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#### A simple algorithm for the direct extraction of the two-dimensional surface image spectrum from the return signal of a synthetic aperture radar

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##### Citation

Hasselmann, K. F. (1980). A simple algorithm for the direct extraction of the two-dimensional
surface image spectrum from the return signal of a synthetic aperture radar.* International Journal
of Remote Sensing,* *1*, 219-240. doi:10.1080/01431168008948234.

Cite as: http://hdl.handle.net/21.11116/0000-0008-845C-C

##### Abstract

A simple method is derived for 1the determination of the two-dimensional surface image spectrum from the return signal of a synthetic aperture radar (SAR) without explicitly producing an image. The algorithm is similar in structure to a two-dimensional Fourier transformation, but the transformed function also depends explicitly on the wavenumber components. The function consists of the quadratic product of the complex return signal amplitude and the time-delayed complex conjugate return signal, the time delays with respect to range and azimuth time being proportional to the corresponding wavenumber components in these directions. The algorithm appears to be sufficiently simple to be implemented in real time aboard a satellite. Because of the considerable data compression achieved in reducing the original image to its statistical variance spectrum, this opens the possibility of obtaining global surface-wave spectral data from satellites without the excessive costs of sophisticated telemetry and a ground station network required for real-time line-of-sight transmission of the unprocessed signal data. The contracted signal-image-Fourier-transform (SIFT) algorithm may be interpreted as the application of the SAR as a continuous, two-dimensional Δκ (dual frequency) scatterometer. The difference frequencies arise through the multiplication of the chirped return signal with the time-lagged (i.e. frequency shifted) complex return signal. Since the return signal is chirped with respect to both range time and azimuth time, the introduction of two time lags corresponds to a two-dimensional Δκ modulation. The standard dual frequency scatterometer yields a modulation wave only in the radar propagation direction, but the basic principle of producing a beat wave by multiplying two chirped signals which are displaced in time relative to each other can be applied also to the azimuthal Doppler chirp of a broad beam dual frequency scatterometer. The generalized two-dimensional Δκ-scatterometer obtained in this manner differs from a SAR-SIFT processor only in the manner in which the spectrum of difference frequencies is generated: in a Δκ-scatterometer, the duration of the difference frequency sweep is normally large compared with the signal travel time, whereas in a SAR, the entire spectrum of difference frequencies is generated within a single pulse in a time shorter than the average signal travel time. The relative operational advantages or disadvantages of the two methods of obtaining microwave surface image spectra will need to be clarified in a more detailed technical analysis.