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Synthetic Aperture Radar Interferometry

IMPLEMENTATION OF INTERFEROMETRIC TOPOGRAPHIC PROCESSING ALGORITHMS

Interferometric topographic processing algorithms generally segregate into two classes:

those designed for single-pass dual-aperture systems, characterized by

accurate attitude and position measurements well suited to automatic processing of the data

those designed for repeat-pass single-aperture systems, characterized by

relatively inaccurate position measurements poorly-suited to automatic processing of the data

These characteristics tend to be a independent of platform venue,

i.e. spaceborne or airborne.

Synthetic Aperture Radar Interferometry

PROCESSOR ARCHITECTURES

Single-pass two aperture implementations tend to segregate further into several classes of processors

Fully automated strip mode processors that carry raw data to an arbitrarily long height strip (e.g. ARPA TOPSAR)

relatively inflexible
usually tailored to problem at hand
optimized for speed and efficiency (minimal disk; much RAM)

Fully automated but scene-oriented processors that produce height maps on a scene by scene basis (NASA AIRSAR; CCRS?)

more flexible; less efficient

Modular scene oriented processors that work well for single pass or repeat pass

very flexible; least efficient

Synthetic Aperture Radar Interferometry

INTERFEROMETRIC PROCESSING FLOW DIAGRAM

Synthetic Aperture Radar Interferometry

PROCESSOR IMPLEMENTATIONS

Modern SAR interferometry implementations are typically

software processors, with the possible exception of front-end data decoding hardware
coded in high level languages such as C or Fortran
parallelized for multiprocessor environments
workstation compatible
protected by institutional patents and profit motives

Real-time on-board implementations will be feasible in the near future

with military, commercial, and planetary applications
sustained processor speed required: roughly 1-4 GFlOp/sec
commercial multiprocessors are approaching these rates

Synthetic Aperture Radar Interferometry

PROCESSING REQUIREMENTS

Synthetic Aperture Radar Interferometry

Motion measurement systems of the NASA DC-8 are commercially available (though expensive) and support automated processing

Six-Gun GPS receiver used to time tag pulse data to 1/2 PRI
Honeywell H764 P-code combined GPS/INU

1 cm/s velocity
3 m position long term
0.001�- 0.005� attitude
motion data time tagged with GPS time

Turbo-Rogue differential GPS data (10 cm accuracy at 1 Hz)

Motion processing entails straightforward transformation of the position and velocity measured in WGS-84 and attitude angles measured in local geoid coordinates (TCN) to the (s,c,h) coordinate system.
Transformations provided in Coordinate Systems module

MOTION PROCESSING

Synthetic Aperture Radar Interferometry

ATTITUDE MOTION PROCESSING

Attitude angles define a transformation taking the fixed interferometric baseline in the platform body-fixed frame, or more precisely the lever arms from the inertial reference to the antenna phase centers, into their orientation with respect to the local (s,c,h) system.

Details of the sign conventions for the rotation angles and axis scaling depends on the specific choice of reference frames
Transformation matrix M is a function of along track position, and is conveniently pre-tabulated against position for use by the interferometric processor

Where D is a diagonal matrix with +1 or =1 on diagonals, and

Y, P and R are Yaw, Pitch, and Roll rotation matrices, respectively

Synthetic Aperture Radar Interferometry

RANGE COMPRESSION AND PRESUMMING

Radar systems typically operate with

constant Pulse Repetition Frequency (PRF) (variable ground sample spacing depending on platform velocity variations)
constant ground sample spacing (variable PRF depending on velocity variations)

To regularize the image coordinates to match the (s,c,h) system, or to simply resample data to some desired reference spacing, variable rate presumming can be invoked
Presumming of pulses most conveniently performed after range compression of pulses. After compression,

data are represented as single-sided spectrum (analytic signal representation), allowing convenient band-pass filtering
scatterer energy is localized in range

Synthetic Aperture Radar Interferometry

PRESUMMING FORMULATION

Synthetic Aperture Radar Interferometry

MOTION COMPENSATION

Motion compensation of radar data is required for accurate airborne interferometers
Uncompensated motion errors lead to image defocusing and mis-registration, reducing interferometric correlation and height acuity
Motion compensation consists of adjusting the range and phase of each image sample to seem as though the imaging positions lie on the reference track(s)

Synthetic Aperture Radar Interferometry

ERRORS RELATED TO MOTION COMPENSATION

If motion compensation is not performed,

Image focusing errors

Common FFT-based focusing algorithms assume platform moves rectilinearly
Without motion compensation, the chirped matched filter assumed in azimuth processing is not matched.

