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Spatial Optimal Interpolation of Aquarius Sea Surface Salinity: Algorithms and Implementation in the North Atlantic* [Journal of Atmospheric and Oceanic Technology](Journal of Atmospheric and Oceanic Technology Via Acquire Media NewsEdge) ABSTRACT A method is presented for mapping sea surface salinity (SSS) from Aquarius level-2 along-track data in order to improve the utility of the SSS fields at short length [O(150 km)] and time [O(1 week)] scales. The method is based on optimal interpolation (OI) and derives an SSS estimate at a grid point as a weighted sum of nearby satellite observations. The weights are optimized to minimize the estimation error variance. As an initial demonstration, the method is applied to Aquarius data in the North Atlantic. The key element of the method is that it takes into account the so-called long-wavelength errors (by analogy with altimeter applications), referred to here as interbeam and ascending/descending biases, which appear to correlate over long distances along the satellite tracks. The developed technique also includes filtering of along-track SSS data prior to OI and the use of realistic correlation scales of mesoscale SSS anomalies. All these features are shown to result in more accurate SSS maps, free from spurious structures. A trial SSS analysis is produced in the North Atlantic on a uniform grid with 0.25° resolution and a temporal resolution of one week, encompassing the period from September 2011 through August 2013. A brief statistical description, based on the comparison between SSS maps and concurrent in situ data, is used to demonstrate the utility of the OI analysis and the potential of Aquarius SSS products to document salinity structure at ~150-km length and weekly time scales. (ProQuest: ... denotes formulae omitted.) 1. Introduction Sea surface salinity (SSS) is a key parameter that re- flects the intensity of the marine hydrological cycle (U.S. CLIVAR Office 2007). Aquarius/Satelite de Aplicaciones Cientificas-D (SAC-D) satellite observations provide an opportunity to observe near-global SSS with space and time resolution not available by other components of the Global Ocean Observing System (GOOS). Aquarius/SAC-D is a collaborative space mission be- tween the National Aeronautics and Space Adminis- tration (NASA) and Argentina's space agency. Since its launch in June 2011 and the onset of data delivery in late August 2011, the Aquarius/SAC-D satellite has been providing space-based observations of SSS with a com- plete global coverage every 7 days. The satellite is po- sitioned on a polar sun-synchronous orbit crossing the equator at 1800 local time (LT) (ascending) and 0600 LT (descending). The Aquarius instrument consists of three microwave radiometers that generate three beams at different angles relative to the sea surface. The beams form three elliptical footprints on the sea surface (76 km 3 94 km, 84km3 120 km, and 96 km 3 156 km), aligned across a ;390-km-wide swath. The emission from the sea surface, measured by the radiometers as an equiva- lent brightness temperature in kelvins, is converted to SSS, subject to corrections for various geophysical ef- fects. A detailed description of the Aquarius/SAC-D satellite mission and the Aquarius instrument can be found in Le Vine et al. (2007) and Lagerloef et al. (2008). Since the availability of Aquarius on-orbit data, the calibration/validation team has been actively identifying problems and errors, improving algorithms, and updat- ing the versions of available data. With respect to SSS, significant sources of errors are temporal sensor drift, ascending/descending biases, and interbeam biases (Lagerloef et al. 2013). The latter biases are the focus of the present study. Although there has been steady im- provement in the level-2 SSS versions over the past two years, both the ascending/descending biases and inter- beam biases continue to have significant space-time variability globally, and are the primary source of re- sidual calibration errors in Aquarius SSS retrievals that manifest themselves as artificial north-south-striped patterns in mapped SSS fields. Figure 1, showing global maps of interbeam differ- ences averaged over the month of September 2012, illus- trates the problem. The differences are shown separately for ascending (from southeast to northwest) and de- scending (from northeast to southwest) Aquarius passes. In many areas, the interbeam differences are much larger than 0.2 psu and do not represent the true ocean signal. Note the large-scale structure of the interbeam differences and the differences between the ascending and descending patterns. The differences also have large amplitude tem- poral variations with an annual cycle (not shown). The primary objective of this investigation is to test the possibility of correcting errors in Aquarius SSS data by incorporating available statistical information about the signal and noise into the mapping procedure com- monly known as optimal interpolation (OI). OI is a fairly straightforward but powerful method of data analysis, extensively used by oceanographers and meteorologists for estimating values of geophysical variables on a regular grid from irregularly sampled ob- servations. The method is based on the Gauss-Markov theorem (Gandin 1965; Bretherton et al. 1976; McIntosh 1990) and determines a pointwise estimate of the in- terpolated field with minimum ensemble mean-square error, given prior information about the variances and correlation functions of the estimated field and the data. The latter requirement is probably the hardest step in practical implementation of the method to the problem of mapping the Aquarius SSS. This is partly because in many parts of the ocean, there are insufficient high- resolution observations to confidently specify the re- quired statistics of the field (Bingham et al. 2002). The attractive feature of OI, however, is that it affords a very convenient way of taking into account error information specific to a given observational platform. This is par- ticularly relevant to the satellite SSS data, since errors in the satellite retrievals are of different types and are spatially correlated (Lagerloef et al. 2013). Finally, the OI formalism has been successfully applied for mapping various satellite data, such as sea surface temperature (e.g., Reynolds and Smith 1994; Reynolds et al. 2007; Thiebaux et al. 2003) and sea level anomaly (Le Traon et al. 1998; Ducet et