Multiple SAR interferograms are used to solve for the full 3-dimensional displacement of the surface of the Henrietta Nesmith Glacier in northern Ellesmere Island Canada. The approach exploits the incidence angle flexibility of different ascending and descending beam modes, and the azimuth angle diversity which occurs when different incidence angles are used at high latitudes. Line-of-sight displacements are estimated in 3 different orientations from the terrain corrected unwrapped differential phase and used to solve for the displacements in a local East, North, Up frame. Uncertainty in the absolute line-of-sight displacements limits the absolute accuracy for vertical displacement, however the relative solution coupled with knowledge of the surface slope allows spaceborne mapping of relative glacier thickening or thinning with unprecedented spatial resolution and with an accuracy of order a millimeter per day.
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 Remote sensing has shown that the speed of glaciers on the large polar ice sheets can change on a time scale much faster than that anticipated prior to the widespread use of satellite data [e.g., Rignot and Kanagaratnam, 2006; Joughin et al., 2010]. And, as documented in the latest IPCC report (http://www.ipcc.ch/ipccreports/ar4-wg1.htm), this limited understanding of ice dynamics and how it may change in a warming climate precludes a good forecast of the glacial ice contribution to future sea level rise.
 Satellite repeat-pass interferometry and speckle tracking are now standard techniques for measuring ice movement and, coupled with ice thickness, ice flux [e.g., Joughin et al., 1996; Gray et al., 1998; Michel and Rignot, 1999; Joughin, 2002]. The problem is that normally at most 2 displacements are measured and the solution is usually achieved by assuming that the vector describing the ice movement runs parallel to the ice surface [Mattar et al., 1998; Joughin et al., 1996; Mohr et al., 1998]. This approach essentially assumes that the ice flow is in equilibrium with the gravitational driving stress and precludes measuring any subtle changes in surface slope and elevation which would accompany changing basal drag or significant changes in subglacial water distribution.
 Results will be presented from an experiment in which RADARSAT-2 InSAR pairs are combined to give a unique 3-dimensional, East, North, Up (ENU) solution for the movement of part of the surface of the Henrietta Nesmith Glacier in northern Ellesmere Island, Canada. This polythermal, land terminating glacier was part of a study of Arctic glaciers [Short and Gray, 2005] and selected because the motion is such that interferometric phase could be unwrapped along the central portion of the 50 km long glacier. While prior work [Gray et al., 2005] combined 2 line-of-sight displacements with azimuth shift from speckle tracking to solve for the vertical displacement, the current work is more accurate and reports the first combination of three temporally- and spatially-overlapping interferograms to create a full 3-dimensional picture of surface movement. It is particularly important in glaciology to be able to detect subtle vertical movement when there is also significant translational movement.
2. Data and Processing Overview
 The RADARSAT-2 fine mode data used for the current work was collected in the early spring of both 2009 and 2010. Figure 1 shows the position of the Henrietta Nesmith Glacier in northern Ellesmere Island. The multiple right-looking acquisitions for each mode occurred every 24 day repeat cycle and are listed in Table 1.
Table 1. Fine Mode RADARSAT 2 Acquisitions During 4 Periods in the Early Spring of 2009 and 2010a
Mode and Year
The two columns of numbers on the right of the table are respectively the nominal glacier incidence angle and azimuth angle with respect to North.
F23 mode ascending 2009
F23 mode descending 2009
F6N mode descending 2009
F6F mode ascending 2010
F6F mode descending 2010
F22N mode ascending 2010
F22N mode descending 2010
 Three temporally- and spatially-overlapping interferograms are required to solve for the 3-dimensional motion and the quality of the solution depends on the difference in the geometry of the 3 data acquisitions [Wright et al., 2004]. As the three interferograms will be offset in time we require that there be a small change in any component of surface velocity over the few days that may be required to acquire the three different geometries. By using incidence angles at the extremes of the fine mode incidence angle range the azimuth angle will also be different for either ascending or descending passes. For example, a difference in azimuth angle of ∼20° can be obtained by using two fine mode acquisitions with incidence angles of ∼33° and ∼49° at the latitude of the Henrietta Nesmith Glacier. The second insert image in Figure 1 illustrates the swathes of the four acquisitions taken in early February 2010. Figure 2 also includes an insert diagram which uses a perspective view to show the four possible line-of-sight vectors from one surface footprint for these four acquisitions. Interferometry provides estimates of the displacements in these directions which are then combined with the geometry to get the ENU movements.
