There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Synthetic aperture radar interferometry is an imaging technique for measuring the topography of a surface, its changes over time, and other changes in the detailed characteristic of the surface. By exploiting the phase of the coherent radar signal, interferometry has transformed radar remote sensing from a largely interpretive science to a quantitative tool, with applications in cartography, geodesy, land cover characterization, and natural hazards. This paper reviews the techniques of interferometry, systems and limitations, and applications in a rapidly growing area of science and engineering.

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Synthetic aperture radar interferometry

The use of SAR interferometry is often impeded by decorrelation from thermal noise, temporal change, and baseline geometry. Power spectra of interferograms are typically the sum of a narrow-band component combined with broad-band noise. We describe a new adaptive filtering algorithm that dramatically lowers phase noise, improving both measurement accuracy and phase unwrapping, while demonstrating graceful degradation in regions of pure noise. The performance of the filter is demonstrated with SAR data from the ERS satellites over the Jakobshavns glacier of Greenland.

https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/1998GL900033

Radar interferogram filtering for geophysical applications

Synthetic aperture radar (SAR) is a coherent active microwave imaging method. In remote sensing it is used for mapping the scattering properties of the Earth's surface in the respective wavelength domain. Many physical and geometric parameters of the imaged scene contribute to the grey value of a SAR image pixel. Scene inversion suffers from this high ambiguity and requires SAR data taken at different wavelength, polarization, time, incidence angle, etc. Interferometric SAR (InSAR) exploits the phase differences of at least two complex-valued SAR images acquired from different orbit positions and/or at different times. The information derived from these interferometric data sets can be used to measure several geophysical quantities, such as topography, deformations (volcanoes, earthquakes, ice fields), glacier flows, ocean currents, vegetation properties, etc. This paper reviews the technology and the signal theoretical aspects of InSAR. Emphasis is given to mathematical imaging models and the statistical properties of the involved quantities. Coherence is shown to be a useful concept for system description and for interferogram quality assessment. As a key step in InSAR signal processing two-dimensional phase unwrapping is discussed in detail. Several interferometric configurations are described and illustrated by real-world examples. A compilation of past, current and future InSAR systems concludes the paper.

https://iopscience.iop.org/article/10.1088/0266-5611/14/4/001/pdf

TanDEM-X (TerraSAR-X add-on for digital elevation measurements) is an innovative spaceborne radar interferometer that is based on two TerraSAR-X radar satellites flying in close formation. The primary objective of the TanDEM-X mission is the generation of a consistent global digital elevation model (DEM) with an unprecedented accuracy, which is equaling or surpassing the HRTI-3 specification. Beyond that, TanDEM-X provides a highly reconfigurable platform for the demonstration of new radar imaging techniques and applications. This paper gives a detailed overview of the TanDEM-X mission concept which is based on the systematic combination of several innovative technologies. The key elements are the bistatic data acquisition employing an innovative phase synchronization link, a novel satellite formation flying concept allowing for the collection of bistatic data with short along-track baselines, as well as the use of new interferometric modes for system verification and DEM calibration. The interferometric performance is analyzed in detail, taking into account the peculiarities of the bistatic operation. Based on this analysis, an optimized DEM data acquisition plan is derived which employs the combination of multiple data takes with different baselines. Finally, a collection of instructive examples illustrates the capabilities of TanDEM-X for the development and demonstration of new remote sensing applications.

/pdf/tandem-x-a-satellite-formation-for-high-resolution-sar-z54xhync6a.pdf

TanDEM-X: A Satellite Formation for High-Resolution SAR Interferometry

Polarimetric and interferometric SAR data are frequently multilook processed for speckle reduction and data compression. The statistical characteristics of multilook data are quite different from those of single-look data. The authors investigate the statistics of their intensity and phase. Probability density function (PDF's) of the multilook phase difference, magnitude of complex product, and intensity and amplitude ratios between two components of the scattering matrix are derived, and expressed in closed forms. The PDF's depend on the complex correlation coefficient and the number of looks. Comparisons of these theoretically derived PDF's are made to measurements from NASA/JPL AIRSAR data. The results of this paper can be applied to feature classification using polarimetric SAR and to the estimation of decorrelation effects of the interferometric SAR. >

Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery

There have long been threads of investigation into covert channels, and threads of investigation into anonymity, but these two closely related areas of information hiding have not been directly associated. This paper represents an initial inquiry into the relationship between covert channel capacity and anonymity, and poses more questions than it answers. Even this preliminary work has proven difficult, but in this investigation lies the hope of a deeper understanding of the nature of both areas. MIXes have been used for anonymity, where the concern is shielding the identity of the sender or the receiver of a message, or both. In contrast to traffic analysis prevention methods which conceal larger traffic patterns, we are concerned with how much information a sender to a MIX can leak to an eavesdropping outsider, despite the concealment efforts of MIXes acting as firewalls.

/pdf/covert-channels-and-anonymizing-networks-4jt01iavxl.pdf

Covert channels and anonymizing networks

The phase difference and the product of two complex Gaussian signals are important parameters in the study of interferometry and polarimetry. To reduce statistical variations, polarimetric and inte...

Statistics of phase difference and product magnitude of multi-look processed Gaussian signals

We discuss the different ways of defining channel capacity for certain types of illicit communication channels. We also correct some errors from the literature, offer new proofs of some historical results, and give bounds for channel capacity. Special function techniques are employed to express the results in closed form. We are interested in a specific type of covert channel, a timing channel. A timing channel exists if it is possible for High to interfere with the system response time to an input by Low. Therefore, a timing channel is a communication channel where the output alphabet is constructed from different time values. However, the thrust of the paper is the analysis of timing channels that are discrete, memoryless, and noiseless. We call such a timing channel a simple timing channel (STC). >

Simple timing channels

The effect of noise on a simple covert timing channel is investigated. Shannon's information theory is used to quantify the resulting information flow across the channel. In particular, how a probabilistic response time to a query by the receiver affects the mutual information and channel capacity is studied. The channel capacity is expressed in terms of the critical probability for the mutual information function which is given in closed form in terms of Wright's hypergeometric function. >

/pdf/the-channel-capacity-of-a-certain-noisy-timing-channel-tn2jm9pakm.pdf

Allen R. Miller

Papers

Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery

Covert channels and anonymizing networks

Statistics of phase difference and product magnitude of multi-look processed Gaussian signals

Simple timing channels

The channel capacity of a certain noisy timing channel