Combining SAR interferometry and the equation of continuity to estimate the three-dimensional glacier surface-velocity vector

1999 ◽  
Vol 45 (151) ◽  
pp. 533-538 ◽  
Author(s):  
Niels Reeh ◽  
Søren Nørvang Madsen ◽  
Johan Jakob Mohr
Keyword(s):  
Steady State ◽  
Mass Balance ◽  
Vertical Velocity ◽  
Velocity Vector ◽  
Thickness Distribution ◽  
Vertical Surface ◽  
Horizontal Velocity ◽  
Sar Interferometry ◽  
Surface Velocity ◽  
Ice Thickness

AbstractUntil now, an assumption of surface-parallel glacier flow has been used to express the vertical velocity component in terms of the horizontal velocity vector, permitting all three velocity components to be determined from synthetic aperture radar interferometry. We discuss this assumption, which neglects the influence of the local mass balance and a possible contribution to the vertical velocity arising if the glacier is not in steady state. We find that the mass-balance contribution to the vertical surface velocity is not always negligible as compared to the surface-slope contribution. Moreover, the vertical velocity contribution arising if the ice sheet is not in steady state can be significant. We apply the principle of mass conservation to derive an equation relating the vertical surface velocity to the horizontal velocity vector. This equation, valid for both steady-state and non-steady-state conditions, depends on the ice-thickness distribution. Replacing the surface-parallel-flow assumption with a correct relationship between the surface velocity components requires knowledge of additional quantities such as surface mass balance or ice thickness.

Combining SAR interferometry and the equation of continuity to estimate the three-dimensional glacier surface-velocity vector

1999 ◽  
Vol 45 (151) ◽  
pp. 533-538 ◽  
Author(s):  
Niels Reeh ◽  
Søren Nørvang Madsen ◽  
Johan Jakob Mohr
Keyword(s):  
Steady State ◽  
Mass Balance ◽  
Vertical Velocity ◽  
Velocity Vector ◽  
Thickness Distribution ◽  
Vertical Surface ◽  
Horizontal Velocity ◽  
Sar Interferometry ◽  
Surface Velocity ◽  
Ice Thickness

AbstractUntil now, an assumption of surface-parallel glacier flow has been used to express the vertical velocity component in terms of the horizontal velocity vector, permitting all three velocity components to be determined from synthetic aperture radar interferometry. We discuss this assumption, which neglects the influence of the local mass balance and a possible contribution to the vertical velocity arising if the glacier is not in steady state. We find that the mass-balance contribution to the vertical surface velocity is not always negligible as compared to the surface-slope contribution. Moreover, the vertical velocity contribution arising if the ice sheet is not in steady state can be significant. We apply the principle of mass conservation to derive an equation relating the vertical surface velocity to the horizontal velocity vector. This equation, valid for both steady-state and non-steady-state conditions, depends on the ice-thickness distribution. Replacing the surface-parallel-flow assumption with a correct relationship between the surface velocity components requires knowledge of additional quantities such as surface mass balance or ice thickness.

scholarly journals Ice thickness and volume of the Renland Ice Cap, East Greenland

2021 ◽  
pp. 1-13
Author(s):  
Iben Koldtoft ◽  
Aslak Grinsted ◽  
Bo M. Vinther ◽  
Christine S. Hvidberg
Keyword(s):  
Surface Topography ◽  
Climate Model ◽  
Thickness Distribution ◽  
High Elevation ◽  
Surface Velocity ◽  
Ice Thickness ◽  
Model Parameters ◽  
East Greenland ◽  
Ice Volume ◽  
Ice Cap

Abstract To assess the amount of ice volume stored in glaciers or ice caps, a method to estimate ice thickness distribution is required for glaciers where no direct observations are available. In this study, we use an existing inverse method to estimate the bedrock topography and ice thickness of the Renland Ice Cap, East Greenland, using satellite-based observations of the surface topography. The inverse approach involves a procedure in which an ice dynamical model is used to build-up an ice cap in steady state with climate forcing from a regional climate model, and the bedrock is iteratively adjusted until the modelled and observed surface topography match. We validate our model results against information from airborne radar data and satellite observed surface velocity, and we find that the inferred ice thickness and thereby the stored total volume of the ice cap is sensitive to the assumed ice softness and basal slipperiness. The best basal model parameters for the Renland Ice Cap are determined and the best estimated total ice volume of 384 km3 is found. The Renland Ice Cap is particularly interesting because of its location at a high elevation plateau and hence assumed low sensitivity to climate change.

scholarly journals The ice thickness distribution of Flask Glacier, Antarctic Peninsula, determined by combining radio-echo soundings, surface velocity data and flow modelling