Focussing error reduces the effective number of looks available for processing, and adds sidelobe energy to unweighted processing schemes
No interferometric biases or variance increase if channel processing is identical!

If motion compensation is performed, there are three categories of compensation-induced errors:

Image processing induced phase errors
Unknown terrain errors
Intrinsic measurement errors

Synthetic Aperture Radar Interferometry

MOTION COMPENSATION APPROACHES

Dual reference track approach - range and phase corrections applied to each channel to independent references

Synthetic Aperture Radar Interferometry

MOTION COMPENSATION APPROACHES

Single reference track approach - range and phase corrections applied to each channel to common reference

Synthetic Aperture Radar Interferometry

MOTION COMPENSATION AND
INTERFEROGRAM FLATTENING

The equations describing motion compensation and baselines for interferometry are identical.
The range-dependent phase shift introduced in motion compenstion is nothing more than than an interferometric phase due to a curved earth for a fictitious baseline from the reference track to one antenna
In the single reference track approach, the difference of the range-dependent phase shifts between the two channels is then the true “curved-earth” interferometric phase. Thus, motion compensation to a single track automatically removes the “curved-earth” fringes
This process (nearly linear phase ramp in range) shifts the spectra of the two channels in opposite directions, essentially aligning the common parts of their spectra
Suitable low-pass filtering of the common spectral band will enact the spectral shift filtering for the curved-earth

Synthetic Aperture Radar Interferometry

MOTION COMPENSATION STRATEGY

Patch-oriented motion compensation allows update of reference track(s) to ensure the actual track is nearby

usual method: pick a patch-local track intersecting at least one point on the actual track and parallel to a global track
reduces unknown terrain errors
usually required for airborne systems subject to variable winds
only cost is in local patch-oriented book-keeping

Global motion compensation defines reference track(s) for entire processing run

usually preferred for spaceborne systems, where motion is smooth
often spaceborne “motion compensation” is simply SLC image co-registration

Synthetic Aperture Radar Interferometry

RANGE MIGRATION AND AZIMUTH COMPRESSION

No interferometric issues are peculiar to azimuth processing
In azimuth processing, a choice must be made as to whether or not the pointing will be corrected for squinted imaging
Deskewing generally adds complication to the buffering scheme employed in the processor, but renders the processed imagery in the desired output along track coordinate frame (if the centroid is known precisely)
Skewed processing leaves imagery in natural range-doppler coordinates and allows straightforward height reconstruction (see Height Reconstruction module)
ARPA processor employs squinted processing

Synthetic Aperture Radar Interferometry

INTERFEROGRAM FORMATION AND CORRELATION

In implementations with proper automated motion compensation, interferogram and correlation formation consists of simple point-by -point cross multiplication and look summation

In this example, look summation was performed only in the along track direction. Typically, azimuth SLC resolution is finer than range resolution, so looks taken in azimuth “square” the pixels.

Synthetic Aperture Radar Interferometry

INTERPOLATION FILTERS

Interpolation filters are employed in

presumming
motion compensation
range migration correction
absolute phase determination

Interpolation filters for all of these critical operations must have the following characteristics

bandwidth preserving, especially the three range dependent interpolators

low order polynomials won’t do!

phase and time delay preserving, especially in motion compensation where misalignment amounts to decorrelation

nearest neighbor won’t do!

Interpolator filter design a rich subject

Synthetic Aperture Radar Interferometry

PHASE UNWRAPPING

Elements of the phase unwrapping problem: From the measured, wrapped phase, unwrap the phase from some arbitrary starting location, then determine the proper ambiguity cycle.

Actual phase

Wrapped (measured) phase

Typical unwrapped phase

Synthetic Aperture Radar Interferometry

TWO-DIMENSIONAL PHASE UNWRAPPING

Two dimensional phase field values below are in units of cycles
One-dimensional unwrapping criterion of half-cycle proximity is inconsistent in two dimensions

Residues, marked with + and -, define ambiguous boundaries.

Synthetic Aperture Radar Interferometry

RESIDUES IN PHASE UNWRAPPING

The wrapping operator delivers the true phase modulo 2 , in the interval - <  < .
The true phase gradient is conservative:
The wrapped gradient of the measured, wrapped phase, however, may not be conservative:
When this function is non-conservative, its integration becomes path dependent.
Residues occur at locations of high phase noise and/or phase shear such that the wrapped gradient of the measured, wrapped phase is no longer conservative.

