Subglacial Geodesy Using GPS and Ground-Penetrating Radar
by William Menke and John Labreque
19 July 1999
Just as in the non-glaciated parts of the earth, geological processes cause the rock beneath the world's major ice sheets to deform. Faulting, folding, thermal subsidence, magma migration and isostatic rebound can all cause horizontal motions and elevation changes in the surface of the rock. However, beneath glaciers the motion of the rock subsurface is obscured by the presence of the glacial ice, whose thickness changes due to glacial flow (which can thicken or thin the ice sheet as a whole), and by snowfall and ablation (which add or subtract material from its upper surface).
Standard geodetic positioning techniques, including those that employ the Global Positioning System (GPS), can only measure the position of the ice surface, and therefore cannot provide unabiguous evidence of the motion of the rock below. On the other hand, ground-penetrating radar observations, made both from airplanes and on the surface, have proved a very useful tool in providing static images of both the internal structure of the ice sheet and the surface of the rock underlying it. In particular, the thickness of the ice sheet can be determined from the traveltime of the radar reflection from the rock subsurface. We therefore ask whether a combination of GPS and ground-penetrating radar could provide geodetic-quality positioning of the rock subsurface. Repeated measurements, made at intervals of months or years, could then be used to detect tectonic deformation, just as it done in the unglaciated parts of the world.
The senario that we envision is a set of GPS-enhanced ground-penetrating radar antennae. The purpose of the GPS is two-fold: First, it provides 3D positioning of the antenna relative to a worldwide GPS coordinate system to an accuracy of a few cm; Second, it provides absolute timing to sub-nanosecond accuracy. This timing capability is essential when several widely-spaced antennae are deployed, since it permits the accurate measurement of the traveltime of radar waves propagating from one antennae to another. GPS positioning requires the stationary deployment of the antennae at a single site for period of between a few hours or a few days. Thus the radar need not be as high-powered as an airbourne unit (which is constantly on the move). It could transmit at a much lower power, with the received signals being stacked at the receiver in order to improve signal-to-noise ratio. The accurate timing provided by GPS ensures that such stacking can ber performed, even over long periods of time. Furthermore, the radar would not need to transmit actual pulses, but instead could transmit continuous, low-power "chirped" or "random telegraph" signals that could be post-processed into a "pulse" by standard signal processing techniques.
In the discussion that follows, we will assume that the radar has a carrier frequency of 100 MHz (wavelength of 1.68 m in ice), a bandwidth of 10 MHz and is digitized at a rate of 1 sample per ns. The effective pulse is about 100 ns long (i.e. about ten oscillations of the carrier, or 16.8 m in ice). The time needed to make a single radar "observation" (meaning a radargram long enough to contain a reflection from the rock subsurface) is governed by the two-way traveltime through the ice-sheet, and is of the order of 105 ns. Thus about 109 radar "observations" can be made per day, allowing a noise reduction of about 30000 relative to the signal-to-noise ratio of a single observation.
The underlying principle in the measurement of deformation is that the traveltime of a pulse reflected from the rock subsurface (i.e. the "basal reflector") is related to the position of that surface, and that changes in that traveltime (between repeated observations made months or years apart) can be used to infer changes in its position. Geodesy thus requires the precise relative timing of pairs of reflectors. Indeed, relative timing with an accuracy of about 0.1 ns is needed to achieve a positioning accuracy of 2 cm. Computer simulations using standard signal-correlation techniques that we have performed (see below) indicate that a signal-to-noise ratio of 10:1 is needed to achieve this accuracy. Thus the signal-to-noise ratio of a single observation could be conceivably be as poor as 1:3000 (given 1 day of observation time).