2013 ◽  
Vol 54 (63) ◽  
pp. 18-24 ◽  
Author(s):  
Daniel Farinotti ◽  
Hugh Corr ◽  
G.Hilmar Gudmundsson
Keyword(s):  
Antarctic Peninsula ◽  
Thickness Distribution ◽  
Surface Velocity ◽  
Ice Thickness ◽  
Ice Shelf ◽  
Ice Flow ◽  
Ice Volume ◽  
Flow Modelling ◽  
Bedrock Topography ◽  
Sparse Measurements

AbstractAn interpolated bedrock topography is presented for Flask Glacier, one of the tributaries of the remnant part of the Larsen B ice shelf, Antarctic Peninsula. The ice thickness distribution is derived by combining direct but sparse measurements from airborne radio-echo soundings with indirect estimates obtained from ice-flow modelling. The ice-flow model is applied to a series of transverse profiles, and a first estimate of the bedrock is iteratively adjusted until agreement between modelled and measured surface velocities is achieved. The adjusted bedrock is then used to reinterpret the radio-echo soundings, and the recovered information used to further improve the estimate of the bedrock itself. The ice flux along the glacier center line provides an additional and independent constraint on the ice thickness. The resulting bedrock topography reveals a glacier bed situated mainly below sea level with sections having retrograde slope. The total ice volume of 120 ±15 km3 for the considered area of 215 km2 corresponds to an average ice thickness of 560 ± 70 m.

scholarly journals Dynamics of the Law Dome Ice Cap, Antarctica, as Found From Bore-Hole Measurements

1989 ◽  
Vol 12 ◽  
pp. 46-50 ◽  
Author(s):  
D.M. Etheridge
Keyword(s):  
Shear Strain ◽  
Vertical Velocity ◽  
Flow Line ◽  
Velocity Profiles ◽  
Surface Velocity ◽  
Ice Thickness ◽  
Shear Strain Rate ◽  
Orientation Data ◽  
Ice Cap ◽  
The Law

The internal dynamics of the Law Dome ice cap have been investigated by measuring the deformation of three bore holes located on an approximate flow line. Bore holes BHC1 (300 m deep) and BHC2 (344 m) were drilled in the coastal area to within several metres of bedrock and BHQ (418 m) was drilled about half-way towards the dome centre to about 50% of the ice thickness. Detailed measurements of orientation (inclination and azimuth), diameter, and temperature were taken through each bore hole over a 1 year span for BHC1 and BHC2 and a 10 year span for BHQ. The orientation data were reduced to obtain ∂u/∂z, a measure of the shear strain-rate. Changes in the depth of features located by bore-hole diameter measurements were used to obtain vertical velocity profiles. Other measurements discussed are temperatures, oxygen isotopes, crystal structure, surface velocities, and surface and bedrock topography.At the coastal sites, the ∂u/∂z profiles show two maxima in the lower third of the ice sheet. Flow due to the measured deformation accounts for about 55% of the surface velocity, the remainder being due to deformation and sliding in the basal zone. The vertical velocity profiles show mostly firn compression. The deeper ∂u/∂z maximum occurs in ice from the Wisconsin period which appears to deform more rapidly than the Holocene ice immediately above. The upper ∂u/∂z maximum may be related to the stress history of the ice, which can also explain the presence of significant shear strain and crystal-fabric development at only half the ice thickness at the BHQ site.

scholarly journals The Equation of Continuity and its Application to the Ice Sheet Near “byrd” Station, Antarctica

1977 ◽  
Vol 18 (80) ◽  
pp. 359-371 ◽  
Author(s):  
I. M. Whillans
Keyword(s):  
Steady State ◽  
Mass Balance ◽  
Ice Sheet ◽  
Ice Thickness ◽  
Surface Mass Balance ◽  
Velocity Measurements ◽  
Ice Flow ◽  
Measured Velocity ◽  
Surface Mass ◽  
Ice Velocity

Abstract The continuity relationship that is often used in the study of ice sheets and ice shelves is developed by integrating the equation of continuity through the ice thickness. This equation is then integrated again with respect to horizontal distance from an ice divide, showing that the difference between the true ice velocity and the balance velocity, which is defined, is a measure of the time chance of the mass of a column through the ice thickness. The relationship is applied using data from along the “Byrd” station strain network, Antarctica. This region is found to be thinning slowly (0.03 m a−1 of ice of mean density) and uniformly, but it is still close to steady-state. The calculations would show a larger thinning rate if bottom sliding contributed more to the ice movement and integral shear contributed less, but the “Byrd” station bore-hole tilting results of Garfield and Ueda (1975, 1976), together with surface velocity measurements at “Byrd” station, indicate that most of the ice flow is by deformation within the ice mass. This large amount of internal deformation is more than that predicted by most “flow laws”, probably because of the strongly oriented ice-crystal fabric in the ice sheet. The cause of ice thinning is probably decreased surface mass balance beginning before A.D. 1550. The consistent relationship between measured velocity and balance velocity indicates that the ice flow is simple and that flow lines are in the same direction at depth as at the surface when considered smoothed over a distance of 10 km. Because the ice sheet is at present thinning, the balance velocity, calculated only from flow line and surface mass-balance data, and the somewhat mistaken assumption of steady-state is 15% less than the true ice velocity. This rather small difference confirms the use of balance-velocity estimates where velocity measurements are not available.

