The Effect of Atmospheric Water Vapor Content on the Performance of Future Wide-Swath Ocean Altimetry Measurement

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CLEMENT UBELMANN,LEE-LUENG FU,SHANNON BROWN,EVA PERAL, AND DANIEL ESTEBAN-FERNANDEZ Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California

(Manuscript received 5 September 2013, in ﬁnal form 26 November 2013)

ABSTRACT

Measurement of sea surface height (SSH) over a finite swath along satellite tracks has been planned for future space missions. The effect of water vapor in the troposphere on the delay of radar signal must be corrected for in the SSH measurement. The efficacy of a nadir-looking radiometer that has been the approach for conventional altimetry is examined in the study. The focus is placed on the cross-track variability of water vapor that is not measured by the nadir-looking radiometer. Simulations of the 2D field of water vapor were performed by spectral analysis of existing radiometer data. The residual error from the application of the correction made by a nadir-looking radiometer was computed over the global ocean and compared to the SSH signal estimated from satellite altimeter data. Global maps of the signal-to-error ratio (the square root of spectral variance 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. Improvement in the correction by using a two-beam radiometer looking off nadir for measuring the slope of the cross-track variability was also explored, leading to a reduction of the error to below 1 cm at wavelengths of 10–500 km.

1. Introduction a finite swath along the satellite’s ground tracks, the issue of the adequacy of a nadir-looking radiometer for making Correction for the path delay in satellite altimetry the correction over the swath arises. What are the char- caused by water vapor in the troposphere is an important acteristics of the variability of water vapor across the factor in making precise measurement of the sea surface swath? How is the variability compared to that of SSH? height (SSH) (Chelton et al. 2001). It has been standard What can be done to address the cross-swath water vapor practice to carry a multifrequency microwave radiometer variability, if necessary? These questions are addressed in on board an altimetric satellite to make simultaneous the present study. In particular, the study is configured for measurement of the tropospheric water vapor content for assessing the expected performance of the wet tropo- the so-called wet tropospheric correction. This approach spheric correction over the ocean for the Surface Water is proven effective in the application of satellite altimetry, and Ocean Topography (SWOT) mission (Fu and Ferrari especially in the open ocean, where the finite footprint of 2008; Durand et al. 2010). radiometer does not cover any land. Instead of nadir-only measurements, SWOT will be The conventional radar altimeter is a profiling in- able to measure SSH over a 50-km-wide swath 10 km strument making SSH measurements along the satel- away from nadir on each side. The current baseline lite’s ground tracks. An onboard microwave radiometer design of the mission is to carry a conventional radi- serves the purpose of making the wet tropospheric cor- ometer like the one flown on the Ocean Surface To- rection (Keihm et al. 1995; Brown et al. 2004). While we pography Mission (OSTM) on Jason-2, called the develop wide-swath altimetry for observing SSH over 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 Corresponding author address: Clement Ubelmann, Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove are investigated in the study. The error of the water Dr., Pasadena, CA 91109. vapor measurement itself is not considered in the main E-mail: clement.ubelmann@jpl.nasa.gov analysis, but it will be presented and discussed in

