The Effects of Altimeter Instrument Noise on the Estimation of the Wavenumber Spectrum of Sea Surface Height

DECEMBER 2012 X U A N D F U 2229

YONGSHENG XU Key Laboratory of Ocean Circulation and Waves, Institute of Oceanology, Chinese Academy of Sciences, Qingdao, China

LEE-LUENG FU Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California

(Manuscript received 16 June 2012, in ﬁnal form 5 August 2012)

ABSTRACT

The wavenumber spectrum of sea surface height (SSH) observed by satellite altimetry was analyzed by Xu and Fu. The spectral shape in the wavelength range of 70–250 km was approximated by a power law, representing a regime governed by geostrophic turbulence theories. The effects of altimeter instrument noise were assumed insignificant at wavelengths longer than 70 km. The authors reexamined the assumption in the study. Using nearly simultaneous observations made by Jason-1 and Jason-2 during their cross-calibration phase, this study found that the white noise level of altimetry measurement was best estimated from the spectral values at wavelengths from 25 to 35 km. After removing a white noise level based on such estimate from the SSH spectrum, the spectral slope values changed significantly over most of the oceans. A key finding 2 is that the spectral slopes are generally steeper than k 2 (k is wavenumber) poleward of the 208 latitudes, where flatter spectral slopes in some regions have previously caused problems for dynamic interpretations. The new results indicate that the spectral slopes in the core regions of the major ocean current systems have values between the original geostrophic turbulence theory and the surface quasigeostrophic theory. The near 2 k 4 spectrum suggests that the sea surface height variability at these wavelengths in the high eddy energy regions might be governed by frontogenesis.

1. Introduction the selected wavelengths. The validity of this assumption is examined in this paper. Instrument noise is mixed with oceanic sea surface The characteristics of the instrument noise are studied height (SSH) signals in altimetry measurements. The by comparing nearly simultaneous observations made signal-to-noise ratio varies with the strength of the sig- by the Jason-1 and Jason-2 altimeters during their cross- nals, affecting the spatial resolution of the signals and calibration phase when the two altimeters flew over their spectrum. Xu and Fu (2011) made a global survey the same spot of the ocean about 1 min apart in time of wavenumber spectral slope in the wavelength range (Bonnefond et al. 2010). The estimated white noise floor of 70–250 km using Jason-1 SSH observations. The is then subtracted from the altimetry spectrum. The wavelength range was selected to represent the scales of spectral slopes after the noise removal are mapped the geostrophic turbulence theories. To minimize the globally. The differences between the new map and the influence of instrument noise, they chose 70 km as the previous map are discussed in terms of ocean dynamics lowest limit of the examined wavelength. This choice and altimetry measurement errors. was made with the assumption of insignificant effect of the instrument noise on the spectral slope estimates at 2. Altimetry noise The instrument noise of altimeter measurement is Corresponding author address: Yongsheng Xu, Key Laboratory of Ocean Circulation and Waves, Institute of Oceanology, Chinese manifested as a near-white spectrum at the high wave- Academy of Sciences, Qingdao, China. number end of a wavenumber spectrum of SSH. The E-mail: [email protected] approach we took was to compute the difference in