al. 2000). Many ideas originally developed for these applications are found to be fruitful for the present study as well. In this paper we focus on the North Atlantic between 08 and 408N. The choice of this particular region is mo- tivated by the ongoing field experiment Salinity Pro- cesses in the Upper Ocean Regional Study (SPURS) to study the physical processes that are responsible for the maintenance and magnitude of the subtropical At- lantic salinity maximum. The overall region includes substantial space-time variability of SSS as well as sig- nificantly enhanced near-surface, in situ salinity obser- vations during SPURS. The rest of the paper is organized as follows. Section 2 provides an overview of the satellite SSS data. Section 3 provides a general description of the algorithm; specifics are given in section 4. Section 5 presents results that for- mally validate the use of the long-wavelength error model to correct Aquarius SSS data for interbeam biases. An intercomparison of SSS analyses is presented in section 6. Section 7 provides the main conclusions and a brief dis- cussion of possible improvements of the analysis. 2. Aquarius SSS data In the present study, we use level-2 (L2) version 2.0 Aquarius data produced by the NASA Goddard Space Flight Center's Aquarius Data Processing System (ADPS). The L2 data files, distributed by the Physical Oceanography Distributed Active Archive Center (PO. DAAC) of the Jet Propulsion Laboratory (JPL), con- tain retrieved SSS, navigation data, ancillary fields, con- fidence flags, and other related information such as surface winds. The data are structured as a sequence of files, each corresponding to one orbit of Aquarius.An orbit is defined as starting when the satellite passes the South Pole. Individual observations along each orbit consist of a sequence of data points sampled at a 1.44-s (;10 km) interval. Each individual observation represents the average salinity in the upper 1-2-cm layer and over a ;100-km footprint (Le Vine et al. 2007; Lagerloef et al. 2008). The ancillary SSS data are provided from the global 1/128 data-assimilative Hybrid Coordinate Ocean Model (HYCOM). The model assimilates satellite altimeter ob- servations, satellite, and in situ SST as well as vertical temperature/salinity profiles from Argo floats and moored buoys. More details on HYCOM can be found in Chassignet et al. (2009).InAquarius L2 data files, the HYCOM SSS is interpolated to the time and location of every Aquarius 1.44-s sample interval (Lagerloef et al. 2013). Figure 2 shows the Aquarius ground tracks over the North Atlantic between the equator and 408N. Each track represents three radiometer beams shown by dif- ferent colors. The Aquarius sampling pattern is quite dense, implying that a variety of commonly used in- terpolation techniques can be applied to construct a spatially mapped product. The problem, however, lies in the relatively large retrieval errors in the satellite SSS data, which, if not corrected, result in spurious structures in the corresponding SSS maps. An example of L2 SSS data is shown in Fig. 3, illus- trating that there are at least two types of errors in the SSS retrievals. A significant source of error is the accuracy of individual measurements along the satellite tracks. An important aspect of this error is its random character and a very short wavelength. As will be shown later, this short-wavelength noise is essentially ''white'' in nature and can effectively be suppressed by averaging over a sufficient number of observations or by filtering the data along track, such as shown in Fig. 3 (heavy lines). Of much greater concern are differences between the three beams, which can be as large as 0.5-0.8 psu and appear to be correlated over large distances along the satellite tracks. This type of error is also illustrated by Fig. 3. During the satellite pass over the North Atlantic on 14 September 2012, the middle beam (red) delivered systematically lower SSS as compared to the other two beams. Such interbeam biases are likely a manifestation of residual geophysical corrections. Since the three ra- diometer beams view the ocean surface at slightly dif- ferent angles, each beam is affected by geophysical errors differently (Lagerloef et al. 2013). 3. Interpolation procedure In the interpolation procedure, it is desirable not only to extract all available information from the satellite data but to simultaneously correct for various errors. The ultimate goal is to produce the best possible estimate of the evenly gridded SSS field. The OI anal- ysis attempts to accomplish this goal by minimizing the mean-square interpolation error for an ensem- ble of analysis realizations (Gandin 1965; Bretherton et al. 1976). a. General description of algorithm The interpolation expression for OI with N observa- tions can be written as (Bretherton et al. 1976; McIntosh 1990; Le Traon et al. 1998) ... (1) where S^x is the interpolated value (or estimate) at the grid point x, S0x is the forecast (or ''first guess'') value at the grid point x, Siobs is the measured value at the obser- vation point i, Si0 is the forecast value at the observation point i, A is the N 3 N covariance matrix of the data ... (2) and C is the joint covariance matrix of the data and the field to be estimated, where ... (3) It is generally assumed that the field Si is imperfectly measured at observation points, yielding values with random errors «i : Siobs 5 Si 1 «i . It is also assumed, as is usually reasonable, that the errors and the field are not correlated. Then the general elements of the covariance matrices (2) and (3) can be written as ... (4) ... (5) The analysis is determined relative to the ''first guess'' field, which is assumed to be a good approximation of the true state. The estimate and the observations are then equal to the first guess plus small increments. In this way, the gridpoint analysis consists of interpolation of the first-guess field to the observation points followed by interpolation of the differences between the observed and first-guess values back to the grid point according to (1). The following a priori information is required for construction of a successful OI scheme: * A background or first-guess field with location- dependent values S0x , which may be a field of climato- logical means or continually updated running averages or forecasts (e.g., Clancy et al. 1990; Reynolds and Smith 1994). * Covariance of the field to be analyzed. In practice, it is often expressed in a simple analytical form with a few degrees of freedom, allowing for a practical estimation of parameters from observations. * Covariance of the measurement noise, which can be estimated from an ensemble of realizations of the data, in particular, from a long time series of the data. Specific choices of parameters used to construct gridded SSS fields from Aquarius L2 data in the North Atlantic are addressed in the following section. b. Specifics 1) PREPARATION OF INPUT DATA To produce the gridded product, the L2 SSS data are first checked for quality. Data points contaminated by land (land fraction . 