## 1 Introduction

[2] 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.

[3] 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* [1982]. 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* [1995]. 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* [1989] 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.* [2001] arrive at a similar conclusion.

[4] 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* [1995]. 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.

[5] 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* [1999]. 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.