Synthetic Aperture Radar Interferometry

BRANCH CUTS IN PHASE UNWRAPPING

Branch-cut algorithms (Goldstein, Zebker, and Werner 1986) seek to neutralize these regions of inconsistency by connecting residues of opposite solenoidal sense with cuts, across which integration may not take place.
Branch cut connections force path independence in the integration of the wrapped phase gradient.
If done properly, the integrated phase field will be correct. But which is correct?

Synthetic Aperture Radar Interferometry

ILLUSTRATION OF BRANCH CUT ALGORITHM

Connection of simple tree

Synthetic Aperture Radar Interferometry

BRANCH CUT STRATEGY

The standard GZW algorithm is designed to connect residues into a neutral network into the shortest possible connection tree, i.e. to minimize the length of the individual branch cuts comprising a tree
This will not necessarily create the shortest possible tree, since GZW makes many unnecessary connections in its search for neutrality
Various criteria have been devised to place guiding centers (unsigned residues) along expected paths to facilitate the right choice in branch cut connection

Residues

Guiding Centers

Synthetic Aperture Radar Interferometry

GUIDING CENTER CRITERIA

A number of criteria have been devised for selecting guiding centers, each more or less tailored to characteristics of SAR data:

when phase slope exceeds threshold (implies layover)
when derivative of phase slope exceeds threshold
when radar brightness exceeds threshold (implies layover)
when decorrelation estimator exceeds threshold (implies noise and/or layover

Some guiding center selections help in some cases
Difficult to assess performance in a qauntitative way

Synthetic Aperture Radar Interferometry

LEAST SQUARES PHASE UNWRAPPING

In recent years, linear least squares algorithms (Ghighlia et al, 1994) have become popular
Numerous challenges to primacy of branch cut algorithms have been made, because branch cut algorithms are not robust in noise and performance is difficult to assess.
Premise of least squares algorithm:

Model the non-conservative component of as noise term in a least squares problem:

Problem can expressed in terms of Poisson’s equation which lends

itself to fast numerical solution (when unweighted)

Synthetic Aperture Radar Interferometry

LEAST SQUARES PHASE UNWRAPPING

Numerous researchers have been finding significant biasing of the phase estimates from the least squares method.

Slopes tend to be underestimated
Large-scale warping of the phase field

Cause of distortion (attributed to Bamler et al, 1996 preprint)

Least square condition of zero-mean uncorrelated gaussian noise is violated.
Error function is signal dependent, and a nonlinear function of the original noise corrupting the signal. Mean of error function is related to the signal, biasing result.

Recent work by Zebker et al is showing that least squares works well when combined with branch cuts, as it must.
Other algorithms exist but are generally variants on branch cuts or least squares

Synthetic Aperture Radar Interferometry

PHASE UNWRAPPING IMPLEMENTATION

In patch oriented processing schemes, it is convenient to “bootstrap” the unwrapped phase from one patch to the next
Occasionally, when unwrapping is difficult, “bootstrapping” can be essential to success

Synthetic Aperture Radar Interferometry

ABSOLUTE PHASE DETERMINATION

Actual phase

Wrapped (measured) phase

Typical unwrapped phase

There are at least three methods for determining the absolute phase ambiguity

Synthetic Aperture Radar Interferometry

ABSOLUTE PHASE DETERMINATION METHODS

Ground control reference points can be used to determine the absolute phase ambiguity

Synthetic Aperture Radar Interferometry

ABSOLUTE PHASE DETERMINATION METHODS

The spectral method of absolute phase determination (Madsen et al., 1992) divides the imagery into two range sub-bands, forming one lower resolution interferogram in each sub-band
The difference of sub-banded interferograms gives a new interferogram with an effective fringe frequency scaled down from the original full resolution interferogram by
The fringe frequency is so low that unwrapping is not necessary
This phase scaled up by the inverse frequency ratio is very noisy
The difference between this phase and the unwrapped phase, averaged over a large area to reduce noise, gives the ambiguity

Synthetic Aperture Radar Interferometry

ABSOLUTE PHASE DETERMINATION METHODS

The range-correlation method of absolute phase detrminion (Madsen, 1995) exploits spatial domain misregistration
Interferometric phase is a record of pixel mis-registration:

Topographic phase also contributes to small variable mis-registration
Unwrapped phase can be used to remove the topographic mis-registration by resampling one image to match the other through above prescription
If unwrapped phase is on the correct ambiguity, then images will align precisely - no mis-registration will be detected in cross-correlation of imagery
If ambiguity is incorrect, cross-correlation will measure a constant offset throughout the image. Average the estimates over the image yields the scaled ambiguity estimate.