DOI: 10.1175/JPO-D-12-0106.1

Ó 2012 American Meteorological Society Unauthenticated | Downloaded 09/26/21 01:36 PM UTC 2230 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 42 altimeter measurements made during the Jason-1/Jason-2 cross-calibration phase when the two satellites flew over all the same ground tracks with one minute apart in time for six months. During such a short time span, the con- ditions of the atmosphere and the ocean were nearly identical. Therefore the geophysical signals in the two altimeter measurements were nearly canceled in the difference between them, leaving only the random instrument noise dominating the spectrum of the difference. Shown in Fig. 1 are typical results of SSH spectral calculations from a ground track of the altimetry observations along pass 132, which is a descending pass from the central North Pacific traversing the eastern tropical Pacific to the eastern South Pacific. The observations along this pass are representative of altimetry measurements in the open ocean. As expected the two spectra from Jason-1 and Jason-2 (the black and red curves) are indistinguishable to the extent of the spectral FIG. 1. SSH spectrum from Jason-1 (black) and Jason-2 (red) estimate error, which is represented by the magnitudes altimeter observations from the central North Pacific traversing the of the wiggles at short wavelengths less than 50 km. The eastern tropical Pacific to the eastern South Pacific. Superimposed is the spectrum of the difference between the two observations spectrum of the SSH difference between the two ob- (blue). The spectral range of 25–35 km is marked by the gray bar. servations (blue curve) essentially represents the instrument noise with time scales shorter than 1 min. The six-month duration of the data is apparently sufficient this wavelength range as the estimated noise level. Be- for estimating the noise spectrum as shown by the rel- cause the instrument noise is not correlated to any of the atively small wiggles at the high wavenumbers. geophysical signals in the altimeter measurement, the Surprisingly it is not exactly the spectrum of a pure estimated white noise spectrum can simply be sub- white noise. At wavelengths shorter than 20 km, the tracted from the SSH spectrum to obtain an estimate of rapid drop of spectral energy is caused by the inter- the signal spectrum. Note that by simply applying a low- polation of the slightly irregularly sampled observations pass filter as most analyses performed on along-track to a common uniform grid along track. The spectral level altimeter data, one cannot remove the effects of a white of the difference is significantly higher than those of the noise at low wavenumbers where the spectral slopes are individual SSH spectrum at wavelengths shorter than affected. 50 km, by about a factor of 2 at wavelengths shorter than about 35 km. This factor-of-2 difference indicates that 3. A new survey of global altimeter wavenumber the difference represents primarily the sum of the ran- spectrum dom noise from the two measurements. The difference spectrum is close to being that of white noise at wave- To remove the effects of the instrument noise at all lengths shorter than 50 km and slowly rising at longer wavenumbers we subtracted the estimated noise level wavelengths, where there are apparent systematic dif- from the SSH spectral values before computing the ferences with unknown origins. spectral slope. The noise level was estimated from av- At wavelengths longer than about 35 km, the two SSH eraging the spectral values in the wavelength band of spectra become closer to the difference spectrum until 25–35 km at every location for computing spectral slope. about 100 km, beyond which the SSH spectra become Using this approach, we recomputed the spectral slopes progressively higher than the difference spectrum. This from the wavelength range of 70–250 km and mapped characteristic reflects that the altimetry observations the results globally for comparison to the results of Xu made by the two satellites represent essentially the same and Fu (2011). An example of SSH spectrum and slope signals at wavelengths longer than 100 km. At shorter estimates with and without removing the effects of noise wavelengths, the random noise becomes dominant. At is shown in Fig. 2, revealing an increase of the spectral wavelengths of 25–35 km, the SSH spectral level, being slope by 23% after removing the noise. The spectrum a factor of 2 less than the difference spectral level, rep- becomes more linear in the range of 70–140 km after resents essentially that of the altimetry instrument noise. removing the noise, making the slope estimate more Therefore we use the average of the spectral values in robust. This indicates that the curvature in the original

Unauthenticated | Downloaded 09/26/21 01:36 PM UTC DECEMBER 2012 X U A N D F U 2231

FIG. 2. The wavenumber spectrum from Jason-1 altimeter observations before (black) and after (blue) removing the noise. The spectra were calculated from the data within a box 1083108 box centered at (2528N, 2408E). The red lines are linear fits in the wavelength range of 70–250 km. The values of spectral slope are noted. spectrum at these wavelengths is caused by the effect of the instrument noise. To our surprise, the effect of the instrument noise is substantial even in a region of high eddy energy with steep spectral slope as illustrated by this case. Using the method of Xu and Fu (2011) we remapped the global SSH spectral slope after removing the noise and compared it with the original calculation (Fig. 3). As in Xu and Fu (2011), the areas poleward of 608S and 608N are excluded from the study to avoid the ice in- FIG. 3. The global distribution of the spectral slopes of SSH fluence on the SSH measurements. The wavelength wavenumber spectrum in the wavelength band of 70–250 km esti- range of 70–250 km was selected for computing the mated from the Jason-1 altimeter measurements (a) before and (b) spectral slope. Despite the differences in the values of after removing the noise. The sign of the slopes was reversed to spectral slope, the geographic patterns of the two maps make the values positive. are similar to each other even in some details. After removing the noise, the spectral slopes have generally Confluence, and the Agulhas Current. The steepest become steeper than the previous estimates, especially spectral slope of the global ocean is 24.5 6 0.12 (see in regions of low eddy energy away from the major error estimation method in Xu and Fu 2011), which is 2 ocean currents. The most important new result is that significantly flatter than the k 5 power law predicted by 2 the spectral slopes are generally steeper than k 2 pole- the original geostrophic turbulence theory (Charney ward of the 208 latitudes. The previous results in some 1971), suggesting that the observed SSH spectral slopes high-latitude regions such as the northeast Pacific and are flatter than the prediction of the geostrophic turbu- 2 southeast Pacific show spectral slopes flatter than k 2, lence theory everywhere in the ocean. implying ‘‘blue’’ spectra in geostrophic velocity, which Recent theoretical work has suggested the rele- are unphysical. These features have been removed in vance of the surface quasigeostrophic (SQG) theory for 2 the new map. Spectral slopes flatter than k 2 are present interpreting altimeter observations (Held et al. 1995; only in low-latitude regions at places where ageostrophic Capet et al. 2008; Le Traon et al. 2008). This theory 2 effects may become important. predicts k 11/3 power law for SSH spectrum. To explore In the high eddy energy regions associated with the consistency of the observed spectral slopes with the the major ocean currents, the slopes become slightly SQG theory, Fig. 4 exhibits the distribution of four steeper. These regions include the core regions of the categories of regions according the spectral slopes. The Gulf Stream, the Kuroshio Extension and the Antarctic type-1 regions are the areas where the spectral slopes are Circumpolar Current (ACC) systems, the Brazil–Malvinas indistinguishable from 211/3 within the 95% confidence

Unauthenticated | Downloaded 09/26/21 01:36 PM UTC 2232 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 42