DOI: 10.1175/JTECH-D-13-00179.1

Ó 2014 American Meteorological Society Unauthenticated | Downloaded 09/26/21 06:07 PM UTC JUNE 2014 U B E L M A N N E T A L . 1447 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 FIG. 1. (a) AMSR-E data converted into wet tropospheric term The key issue is the cross-track variability of the wet (cm) for radar altimetry correction. The 11th period on 25 Jun 2007 is tropospheric correction that cannot be accounted for by represented (98 min of observations). (b) Zoom off the East Coast. a nadir-looking instrument like AMR. To evaluate the cross-track variability, we analyzed the AMSR-E data in the power spectrum shown in Fig. 2 for the Gulf that offer a quasi-daily global coverage. The AMSR-E Stream region. However, the high-resolution water va- brightness temperature data are acquired from the Ja- por measurement from the airborne instrument the pan Aerospace Exploration Agency. The spatial reso- High-Altitude Monolithic Microwave Integrated Cir- lution of the AMSR-E brightness temperatures (TBs) cuit (MMIC) Sounding Radiometer (HAMSR) (Brown required for computing path delay have a spatial reso- et al. 2011) has revealed that the power spectrum at short lution of #32 km. The top panel of Fig. 1 shows a swath wavelength is roughly a linear extension (in the log do- of the AMSR-E observations converted to wet tropo- main) of the power spectrum observed by AMR at longer spheric correction (in centimeters). The wet path delay wavelengths (Fig. 3). To obtain an estimate of the water was computed from the AMSR-E TB observations in vapor variability beyond the resolution of AMSR-E, we a two-step process. First, a polynomial expression de- simply discarded the white noise portion of the AMSR-E scribed in Brown (2013) was used to convert the AMSR- spectrum and extended the long-wavelength part of the E TBs to equivalent TBs that would have been observed spectrum linearly to 1-km wavelength as shown in Fig. 2 by the nadir-looking AMR. Then, the AMR path delay by the dotted line, which is a linear ﬁt to the spectrum at retrieval algorithm described in Keihm et al. (1995) was wavelengths of 100–200 km. used to retrieve path delays from these AMR-equivalent The wavenumber spectra of water vapor were com- TBs derived from AMSR-E. This approach had the puted in all regions using 5 years’ worth of the AMSR-E beneﬁt of producing path delays from AMSR-E that data on 28328 grids. At each grid node, the data in were intercalibrated in an absolute sense with those a 1500-km-long and 200-km-wide swath centered on produced from the AMR. the node were used for the computation of along-track The wet tropospheric correction varies from a few wavenumber spectrum (averaged across the swath). centimeters in dry mid- and high-latitude regions to about The spectra were adjusted with the linear extension 40 cm in wet tropical regions. From the bottom panel of described above for estimating the short-wavelength Fig. 1 (zoomed into the Gulf Stream region), the highest part of the spectrum. Such a database for the water resolution visually detectable is of the order of 30 km. vapor wavenumber spectrum was constructed for the However, this maximum resolution is affected by the noise study of the cross-track variability of the wet tropo- in the brightness temperature. This is clearly illustrated spheric correction. Because the swath width of SWOT is

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FIG. 3. Wavenumber spectra of the wet tropospheric range delay FIG. 2. The thick line is the averaged (from 5 years of AMSR-E from the AMR observations (black) and the HAMSR observations data) power spectral density in the Gulf Stream region. The dashed in summer (red) and winter (blue). line is the estimation of the same power spectrum for the un- resolved wavelength (below 50 km). is absent here. With this approach, the resolution limi- tation of AMSR-E has been mitigated. only 120 km, the assumption of isotropy was made in the The baseline design of SWOT is to carry a microwave assessment of cross-track variability. This assumption is radiometer like the AMR on Jason-2. The AMR mea- valid because the atmosphere can be described by iso- surement at nadir will be applied across the swath for the tropic geostrophic turbulence at the scale of 120 km wet tropospheric correction. In our simulation, the AMR (Charney 1971). We also verified that the power spectra measurement was produced by applying an 18-km foot- computed from AMSR-E in two orthogonal directions print at the center of the swath with a two-dimensional were fairly similar for wavelengths below 1000 km. Gaussian function (18-km diameter for midvalues) to the simulated water vapor field. The resulting wet tropospheric correction was then subtracted uniformly from 3. The cross-track errors in the wet tropospheric the two-dimensional field of wet tropospheric correction. correction The residual values then represent the residual errors for Using the database of the water vapor spectrum de- the wet tropospheric correction caused by the cross-track scribed in the preceding section, we performed simula- variability of water vapor. It should be noted that there tions of the two-dimensional field of the wet tropospheric will be additional instrument measurement error associ- correction over the swath of the SWOT mission. The ated with the nadir measurement, but only the error simulations were used to evaluate the cross-track residual component due to the residual cross-swath variability is errors after the application of a nadir-looking radiometer. considered here. Figure 4b shows the residual errors over Instead of using the actual AMSR-E data, which are the swath shown in Fig. 4a. Near the nadir, the error is noisy at small scales, to represent the 2D water vapor very small, as the region is covered by the radiometer. 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 788 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 FIG. 4. (a) Two-dimensional [x and y axes (km)] random re- shows a random realization of the simulated wet tro- alization of water vapor anomaly (cm) following the spectral pospheric correction in the Gulf Stream region. Note characteristics in the Gulf Stream region. (b) Residual error (cm) that the small-scale noise in the AMSR-E data on Fig. 1b after nadir radiometer correction.