scholarly journals Estimating the ice thickness of mountain glaciers with a shape optimization algorithm using surface topography and mass-balance

2014 ◽  
Vol 22 (6) ◽  
Author(s):  
Laurent Michel ◽  
Marco Picasso ◽  
Daniel Farinotti ◽  
Andreas Bauder ◽  
Martin Funk ◽  
...  
Keyword(s):  
Shape Optimization ◽  
Mass Balance ◽  
Surface Topography ◽  
Optimization Algorithm ◽  
Moving Boundary ◽  
Thickness Distribution ◽  
Ice Thickness ◽  
Surface Geometry ◽  
Mountain Glacier ◽  
Surface Mass

AbstractWe present a shape optimization algorithm to estimate the ice thickness distribution within a two-dimensional, non-sliding mountain glacier, given a transient surface geometry and a mass-balance distribution. The approach is based on the minimization of the surface topography misfit at the end of the glacier's evolution in the shallow ice approximation of ice flow. Neither filtering of the surface topography where its gradient vanishes nor interpolation of the basal shear stress is involved. Novelty of the presented shape optimization algorithm is the use of surface topography and mass-balance only within a time-dependent Lagrangian approach for moving-boundary glaciers. On real-world inspired geometries, it is shown to produce estimations of even better quality in smaller time than the recently proposed steady and transient inverse methods. A sensitivity analysis completes the study and evinces the method's higher susceptibility to perturbations in the surface topography than in surface mass-balance or rate factor.

Changes in mass balance, velocity, and surface profile along a flow line on Barnes Ice Cap, 1970–1984

1987 ◽  
Vol 24 (8) ◽  
pp. 1550-1561 ◽  
Author(s):  
Roger LeB. Hooke ◽  
Gerald W. Johnson ◽  
Keith A. Brugger ◽  
Brian Hanson ◽  
Gerald Holdsworth
Keyword(s):  
Mass Balance ◽  
Flow Line ◽  
Surface Profile ◽  
Horizontal Velocity ◽  
Surface Elevation ◽  
Surface Velocity ◽  
Transverse Strain ◽  
Average Mass ◽  
Ice Cap ◽  
Hydrostatic Approximation

On a number of occasions between 1970 and 1984, measurements of mass balance, surface velocity, and surface elevation were made along a 10.2 km flow line extending from the divide to the margin on Barnes Ice Cap, Baffin Island, Canada. The average mass balance rate on this Arctic ice cap appears to be controlled principally by summer temperature. Years of negative mass balance dominated during the 14 year period of the study; according to our measurements, this should have resulted in a thinning of an average of 2.65 m along the flow line. Surface-elevation measurements, however, suggest that the glacier thinned only ~ 1.7 m during this time period. The slightly compressive mean transverse strain can account for ~ 0.20 m of the discrepancy between these two figures. The rest is attributed to errors in mass balance. In particular, increases in density of old firn and freezing of water in crevasses could not be evaluated adequately from our measurements.The thinning was not uniform. Near the margin the glacier thinned ~ 5 m and near the divide, ~ 2.5 m, but between 3.5 and 5.5 km from the divide there seems to have been no substantial change in thickness. Thus, along the down-glacier half of the flow line the decrease in thickness was accompanied by an increase in slope.Within 4 km of the margin the decrease in thickness resulted in a decrease in horizontal velocity of up to ~ 15%, despite the increase in slope. Higher on the glacier, however, the horizontal velocity increased ~ 5%. These changes are broadly consistent with those estimated from the changes in shear stress calculated using the hydrostatic approximation.

scholarly journals The Equation of Continuity and its Application to the Ice Sheet Near “byrd” Station, Antarctica

1977 ◽  
Vol 18 (80) ◽  
pp. 359-371 ◽  
Author(s):  
I. M. Whillans
Keyword(s):  
Steady State ◽  
Mass Balance ◽  
Ice Sheet ◽  
Ice Thickness ◽  
Surface Mass Balance ◽  
Velocity Measurements ◽  
Ice Flow ◽  
Measured Velocity ◽  
Surface Mass ◽  
Ice Velocity