FIG. 4. Four types of regions in terms of the spectral power law (left) before and (right) after removing the noise. 2 The black color represents the type-1 regions where the spectral slopes are consistent with the k 11/3 power law, the 2 dark gray color represents the type-2 regions where the spectral slopes are steeper than k 11/3, the light gray color 2 2 represents the type-3 regions where the spectral slopes are between k 11/3 and k 2, and the white areas are type 4 2 where the spectral slopes are flatter than k 2. level. They are denoted by black color, representing 4. Conclusions areas where the SQG theory is consistent with the observations. The type-2 regions, denoted by dark gray Using nearly simultaneous observations from two al- color, are the places where the spectral slopes are sig- timeters we found that the white noise level of altimeter nificantly steeper than 211/3. The type-3 regions, denoted instrument noise was best estimated from the SSH by light gray, are the places where the spectral slopes are spectral values at wavelengths of 25–35 km. The white significantly flatter than 211/3 but steeper than 22. The noise spectrum can simply be subtracted from the SSH type-4 regions, denoted by white, are the places where wavenumber spectrum to minimize the effects of in- the spectral slopes are flatter than 22. strument noise in the spectral estimates. Using this ap- The type-1 regions are generally near the edge of the proach, we have recomputed the spectral slopes of SSH core of the major current systems. After removing the wavenumber spectrum and obtained a new map showing noise, the areas of the type-1 regions have largely in- the global variability of SSH wavenumber spectral creased around the Gulf Stream and the Kuroshio cur- slopes. Although the geographic pattern of the vari- rent systems, and they appear more connected around ability is similar to the previous results reported in Xu the ACC. Note that much of the type-1 regions in the and Fu (2011), the spectral slopes have become signifi- ACC have changed to type-2 after removing the noise. cantly steeper everywhere. The type-2 regions have significantly expanded in the new A major new result is that the spectral slopes are gen- 2 map, mainly in the core regions of the major ocean current erally steeper than k 2 poleward of the 208 latitudes, 2 systems. The spectral slopes in these regions are steeper where spectral slopes flatter than k 2 have been previously than the SQG theory but flatter than the original geo- found in many regions, causing problems in interpreting 2 strophic turbulence theory. They are fairly close to k 4, the resultant ‘‘blue’’ geostrophic velocity spectrum. In the 2 consistent with a k 2 kinetic energy spectrum of an core regions of the major ocean current systems where the ocean dominated by the presence of fronts (Boyd 1992). eddy energy is high, the spectral slopes have values be- The type-3 regions, covering the extratropical areas tween the predictions of the original geostrophic turbu- 2 outside the major current systems, have expanded after lence theory (k 5) and the surface quasigeostrophic theory 2 removing the noise, with slopes steeper than 22 every- (k 11/3). This new result suggests that the ocean dynamics where poleward of the 208 latitudes. The type-4 regions in these regions at wavelengths of 70–250 km may be have retreated to topical areas within the 20 latitude governed by frontogenesis. The spectral slopes in the re- degrees, where the slopes are much steeper than the gions equatorward of 208 are also steeper than the pre- original values after removing the noise, changing from vious estimates. Ageostrophic dynamics may be required the range of [20.7–2] to the range of [21.5–2.5]. to account for the relatively flat spectral slopes.

Unauthenticated | Downloaded 09/26/21 01:36 PM UTC DECEMBER 2012 X U A N D F U 2233

Acknowledgments. The research presented in the Boyd, J. P., 1992: The energy spectrum of fronts: Time evolution of paper was partly (LLF) carried out at the Jet Pro- shocks in Burger’s equation. J. Atmos. Sci., 49, 128–139. pulsion Laboratory, California Institute of Technol- Capet, X., P. Klein, B. Hua, G. Lapeyre, and J. C. McWilliams, 2008: Surface kinetic energy transfer in surface quasi-geostrophic ogy, under contract with the National Aeronautic and ﬂows. J. Fluid Mech., 604, 165–174. Space Administration. Support from the Jason-1 and Charney, J. G., 1971: Geostrophic turbulence. J. Atmos. Sci., 28, OSTM/Jason-2 Projects is acknowledged. This work 1087–1094. was partly supported by CAS Innovation Program Held, I. M., R. T. Pierrehumbert, S. T. Garner, and K. L. Swanson, Y22114101Q and the China 973 Project 2012CB956000. 1995: Surface quasi-geostrophic dynamics. J. Fluid Mech., 282, 1–20. Le Traon, P.-Y., P. Klein, and B. L. Hua, 2008: Do altimeter REFERENCES wavenumber spectra agree with the interior or surface quasigeostrophic theory? J. Phys. Oceanogr., 38, 1137–1142. Bonnefond, P., P. Exertier, O. Laurain, and G. Jan, 2010: Absolute Xu, Y., and L.-L. Fu, 2011: Global variability of the wavenumber calibration of Jason-1 and Jason-2 altimeters in Corsica during spectrum of oceanic mesoscale turbulence. J. Phys. Oceanogr., the formation ﬂight phase. Mar. Geod., 33 (S1), 80–90. 41, 802–809.

Unauthenticated | Downloaded 09/26/21 01:36 PM UTC