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FIG. 5. (a) Water vapor and SSH spectra in the Gulf Stream region (at 358N, 3008E). 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 108S, 2008E.

However, the error can exceed 2 cm at the edge of the underestimation of the spectral slope, its exact value at swath. 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 4. Comparison to SSH signals spectrum for the short unobserved wavelengths was es- The major objective of the study is to quantify the re- timated by the linear extension of the slope estimated sidual errors in the wet tropospheric correction for the for 70–250-km wavelengths, as shown by the thin black SWOT mission with the use of a nadir radiometer. The line in Fig. 5. At wavelengths shorter than 250 km, the residual errors estimated using the approach described SSH spectrum was replaced by the linear fit. The re- in the preceding section were evaluated against SSH sulting SSH spectrum was directly compared to the signals derived from altimeter data on 28328 grids spectrum of the residual wet tropospheric error. globally. At each point of the 28328 grids, an ensemble In the energetic Gulf Stream region, the SSH spectral of residual errors was generated. Along-track spectra level is higher than that of the residual error by at least of the residual error were then computed everywhere a factor of 100 for wavelengths longer than 100 km. At across the swath. This analysis therefore represents the wavelengths of 100–10 km (at these scales the SSH swath-averaged performances, which are actually non- spectrum is from extrapolation, not directly measured), homogeneous across the swath. At each location, a sin- the SSH is higher than the error by a factor larger than gle spectrum was obtained after averaging the spectra 30. However, in the eastern tropical region (Fig. 5b), the from 100 realizations. An example of this spectrum is ratio is close to only 10–20 at wavelengths between 250 represented by the blue curve in Fig. 5 for two different and 50 km with a minimum at around 100 km, which is regions: the Gulf Stream region, where the variability of a wavelength of interest for SWOT. SSH is large, and the eastern tropical Pacific, where the Figure 5 has revealed possible geographic variability SSH variability is low. The blue dotted line represents in the ratio of the SSH signal to the residual error of the the spectrum of the residual error after the nadir correc- wet tropospheric correction. To explore the spatial vari- tion. The thick blue line is the water vapor spectrum from ability, we mapped the magnitude of both the signal and the AMSR-E observations. The thin blue line represents error by computing the square root of the variance in- the linear fit to the AMSR-E spectrum at 100–250-km tegrated over wavelengths of 1–500 km. The results are wavelengths for the approximation of the spectrum at shown in Fig. 6. As both signal and error have ‘‘red wavelengths shorter than 100 km. spectra’’ with spectral density increasing with wavelength, Using the along-track altimeter data from the Jason-1 the spectral integral is dominated by long wavelengths satellite, the SSH spectra were computed on the same close to 500 km. The regions with high residual errors are 28328 grids. As the white noise floor can lead to equatorward of 408N/8S with a maximum close to 0.8 cm

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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 2 2 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 ra- tios 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