Synthetic Aperture Radar Interferometry

ABSOLUTE PHASE METHOD ASSESSMENT

The performance of all methods is highly dependent on the system performance characteristics
Both spectral and correlation methods rely on averaging of noisy data
Estimator performance characteristics can be modeled and a system designed to meet the required specification
Ground control point method requires access to DEM or ground control wherever the data are taken
All methods require system stability to a few degrees of phase

Synthetic Aperture Radar Interferometry

REGRIDDING IN GEOLOCATION

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Interpolator

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Position vectors are not uniformly

distributed in the plane following

height reconstruction process.

Use some interpolator to resample to

a uniform grid.

Synthetic Aperture Radar Interferometry

REGRIDDING INTERPOLATORS IN GEOLOCATION

Interpolating noisy, irregularly spaced data to a uniform grid is a difficult problem.
Several interpolation algorithms have been implemented

Aikima interpolator - triangulates the surface. Fits a 5th degree polynomial over each triangle, forcing continuity of the first and second partials along all triangle edges. Method is sens-itive to noise and requires a great deal of memory and time.
Nearest neighbor - Fast and easy, but shows some artifacts in shaded relief images. Potential exists for biasing height estimates when data are noisy but regularly laid out on output grid.

Synthetic Aperture Radar Interferometry

MORE REGRIDDING INTERPOLATORS

Another implementation:

First or second order surface fitting - Uses the height data centered in a box about a given point and fits a surface by weighted least squares.

Fit weighted by distance from point and height noise (as determined from the interferometric correlation).
Box size determined by local topography: small height variations large box - large variations use small box.
Another benefit to the surface fitting method is slope and/or surface curvature maps are readily available from the fit coefficients directly using the unresampled data.
Efficient implementation achieved by storing the position vectors in an array structured a(3,M,N) where the location in the array is determined using a nearest neighbor algorithm.

Synthetic Aperture Radar Interferometry

SURFACE FITTING BENEFITS

Slope and Curvature Images of Galapagos Islands

Synthetic Aperture Radar Interferometry

REPEAT-PASS SINGLE-APERTURE INTERFEROMETER IMPLEMENTATIONS

Repeat pass implementations are distinguished by insufficient position accuracy to allow automated processing

For spaceborne platforms, orbits traditionally have been determined to no better than tens to hundreds of meters. In this case, the data themselves must be used to determine the interferometric baseline
Modern spaceborne systems such as ERS, SIR-C are producing reconstructed orbit solutions that approach meter -level accuracy or better. Some automation, through interferogram formation at least, is possible when the orbit accuracy is sub-pixel.
Airborne systems can achieve cm-level track accuracy if equipped with differential kinematic GPS processing. However, geometry usually requires mm level accuracy for accurate topographic mapping.

Synthetic Aperture Radar Interferometry

REPEAT-PASS PROCESSOR ARCHITECTURES

Repeat pass interferometric processors are typically “scene-” or “frame-based”, working from data processed to single look complex (SLC) imagery at the agency processing facilities, or from raw data processed locally. Interferometric processing could be done in GIS-based software or with an image processing tool suite, though perhaps not efficiently.
With accurate interferometric baseline knowledge (or equivalently the SLC coregistration offset field), it is possible to process from raw signal samples to as far down the chain as accuracy and data quality permit

Arbitrarily long swaths can be processed efficiently through the interferogram level.
Further processing usually an iterative process

Synthetic Aperture Radar Interferometry

FLEXIBLE REPEAT PASS INTERFEROMETRY

Procedure to pre-determine

offsets: process first and last

piece of data, then interpolate

offsets between

Synthetic Aperture Radar Interferometry

SCENE MATCHING AND COREGISTRATION

Cross-correlate images by expedient method

Synthetic Aperture Radar Interferometry

There are a plethora of algorithms in existence for automatic scene matching, each with particular strengths and weaknesses
SAR scenic matching for interferometry applications is difficult for several reasons:

Thermal noise causes the fine structure of two SAR images to differ, adding noise to the correlation measurements.
Geometric speckle noise is similar in images acquired in nearly the same geometry, but as the interferometric baseline increases, differing speckle noise corrupts the matching correlation.
Scene decorrelation is another form of speckle noise difference that corrupts matching correlation.

SAR scenic matching for mosaicking applications involves greater challenges, including severe speckle noise differences, layover and shadow effects.

SCENE MATCHING ON IMAGERY

Synthetic Aperture Radar Interferometry

AUTOMATIC SCENE MATCHING

Find overlap region and sample points at specified spacing in along track and cross track direction.
Typical window sizes are 64x64 pixels for image data and 64x64 or 128x128 pixels for height data.
Uses a modified Frankot's method for rejecting bad matches and to provide an estimate of the match covariance matrix.
Cross correlation uses a normalized mean correlation function, mean of search window is calculated over the region in common with the reference window.

