.
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 Many meteorological and climatological applications rely on the estimation of mean geophysical fields and their derivative quantities, e.g., wind, and its derivatives vorticity and divergence, on scales ranging from the smallest resolvable to climatological. With the ever increasing availability of satellite measurements, it is desirable to estimate such mean fields from satellite combined with in situ observations rather than from numerical weather forecast (NWF) model analyses alone. Granted, most NWF models now assimilate satellite measurements. They do not always assimilate the full range of information available in the measurements, however. This is especially true of satellite wind measurements on the mesoscale [Patoux et al., 2010]. Optimizing the use of assimilated data, especially winds, from heterogeneous sources in the assimilation cycle holds a potential for improved forecasting [Kalnay, 2003]. Conversely, suboptimal assimilation of such data generates noise which propagates forward and impairs forecasts [Levy et al., 2010]. It is therefore of interest to investigate the possibility of computing average geophysical fields in a manner that preserves a maximum amount of the information contained in the satellite observations. Here we propose a method that combines the concept of subgrid-scale variability as a measure of matching time and space scales, and spectral analysis. We illustrate it with SeaWinds-on-QuikSCAT (QS) scatterometer wind measurements and the estimation of mean divergence fields in the tropics. Total horizontal divergence, hereafter divergence, is one kinematic and dynamical field that has widespread use in meteorology and climatology (convergence lines, fronts, convective cells, Intertropical Convergence Zone (ITCZ), tropical instability waves, inertial instabilities, …); vertical vorticity, hereafter vorticity, is another (cyclones and frontal waves, tropical instability waves, tropical depressions, inertial instabilities, …). These second derivatives are sensitive to noise and errors, and a better estimation can be very useful in various contexts. We concentrate on the tropics as they remain a region that is not so well sampled by in situ or polar-orbiting satellite observations, yet has a profound effect on the general circulation and climate. Thus, they are an ideal region where the proposed methodology could have a large impact.
 When averaging measurements from polar-orbiting satellites, major difficulties may result from the uneven sampling of the Earth surface and atmosphere and from the time lag between successive satellite swaths. In the case of QS, the wind field is sampled spatially over large swaths of about 1700 km width, but there remain unsampled gaps between swaths (about 1000 km in width at the equator, where they are the largest). Moreover, the measurements in two successive (adjacent) orbits are about 100 min and 2800 km apart (at the equator). The satellite returns to the same general geographic area roughly every 12 h, but the true return period (after which the satellite returns to the exact same location) is of 4 days because the orbits of sun-synchronous polar-orbiting satellites process with time. These engineering constraints result in an uneven sampling of certain areas in time for a given averaging window. Some of the special problems stemming from the sampling characteristics of a polar orbiter that can occur while performing time series analysis for diurnal cycle retrieval have been demonstrated by Salby . Simple “binning and averaging” undersamples some areas with respect to others, creating artificial banded structures and aliasing in the mean fields, as shown by Zeng and Levy . This is a classic problem in optimal analysis and optimal interpolation that has been addressed with various methods [e.g., Bentamy et al., 2003; Bentamy and Fillon, 2011; Bentamy et al., 2013; Grodsky et al., 2009; Santorelli et al., 2011]. Salby  discusses how higher-frequency signals undersampled by polar-orbiting satellites can be aliased into scales that are of interest in climate studies. He concludes that time mean quantities, adequate for many purposes, are immune to aliasing from unresolved random variability because of cancelation of aliases in the averaging process provided that the undersampled variability is stationary. He also points out, however, that achieving this result in practice is limited by the length of the averaging period and that nonstationarity usually needs to be considered. Bell et al.  arrive at a similar conclusion.