FIG. 6. The square root of the integrated spectra (cm) below color scale) in these regions. At smaller wavelengths 500 km for (a) wet tropospheric residual error and (b) for SSH. (100 and 50 km), the ratio is more homogeneous, but it (c) Ratio of (a) by (b). Units are expressed in the log of values: still tends to be systematically higher than 10. 0 corresponds to a ratio of 1, 1 to a ratio of 10, 2 to a ratio of 100. These results are based on power spectra computed over 5 years of data without distinguishing any possible in the western tropical basins (Fig. 6a). Note the pres- seasonal variations. We have performed some experi- ence of a clear pattern following the intertropical con- ments where we selected summer months only and winter vergence zone (ITCZ) across the Paciﬁc. The residual months only. In the tropics, very small seasonal differences error is lower at high latitudes off the west coasts of have been observed. At higher latitudes, the wet tro- North and South America and Africa, where the atmo- pospheric signal has more signiﬁcant seasonal variability sphere is dryer with less spatial variability. with a factor up to 2.5 higher in summer than the yearly

FIG. 7. The ratio of the SSH signal to the residual wet tropospheric error. Results are shown for different wavelengths of interest for SWOT.

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FIG. 8. (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. average. However, in these regions, the 5-yr averaged vapor variability, which tends to reduce the residual ratio of signal to error is above 40 at all wavelengths error. After making the correction from the two-beam (Fig. 7), suggesting that the ratio is well above 10 even in estimate, the residual error in the same swath of the Gulf the worst case of seasonal variability. 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 5. Examination of a multibeam radiometer error does not exceed 1 cm anywhere across the swath. Even though the residual wet tropospheric error is The same analysis of signal-to-error ratio as Fig. 7 is expected to remain well below the SSH signal at all presented in Fig. 9 for the two-beam case. For long wavelengths of interest, the signal-to-error ratio becomes wavelengths (500 and 250 km), the improvement from close to 10 in some low-latitude regions. This being swath- one-beam to two-beam correction is quite significant. averaged performance, this ratio is certainly less than 10 The ratio of signal to error exceeds 30 everywhere, and it at the edge of the swath. We have examined the efficacy is higher than 100 in most of the extratropical regions. of a multibeam radiometer to improve the signal-to-error However, for short wavelengths, the improvement is less ratio. In fact, the 2-cm residual error occurring at the pronounced, because the water vapor variability be- edges of the swath in high water vapor variability regions comes decorrelated across the swath. (Fig. 4b) is not negligible for many applications. To illustrate the effects of wet tropospheric correction The case in which an onboard two-beam radiometer in the low-latitude regions, where the signal-to-error looking at each side of the nadir has been studied. As ratio is the lowest, we compare the corrected SSH from a first simple case of study, we set each beam of the ra- the two approaches to the reference SSH signals. We diometer looking at 35 km away from the nadir (i.e., the used a typical SSH spectrum and a wet tropospheric middle of the swath) with the same 18-km footprint. The error spectrum from low latitudes (108S, 2008E) to sim- optimization of the distance and the footprint diameter ulate the reference SSH signals and the SSH after wet would be done in a separate study if the two-beam radi- tropospheric corrections (from one- and two-beam ra- ometer option is considered for implementation. The diometer simulations), at the edge of a swath. The re- configuration of the two-beam radiometer is shown in sults are shown in Fig. 10. The improvement with the Fig. 8 in comparison to the standard one-beam radiom- two-beam radiometer is about 1–2 cm, and the RMS is eter. From the two simultaneous measurements across reduced from 1.1 cm with one-beam correction to 0.4 cm the swath, the wet tropospheric correction is estimated with two-beam correction. by a constant plus a slope as shown at the top right of Finally, we would like to present a global comparison Fig. 8. This configuration provides a linear estimation of the wavenumber spectrum of the residual error to that (rather than a constant value) of the cross-track water of SSH. Shown in Fig. 11 are the globally averaged

