1. Introduction
Correction for the path delay in satellite altimetry caused by water vapor in the troposphere is an important factor in making precise measurement of the sea surface height (SSH) (Chelton et al. 2001). It has been standard practice to carry a multifrequency microwave radiometer on board an altimetric satellite to make simultaneous measurement of the tropospheric water vapor content for the so-called wet tropospheric correction. This approach is proven effective in the application of satellite altimetry, especially in the open ocean, where the finite footprint of radiometer does not cover any land.
The conventional radar altimeter is a profiling instrument making SSH measurements along the satellite’s ground tracks. An onboard microwave radiometer serves the purpose of making the wet tropospheric correction (Keihm et al. 1995; Brown et al. 2004). While we develop wide-swath altimetry for observing SSH over a finite swath along the satellite’s ground tracks, the issue of the adequacy of a nadir-looking radiometer for making the correction over the swath arises. What are the characteristics of the variability of water vapor across the swath? How is the variability compared to that of SSH? What can be done to address the cross-swath water vapor variability, if necessary? These questions are addressed in the present study. In particular, the study is configured for assessing the expected performance of the wet tropospheric correction over the ocean for the Surface Water and Ocean Topography (SWOT) mission (Fu and Ferrari 2008; Durand et al. 2010).
Instead of nadir-only measurements, SWOT will be able to measure SSH over a 50-km-wide swath 10 km away from nadir on each side. The current baseline design of the mission is to carry a conventional radiometer like the one flown on the Ocean Surface Topography Mission (OSTM) on Jason-2, called the advanced microwave radiometer (AMR) (Brown 2013). The residual errors in SSH due to the cross-swath variability of water vapor not sampled by the AMR are investigated in the study. The error of the water vapor measurement itself is not considered in the main analysis, but it will be presented and discussed in the last section. These errors are also added to other instrumental and algorithm errors not accounted for in this study, as discussed in the conclusions.
The 2D field of water vapor sampled by the scanning microwave radiometer on board the Aqua satellite called the Advanced Microwave Scanning Radiometer for Earth Observing System (EOS; AMSR-E) (Kawanishi et al. 2003) is used for estimating the sampling errors of AMR of the two-dimensional field along the SWOT orbit nadir tracks. The magnitude of the residual range delay error after applying the AMR correction is mapped globally and compared to the magnitude of the SSH signals obtained from Jason-2 altimeter. The ratio of the SSH signal to the error provides a basis for assessing the adequacy of the SWOT baseline design. The signal-to-error ratio is also examined in a spectral domain at different wavelengths. Potential improvement of the approach to the error correction by flying a multibeam radiometer is also examined.
2. The spatial variability of atmospheric water vapor
The key issue is the cross-track variability of the wet tropospheric correction that cannot be accounted for by a nadir-looking instrument like AMR. To evaluate the cross-track variability, we analyzed the AMSR-E data that offer a quasi-daily global coverage. The AMSR-E brightness temperature data are acquired from the Japan Aerospace Exploration Agency. The spatial resolution of the AMSR-E brightness temperatures (TBs) required for computing path delay have a spatial resolution of ≤32 km. The top panel of Fig. 1 shows a swath of the AMSR-E observations converted to wet tropospheric correction (in centimeters). The wet path delay was computed from the AMSR-E TB observations in a two-step process. First, a polynomial expression described in Brown (2013) was used to convert the AMSR-E TBs to equivalent TBs that would have been observed by the nadir-looking AMR. Then, the AMR path delay retrieval algorithm described in Keihm et al. (1995) was used to retrieve path delays from these AMR-equivalent TBs derived from AMSR-E. This approach had the benefit of producing path delays from AMSR-E that were intercalibrated in an absolute sense with those produced from the AMR.
(a) AMSR-E data converted into wet tropospheric term (cm) for radar altimetry correction. The 11th period on 25 Jun 2007 is represented (98 min of observations). (b) Zoom off the East Coast.
Citation: Journal of Atmospheric and Oceanic Technology 31, 6; 10.1175/JTECH-D-13-00179.1
The wet tropospheric correction varies from a few centimeters in dry mid- and high-latitude regions to about 40 cm in wet tropical regions. From the bottom panel of Fig. 1 (zoomed into the Gulf Stream region), the highest resolution visually detectable is of the order of 30 km. However, this maximum resolution is affected by the noise in the brightness temperature. This is clearly illustrated in the power spectrum shown in Fig. 2 for the Gulf Stream region. However, the high-resolution water vapor measurement from the airborne instrument the High-Altitude Monolithic Microwave Integrated Circuit (MMIC) Sounding Radiometer (HAMSR) (Brown et al. 2011) has revealed that the power spectrum at short wavelength is roughly a linear extension (in the log domain) of the power spectrum observed by AMR at longer wavelengths (Fig. 3). To obtain an estimate of the water vapor variability beyond the resolution of AMSR-E, we simply discarded the white noise portion of the AMSR-E spectrum and extended the long-wavelength part of the spectrum linearly to 1-km wavelength as shown in Fig. 2 by the dotted line, which is a linear fit to the spectrum at wavelengths of 100–200 km.