Search Window

Mean of reference

is constant

Mean of search

varies with position

Synthetic Aperture Radar Interferometry

MATCH CORRELATION ESTIMATE

Correlation is computed in the spatial domain using a normalized cross correlation algorithm.
Let be the image values at a point in the reference window in the first data set, , the mean of the intensities in that window.
Let be the image value at point in the search window of the second data set, and the mean of the intensities in that window.
Viewing each image as a vector in an n-dimensional vector space then the cross correlation is computed as

where and are the standard deviations of the image

intensities in data sets 1 and 2, respectively.

Space-domain correlation used for speed and because large

irregular data gaps are trouble free.

Synthetic Aperture Radar Interferometry

BASELINE ESTIMATION FROM OFFSETS

Baseline estimation in repeat pass interferometry is a rich subject, and will be covered completely in a separate module. Briefly:
Offset field determined from scene matching carries information sufficient to reconstruct the baseline to an accuracy com-mensurate with ability to co-register images.
For nearly parallel, smoothly varying, orbit tracks, the baseline can be modeled as a simple function of the range

Solve two equations for two unknowns:
Evaluating baseline at several widely-spaced along-track (azimuth) locations gives azimuth history of the baseline.

Synthetic Aperture Radar Interferometry

BASELINE ESTIMATION FROM OFFSETS

For orbit tracks with substantial convergence, azimuth offsets strongly characterize convergence rates.

Observation vector:

where  is an image scaling factor resulting from velocity differences

between the tracks, and (s,c,h) are local coordinates defined by the

orbit track (see notes on coordinate systems).

Parameter vector:

Synthetic Aperture Radar Interferometry

BASELINE ACCURACY FROM OFFSETS

Baselines determined from offset fields are typically accurate to a fraction of the scene matching accuracy that depends on the model function and the number of offset estimates. Example:

Zero baseline image pair with 10 match points across range.
Scene matching accuracy of 1/20 pixel resolution at typical single look resolution of 15 meters: 75 cm / 25 cm

This accuracy is insufficient for topographic mapping applications

These geometric parameters are typical of ERS with baselines

suitable for topography. Generally, ground control points tied to

the unwrapped interferometric phase are required for mapping.

Synthetic Aperture Radar Interferometry

INTERFEROGRAM FORMATION

From interferogram and

detected imagery, corr-

lation can be formed

properly at any resolution:

i.e. average of correlation

is not correlation of average

Doppler spectra

matched during

processing

Synthetic Aperture Radar Interferometry

REGISTRATION IMPLEMENTATION

In resampling single look complex image to register properly with the reference image, care must be taken in interpolation of complex data

The azimuth spectrum of squinted SAR data is centered at the Doppler centroid frequency - a band-pass signal
Simple interpolators, such as linear or quadratic interpolators, are low-pass filters and can destroy band-pass data characteristics

Band-pass interpolators or spectral methods preserve phase fidelity

Synthetic Aperture Radar Interferometry

INTERFEROGRAM RANGE OVERSAMPLING

Several references recommend oversampling single look complex data in range before forming the interferogram, arguing

each SLC is a full bandwidth signal
product of two signals forms a new signal whose spectrum is the convolution of individual spectra, hence has spectral content beyond the Nyquist sampling frequency in range.

In practice, oversampling is unnecessary because

there is no appreciable improvement in performance
each SLC usually utilizes less than 80% of the bandwidth
spectral shift filtering reduces the bandwidth further
noise from aliased energy usually inconsequential relative to decorrelation noise

Synthetic Aperture Radar Interferometry

REPEAT PASS INTERFEROMETRY BACK-END PROCESSING

Synthetic Aperture Radar Interferometry

PHASE UNWRAPPING FOR REPEAT-PASS TOPOGRAPHY APPLICATIONS

Unlike single-pass dual-aperture systems, phase unwrapping in repeat pass interferometry is a critical unsolved problem
C-band data suffer from temporal decorrelation over all repeat time scales, destroying interferometric fringe quality in vegetated terrain environments
L-band data suffer from temporal decorrelation over long repeat time scales, and over heavy vegetation (JERS also suffers from poor SNR)
Phase uwrapping challenges include methods of robustly masking regions of poor fringe quality and decoupling them from areas of phase shear near layover
The promise of repeat pass interferometry is in differential techniques for surface deformation studies, not in topographic mapping applications. There, strategies exist for reducing the reliance on accurate phase unwrapping

UCLA SAR Interferometry

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