0.005) are excluded from the analysis. Also excluded from the analysis are data points that are flagged as severely contaminated by radio fre- quency interference (RFI), and/or sampled during high wind (wind speed . 15 m s21). The next step consists of smoothing the along-track SSS data (each beam separately) with a running Han- ning filter of half-width of about 60 km to suppress high- frequency instrument noise (e.g., Fig. 3). With the Aquarius ;10-km along-track sampling, the filter weights 12 adjacent observations, which has been found to be quite sufficient to significantly reduce the noise level, yet preserve the ocean signal from oversmoothing. The effect of filtering of the along-track data is dem- onstrated in Fig. 4, which displays the mean wave- number spectra of SSS representing the unfiltered and filtered data from the Aquarius repeat track passing through the North Atlantic (see Fig. 2 for location). The spectrum of the unfiltered data (blue line) is character- ized by a pronounced transition from ''red'' to ''white'' shape at the wavelength near 100 km. The white spec- trum at wavelengths shorter than 100 km is primarily due to the instrument noise. At wavelengths longer than 100 km, the oceanic signal starts to emerge and the power level rises toward the longest wavelength re- solved by the spectral analysis. Integrating the power of the white noise over the wavenumber domain yields a root-mean-square error of ; 0.21 psu. The signal-to- noise ratio, defined as the ratio of the low-wavenumber signal variance to the high-wavenumber noise variance, is about 40 at 1000-km wavelength and only 10 at 500-km wavelength. After applying the filter procedure (red line) most of the short-wavelength noise is eliminated while leaving the ocean signal practically unchanged. [this can be shown, e.g., by subtracting a flat variance of white noise (0.00025 psu2) from the blue curve; the result is the green curve]. It is likely, however, that residual noise effects are still present in the filtered data, par- ticularly in the form of long-wavelength errors, which are treated separately. 2) FIRST GUESS The first-guess fields, from which deviations are computed by the OI analysis, are derived from monthly- mean SSS fields obtained with variational interpolation of Argo buoy measurements. The Argo product is de- veloped at the Asia-Pacific Data-Research Center (APDRC), which provides salinity maps on standard depth levels on a monthly basis (http://apdrc.soest. hawaii.edu/projects/argo/). Figure 5 shows an example of the Argo-derived monthly-mean SSS field in the North Atlantic. The advantage of using Argo-derived SSS fields as the first guess is twofold. First, Argo-derived SSS fields are independent of the analysis of the satellite data. Therefore, the data increments, defined as the differ- ence between the data and the first guess, are also in- dependent of the analysis and can be used to compute the error statistics required by OI (Reynolds and Smith 1994). Second, Argo-derived SSS fields, since they are based on concurrent data, provide unbiased estimates of the first guess as compared to, say, climatological fields, which can be biased at large-scales due to the presence of significant trends related to climate change (e.g., Durack and Wijffels 2010) and/or their reliance on highly inhomogeneous multitype-instrument historical data (Gouretski and Koltermann 2007; Roemmich and Gilson 2009). 3) SIGNAL STATISTICS The OI analysis is determined in terms of data in- crements relative to a first guess. Therefore, the signal statistics, required by OI, must be derived for the data increments relative to the specified first guess (Reynolds and Smith 1994). However, the Aquarius along-track data are contaminated by long-wavelength correlated errors, which may result in correlation functions domi- nated by these errors. To overcome this problem, the spatial correlation structure of mesoscale SSS anomalies is derived from Aquarius data by dividing the along- track observations into shorter 108 latitude segments. The basic assumption here is that the dominant wave- lengths of the correlated errors are long enough (half wavelength . 108 in latitude; Fig. 1) such that the effect of these errors can significantly be reduced by removing liner trends fitted to the along-track SSS data. The spatial correlation scales of SSS anomalies were computed from Aquarius data as follows. The L2 SSS data [low-pass filtered as described in section 3b(1)] were split into four subregions, each spanning 108 in latitude: 08-108,108-208,208-308, and 308-408N. The first-guess values of SSS were subtracted from the data to obtain the data increments. Here, the first-guess values of SSS at observation locations at any given time were obtained by the space-time interpolation of the Argo-derived monthly-mean SSS fields [section 3b(2)]. To estimate autocorrelation functions of SSS, linear trends were first removed for each 108 ground-track seg- ment to produce SSS anomalies, presumably free from long-wavelength errors. The along-track autocorrelation functions of SSS anomalies were then estimated for the fractions of ascending and descending paths that span individual 108 subregions, assuming that the correlation between two points on a given track is a function only of a distance between the points. Finally, the ensemble mean autocorrelation functions in each subregion were estimated by averaging over all the corresponding in- dividual autocorrelations. Figure 6 illustrates the procedure described above. Displayed are ensemble-mean autocorrelations of SSS for the repeat swath shown by the heavy lines in Fig. 2. Each color in Fig. 6 represents a group of ground-track segments within a particular latitude band. For com- parison, autocorrelation functions of ancillary SSS are shown by the dashed lines. [The model-derived L2 an- cillary data were processed in exactly