The thick line is the averaged (from 5 years of AMSR-E data) power spectral density in the Gulf Stream region. The dashed line is the estimation of the same power spectrum for the unresolved wavelength (below 50 km).
Citation: Journal of Atmospheric and Oceanic Technology 31, 6; 10.1175/JTECH-D-13-00179.1
Wavenumber spectra of the wet tropospheric range delay from the AMR observations (black) and the HAMSR observations in summer (red) and winter (blue).
Citation: Journal of Atmospheric and Oceanic Technology 31, 6; 10.1175/JTECH-D-13-00179.1
The wavenumber spectra of water vapor were computed in all regions using 5 years’ worth of the AMSR-E data on 2° × 2° grids. At each grid node, the data in a 1500-km-long and 200-km-wide swath centered on the node were used for the computation of along-track wavenumber spectrum (averaged across the swath). The spectra were adjusted with the linear extension described above for estimating the short-wavelength part of the spectrum. Such a database for the water vapor wavenumber spectrum was constructed for the study of the cross-track variability of the wet tropospheric correction. Because the swath width of SWOT is only 120 km, the assumption of isotropy was made in the assessment of cross-track variability. This assumption is valid because the atmosphere can be described by isotropic geostrophic turbulence at the scale of 120 km (Charney 1971). We also verified that the power spectra computed from AMSR-E in two orthogonal directions were fairly similar for wavelengths below 1000 km.
3. The cross-track errors in the wet tropospheric correction
Using the database of the water vapor spectrum described in the preceding section, we performed simulations of the two-dimensional field of the wet tropospheric correction over the swath of the SWOT mission. The simulations were used to evaluate the cross-track residual errors after the application of a nadir-looking radiometer. Instead of using the actual AMSR-E data, which are noisy at small scales, to represent the 2D water vapor field, we used the wavenumber spectrum to simulate the field with better representation of the small-scale variability that is important to the SWOT mission.
The present baseline orbit of the SWOT mission has an inclination of 78° with a swath 120 km wide with a gap of 20 km centered on the nadir. The 2D wet tropospheric correction over the swath was simulated using the water vapor spectrum over the global coverage of the SWOT mission. The method of the simulation from the wavenumber spectrum is described in the appendix. Figure 4a shows a random realization of the simulated wet tropospheric correction in the Gulf Stream region. Note that the small-scale noise in the AMSR-E data on Fig. 1b is absent here. With this approach, the resolution limitation of AMSR-E has been mitigated.
(a) Two-dimensional [x and y axes (km)] random realization of water vapor anomaly (cm) following the spectral characteristics in the Gulf Stream region. (b) Residual error (cm) after nadir radiometer correction.
Citation: Journal of Atmospheric and Oceanic Technology 31, 6; 10.1175/JTECH-D-13-00179.1
The baseline design of SWOT is to carry a microwave radiometer like the AMR on Jason-2. The AMR measurement at nadir will be applied across the swath for the wet tropospheric correction. In our simulation, the AMR measurement was produced by applying an 18-km footprint at the center of the swath with a two-dimensional Gaussian function (18-km diameter for midvalues) to the simulated water vapor field. The resulting wet tropospheric correction was then subtracted uniformly from the two-dimensional field of wet tropospheric correction. The residual values then represent the residual errors for the wet tropospheric correction caused by the cross-track variability of water vapor. It should be noted that there will be additional instrument measurement error associated with the nadir measurement, but only the error component due to the residual cross-swath variability is considered here. Figure 4b shows the residual errors over the swath shown in Fig. 4a. Near the nadir, the error is very small, as the region is covered by the radiometer. However, the error can exceed 2 cm at the edge of the swath.