 The speckle tracking technique was used to create an azimuth and slant range shift database such that the slave image could be resampled and optimum coherence preserved. Orbit data and data from the Global ASTER DEM were used to create a pseudo interferogram which would reflect a terrain elevation-only phase pattern. While the vertical accuracy is nominally ∼9 m RMSE (Aster GDEM Validation Summary Report at http://asterweb.jpl.nasa.gov/gdem.asp) comparisons between ICESAT GLAS passes and the DEM across the lower glacier yielded 5 and 3.3 m RMS difference but increased to over 10 m RMS due to DEM height artifacts in the ice field above ∼1100 m elevation. Consequently, results are presented here only for the lower percolation and ablation zone of the glacier. The phase corresponding to the terrain elevation was removed from the actual interferogram to obtain a wrapped phase pattern reflecting predominantly the ice motion. However, there are limitations in the accuracy of the orbit data and the DEM which when coupled with propagation variations related to gradients in tropospheric water vapor can lead to gradients in the differential phase across the image for regions which are not moving [Joughin et al., 1996]. Using mountains adjacent to the glacier as a zero motion reference, a linear phase gradient was used in both range and azimuth to try to minimize these problems. Averaging of the full resolution complex interferogram was done to reduce phase noise and ease the phase unwrapping task. This was accomplished by creating overlapping windows of typically 5 pixels in slant range by 7 in azimuth which were sampled at 3 pixels in range and 4 in azimuth. Even after the phase is unwrapped and corrected there is still the problem of estimating the line-of-sight displacement corresponding to zero motion. The problem arises because often we can't unwrap reliably to an area of zero motion. The bias error in the resulting line-of-sight displacement can be reduced by using the terrain corrected range shift from speckle tracking, which is an absolute but noisy estimate. The error in the speckle tracking range shift can be estimated [Bamler, 2000] from the size of the image chips used in the cross-correlation (∼250 m slant range by 250–500 m azimuth), coherence (∼0.3 to 0.7), the pixel spacing (5.2 m azimuth by 4.7 m slant range) and resolution (7.7 m azimuth by 5.2 m slant range). Because the bias error is common to many cross-correlation estimates the resulting error in the line-of-sight displacement is in the range 5 to 25 cm.
 The incidence and azimuth angles are calculated for the master image in each of the overlapping pairs at the stage at which the pseudo interferogram is calculated. Geo-referencing and calculating the incidence angle for each interferogram pixel was accomplished using the satellite orbit state vectors and the geocoded DEM. Each data set is then resampled to a UTM zone 18X projection with pixel spacing of 25 m. The azimuth angle can be calculated after geocoding or from the control points provided in the accompanying ancillary data file. A small amount of smoothing was applied to each estimate of the geocoded line-of-sight displacement prior to the application of the 3 d algorithm outlined below. This resulted in a final pixel resolution of ∼50 × 50 m.
3. Measuring Vertical Displacements With InSAR
 ENU displacements can be derived from three temporally overlapping, co-registered geocoded data sets where each point contains three line-of-sight displacements and the associated incidence and azimuth angles. For each point we select the origin to be the pixel location observed by the first image then 24 days later due to its relative motion that pixel can be at position de, dn, du in a local ENU frame. Details of the derivation of the orthogonal displacements are given in the auxiliary material. Using the superscripts a, b and c to indicate the three registered data sets with different geometries we have the following relationships for the vertical displacement du in terms of the various slant range displacements; δr, incidence; θ, and azimuth angles; χ.
 In 2009 one data acquisition was missed and consequently only 2 interferograms were available for two 24 day time periods, each with one ascending pass interferogram and one descending pass interferogram. To solve for the vertical displacements in this case the orientation of the projection of the ice flow in the east-north plane was used from the prior 3 interferogram solution. Using the superscripts a and d to indicate measurements from ascending and descending pass data and assuming that the net ice motion is in a vertical plane at an angle ϕ to East leads to an expression for du, the vertical displacement, as
In this work we concentrate on the vertical displacement du as it is the most revealing in the glaciological context, however, the auxiliary material also contains expressions for the local east and north displacements.
 The 2010 data for the first 2 acquisition periods consisted of 2 ascending and 2 descending passes so that there 4 possible combinations of 3 different geometries and thus 4 solutions are possible. Clearly these solutions are not independent but the additional fourth geometry does reduce the problem of the uncertainty in the zero position in some of the line-of-sight displacements. However, it is important to recognize that a bias error in the line-of-sight estimates from one interferogram will lead primarily to a bias error in the resulting vertical displacement. This is a consequence of the relatively slow variation across- and along-track in the incidence and azimuth angles but it does mean that the results are essentially relative. The solution is constrained by the fact that over the whole glacier the average surface position is not moving up and the surface settling due to snow and firn compaction must be very small at this time of the year. The solution for the orientation of the ice displacement in the local east-north plane was used subsequently for the two situations in 2009 when only 2 interferograms were available.
 The vertical component of the 3-dimensional solution for ice surface displacement includes a component due to the product of the horizontal displacement and the average glacier slope in the direction of movement. To detect non-steady state glacier thickening or thinning we need to remove the down-slope component of the vertical ice displacement. A smoothed version of the DEM was used to ensure that smaller scale surface height variations or DEM noise did not influence the results. Figure 1 illustrates the horizontal direction of motion from the 3-dimensional solution (the white vectors) and the smoothed DEM slope in this direction as a color overlay on the SAR image. Subtraction of the down-slope vertical component from the vertical solution gives an estimate of the glacier surface movement in the vertical direction at each position on the glacier where the solution is possible.