the same way as Aquarius data (including along-track filtering) except for replacing the first guess by the time mean over the period of Aquarius observations.] The space-lagged correlations computed from the Aquarius along-track data agree well with the correlations computed from ancillary SSS, providing additional confidence in our approach. Note that ancillary SSS, since it comes from a HYCOM model solution, is free from ''measurement'' errors, including long-wavelength errors. Figure 6 indicates that the structure of the correlation functions is very similar in all latitude bands. The spatial (meridional) scales of mesoscale SSS variability, de- termined here as the lag of the first zero crossing of the corresponding correlation function, vary little with lat- itude.They are ;180km in the zonal band 08-108N and ;150 km in the zonal band 308-408N. Because the dif- ferences are relatively small, it is reasonable to model SSS variability with a constant spatial decorrelation scale, independent of latitude (see also Table 1). To approximate the observed correlation array, we choose to use a simple Gaussian curve given by ... (6) where r is the spatial lag and R 5 90 km is the e-folding decay scale. The Gaussian function with the e-folding scale R 5 90 km (green curve in Fig. 6a) was found to best repre- sent the shape of the ensemble-mean autocorrelation function over the distance range 0-180 km. The corre- sponding wavenumber spectra are displayed in Fig. 6b. In the wavelength range from about 60 to 300 km, the empirical spectrum follows a power law of the form ;k22, where k is the wavenumber. Note that the Gaussian-shape autocorrelation function has the decay rate for k that matches that of the observed spectra. The apparent shortcoming of the Gaussian function, which we select as a statistical model for interpolation of Aquarius SSS, is that it fails to accommodate the nega- tive (oscillatory) lobe of the sample correlation array. Although it is possible, in principle, to utilize a more sophisticated analytical function to fit the estimations, the simpler Gaussian model has been selected for the following reasons. First, one of the strict requirements on the choice of a possible analytical form of the cor- relation function in the OI analysis is that such a func- tion must be positive definite; that is, the eigenvalues of each resulting correlation matrix must be nonnegative (Gandin 1965; Bretherton et al. 1976; Thiebaux and Pedder 1987; Weber and Talkner 1993). This is difficult to test for an arbitrary correlation model in two di- mensions (Weber and Talkner 1993). In this regard, the correlation model given by the Gaussian function is proven to be positive definite on every Euclidian space and on the sphere (Yaglom 1986; Weber and Talkner 1993), which warrants stability of the algorithm. This choice may not be truly optimal; nonetheless it is suit- able, since the decorrelation scales and the major structure of the observed correlations are well repro- duced by the Gaussian model (see also the appendix). Second, interpolation with the Gaussian function can be considered as a general form of a low-pass filter acting on the data (McIntosh 1990; Sokolov and Rintoul 1999). Consideration of the assumptions used to compute correlations from the along-track satellite data suggests that such a low-pass filtration would be more preferable than the case of a bandpass filter, which would corre- spond to the oscillatory correlation model (Sokolov and Rintoul 1999). More sophisticated functional forms could be utilized when more precise data on the SSS correlation structure become available. The analysis of along-track data gives some useful information about the characteristic meridional scales of SSS variability, but it tells us virtually nothing about the zonal scales. One way to overcome this problem is to assume that the spatial correlations are isotropic. This might be true in some areas but unlikely, for example, in the tropical region, where both atmospheric forcing and ocean dynamics are strongly anisotropic (Delcroix et al. 2005; Reverdin et al. 2007). Yet, limited information exists on the characteristic time and space scales of SSS variability in the ocean (Delcroix et al. 2005; Reverdin et al. 2007). Studying seasonal variability of SSS in the North Atlantic, Reverdin et al. (2007) found that in most regions outside of the equatorial belt, the zonal and meridional scales are comparable, while near the equa- torthezonalscalesare;1.5-2 times larger than the meridional scales. To add to the realism of our OI analysis, we also as- sume that in the tropical region (08-158N) the zonal scales are larger than the meridional scales and modify (6) to take an anisotropic form ... (7) where rx and ry are spatial lags in the zonal and merid- ional directions, respectively; and Rx and Ry are the associated zonal and meridional decorrelation scales. The meridional scale is set as Ry 5 90 km (the same as in the subtropical region), while the zonal scale varies from Rx 5 180 km at the equator to Rx 5 90 km at 158N as follows: ... (8) where y is latitude in degrees. Near the equator, the aspect ratio Rx /Ry equals 2 (following Reverdin et al. 2007) and gradually decreases toward higher latitudes. At latitude 158N, the correlation function (7) becomes isotropic (Rx 5 Ry 5 90 km) and matches the correlation function given by (6). We note, however, that our as- sumptions of the zonal decorrelation scales are somewhat arbitrary due to the lack of appropriate high-resolution SSS data. (It has been determined a posteriori that the use of the anisotropic correlation in the tropics results in slight improvement of the OI SSS analysis.) 