4. Comparison to SSH signals
The major objective of the study is to quantify the residual errors in the wet tropospheric correction for the SWOT mission with the use of a nadir radiometer. The residual errors estimated using the approach described in the preceding section were evaluated against SSH signals derived from altimeter data on 2° × 2° grids globally. At each point of the 2° × 2° grids, an ensemble of residual errors was generated. Along-track spectra of the residual error were then computed everywhere across the swath. This analysis therefore represents the swath-averaged performances, which are actually nonhomogeneous across the swath. At each location, a single spectrum was obtained after averaging the spectra from 100 realizations. An example of this spectrum is represented by the blue curve in Fig. 5 for two different regions: the Gulf Stream region, where the variability of SSH is large, and the eastern tropical Pacific, where the SSH variability is low. The blue dotted line represents the spectrum of the residual error after the nadir correction. The thick blue line is the water vapor spectrum from the AMSR-E observations. The thin blue line represents the linear fit to the AMSR-E spectrum at 100–250-km wavelengths for the approximation of the spectrum at wavelengths shorter than 100 km.
(a) Water vapor and SSH spectra in the Gulf Stream region (at 35°N, 300°E). The thick blue line is the reference water vapor spectrum from AMSR-E. The thin blue line is the estimation of the same spectrum for short wavelengths (below 100 km). The blue dashed line is the residual error spectrum after the nadir radiometer correction. The thick black line is the along-track SSH spectrum from Jason-1, at the same location. The thin black line is the estimation of the same spectrum for short wavelengths (below 100 km). (b) As in (a), but in the eastern tropical Pacific at 10°S, 200°E.
Citation: Journal of Atmospheric and Oceanic Technology 31, 6; 10.1175/JTECH-D-13-00179.1
Using the along-track altimeter data from the Jason-1 satellite, the SSH spectra were computed on the same 2° × 2° grids. As the white noise floor can lead to underestimation of the spectral slope, its exact value at each location (estimated as an averaged value from 25- to 35-km wavelengths) was subtracted from the spectral estimates, as performed in Xu and Fu (2012). Then, the spectrum for the short unobserved wavelengths was estimated by the linear extension of the slope estimated for 70–250-km wavelengths, as shown by the thin black line in Fig. 5. At wavelengths shorter than 250 km, the SSH spectrum was replaced by the linear fit. The resulting SSH spectrum was directly compared to the spectrum of the residual wet tropospheric error.
In the energetic Gulf Stream region, the SSH spectral level is higher than that of the residual error by at least a factor of 100 for wavelengths longer than 100 km. At wavelengths of 100–10 km (at these scales the SSH spectrum is from extrapolation, not directly measured), the SSH is higher than the error by a factor larger than 30. However, in the eastern tropical region (Fig. 5b), the ratio is close to only 10–20 at wavelengths between 250 and 50 km with a minimum at around 100 km, which is a wavelength of interest for SWOT.
Figure 5 has revealed possible geographic variability in the ratio of the SSH signal to the residual error of the wet tropospheric correction. To explore the spatial variability, we mapped the magnitude of both the signal and error by computing the square root of the variance integrated over wavelengths of 1–500 km. The results are shown in Fig. 6. As both signal and error have “red spectra” with spectral density increasing with wavelength, the spectral integral is dominated by long wavelengths close to 500 km. The regions with high residual errors are equatorward of 40°N/°S with a maximum close to 0.8 cm in the western tropical basins (Fig. 6a). Note the presence of a clear pattern following the intertropical convergence zone (ITCZ) across the Pacific. The residual error is lower at high latitudes off the west coasts of North and South America and Africa, where the atmosphere is dryer with less spatial variability.
The square root of the integrated spectra (cm) below 500 km for (a) wet tropospheric residual error and (b) for SSH. (c) Ratio of (a) by (b). Units are expressed in the log of values: 0 corresponds to a ratio of 1, 1 to a ratio of 10, 2 to a ratio of 100.
Citation: Journal of Atmospheric and Oceanic Technology 31, 6; 10.1175/JTECH-D-13-00179.1
The SSH variability (Fig. 6b) shows the well-known patterns of the ocean mesoscale circulation with a maximum in the regions of the boundary currents and the Antarctic Circumpolar Current. The signal-to-error ratio (Fig. 6c) is highest in the regions of high SSH variability with values from 20 to 50. The ratio reaches its lowest values of 3–5 in the regions of the lowest SSH variability.