 A conservative approach would dictate that, if the return period is of 4 days, then one should not attempt to interpolate and average with a temporal resolution shorter than 8 days (i.e., the Nyquist frequency/period). This would result in suboptimal use of available information in critical applications, e.g., synoptic forecasting. Because of the width of the swath and the overlapping pattern, there is redundancy of coverage for certain areas at scales much shorter than 8 days, and it is desirable to take advantage of this redundancy to create interpolated fields at higher temporal resolution. Similarly, temporal resolution from in situ observations that provide time series measurements at fixed locations can be exploited [Levy and Vickers, 1999]. Here we propose to do so for estimating average fields of vorticity and divergence in the tropics from scatterometer wind measurements by optimizing the space-time interpolation method developed by Zeng and Levy . The temporal and spatial scales used in the interpolator will be determined by matching the subgrid-scale variability in time and space, as determined from scatterometer and buoy wind measurements. The methodology is demonstrated with QS wind measurements and the Tropical Atmosphere Ocean (TAO) buoy array in the tropical Pacific Ocean (http://www.pmel.noaa.gov/tao), but is relevant for any type of satellite satellite wind measurement, and in fact, for any variable for which subscale variability can be and temporal measurements.
 In section 2, the space-time interpolation method is described. In section 3, a method is proposed for optimizing the spatial and temporal ranges to be used in the space-time interpolation, based on the concept of subgrid velocity scale described by Levy and Vickers . As we will see, this method pairs the spatial and temporal ranges, but does not fully constrain the problem. To choose an adequate pair of ranges, we perform a spectral analysis of the interpolated vorticity and divergence fields, as described in section 4. We discuss the results in section 5.
2 Space-Time Interpolation
 Many climate studies require the analysis of global fields of surface vorticity or divergence, averaged over varying temporal scales. Levy and Patoux , for example, investigated cross-equatorial excursions of absolute vorticity in the Indian Ocean using 10 day averages of absolute vorticity. Here, for illustration purposes, we concentrate on the regions of convergence in the equatorial Pacific Ocean, which have implications for our understanding of the ITCZ, the cold tongue, and El Niño/Southern Oscillation (ENSO) dynamics [Levy et al., 2011]. We are interested in computing average fields of divergence from QS surface wind measurements. To circumvent the uneven sampling of the ocean surface by the satellite, we seek an interpolation solution that will make optimal use of the scatterometer measurements inside the swaths while limiting the adverse effect of missing data between the swaths. Zeng and Levy  proposed an interpolation scheme in which temporal information (from earlier or later swaths) is substituted where spatial information is missing (in the gaps between successive swaths) and vice versa. At a certain location x0, y0, and time t0, an average is estimated by a scheme that searches for nearby observations within a horizontal range D and a temporal range T. The value at (x0, y0, t0) is then estimated as a combination of the N observed values found, namely,
where wk is a weight function
and (xk, yk) and tk are the location and time of the scatterometer observations within the spatial range D and temporal range T. Zeng and Levy  chose D and T subjectively on the basis of the satellite orbit and selected 300 km and 1 day for the European Remote Sensing (ERS-1) satellite. Here we seek an objective method to determine D and T by matching spatial and temporal variability and performing a spectral analysis of the interpolated fields.
3 Spatial Range and Temporal Range
 The space-time interpolator is based on the premise that there is a correspondence between averaging time and space scales such that measurements within time interval T and space range D can be properly weighted and used to estimate a mean value at a given location (x0, y0) and time t0. The corresponding space and time scales are found by matching the spatial and temporal variability. This supposes that the temporal variability of the wind on a timescale of T is similar to the spatial variability of the wind on a spatial scale of D. Thus, we propose to determine this common temporal and spatial variability using a method similar to that of Levy and Vickers  who defined the magnitude of the wind speed U as
where V is the magnitude of the (mean) resolved wind velocity at a given scale and Vsg is a subgrid velocity scale—a correction due to the spatial or temporal variability unresolved by the mean, V. Levy and Vickers  model Vsg based on the difference between the scalar mean U and the vectorial mean, V, for both spatial averages of satellite observations and temporal (time series) means from buoy measurements. An example is shown in Figure 1a where the spatial subgrid velocity scale was calculated for increasingly larger spatial windows around the location of a TAO buoy (2°N, 165°E) by averaging available QS measurements around the buoy at times when there were colocated QS and buoy measurements (year 2005). Similarly, Figure 1b shows the temporal velocity scale obtained by averaging available buoy measurements around the time of colocation for increasingly longer time windows. In both cases, one can observe two different ranges where the relationship is relatively linear: a first range between about 100 and 1000 km and a second range above 1000 km for the spatial velocity scale; a first range between 6 and 48 h and a second range above 48 h for the temporal velocity scale. The velocities also drop off more rapidly at very small scales (i.e., below 100 km and below 6 h).
