Ground deformation caused by a moderate earthquake in California. The deformation was measured by interferometric analysis of synthetic aperture radar (InSAR) data, as shown in panels (a) and (e). The deformation field was also simulated by a theoretical model, as shown in panels (b) and (f). The parameters in the model were initially estimated by a trial-and-error process (b) and finally (f) by the General Inversion of Phase Technique (GIPhT) developed by Feigl and Thurber [2009]. The final estimate (right column) fits the data better than does the initial estimate (left column). The top three rows are interference patterns that show the difference in phase between a pair of images spanning the time interval from 24 April 1992 to 18 June 1993. The concentric fringes show that the magnitude 5 Fawnskin earthquake on 4 December 1992 raised the ground surface by as much as 12 cm over an area several kilometers wide.

The panels include (a) observed phase values ; (b) modeled phase values calculated from the initial estimate; (c) initial residual phase values formed by subtracting the initial modeled phase values from the observed phase values; (d) angular deviations for the initial estimate; (e) observed phase values , repeated for convenience; (f) modeled phase values calculated from the final estimate; (g) final residual phase values formed by subtracting the final modeled values from the observed phase values; and (h) angular deviations for the final estimate. In the upper three rows, one colored fringe corresponds to one cycle of phase change, or 28 mm of range change. In the lowermost (fourth) row, the colors denote the angular deviation in phase between 0 and ½ cycle. Coordinates are Universal Transverse Mercator easting and northing in km outside the frames; latitude and longitude in degrees inside. In panel (f), the black rectangle shows the corners of the south-dipping fault patch that slipped during the Fawnskin earthquake at epoch 1992.9 calculated from the final estimate.

Feigl, K. L. and C. H. Thurber (2009). "A method for modelling radar interferograms without phase unwrapping: application to the M 5 Fawnskin, California earthquake of 1992 December 4." Geophysical Journal International 176(2): 491-504. [PDF]

Abstract: Interferometric analysis of synthetic aperture radar images (InSAR) measures the phase shifts between two images acquired at two distinct times. These ambiguous 'wrapped' phase values range from -1/2 to +1/2 cycles. The standard approach interprets the phase values in terms of the change in distance between the ground and the radar instrument by resolving the integer ambiguities in a process known as 'unwrapping'. To avoid unwrapping, we have developed, validated and applied a new method for modelling the wrapped phase data directly. The method defines a cost function in terms of wrapped phase to measure the misfit between the observed and modelled values of phase. By minimizing the cost function with a simulated annealing algorithm, the method estimates parameters in a non-linear model. Since the wrapped phase residuals are compatible with a von Mises distribution, several parametric statistical tests can be used to evaluate the fit of the model to the data. The method, named General Inversion for Phase Technique (GIPhT), can handle noisy, wrapped phase data. Applying GIPhT to two interferograms in the area of Fawnskin, California, we estimate a set of model parameters describing a magnitude 5 aftershock of the 1992 Landers earthquake. The resulting simulation fits the data well. The phase final residuals have a circular mean deviation less than 0.15 cycles per datum. Sampling the final residuals, we find the circular standard deviation of a phase measurement to be approximately 0.2 cycle, corresponding to 6 mm in range.

http://dx.doi.org/10.1111/j.1365-246X.2008.03881.x

Link to KMZ
file corresponding to Figure 2a from Massonnet et al. 1994.

Link to GEOTIF file
corresponding to Figure 2a from Massonnet et al. 1994.

Massonnet,
D., M. Rossi, et al. (1993). "The displacement field of the Landers
earthquake mapped by radar interferometry." Nature 364: 138-142. PDF.

Geodetic data, obtained by ground- or space-based
techniques, can be used to infer the distribution of slip on a fault
that has ruptured in an earthquake. Although most geodetic techniques
require a surveyed network to be in place before the earthquake
satellite images, when collected at regular intervals, can capture
co-seismic displacements without advance knowledge of the earthquakeÕs
location. Synthetic aperture radar (SAR) interferometry, first
introduced in 1974 for topographic mapping can also be used to detect
changes in the ground surface, by removing the signal from the
topography. Here we use SAR interferometry to caputure the movements
produced by the 1992 earthquake in Landers, California. We construct an
interferogram by combining topographic information with SAR images
obtained by the ERS-1 satellite before and after the earthquake. The
observed changes in range from the ground surface to the satellite
agree well with the slip measured in the field, with the displacements
measured by surveying, and with the results of an elastic dislocation
model. As a geodetic tool, the SAR interferogram provides a denser
spatial sampling (100 m per pixel) than surveying methods and a
better precisiοn (~3 cm) than previous space imaging techniques.

Massonnet, D., K. L. Feigl, et al. (1994). "Radar
interferometric mapping of deformation in the year after the Landers
earthquake." Nature 369: 227Ð230. PDF

Although the 1992 Landers, California earthquake
sequence occurred in an area well sampled by geodetic networks, the
postseismic deformation in the months following the earthquake has been
measured at only 15 geodetic stations. Another shortcoming in the
geodetic coverage occurs west of the primary rupture, where the
existing geodetic observations suggest, but cannot resolve, sympathetic
slip on secondary faults. Such measurements, which are needed to place
the Landers earthquake sequence in the context of a recurring seismic
cycle in California, can be obtained with the dense spatial coverage
provided by satellite radar interferometry. Here we present radar maps
of the surface deformation field which reveal features that would
otherwise have been poorly sampled, particularly if the earthquake had
occurred in a less accessible area. We see triggered slip at the level
of several centimeters as far as 100 km from the primary rupture and
can resolve the geodetic signal of at least one small (magnitude ~5)
aftershock. The amount of postseismic deformation following the main
shock is less than a decimeter at the surface and is consistent with an
exponential decay time of several months.

Feigl, K. L. and C. H. Thurber (2009). "A method for
modelling radar interferograms without phase unwrapping: application to
the M 5 Fawnskin, California earthquake of 1992 December 4."
Geophysical Journal International 176(2): 491-504. PDF

Interferometric analysis of synthetic aperture radar
images (InSAR) measures the phase shifts between two images acquired at
two distinct times. These ambiguous 'wrapped' phase values range from
-1/2 to +1/2 cycles. The standard approach interprets the phase values
in terms of the change in distance between the ground and the radar
instrument by resolving the integer ambiguities in a process known as
'unwrapping'. To avoid unwrapping, we have developed, validated and
applied a new method for modelling the wrapped phase data directly. The
method defines a cost function in terms of wrapped phase to measure the
misfit between the observed and modelled values of phase. By minimizing
the cost function with a simulated annealing algorithm, the method
estimates parameters in a non-linear model. Since the wrapped phase
residuals are compatible with a von Mises distribution, several
parametric statistical tests can be used to evaluate the fit of the
model to the data. The method, named General Inversion for Phase
Technique (GIPhT), can handle noisy, wrapped phase data. Applying GIPhT
to two interferograms in the area of Fawnskin, California, we estimate
a set of model parameters describing a magnitude 5 aftershock of the
1992 Landers earthquake. The resulting simulation fits the data well.
The phase final residuals have a circular mean deviation less than 0.15
cycles per datum. Sampling the final residuals, we find the circular
standard deviation of a phase measurement to be approximately 0.2
cycle, corresponding to 6 mm in range.