While Fig. 6 reveals a favorable impression of the efficacy of the wet tropospheric correction based on a nadir radiometer for the SWOT mission, the result is primarily dominated by wavelengths larger than the main objectives of SWOT. We would also like to evaluate the residual error at various wavelengths. Figure 7 shows the maps of the ratio of signal to error [in terms of spectral density (cm2 cycles−1 km−1)] at four wavelengths: 500, 250, 100, and 50 km. Note this is a ratio of power and therefore not directly comparable to the ratios of magnitude in Fig. 6c. As expected, for the large mesoscale wavelengths (500 or 250 km), we clearly see the signature of the boundary currents and the Southern Ocean, where the ratio is very high. The smallest values are close to 10 (corresponding to 1 on the logarithmic color scale) in these regions. At smaller wavelengths (100 and 50 km), the ratio is more homogeneous, but it still tends to be systematically higher than 10.
The ratio of the SSH signal to the residual wet tropospheric error. Results are shown for different wavelengths of interest for SWOT.
Citation: Journal of Atmospheric and Oceanic Technology 31, 6; 10.1175/JTECH-D-13-00179.1
These results are based on power spectra computed over 5 years of data without distinguishing any possible seasonal variations. We have performed some experiments where we selected summer months only and winter months only. In the tropics, very small seasonal differences have been observed. At higher latitudes, the wet tropospheric signal has more significant seasonal variability with a factor up to 2.5 higher in summer than the yearly average. However, in these regions, the 5-yr averaged ratio of signal to error is above 40 at all wavelengths (Fig. 7), suggesting that the ratio is well above 10 even in the worst case of seasonal variability.
5. Examination of a multibeam radiometer
Even though the residual wet tropospheric error is expected to remain well below the SSH signal at all wavelengths of interest, the signal-to-error ratio becomes close to 10 in some low-latitude regions. This being swath-averaged performance, this ratio is certainly less than 10 at the edge of the swath. We have examined the efficacy of a multibeam radiometer to improve the signal-to-error ratio. In fact, the 2-cm residual error occurring at the edges of the swath in high water vapor variability regions (Fig. 4b) is not negligible for many applications.
The case in which an onboard two-beam radiometer looking at each side of the nadir has been studied. As a first simple case of study, we set each beam of the radiometer looking at 35 km away from the nadir (i.e., the middle of the swath) with the same 18-km footprint. The optimization of the distance and the footprint diameter would be done in a separate study if the two-beam radiometer option is considered for implementation. The configuration of the two-beam radiometer is shown in Fig. 8 in comparison to the standard one-beam radiometer. From the two simultaneous measurements across the swath, the wet tropospheric correction is estimated by a constant plus a slope as shown at the top right of Fig. 8. This configuration provides a linear estimation (rather than a constant value) of the cross-track water vapor variability, which tends to reduce the residual error. After making the correction from the two-beam estimate, the residual error in the same swath of the Gulf Stream region (Fig. 4) is shown at the bottom of Fig. 8. The improvement is significant in comparison to the one-beam simulation shown in Fig. 4b. The residual error does not exceed 1 cm anywhere across the swath.
(top left) Scheme of the classical single-nadir radiometer correction, and (top right) scheme of the two-beam radiometer correction targeting at 35 km away from nadir. (bottom) The residual error (cm) after two-beam correction for the random field of Fig. 4a.
Citation: Journal of Atmospheric and Oceanic Technology 31, 6; 10.1175/JTECH-D-13-00179.1
The same analysis of signal-to-error ratio as Fig. 7 is presented in Fig. 9 for the two-beam case. For long wavelengths (500 and 250 km), the improvement from one-beam to two-beam correction is quite significant. The ratio of signal to error exceeds 30 everywhere, and it is higher than 100 in most of the extratropical regions. However, for short wavelengths, the improvement is less pronounced, because the water vapor variability becomes decorrelated across the swath.
As in Fig. 7, but for the two-beam configuration.
Citation: Journal of Atmospheric and Oceanic Technology 31, 6; 10.1175/JTECH-D-13-00179.1
To illustrate the effects of wet tropospheric correction in the low-latitude regions, where the signal-to-error ratio is the lowest, we compare the corrected SSH from the two approaches to the reference SSH signals. We used a typical SSH spectrum and a wet tropospheric error spectrum from low latitudes (10°S, 200°E) to simulate the reference SSH signals and the SSH after wet tropospheric corrections (from one- and two-beam radiometer simulations), at the edge of a swath. The results are shown in Fig. 10. The improvement with the two-beam radiometer is about 1–2 cm, and the RMS is reduced from 1.1 cm with one-beam correction to 0.4 cm with two-beam correction.
Random realization of SSH (black) and the estimation of SSH after wet tropospheric correction in the one-beam (blue) and two-beam (green) configurations at the edge of the swath. The x axis is the distance along the swath. The random realization follows the spectral characteristics in the tropical Pacific at 10°S, 200°E for both SSH and water vapor.