4) ERROR STATISTICS Analysis of Aquarius along-track SSS data (e.g., Fig. 3) reveals that there are long-wavelength errors (inter- beam biases) that are correlated over long distances along the satellite tracks. To incorporate statistical in- formation on these errors into our OI scheme, we adopt the idea that has originally been developed for altimeter applications (e.g., Blanc et al. 1995; Le Traon et al. 1998) and introduce the error covariance model for the Aquarius data in the form ... if data points i, j are on the same track and beam and in the same cycle, and ... otherwise, where dij is the Kronecker delta, s2w is the variance of the uncorrelated (white) noise, and s2L is the variance of the long-wavelength (along track) error. Thus, the algorithm allows two types of random errors to contribute to the elements of the error covariance matrix: the white noise (diagonal elements), represent- ing uncorrelated errors; and the long-wavelength error (off-diagonal elements), representing interbeam biases that correlate over long distances along the satellite tracks. Each beam is modeled as having independent errors. Taking into account prior filtering of the along-track SSS, the variance of the white noise in the input data is assumed to be 10% of the signal variance, independent of the geographical location. It is thus assumed that uncorrelated errors, although relatively small, are still present in the data, allowing for some additional smooth- ing during the OI procedure. The long-wavelength error in Aquarius observations of SSS is difficult to assess in a direct way due to the lack of a proper reference or ''ground truth.'' To infer the statistical structure of the correlated portion of the re- trieval error in Aquarius data, we compare statistics of the interbeam differences as seen by HYCOM (ancillary SSS) and those evaluated from Aquarius observations. In this way, we diminish the effects of large-scale biases that may simultaneously be present in both the Aquarius and HYCOM data. The statistics of the interbeam differences are evalu- ated using Aquarius ground-track segments that span the entire domain from 08 to 408N. To eliminate con- tributions from mesoscale SSS anomalies (Fig. 6), the along-track SSS data are low-pass filtered with a running Hanning filter of half-width of ;600 km. The interbeam differences are computed for each ground track as SSS of the middle beam (red lines in Fig. 2) minus SSS of the two other beams (green and blue lines in Fig. 2). The covariances of the interbeam differences are com- puted as a function of along-track separation and then averaged over all tracks to obtain the ensemble statis- tics. The ancillary SSS data are processed in exactly the same way. The estimation of the long-wavelength error statistics is accomplished by comparing the co- variances of the interbeam differences for Aquarius and ancillary SSS. Figure 7a shows covariances of the interbeam differ- ences as a function of along-track separation distance for Aquarius (red) and HYCOM (blue) SSS. Notice that the variance of the Aquarius SSS interbeam differences is consistently larger than its HYCOM counterpart at all lags, presumably due to correlated errors in Aquarius SSS retrievals. Assuming that the interbeam differences in Aquarius and HYCOM data are not correlated, we can estimate the statistical structure of the long-wavelength retrieval error in Aquarius SSS data as the difference between the Aquarius and HYCOM interbeam differ- ence covariances (black). The corresponding variance spectrum is shown in Fig. 7b (black). Both the covariance function and the spectrum of the long-wavelength error demonstrate that this error has a complex spatial structure. The spectrum is red with more energy concentrated at longer wavelengths with no significant peaks. To obtain a functional form for the long-wavelength error correlation to use in the OI al- gorithm, we utilize a simple analytical model given by the exponential function of the form ... (9) where l is the along-track separation distance and RL 5 500 km is the exponential decay scale. The estimate of RL is obtained by fitting the curve (9) to the interbeam bias statistics, as shown in Fig. 7 by the green curve. The model (9) is chosen to represent the error corre- lation structure because this is the simplest model con- sistent with the data. It provides a good fit to the error correlation array over the distance range 0-600 km over which the correlation is significant, and it satisfies the functional requirements of OI (Weber and Talkner 1993). The variance of the long-wavelength error is assumed to be independent of the geographical location (s2L ' 0.085 psu2; Fig. 7a, black curve at zero spatial lag). However, the ratio of the error variance to the signal variance is allowed to vary with latitude, following the associated changes in the signal variance (Table 1). These variations are modeled as follows: ... (10) where h is the ratio of the long-wavelength error vari- ance to signal variance. Thus, the relative long-wavelength error variance varies from 30% in the near-equatorial region, where the signal variance is large, to about 100% at midlatitudes, where the signal variance is relatively low (Table 1). 5) IMPLEMENTATION The OI SSS analysis is computed weekly on a 0.258 longitude 3 0.258 latitude grid in the North Atlantic between 08 and 408N, covering the period from September 2011 through August 2013. The weeks are defined to correspond to the standard level-3 product produced by ADPS. The OI SSS analysis is run in a local approximation; namely, only data points in a smaller subdomain around the analysis grid point are used. The radius of the subdomain is set to 600 km to accommo- date the long-wavelength correlation structure (Fig. 7a). This approach seems to be reasonable. Data points be- yond this radius contribute very little to the gridpoint analysis, since the decay length scales for both the signal and error are shorter than 600 km. The local approxi- mation also helps to reduce effects of spatial in- homogeneity in the signal and error statistics (Weber and Talkner 1993). Finally, taking into account prior filtering of along-track SSS data and to reduce compu- tational load, only one data point out of three (for each track/beam) is retained. 4. Mapping results The following examples demonstrate the utility of the OI algorithm described above. Figure 8 compares SSS maps in the North Atlantic for the week 26 August-1 September 2012 produced by three different analyses, including 1) the standard 7-day level-3 analysis currently produced by ADPS; 2) the conventional OI analysis (COI), which does not take into account the long-wavelength error (s2L 5 0); and 3) the advanced OI scheme (AOI), which takes into account the long-wavelength error as discussed in section 3b(4). The standard 7-day level-3 product is constructed by bin averaging of Aquarius L2 SSS data within 18 lon- gitude 3 18 latitude spatial bins centered on a regular 18 resolution grid. The two OI analyses differ only in the way they treat the long-wavelength error; all other pa- rameters are kept the same. The bin-average procedure in the standard level-3 product effectively eliminates high-frequency (white) instrument noise. Yet, it fails to correct for correlated errors (interbeam