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FIG.9.AsinFig. 7, but for the two-beam configuration. spectra for the SSH and the residual error from the wavelength), indicating that any improvement beyond the different beam configurations. Also shown are the 95% three-beam configuration would not be useful. quantiles. Note that the two-beam correction leaves residual errors more than one-order magnitude less than 6. Conclusions the SSH signals at all wavelengths. The three-beam case (one additional beam at nadir) is presented as well. In- This study addresses the issue of the adequacy of using deed, if a nadir altimeter were added to SWOT, then such a nadir-looking microwave radiometer for making the a three-beam configuration would allow a direct mea- wet tropospheric correction in the SSH measurement surement of path delay at nadir. In this configuration, the from a wide-swath altimeter. We focused the study for additional improvements are significant for all wave- applications to the SWOT mission under development. lengths longer than 50 km. Below 50 km, the spectra A global study has been conducted to evaluate the error from all configurations converge to the same values as from the wet tropospheric correction owing to the cross- the water vapor variability becomes decorrelated across track variability of the water vapor content in the atmo- the swath, making the across-swath estimation inefficient sphere. The measurement of water vapor from AMSR-E at these wavelengths. However, for these wavelengths on board the Aqua mission was used for the analysis of the shorter than 50 km, the cross-track residual error would spatial variability. Along-track wavenumber spectra of the be dominated by the error of the water vapor measure- water vapor content converted to wet tropospheric cor- ment (see green curve on Fig. 11). Note that with the three- rection were computed on global 28328 grids. The mea- beam configuration, the residual error becomes as low as surement noise at short wavelengths was replaced by the measurement error (even smaller beyond the 500-km a linear extension of the spectrum at long wavelengths

FIG. 10. 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 108S, 2008E for both SSH and water vapor.

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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.

FIG. 11. Globally averaged wavenumber spectrum of SSH (black) with 95% quantile (black dashed). Residual wet tropospheric error Acknowledgments. The research presented in the pa- from one-, two-, and three-beam configurations (blue, red, and pink, per was carried out at the Jet Propulsion Laboratory, respectively) with 95% quantile (dashed). Green shows an estima- California Institute of Technology, under contract with tion of the AMR error spectrum from AMR measurement simula- the National Aeronautic and Space Administration. tions that include both representative instrument and retrieval algorithm errors. Government sponsorship is acknowledged. Support from SWOT projects is acknowledged. based on the observed characteristics of an airborne water vapor measurement with resolution much higher than that of the AMSR-E. APPENDIX With the assumption of spatial isotropy at wavelengths shorter than 120 km, the width of the swath of SWOT, the Random Realization of 2D Isotropic Field from two-dimensional wet tropospheric correction was gener- a Predefined 1D Wavenumber Spectrum ated over the entire swath. The residual errors after the The goal is to generate a two-dimensional (2D) ran- application of the nadir correction across the swath (as dom isotropic field H(x, y) from a given wavenumber a constant) were computed for comparison to the SSH spectrum E(k) defined in the interval [ka, kb] and set to spectrum obtained from the Jason-1 altimeter observa- zero outside. Assuming isotropy, the 2D spectrum can tions. Global maps of the signal-to-error ratio (signal was be expressed as a function of the 1D spectrum as follows: computed as the square root of the integration of spectral density at wavelengths shorter than 500 km) were cre- 1 E (k , k ) 5 E(k), ated, showing values of 20–50 in the regions of high SSH 2d x y 2pk variability of the boundary currents and the Antarctic Circumpolar Current, and 3–5 in the regions of low SSH where kx and ky are the two orthogonal wavenumber variability in the tropics. The signal-to-error ratio com- components;qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k is the 1D scalar wavenumber defined as 5 2 1 2 puted from spectral power density was also obtained at k kx ky, H is constructed as the sum of N random various wavelengths. Again, high ratio values were found 2D Fourier components from the annulus in the 2D in the regions of high SSH variability and low values in wavenumber domain bounded by ka and kb; namely, the the regions of low SSH variability. nth component has a 2D random wavenumber (kxn, kyn) At wavelengths shorter than 100 km in the tropical re- in the x and y directions, respectively, selected randomly gions, the signal-to-error ratio is the lowest with values close inside the annulus over which E2d is nonzero. A random u p to 10. Possible improvement in the correction by using phase n is selected in the interval of [0, 2 ]. The am- a two-beam radiometer looking off nadir for measuring the plitude of the Fourier component follows the spectrum slope of the cross-track variability was explored. Significant E2d. The term H can then be written as

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p(k2 2 k2) N 5 b a å 1 1 u H(x, y) 2E2d(kxn, kyn) cos(kxnx kyny n). N n51

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FIG. A1. (left) Theoretical 1D spectrum (red), and the spectrum computed from the H ﬁeld in the y direction and averaged in the x direction. (right) The H ﬁeld constructed from the theoretical 1D spectrum.