 We matched the spatial and temporal scales by identifying the pairs of scales that had similar variability, as shown in Figure 1c. To illustrate, we show in Figure 1b that a temporal scale of 48 h corresponds to a variability of 0.7 m s−1. In Figure 1a, we see that a variability of 0.7 m s−1 corresponds to a spatial scale of 680 km. As a result, 48 h and 680 km are paired in Figure 1c. This matching is repeated for all temporal scales out to 144 h. In the end, this method allows us to pair any particular temporal scale that we might choose with its spatial counterpart and vice versa, such that the corresponding wind variabilities are comparable. We note that there is also a transition between two somewhat linear ranges in Figure 1c.
 Of course, this relationship is only valid for the TAO buoy at (2°N, 165°E). Since we are interested in creating average divergence fields for the entire equatorial Pacific Ocean, we repeated the previous calculations for all available TAO buoys (Figure 2) to obtain Figures 3 and 4. We can see that, while the relationships are similar in shape, they vary in magnitude with both latitude and longitude. The wind variability tends to be larger at higher latitudes, in particular in the Northern Hemisphere, and larger for the westernmost buoys. It also tends to be larger in the northeast corner of the equatorial region covered by the buoys. Comparison with a map of average sea surface temperature for 2005 (Figure 5) suggests that the wind variability is larger over warmer water, typically associated in this area with unstably stratified atmospheric boundary layer (ABL), and reduced over colder water, i.e., characterized in this area with stably stratified ABL. In particular, the wind variability is reduced over the cold tongue (Eastern Pacific) and enhanced over the Pacific warm pool (Western Pacific). This is consistent with the notion that the unstably stratified boundary layer has generally more variable winds and more gustiness than the stably stratified boundary layer [Godfrey and Beljaars, 1991]. It reflects the more efficient downward transfer of momentum of the free troposphere through the more unstable boundary layer found over warmer water, and in our case the variability associated with it, and conversely the less efficient downward transfer through the more stable boundary layer found over the cold tongue [Wallace et al., 1989].
 For our purpose here, we chose to average over all buoys within 5°S and 5°N to produce the mean relationships shown in Figure 6. These have more of an exponential shape, as modeled by Levy and Vickers . In particular, there is no transition between ranges of clearly different characteristics, as was observed in Figure 1.
 Figure 6c informs us on the average correspondence between spatial and temporal scales in the equatorial Pacific Ocean. We look at these matching scales as possible pairs of spatial and temporal ranges (D, T) that we could use in our space-time interpolator (equation (2)). There remains a degree of freedom, however, as we still need to choose a particular pair of values among all possible pairs. Each pair will yield a different interpolated field, increasingly smoother as the spatial and temporal scales increase in magnitude.
 To illustrate, we compute a series of interpolated fields of surface divergence from QS winds (0.5° grid spacing). Figure 7 shows different interpolated divergence fields for 15 April 2000 at 06 UTC based on five possible pairs (D, T) as provided in each panel. As T in these panels increases from 1 to 5 days, the corresponding D is provided by Figure 6c. The original QS swaths closest in time to 15 April 2000 at 06 UTC are shown in the top panel for reference. In the (24 h, 320 km) interpolated field, one can notice a number of suspicious minima and maxima in the vicinity of the swath edges, in particular at about 130°W, 155°W, and 180°W along the interswath region. These extrema shift with and follow the swath pattern with time (not shown). Indeed, as the satellite returns to sample the same geographic area 24 h later, the set of orbits is shifted east by 700 km. The fact that the extrema are also shifted suggests that they are linked to the swath geometry. They are probably due to the sharp transitions between regions where measurements are available (inside the swaths) and regions between swaths where divergence is estimated from remote measurements. Since these extrema follow the swath pattern, it is unlikely that they be the signature of a real geophysical feature, such as tropical instability waves (TIWs).