Citation: Journal of Atmospheric and Oceanic Technology 31, 6; 10.1175/JTECH-D-13-00179.1
Finally, we would like to present a global comparison of the wavenumber spectrum of the residual error to that of SSH. Shown in Fig. 11 are the globally averaged spectra for the SSH and the residual error from the different beam configurations. Also shown are the 95% quantiles. Note that the two-beam correction leaves residual errors more than one-order magnitude less than the SSH signals at all wavelengths. The three-beam case (one additional beam at nadir) is presented as well. Indeed, if a nadir altimeter were added to SWOT, then such a three-beam configuration would allow a direct measurement of path delay at nadir. In this configuration, the additional improvements are significant for all wavelengths longer than 50 km. Below 50 km, the spectra from all configurations converge to the same values as the water vapor variability becomes decorrelated across the swath, making the across-swath estimation inefficient at these wavelengths. However, for these wavelengths shorter than 50 km, the cross-track residual error would be dominated by the error of the water vapor measurement (see green curve on Fig. 11). Note that with the three-beam configuration, the residual error becomes as low as the measurement error (even smaller beyond the 500-km wavelength), indicating that any improvement beyond the three-beam configuration would not be useful.
Globally averaged wavenumber spectrum of SSH (black) with 95% quantile (black dashed). Residual wet tropospheric error from one-, two-, and three-beam configurations (blue, red, and pink, respectively) with 95% quantile (dashed). Green shows an estimation of the AMR error spectrum from AMR measurement simulations that include both representative instrument and retrieval algorithm errors.
Citation: Journal of Atmospheric and Oceanic Technology 31, 6; 10.1175/JTECH-D-13-00179.1
6. Conclusions
This study addresses the issue of the adequacy of using a nadir-looking microwave radiometer for making the wet tropospheric correction in the SSH measurement from a wide-swath altimeter. We focused the study for applications to the SWOT mission under development.
A global study has been conducted to evaluate the error from the wet tropospheric correction owing to the cross-track variability of the water vapor content in the atmosphere. The measurement of water vapor from AMSR-E on board the Aqua mission was used for the analysis of the spatial variability. Along-track wavenumber spectra of the water vapor content converted to wet tropospheric correction were computed on global 2° × 2° grids. The measurement noise at short wavelengths was replaced by a linear extension of the spectrum at long wavelengths based on the observed characteristics of an airborne water vapor measurement with resolution much higher than that of the AMSR-E.
With the assumption of spatial isotropy at wavelengths shorter than 120 km, the width of the swath of SWOT, the two-dimensional wet tropospheric correction was generated over the entire swath. The residual errors after the application of the nadir correction across the swath (as a constant) were computed for comparison to the SSH spectrum obtained from the Jason-1 altimeter observations. Global maps of the signal-to-error ratio (signal was computed as the square root of the integration of spectral density at wavelengths shorter than 500 km) were created, showing values of 20–50 in the regions of high SSH variability of the boundary currents and the Antarctic Circumpolar Current, and 3–5 in the regions of low SSH variability in the tropics. The signal-to-error ratio computed from spectral power density was also obtained at various wavelengths. Again, high ratio values were found in the regions of high SSH variability and low values in the regions of low SSH variability.
At wavelengths shorter than 100 km in the tropical regions, the signal-to-error ratio is the lowest with values close to 10. Possible improvement in the correction by using a two-beam radiometer looking off nadir for measuring the slope of the cross-track variability was explored. Significant improvement was demonstrated, especially for long and intermediate wavelengths (500–100 km) with a minimum signal-to-error ratio of 30. This study suggests that the use of a nadir radiometer may be adequate for making the wet tropospheric correction, but the residual errors must be combined with all additional errors and compared relative to the particular mission requirements for the SSH measurement. The use of a two-beam radiometer does make significant improvement everywhere and reduces the residual error below 1 cm even at the edges of the swath. It is particularly useful in the tropical region, where the signal-to-error ratio is the smallest. Finally, the use of a three-beam radiometer would provide additional improvements of the residual error, reaching values as low as the along-track water vapor measurement error.
Acknowledgments
The research presented in the paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautic and Space Administration. Government sponsorship is acknowledged. Support from SWOT projects is acknowledged.
APPENDIX
Random Realization of 2D Isotropic Field from a Predefined 1D Wavenumber Spectrum
For 
(left) Theoretical 1D spectrum (red), and the spectrum computed from the 
Citation: Journal of Atmospheric and Oceanic Technology 31, 6; 10.1175/JTECH-D-13-00179.1
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