# Covariant errors in ocean retrackers evaluated using along-track cross-spectra

**CoAuthors**

**Event: **2017 Ocean Surface Topography Science Team Meeting

**Session: **Instrument Processing: Measurement and Retracking

**Presentation type: **Type Oral

An altimeter’s waveform displays backscattered power as a function of range. Retrackers fit parametric models to waveforms in order to estimate sea surface height, SWH, and sigma0; some retrackers also fit xisq (off-nadir angle squared). Radar backscatter from the ocean is a random process and the waveform is a random variable, hence the estimated parameters also contain random noise. Previous studies (Sandwell and Smith, GJI, 2005; Quartly, GRSL, 2009; Smith and Scharroo, OSTST, 2011; Zaron and de Carvalho, JTECH, 2016) have found some co-variance among the random errors in the retrieved parameters, and have suggested that parameter estimates could be improved if correlated errors could be reduced.

We evaluate the error characteristics of 5 retrackers (ALES, MLE3, MLE4, PEACHI Nelder-Mead, and PEACHI Newton-Raphson ) applied to 3 altimeters (Jason-2, Jason-3, and SARAL), examining a region of the South Pacific (the SAR box) characterized by low sea surface height spectral variability where unwanted noise (the spectral bump) has previously been identified. The SARAL waveform differs from the others in being beam-limited as well as pulse-limited. The PEACHI Nelder-Mead retracker differs from the others in using a weighted fitting scheme. ALES and MLE3 differ from the others in excluding xisq; ALES also excludes the waveform tail. These different waveforms and approaches to parameter fitting allow us to show how the estimated geophysical parameters are affected by random variability in the pulse-limited and beam-limited footprint areas, factors that need to be understood to best use the various retrackers in specific applications.

We analyze the raw sea surface height anomaly (RSSHA, orbit height minus retracked range minus mean sea surface), SWH, sigma0, xisq, and PPP (pulse peakiness parameter). For each retracker, each possible pairing of analyzed parameters was put through an along-track cross-spectral analysis at the maximum possible sampling rate (20 Hz for the Jason data and 40 Hz for the SARAL data). Auto- and cross-power spectral densities were averaged for several cycles and passes in order to estimate auto-spectral density, magnitude-squared coherence, and cross-spectral admittance.

Auto-spectral densities of all parameters are white at wavelengths equal to and shorter than the pulse-limited footprint diameter. Auto-spectra of PPP and xisq are also white, but at higher power level, at wavelengths longer than 50 km, with transition in between. Auto-spectra of RSSHA, SWH, and sigma0 are red, decaying as wavenumber squared, for long wavelengths, with transition bands often having a spectral bump at intermediate wavelengths. The onset of the non-red and bump-like transition band is at 100 km wavelength for RSSHA, 50 km wavelength for sigma0, and 500–2000 km for SWH. Auto-spectral density levels are different from one retracker to another in the transition (spectral bump) region, differing by a factor of 4 in RSSHA, a factor of 10 in SWH, and a factor of 100 in sigma0.

The cross-spectral analysis exhibits correlations among the parameters as a function of along-track wavelength, and shows that the covariance is a function of along-track wavelength, antenna gain pattern and footprint, and (most importantly): (1) whether or not the waveform tail is fitted (ALES does not); (2) whether or not the xisq is fitted (MLE3 and ALES do not); and (3) whether or not a weighted fit is done (Peachi Nelder-Mead does). Examining the along-track wavelengths where parameter covariance is large exhibits how each retracker contributes to its own sea state bias and its own spectral bump.

Random variance in PPP (speckle noise) couples into xisq if it is estimated, and into sigma0 if it is not (MLE3). Variations in sigma0 are dominated by variations in xisq at wavelengths shorter than 50 km for those retrackers that estimate xisq, with the notable exception of SARAL MLE4 (xisq estimation error is lower at Ka band). Errors in xisq and errors due to PPP co-vary with errors in SWH, sigma0, and RSSHA over various bands.

The sea state bias is quite obviously not a constant percentage of SWH or sigma0, as admittance between these shows a complex behavior with generally increasing covariance as wavenumber increases. SWH:RSSHA ratio grows from about –4% at 1000 km wavelength to about –10% at 10 km wavelength. However, and as expected, weighting the retracker (PEACHI Nelder-Mead) strongly increases this correlation, from less than –1% at 500 km to almost –20% at 10 km.