Two basic geometries of observation are important to the geodetic problem: the vertical-incidence reflection geometry, where the transmitter and receiver are co-located and the radar signal reflects from the rock subsurface at more-or-less normal incidence from a point more-or-less directly below them; and the wide-angle angle reflection geometry, where the transmitter and receiver are separated by several (up to 10) km, and where the radar signal reflects at oblique incidence from a point more-or-less midway between them. These two measurements geometries are complimentary: At vertical-incidence, the radar pulse rapidly tranverses the firn (the high-porosity, high-velocity upper 30-100 m of the ice sheet), but at wide-angle it spends a disproportionate amount of its time in the firn. We first discuss how geodetic measurements can be made from vertical-incidence measurement.
The vertical-incidence radargram contains an initial portion containing reverberations from the firn, and a later portion containing the basal reflector. As long as the basal reflection is changed bewteen two repeat measurements only by changes in the ice above it, the differential traveltime can be measured using standard signal-correlation techniques. (Errors can arise using this technique if the change arises due to a change in the reflecting interface itself, such as might occur when a till layer thickens). The differential traveltime must then be converted to a change in position of the basal reflector, which requires calulating any changes that occured in the velocity structure of the ice. Fortunately, the initial portion of the radargram contains sufficient information to allow perturbations in the velocity structure of the firn (the region where the largest velocity changes can be expected to occur) to be determined.
We have examined two types of firn perturbations: In the first case, the top surface of the ice remains in a fixed position, but the velocity is perturbed by an amount that exponentially decays with depth (with a scale length of a few tens of meter). In the second case, a new layer of firn is added to the top surface. (Note that these two cases can be easlily distinguished, because GPS provides absolute location of the radar antennae, which is presumably resting on the ice). In both cases, the initial part of the radargram contains sufficient information to allow the perturbation to be determined (at least for models that contain only a few adjustable parameters). As long as the sigal-to-noise ratio of the radargram is high (>10:1), the errors in determining changes in firn structure lead to errors in the vertical position of the basal reflector of less than 2-3 cm.
In an actual experiment, we would envision an array of transmitter/receiver stations (either acual separate instruments, or the same instrument moved to different positions on successive days), with a spacing of a 50-100 m between them (the Frenel zone for the basal reflector is about 85 m in diameter). Measurements from each of these stations could be averaged to improve the sinal-to-noise ratio of estimates of the vertical position of the basal reflector. Changes in the horizontal position of the rock subsurface are more difficult to determine, and would require identifying some recognizable "mark" on the surface that leads to a diffraction with a measurable traveltime.
Wide-angle measurements, especially when conducted in a "common depth point" mode (meaning that both transmitter and receiver are moved, so that all the different offsets measured have the same reflection point on the basal surface). The differential traveltime curve for the basal reflector, as a function of transmitter-receiver offset, contains sufficient information that changes in basal reflector depth can be uniquely separated from perturbations in firn structure. Thus wide-angle measurements provide data that significantly complement and constrain position estimates made with the the vertical-incidence geometery. A combined experiment that included both geometries over a 10-20 km long profile (with 100 m station spacing) across the ice would provide sufficient kinds of both kinds of data.
Other Interesting Ideas
This geodetic technique could also be used to track changes in internal reflectors in the ice, such as might be produced by layers of dust or volcanic ash. Such measurements would provide information of the internal deformationof the glacier as it flows.
A collegue, Vadim Levin, pointed out to us that acoustic measurements could usefully complement the radar data. A 1 kHz acoustic signal would have a wavelength similar to the 100 MHz radar waves, and could be processed in much the same way. However, as the acoustic velocity increases strongly through the firn (in contrast to the electromagnetic velocity, which decreases), acoustic measurements would provide complementary data that constrained changes in firn structure. A standard marine "pinger" would probably work fine as a transmitter/receiver.
It is possible that flowing glacial ice might exhibit diaelectric anisotropy, owing to lattice-perferred orientation of the ice crystal or alligned, elongated airbubbles or cracks. Radar birefringence measurements, made by transmitting with one linear polarization and receiving at the orthogonal one, can easily detect such birefringence. Temporal changes in it might be diagnostic of changing flow or stress patten in the ice, which might aid in the interpretation of the geodetic results (besides being interesting in their own right).
Technial Discussion and Supporting Calculations