For N L2/dx2, where L is the domain length and dx Hawk unmanned aerial vehicle: Instrument description and is the grid spacing, the 2D random signal H has a 1D performance. IEEE Trans. Geosci. Remote Sens., 49, 3291– isotropic wavenumber spectrum close to E(k). An ex- 3301, doi:10.1109/TGRS.2011.2125973. 5 5 Charney, J. G., 1971: Geostrophic turbulence. J. Atmos. Sci., 28, 1087– ample is shown in Fig. A1 for L 1000 and dx 1. We 1095, doi:10.1175/1520-0469(1971)028,1087:GT.2.0.CO;2. 6 set E 5 1 for a wavelength of 10 and E 5 10 for Chelton, D. B., J. Ries, B. Haines, L.-L. Fu, and P. Callahan, 2001: a wavelength of 1000 with a linear-log interpolation Satellite altimetry. Satellite Altimetry and Earth Sciences: A (a power law) in between. The slope of the spectrum Handbook of Techniques and Applications, L.-L. Fu and is 22.5. Such a spectrum is shown in red in the left panel A. Cazenave, Eds., International Geophysics Series, Vol. 69, Academic Press, 1–131. of Fig. A1. The result of a random realization H using Durand, M., E. Rodriguez, D. E. Alsdorf, and M. Trigg, 2010: 4 N 5 10 random components is shown in the right panel. Estimating river depth from remote sensing swath inter- The 1D power spectrum of H has been computed in the ferometry measurements of river height, slope, and width. y direction and averaged in the x direction. It is repre- IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens., 3, 20–31, sented in black in the left panel. The consistency be- doi:10.1109/JSTARS.2009.2033453. Fu, L.-L., and R. Ferrari, 2008: Observing oceanic submesoscale tween the two spectra is clearly shown. We have veriﬁed processes from space. Eos, Trans. Amer. Geophys. Union, 89, the consistency in all directions. 488, doi:10.1029/2008EO480003. Kawanishi, T., and Coauthors, 2003: The Advanced Microwave REFERENCES Scanning Radiometer for the Earth Observing System (AMSR-E), NASDA’s contribution to the EOS for global Brown, S., 2013: Maintaining the long-term calibration of the energy and water cycle studies. IEEE Trans. Geosci. Remote Jason-2/OSTM advanced microwave radiometer through inter- Sens., 41, 184–194, doi:10.1109/TGRS.2002.808331. satellite calibration. IEEE Trans. Geosci. Remote Sens., 51, Keihm, S. M., M. A. Janssen, and C. S. Ruf, 1995: TOPEX/Poseidon 1531–1543, doi:10.1109/TGRS.2012.2213262. microwave radiometer (TMR). III. Wet troposphere range ——, C. Ruf, S. Keihm, and A. Kitiyakara, 2004: Jason microwave correction algorithm and pre-launch error budget. IEEE Trans. radiometer performance and on-orbit calibration. Mar. Geod., Geosci. Remote Sens., 33, 147–161, doi:10.1109/36.368213. 27, 199–220, doi:10.1080/01490410490465643. Xu, Y., L.-L. Fu, 2012: The effects of altimeter instrument noise ——, B. Lambrigtsen, R. Denning, T. Gaier, P. Kangaslahti, on the estimation of the wavenumber spectrum of sea sur- B. Lim, J. Tanabe, and A. Tanner, 2011: The High-Altitude face height. J. Phys. Oceanogr., 42, 2229–2233, doi:10.1175/ MMIC Sounding Radiometer (HAMSR) for the Global JPO-D-12-0106.1.

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