If the overall goal is simply to find the best RSSHA performance, then ALES is best for wavelengths longer than 10 km, where it has the lowest noise and most un-bump-like spectrum; it and MLE3 also have the lowest correlation with PPP and sigma0. ALES, MLE4, and Nelder-Mead also show the lowest coherence between SWH and RSSHA at wavelengths longer than 10 km.

We evaluate the error characteristics of 5 retrackers (ALES, MLE3, MLE4, PEACHI Nelder-Mead, and PEACHI Newton-Raphson ) applied to 3 altimeters (Jason-2, Jason-3, and SARAL), examining a region of the South Pacific (the SAR box) characterized by low sea surface height spectral variability where unwanted noise (the spectral bump) has previously been identified. The SARAL waveform differs from the others in being beam-limited as well as pulse-limited. The PEACHI Nelder-Mead retracker differs from the others in using a weighted fitting scheme. ALES and MLE3 differ from the others in excluding xisq; ALES also excludes the waveform tail. These different waveforms and approaches to parameter fitting allow us to show how the estimated geophysical parameters are affected by random variability in the pulse-limited and beam-limited footprint areas, factors that need to be understood to best use the various retrackers in specific applications.

We analyze the raw sea surface height anomaly (RSSHA, orbit height minus retracked range minus mean sea surface), SWH, sigma0, xisq, and PPP (pulse peakiness parameter). For each retracker, each possible pairing of analyzed parameters was put through an along-track cross-spectral analysis at the maximum possible sampling rate (20 Hz for the Jason data and 40 Hz for the SARAL data). Auto- and cross-power spectral densities were averaged for several cycles and passes in order to estimate auto-spectral density, magnitude-squared coherence, and cross-spectral admittance.

Auto-spectral densities of all parameters are white at wavelengths equal to and shorter than the pulse-limited footprint diameter. Auto-spectra of PPP and xisq are also white, but at higher power level, at wavelengths longer than 50 km, with transition in between. Auto-spectra of RSSHA, SWH, and sigma0 are red, decaying as wavenumber squared, for long wavelengths, with transition bands often having a spectral bump at intermediate wavelengths. The onset of the non-red and bump-like transition band is at 100 km wavelength for RSSHA, 50 km wavelength for sigma0, and 500–2000 km for SWH. Auto-spectral density levels are different from one retracker to another in the transition (spectral bump) region, differing by a factor of 4 in RSSHA, a factor of 10 in SWH, and a factor of 100 in sigma0.

The cross-spectral analysis exhibits correlations among the parameters as a function of along-track wavelength, and shows that the covariance is a function of along-track wavelength, antenna gain pattern and footprint, and (most importantly): (1) whether or not the waveform tail is fitted (ALES does not); (2) whether or not the xisq is fitted (MLE3 and ALES do not); and (3) whether or not a weighted fit is done (Peachi Nelder-Mead does). Examining the along-track wavelengths where parameter covariance is large exhibits how each retracker contributes to its own sea state bias and its own spectral bump.

Random variance in PPP (speckle noise) couples into xisq if it is estimated, and into sigma0 if it is not (MLE3). Variations in sigma0 are dominated by variations in xisq at wavelengths shorter than 50 km for those retrackers that estimate xisq, with the notable exception of SARAL MLE4 (xisq estimation error is lower at Ka band). Errors in xisq and errors due to PPP co-vary with errors in SWH, sigma0, and RSSHA over various bands.

The sea state bias is quite obviously not a constant percentage of SWH or sigma0, as admittance between these shows a complex behavior with generally increasing covariance as wavenumber increases. SWH:RSSHA ratio grows from about –4% at 1000 km wavelength to about –10% at 10 km wavelength. However, and as expected, weighting the retracker (PEACHI Nelder-Mead) strongly increases this correlation, from less than –1% at 500 km to almost –20% at 10 km.

If the overall goal is simply to find the best RSSHA performance, then ALES is best for wavelengths longer than 10 km, where it has the lowest noise and most un-bump-like spectrum; it and MLE3 also have the lowest correlation with PPP and sigma0. ALES, MLE4, and Nelder-Mead also show the lowest coherence between SWH and RSSHA at wavelengths longer than 10 km.