biases) that manifest themselves as characteristic north-south-striped patterns aligned with the satellite tracks. These stripes are particularly visible when only ascending (Fig. 8a)ordescending (Fig. 8d) data are used as input data to construct the corresponding SSS maps, but they are also noticeable in the combined data (Fig. 8g). The same is true for the COI analysis. While resulting in better spatial resolu- tion, the COI analysis leaves the long-wavelength error untreated, such that the satellite tracks appear even more visible in the corresponding SSS maps (Figs. 8b, 8e, and 8h). In contrast, the AOI scheme effectively eliminates the along-track correlated errors. The re- sulting SSS maps constructed from either ascending (Fig. 8c)ordescending(Fig. 8f) data are nearly iden- tical and both resemble the true ocean, free from spurious structures. The impact of taking into account the long-wavelength error in the AOI analysis is fur- ther illustrated by comparing the differences between the ascending and descending products (Figs. 8j-l). In the AOI analysis, these differences are significantly reduced. The resolution capabilities as well as limitations of the AOI SSS analysis can be inferred from Fig. 9, which compares the SSS map for the week 9-15 September 2012 with thermosalinograph (TSG) salinity measure- ments taken from 3-m depth by Research Vessel (R/V) Thalassa. The in situ measurements along the ship track reveal numerous small-scale structures with spatial scales smaller than the ;100-km Aquarius footprint. Not surprising, these structures are not resolved in the satellite-derived SSS map. At the same time, it is evident that the analysis is capable of capturing features at scales of at least 150 km (see also the appendix). An example is the tongue of low SSS at ;328-338N followed by the tongue of high SSS to the north (Fig. 9b). Unlike the TSG line, the SSS map from Aquarius provides a de- tailed two-dimensional view on the spatial structure of SSS variability in the region. The high spatial resolution of weekly AOI SSS anal- yses is further illustrated by Fig. 10, which shows ex- ample SSS maps in the tropical North Atlantic for three weeks in July, September, and October 2012. Among the many features represented in Fig. 10 is the plume of low-salinity water that extends far offshore off the coast of South America. The plume is associated with the Amazon River outflow and is present seasonally during summer and fall and weakens or disappears in other months (Muller-Karger et al. 1988; Lentz 1995; Ffield 2007). The Aquarius SSS maps show a very detailed structure of the plume (Lagerloef 2012). Figure 10a shows how the plume starts to spread eastward into the North Atlantic in July 2012, presumably in the retro- flection of the North Brazil Current (Muller-Karger et al. 1988; Lentz 1995). Over time, as the plume extends farther eastward, it becomes less continuous. However, the boundaries of the plume remain well defined and are characterized by strong SSS gradients. Finally, to characterize SSS variability in the North Atlantic in one concise picture, Fig. 11 shows a time- latitude plot of SSS along the meridional section passing through the SPURS domain. The section coincides with the Aquarius track passing through the SPURS domain (heavy red line in Fig. 2 along the ascending pass). SSS values along the section are obtained by linear inter- polation of weekly AOI SSS maps. The analysis dem- onstrates a consistent pattern of seasonal variability that is most pronounced in the tropical region. A narrow belt of low SSS, presumably associated with the intertropical convergence zone (ITCZ), migrates from the south- ernmost position near the equator in early spring to the northernmost position at about 88N in winter. This structure also exhibits rapid temporal changes in some cases and is characterized by strong spatial gradients (see also Fig. 10). The weakest seasonal variability is observed in the subtropics, particularly in the area of the subtropical salinity maximum. The location of the salinity maximum slightly changes during the course of the year from ;268N in fall-winter, when SSS also rea- ches its maximum, to ;248N in late spring, generally consistent with the analysis of historical hydrographic data (A. Gordon 2013, personal communication). 5. Verification statistics and intercomparison of SSS analyses Argo buoy salinity measurements in the near-surface layer are used to provide OI error statistics during the period from September 2011 through August 2013. The error statistics are calculated by comparing buoy mea- surements for a given week with SSS values at the same locations obtained by interpolating the corresponding Aquarius OI SSS maps. To quantify specifically the ef- fect of incorporating error statistics into the OI algo- rithm, two versions of the OI analysis are run: AOI and COI. Also, in order to answer the question whether the OI analysis significantly improves the accuracy of Aquarius-derived SSS maps, the analysis-to-buoy com- parisons are made for the standard level-3 SSS product currently produced by the ADPS. The number of buoy data per each week in the North Atlantic is around 80 with quasi-random geographical distribution (e.g., Fig. 9a), and it remains around this number during the course of Aquarius measurements. The only exception is fall 2012, when a large number of Argo floats were deployed in the SPURS domain. The buoy data are typically drawn at 4-5-m depth and in most cases provide quite accurate representation of SSS. Under certain meteorological conditions, however, the difference between salinity at 5-m depth and the sea surface can be significant and exceed 0.1 psu (Henocq et al. 2010; Lagerloef et al. 2013). Figure 12 compares different SSS analyses using common statistics. The mean average of the differ- ences between each product and buoy data over all buoy locations, shown in Fig. 12a, is a measure of bias. A negative number in this case implies that on average the SSS estimate from Aquarius data is fresher than the Argo buoy data, and vice versa. The weekly time series of the root-mean-square differences (RMSD) between each of the analyses and buoy data are shown in Fig. 12b. Table 2 summarizes the mean, standard deviation, and RMSD of the differences between the analyses and buoy data for the 104-week period of comparison. Several conclusions can be made from Fig. 12 and Table 2. First, the average biases for the three analyses are all smaller than 0.03 psu (Table 2). However, the weekly time series of the biases (Fig. 12a) reveal that there are periods, such as in the fall of 2011, when the biases are significant. For example, the COI analysis and the standard level-3 product are both ;0.08 psu fresher than the buoy data in October 2011 and ;0.1 psu saltier than the buoy data in January 2012. The AOI analysis results in much smaller biases, but it does not completely eliminate them. All three analyses exhibit periods of both negative and positive biases that tend to cancel each other over the 104-week period of comparison. In general, the standard deviation of the weekly biases is the smallest for the AOI analysis as compared to the other two analyses (Table 2). The RMSD differ significantly for the three analyses. On average, the RMSD of the AOI analysis is about 35% less than that of the COI analysis and about 40% less than that of the standard level-3 product (Table 2). Figure 12b demonstrates that the AOI analysis has the lowest RMSD with respect to the buoy data for nearly all weeks. In all three analyses, the buoy-to-analysis comparison has the worst RMSD in spring and summer. This is likely a reflection of the fact that very shallow mixed layers are often formed in spring and summer, so that salinity at 4-5-m depth measured by a typical Argo buoy may differ from that at the sea surface. A detailed comparison (not shown here) indicates that multiple spikes in the RMSD time series, particularly in the standard level-3 product, are caused by a few buoys located in the tropics. The fact that the spikes are ob- served in spring and summer suggests that these spikes are likely due to misrepresentation of SSS by the Argo buoy measurements, as discussed above. It is also important to note that the RMSD of the AOI analysis is smaller than 0.2 for nearly all weeks during the winter sea- son when, due to surface cooling and usually stronger winds, mixing penetrates to greater depths; thus, buoy measurements at 4-5-m depth provide more accurate representation of SSS. The utility of the AOI product is further illustrated by Fig. 13, which compares histograms of the differences between the buoy data in the North Atlantic (08-408N) and the three SSS analyses. The AOI estimates have an overall good agreement with the buoy data, such that the histogram of the differences is quite narrow, with ;55% of the differences falling into the range [20.1, 0.1] psu. For comparison, this number is 36% for the COI anal- ysis and about 34% for the standard level-3 product. The number of outliers, defined here as the differences larger than 0.5 psu, is about 3% in the AOI analysis, 5% in the COI analysis, and 6% in the standard level-3 product. One should keep in mind, however, that the relatively poor performance of the standard level-3 product with respect to the buoy data is partly due to the coarser grid on which the product is constructed. Finally, Fig. 14 shows the scatterplots between the Aquarius SSS (mapped by the three analyses) and Argo buoy data, which clearly demonstrates where most of the close agreement between the AOI SSS analysis and in situ data is achieved. The scatter of points is consid- erably reduced over the regions where SSS is higher than ;35.5 psu (yellow-to-red colors in Fig. 5), but it remains significant over fresher areas, generally in the tropics (blue-to-magenta colors in Fig. 5). There are a few possible explanations for this effect. First, the tropics are characterized by vigorous variability at different space and time scales (Fig. 11), including small-scale vari- ability. In the presence of strong spatial gradients (e.g., Fig. 10), the difference between a point measurement by a buoy and the area-averaged SSS sampled by Aquarius can exceed 0.2 psu (Lagerloef et al. 2010). Another source of discrepancy can be related to strong vertical gradients of salinity in the near-surface layer, such that salinity at 5-m depth, sampled by a typical Argo buoy, differs significantly from the surface salinity, sampled by Aquarius. Vertical salinity differences larger than 0.1 psu (sometimes as large as 1.0 psu) are often ob- served in the tropical belt between the equator and 158N, which coincides with the average position of ITCZ (Henocq et al. 2010). It follows that the observed rela- tively large discrepancies between the Aquarius and buoy data in the tropics are not necessarily errors in Aquarius measurements or errors in the mapping pro- cedure, but may rather reflect the disparity between time and space scales captured by two different obser- vational platforms. 6. Summary and discussion A method has been presented for mapping SSS fields from Aquarius level-2 data. The method is based on optimal interpolation (OI) and estimates SSS at a grid point as a weighted sum of nearby satellite observations with the weights optimized to minimize the estimation error variance. The key element of the proposed ad- vanced OI (AOI) algorithm is that it takes into account statistics of correlated errors in the satellite retrievals, referred to here as interbeam biases that appear to correlate over long distances along the satellite tracks. The inclusion of this type of error information into the AOI algorithm has been shown to result in more accu- rate SSS maps, free from spurious structures. Examples have been presented that suggest that the OI technique can be an effective tool for mapping Aquarius SSS while correcting for various errors in the data. The quality of the AOI analysis has been demonstrated by considering the agreement between synoptic features in the SSS fields and those observed in independent in situ data, particularly high-resolution TSG data. The AOI analysis has been shown to resolve SSS features at scales of ;150 km and larger, consistent with the limited resolution of the input data, and to observe North Atlantic SSS with space and time resolution not available from the present global Argo array. A trial AOI SSS analysis is produced in the North Atlantic (08-408N) on a uniform grid with 0.258 grid resolution and with a temporal resolution of one week. Statistical comparison of the AOI analysis with respect to the Argo buoy data demonstrates its superior per- formance as compared to the standard level-3 product currently produced by the NASA Goddard Space Flight Center's Aquarius Data Processing System (ADPS). In particular, the estimated error of the AOI analysis is ;40% smaller than that of the standard level-3 product. It is worth emphasizing that the analysis