Figure 2 illustrates the vertical thickening or thinning as a color overlay on the SAR image for the first 24 day multiple InSAR period in February 2010. Because the glacier terminus moves only a few tens of centimeters in the 24 day cycle but ∼3 meters at mid glacier, there must be a small vertical thickening in the lower ablation zone of the glacier to balance the downward ice flux. Figure 3 illustrates results for the down glacier profile shown in Figure 2. These were obtained for the February–March combination in 2010 and include in Figure 3 the vertical solution (in blue), the down-slope vertical component (red) and the glacier thickening or thinning (green). The significant area of surface inflation visible as the red area in Figure 2 occurs in all the solutions in both 2009 and 2010 and may be a consequence of the movement of subglacial water from the up-stream blue areas of surface deflation. While details of the glaciological results will be reported elsewhere we note that the shape of the glacier thickening-thinning hardly changes between the 3 time periods in the spring of each year but there are differences year-to-year, e.g., the peak of the surface inflation feature was ∼5 cm larger in 2009 than in 2010. The requirement that the velocity not change rapidly is satisfied for this glacier at this time of the year. Also, it is clear that the vertical displacements must be part of a yearly cycle.
 The expression for the vertical displacement is of the form dz = caδra + cbδrb + ccδrc where the ca,b,c terms are a function of the 3 incidence and 3 azimuth angles. The error in dz is dominated by the errors in the line-of-sight displacements δra,b,c which arise from a combination of interferogram phase noise and lower frequency bias errors including imperfect bias removal for the zero motion, DEM or mapping errors, and variations in atmospheric propagation across the scene, etc. [Joughin et al., 1996]. The random noise component of error in the vertical displacement Δ(dz) can be written as
where Δ(δra,b,c) are the random errors in the line-of-sight displacements. The variance in phase noise, σϕ2, can be estimated from the coherence γ using σϕ2 = (1 − γ2)/2NLγ2 where NL is the number of independent samples averaged [Rosen et al., 2000]. Using appropriate values for coherence and NL, coupled with the typical ca,b,c terms obtained in this experiment, it can be shown that the noise in the vertical solution should be less than 1 cm for the 24 day time separation.
 The lower frequency bias errors are more difficult to characterize and include the possibility of imperfect bias removal for zero line-of-sight displacement. This means that the vertical solution has a potential bias error of order a few centimeters and is therefore essentially relative. This weakness in the current data set could be alleviated by a few accurate reference ground GPS points or airborne laser data. Any of the three interferograms in any solution may be affected by varying tropospheric conditions even in the high Arctic under winter conditions [Mattar et al., 1999]. Variations in water vapor can lead to fractional fringe errors in any one interferogram which will lead to slowly varying bias error in the solution for vertical displacement. The magnitude of the error depends on the gradient of the propagation attenuation (in line-of-sight displacement per kilometer) and the appropriate geometric factor c in the 3-dimensional solution. Gradients in the line-of-sight displacement of order 0.5 cm for 10 km are possible due to variation in tropospheric water vapor; this could lead to a gradient in dz of order ∼2 cm over 10 km. Variations in ionospheric propagation can also impact interferometric phase particularly in the polar auroral zones. However, no instances of significant azimuth shift banding were observed characteristic of changing ionospheric propagation conditions [Gray et al., 2000]. The same DEM errors will occur in the creation of all the interferograms but the resulting error in the line-of-sight displacement depends also on the perpendicular baseline which does vary between interferograms. Consequently, a consistent feature in the vertical displacement results cannot be caused by an error in the DEM.
6. Limitations and Summary
 The requirement for phase unwrapping will restrict the application of this technique to areas with low gradients in speed, e.g. solutions for the high sheer regions at the edges of glaciers and ice streams will not be possible. Also, it is clear that there is a limited latitude region over which this 3 interferogram solution will work well: By utilizing the extra-low and extra-high beams of RADARSAT 2 it should be possible to achieve the different geometries that are required for a ‘good’ solution at latitudes greater than around 80° (with a right-looking configuration in the North and left-looking in Antarctica). Secondly, by combining look-right and look-left data it will be possible to extend the range of latitudes to ∼65° but achieving good temporal overlap becomes more difficult as the latitude is decreased and the accuracy in the north-south direction will be less than in the other two directions. While limited in scope, the relative accuracy and spatial resolution at which the height changes can be mapped with this technique far exceeds that possible with other spaceborne systems.
 RADARSAT 2 data are copyright and provided by MDA, Vancouver. This work was supported by an EOADP contract from the Canadian Space Agency to Noetix Research, Ottawa. The ASTER Global DEM data was provided by USGS and METI, Japan. Help from colleagues at Noetix (John Bennett, Mark Kapfer, Tom Hirose) and the contract Scientific Authority (Dave Burgess, GSC, NRCan) is gratefully acknowledged.