 As we increase the interpolation windows, the suspicious extrema disappear and the divergence field becomes more homogeneous. There is no clear indication, however, of which pair of spatial and temporal ranges (D, T) yields the best result. Since the extrema are suspected to occur at regular intervals dictated by the size of the swaths and interswath gaps, they should appear as spurious peaks of energy in a spectral decomposition of the divergence fields. Therefore, we now carry out a spectral analysis of divergence fields interpolated with increasing spatial and temporal windows in order to determine whether there exists a pair of spatial and temporal ranges beyond which divergence spectra are devoid of those spurious peaks, i.e., the optimum pair (D, T).
4 Spectral Analysis
 We first perform a two-dimensional spectral analysis of all interpolated divergence fields in 2005 and average them to produce Figure 8. All spectra are characterized by higher energy levels at large spatial scales and long temporal scales (in red), with decreasing amounts of energy toward smaller scales (in blue), as expected. In the (24 h, 320 km) spectrum, however, there appear to be unphysically high energy levels at the smallest resolved scales—a typical signature of aliasing, by which artificial features are projected onto the spectrum at specific scales and their harmonics. This is even more obvious in the (48 h, 450 km) spectrum, where this aliased energy “ripples” through the spectrum to create a series of peaks modulated in the zonal and meridional directions. It would be difficult to determine the exact origin of each peak, but we can assume that they are the result of a combination of factors including the sampling of the ocean dictated by the return period of the satellite (4 days), the sharp transitions at the swath edges, and therefore the width of the swaths (about 1700 km), the width of the gaps (about 1000 km), and the distance between consecutive swaths (about 2800 km at the equator). Note that the swaths are at an angle with respect to meridians, due to the tilt of the satellite orbit with respect to the poles, and that the interswath distance changes with latitude, as the swaths converge toward the poles and the gaps narrow. Therefore, the zonal and meridional artifacts combine in complicated ways to produce spurious peaks and “arcs” of energy in the spectra. At 72 h and 510 km, these spurious extrema mostly disappear and interpolating at larger spatial and temporal scales does not further improve the spectrum significantly. While this threshold is determined subjectively here, for demonstration purposes, one could easily set up an objective scheme by which the spectra are calculated for increasingly larger time and space windows, with small incremental steps, and the threshold is chosen as that above which successive spectra do not differ by more than a prescribed percentage, e.g., 5%.
 Because these two-dimensional spectra are more difficult to decipher, we show the corresponding one-dimensional spectra in the meridional and zonal direction in Figure 9. They both support the interpretation that interpolating at 72 h and 510 km greatly reduces the small-scale aliasing, while interpolating at larger scales does not significantly improve the spectra. This suggests that choosing a (T, D) pair around (72 h, 510 km) might be a good compromise by which interpolation artifacts and aliasing are minimized without resorting to a conservative averaging period of 8 days, thus preserving valuable dynamical information. Note that the (T, D) time and space scales are really a function of the satellite sampling, which is in itself constrained by engineering considerations (e.g., orbit, whether or not geosynchronous, or polar-orbiting, swath width, swath separation, inclination, and return period). Dynamically, the process monitored or studied will determine the desired space and time scales, but these engineering considerations will constrain the dynamical processes one can study without the ability to change sampling or data availability. Thus, dynamics determine what scales one might desire, but the preferred scales are determined by engineering constraints.
