# Spectral Windows for Satellite Altimetry

**Event: **2016 Ocean Surface Topography Science Team Meeting

**Session: **Instrument Processing: Measurement and retracking (SAR and LRM)

**Presentation type: **Type Poster

A satellite altimeter’s waveform is a power spectral density (PSD) estimate obtained from discrete Fourier transforms (DFTs) applied to time series produced by the instrument’s analog-to-digital converter (ADC). All altimeters apply a DFT to the digital sample sequence obtained from each pulse echo to synthesize pulse compression; the PSD of the echo displays the variation in backscattered power as a function of range. Delay/Doppler or multi-looked SAR (D/D-SAR) altimetry also applies a DFT across a sequence of echoes; the PSD in this dimension relates backscattered power to the along-track position of scatterers.

PSD estimation from DFTs of discrete sequences inevitably suffer from leakage, except in the special case that the sequence contains only noise-free signals at only those discrete frequencies sampled by the DFT. Altimeter waveforms inevitably have leakage because they contain noise and signal power at a continuum of frequencies, since the Earth surface returns power at a continuum of ranges and along-track positions. Leakage may be mitigated, but not eliminated, by employing a spectral window. For altimeters, windows may be applied to either the range or along-track dimensions, or both, when one has access to the raw ADC output, as in the CryoSat FBR data product.

This presentation examines both classical windows (such as the Hamming window used in the along-track dimension of the CryoSat D/D-SAR processor) and newer windows obtained from the eigenvectors of a discrete PSD estimate optimization problem (such as the DPSS, or Slepian, and other windows). The window literature usually emphasizes reducing the window side lobes to quite low levels (e.g. less than -40 dB for the Hamming window) but this comes at a cost: the main lobe is considerably (about 30%) wider, degrading the resolution (e.g. the Hamming window widens the CryoSat along-track resolution from 300 to 400 m). As the signal-to-noise ratio of altimeters operating over heterogeneous scenes is seldom much higher than 20 dB, one can design a window for altimeters that gives excellent side lobe suppression while widening the main lobe only a few percent. This is the approach I recommend.

PSD estimation from DFTs of discrete sequences inevitably suffer from leakage, except in the special case that the sequence contains only noise-free signals at only those discrete frequencies sampled by the DFT. Altimeter waveforms inevitably have leakage because they contain noise and signal power at a continuum of frequencies, since the Earth surface returns power at a continuum of ranges and along-track positions. Leakage may be mitigated, but not eliminated, by employing a spectral window. For altimeters, windows may be applied to either the range or along-track dimensions, or both, when one has access to the raw ADC output, as in the CryoSat FBR data product.

This presentation examines both classical windows (such as the Hamming window used in the along-track dimension of the CryoSat D/D-SAR processor) and newer windows obtained from the eigenvectors of a discrete PSD estimate optimization problem (such as the DPSS, or Slepian, and other windows). The window literature usually emphasizes reducing the window side lobes to quite low levels (e.g. less than -40 dB for the Hamming window) but this comes at a cost: the main lobe is considerably (about 30%) wider, degrading the resolution (e.g. the Hamming window widens the CryoSat along-track resolution from 300 to 400 m). As the signal-to-noise ratio of altimeters operating over heterogeneous scenes is seldom much higher than 20 dB, one can design a window for altimeters that gives excellent side lobe suppression while widening the main lobe only a few percent. This is the approach I recommend.