presented in this paper is to a large extent experimental, focusing on a limited area in the North Atlantic. The results can be considered only ''suboptimal'' in the sense that the sig- nal and error statistics, required by the analysis, are de- termined approximately. Many assumptions have been made, some of which are not fully justified. In particular, the analysis scheme described here assumes both ho- mogeneity and stationarity of the signal and error sta- tistics, which is certainly one of the weakest aspects of the analysis. This is particularly relevant to the error correlation matrix. The results indicate that incorporating error information into the mapping procedure has a dramatic effect on the quality of resulting SSS maps. Seasonal and geographical variations in the variance and/or length scales of the correlated errors in Aquarius SSS retrievals are likely very important factors to con- sider, but these are beyond the scope of the present paper and will be evaluated in future studies. Users of Aquarius SSS data should also be aware that there are large-scale, space- and time-varying satellite biases relative to the in situ data in the present global products (Lagerloef et al. 2013). This problem seems to be not severe for the North Atlantic between 08 and 408N(Fig. 12a), but it must be addressed in future global and regional analyses. Although the quality of Aquarius level-2 data will surely improve in future data versions as processing algorithms improve, the methodology pre- sented in this paper should continue to provide value- added SSS products for regional, high-resolution studies. Digital data of the weekly AOI SSS analysis in the North Atlantic are currently available online (at http:// iprc.soest.hawaii.edu/users/oleg/oisss/atl/; weekly SSS be- ginning from September 2011). Acknowledgments. This research was supported by the National Aeronautic and Space Administration (NASA) Ocean Salinity Science Team through Grants NNX09AU75G and NNX12AK52G. Additional sup- port was provided by the Japan Agency for Marine- Earth Science and Technology (JAMSTEC), by NASA through Grant NNX07AG53G, and by the National Oceanic and Atmospheric Administration through Grant NA17RJ1230 through its sponsorship of research activi- ties at the International Pacific Research Center (IPRC). 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Springer-Verlag, 526 pp. OLEG MELNICHENKO International Pacific Research Center, School of Ocean and Earth Science and Technology, University of Hawai'i at M^anoa, Honolulu, Hawaii PETER HACKER Hawaii Institute of Geophysics and Planetology, School of Ocean and Earth Science and Technology, University of Hawai'i at M^anoa, Honolulu, Hawaii NIKOLAI MAXIMENKO International Pacific Research Center, School of Ocean and Earth Science and Technology, University of Hawai'i at Ma^noa, Honolulu, Hawaii GARY LAGERLOEF Earth and Space Research, Seattle, Washington JAMES POTEMRA Hawaii Institute of Geophysics and Planetology, School of Ocean and Earth Science and Technology, University of Hawai'i at Ma^noa, Honolulu, Hawaii (Manuscript received 1 November 2013, in final form 5 March 2014) Corresponding author address: Oleg Melnichenko, International Pacific Research Center, University of Hawai'i at M^anoa, POST Bldg., Room 401, 1680 East-West Road, Honolulu, HI 96822. E-mail: [email protected] DOI: 10.1175/JTECH-D-13-00241.1 APPENDIX Impact of Using the Simplified Correlation Model and Assessment of the Resolution Capability of the AOI SSS Analysis To examine the effect of using the simplified corre- lation model for the AOI SSS analysis, we computed correlations of SSS anomalies using the data of weekly AOI SSS maps. To do this in a straightforward manner, the maps were interpolated into locations of actual ob- servations along the satellite tracks. The SSS correla- tions were then computed in exactly the same way as using the original L2 data [section 3b(3)]. Figure A1 illustrates the ensemble-mean autocorre- lations of AOI SSS for the repeat track shown by the heavy lines in Fig. 2. For comparison, autocorrelations computed from the Aquarius L2 data (Fig. 6) are shown by the dashed lines. The figure indicates that the shapes of the space-lagged correlation functions computed from the Aquarius along-track data agree well with those computed from the AOI output. This includes not only positive values prior to the first zero crossings (which are approximated by the Gaussian model) but also the negative lobes at larger lags. The mesoscale SSS variance, however, is much reduced in the AOI SSS fields as compared to the along-track data, consistent with the filtering properties of both the signal and error correlation models used in the analysis. The degree of reduction is about a factor of 1.5 in the tropics and up to 3 at higher latitudes. To assess the spatial resolution capability of the AOI SSS analysis, we follow the spectral approach of Chelton et al. (2011) and compare wavenumber spectra of SSS evaluated from the gridded SSS product and the original Aquarius along-track SSS data. Figure A1b shows zonal (red line) and meridional (blue line) wavenumber spectra of SSS computed from the 104 weekly AOI SSS fields in the subtropical North Atlantic in the region extending from 158 to 358N (2220 km 3 2220 km) and from 508 to 288W. The black line is the composite spectrum derived from the Aquarius along-track mea- surements of SSS. All the spectra are normalized by the variance and scaled to have the same value at wave- number k52:231023km21 (wavelength 5 450km). Figure A1b demonstrates that for wavenumbers smaller than about 2:7 31023 km21 (wavelengths larger than about 370 km), the AOI and Aquarius level-2 SSS spectra are very similar in shape. In fact, the three curves are nearly indistinguishable for wavelengths between 450 and 1100 km. [Different spectral be- havior at the largest scales is due to the area of the subtropical SSS maximum being elongated in the zonal direction (e.g., Fig. 5), so that the large-scale meridi- onal gradients of SSS are larger than the zonal ones.] For wavenumbers higher than about 2:7 3 1023 km21, the AOI SSS spectra quickly roll off with increasing wavenumber, indicating the smoothing effect of the AOI procedure. It is thus apparent that the spatial resolution capability of the AOI SSS analysis is about 370-450 km in terms of a wavelength (scales larger than about 120 km), consistent with our estimates in section 4 (Fig. 9). (c) 2014 American Meteorological Society |