 Finally, note that the last two spectra have a slope close to −3 in the range 500–3000 km (enstrophy cascade characteristic of large-scale 2-D turbulence), and a slope close to −5/3 below 500 km (energy cascade characteristic of 3-D turbulence), which is consistent with both theoretical studies and similar analyses using scatterometer winds [e.g., Patoux and Brown, 2001].
 To illustrate how this methodology might be beneficial to the creation of average fields, we show an example of a 5 day average (“pentad”) divergence field for 13–17 April 2000 in Figure 10, centered around the time of Figure 7, and calculated by averaging interpolated fields obtained with increasingly larger time and space windows. One can see that, even though the effect of individual swaths is reduced due to the averaging, sharp transition features are still visible in the (24 h, 320 km) averaged field, and much less pronounced beyond 72 h and 510 km. The combination of more realistic spectra and average divergence fields beyond (72 h, 510 km) leads us to believe that the spectral analysis optimization is a valuable complement to the spatiotemporal variability matching to identify an optimal pair (T, D) of time and space ranges that will produce adequate interpolated fields when using the space-time interpolator described above. Choosing an appropriate set of parameters can be important, as some of the artifacts could be mistaken for real features of the wind field, such as the signature of TIWs [Chelton et al., 2001].
 Note that variability matching is not the only way to pair space and time ranges. The advective timescale (i.e., the grid scale divided by the mean wind speed) is a commonly used alternative. Computing this scale, however, requires one to make choices and assumptions that are not obvious (e.g., averaging period and steadiness of the wind). The advective timescale is inherently one-dimensional and linear and “sees” only upwind conditions advected past the observing location. Thus, except for very special homogeneous and stationary conditions, the advective scale cannot provide an appropriate conversion between temporal and spatial averages. In general, the temporal unresolved by analyzing time series averages, when converted to a length scale using the advective scale, will not represent the spatial variability of the corresponding two-dimensional area.
 Wind variability is particularly well suited for the type of space-time interpolation analysis we present here because, by invoking a Taylor-type hypothesis, one can argue that temporal wind variability at a point projects onto spatial wind variability through advection, and vice versa. But given any process where time-space correspondence can be assumed, and space and time ranges can be matched through an estimate of spatial and temporal variability, spectral spectral analysis can be used to select the best set of parameters. Thus, the spectral analysis optimization has wide applicability to a number of geophysical quantities.
 We have described a method for interpolating divergence and vorticity fields obtained from satellite and in situ wind measurements in a way that maximizes the retention of resolvable information contained in the actual measurements while minimizing aliasing and artifacts due to sampling. The methodology was described using TAO buoys and QS wind measurements in the equatorial Pacific Ocean, but can be used with any geophysical quantity where time-space correspondence can be assumed based on a common spatiotemporal process for which the variability can be estimated. Regional differences in wind variability were found, in particular with latitude and over the cold tongue and the Pacific warm pool. Since our results are based on the TAO buoy array and are mostly representative of the equatorial Pacific Ocean, an interesting extension of this study will consist of repeating it for other regions of the world where different dynamics prevail and subgrid variability might be different, although an obvious obstacle will be the limited availability of buoy measurements. One can surmise, for example, that D will be a function of latitude or Rossby number. An interesting application will consist of investigating seasonal and interannual differences in wind variability, such as due to the impact of ENSO on the extension of the cold tongue and the associated circulation [e.g., Ramesh Kumar et al., 2011], and therefore the variability in the space and time matching (Figure 6).
 The results presented here will be different for other satellites, and it will be interesting to repeat the analysis with other scatterometers, such as the Advanced Scatterometer (ASCAT) and the Oceansat-2 Scatterometer (OSCAT), as well as future scatterometers and indeed other sensors and variables for which a common process and spatial and temporal variability can be found. An important extension will be to apply this method to the assimilation of heterogeneous (in space and time) data in weather and climate models, since aliasing such as described in this paper, if not properly removed, may adversely affect model forecasts by introducing noise into “analysis fields.”
 The authors acknowledge support from NASA OVWST grant NNX10AO87G and from NSF grant ATM-0741832.