AbstractThe continuity relationship that is often used in the study of ice sheets and ice shelves is developed by integrating the equation of continuity through the ice thickness. This equation is then integrated again with respect to horizontal distance from an ice divide, showing that the difference between the true ice velocity and the balance velocity, which is defined, is a measure of the time chance of the mass of a column through the ice thickness.The relationship is applied using data from along the “Byrd” station strain network, Antarctica. This region is found to be thinning slowly (0.03 m a−1 of ice of mean density) and uniformly, but it is still close to steady-state. The calculations would show a larger thinning rate if bottom sliding contributed more to the ice movement and integral shear contributed less, but the “Byrd” station bore-hole tilting results of Garfield and Ueda (1975, 1976), together with surface velocity measurements at “Byrd” station, indicate that most of the ice flow is by deformation within the ice mass. This large amount of internal deformation is more than that predicted by most “flow laws”, probably because of the strongly oriented ice-crystal fabric in the ice sheet. The cause of ice thinning is probably decreased surface mass balance beginning before A.D. 1550.The consistent relationship between measured velocity and balance velocity indicates that the ice flow is simple and that flow lines are in the same direction at depth as at the surface when considered smoothed over a distance of 10 km. Because the ice sheet is at present thinning, the balance velocity, calculated only from flow line and surface mass-balance data, and the somewhat mistaken assumption of steady-state is 15% less than the true ice velocity. This rather small difference confirms the use of balance-velocity estimates where velocity measurements are not available.

scholarly journals Mass Balance Along Two Transects of the Greenland Ice Sheet (Abstract)

1986 ◽  
Vol 8 ◽  
pp. 198-198
Author(s):  
J.M. Kostecka ◽  
I.M. Whillans
Keyword(s):  
Mass Balance ◽  
Temperature Profile ◽  
Accumulation Rate ◽  
Horizontal Velocity ◽  
Surface Velocity ◽  
The Arctic ◽  
Accumulation Rates ◽  
Mean Velocity ◽  
Enhancement Factors ◽  
The Mean

The mass balance and sensitivity of calculated mass balance to uncertainties in the data and in the model for the variation of ice velocity with depth are addressed, using data from the EGIG transect and from the OSU transect in south Greenland (Drew and Whillans, 1984). The calculation uses a non-steady continuity model with allowances for three-dimensional flow and horizontal velocity variation with depth. Depth variation in horizontal velocity is obtained, using the constitutive relation for ice with calculated temperature profiles and with full allowance for longitudinal stresses and enhancement of flow due to ice anistrophy and texture. Separate calculations are made for different thickening or thinning rates, until a match between observed and calculated surface velocities is obtained.For the EGIG transect, our mass-balance results are in the range reported by Mälzer and Seckel (1975) and Bindschadler (1984). Results for the OSU transect, just south of the Arctic Circle on the south dome, are also reported.Of particular interest is the sensitivity study, which is designed to determine which aspects of the data and flow behavior are most critical to the calculations. Data that are comparatively well-constrained are flow-line definition, surface velocity and thickness. The flow lines and lateral spreading are obtained from satellite radar altimetry (Zwally and others, 1983) and checked against velocity data along the OSU profile. Thicknesses are from aerial radar sounding (Overgaard, personal communication). Surface velocities along the EGIG transect are corrected in a manner similar to that suggested by Robin (1983, figure 2.17b) and possible errors are not critical. Along the OSU transect the surface velocities are well-determined by Doppler satellite tracking in short-arc translocation mode.Accumulation rates are not well-determined and show substantial scatter for reasons that are not understood. Along the OSU transect, accumulation rates were obtained by augering for the nuclear bomb horizons. As reported earlier (Mock, 1967), the accumulation rates in this region show no clear pattern with elevation or slope. Similar, but less severe, problems exist in the EGIG area. The lack of a good model describing the geographic variation in accumulation rate results in calculated surface velocity uncertainties of 9% for the OSU and 6% for the EGIG transect.The calculated results are comparatively insensitive to parameters affecting the temperature profile. This is because, in this model, the mean velocity is determined by continuity and temperature affects only the ratio of horizontal velocity at a given depth to the mean velocity for the entire profile.Longitudinal stresses are included because they affect the viscosity through the effective shear stress. Neglecting this would lead to a 30% error in calculated surface velocity near the ice divide and about 6% away from the divide. Provided longitudinal stresses are included, uncertainties in these stresses are not critical.Enhancement factors are, however, important. Most authors believe that ice fabric, texture, and impurity content can affect strain rates by 100% to 1200% (Shoji and Langway, 1984). As with the temperature profile, this problem does not affect calculated mean velocities, which are based on continuity, but it does affect the ratio of surface to mean velocity. Enhancement factors are more critical than temperature uncertainties and therefore failure to include depth-variable flow enhancement alters calculated surface velocities by 5 to 10%.In summary, the mass balance along two transects has been investigated by comparing measured and calculated surface velocities. The most critical problems are (1) understanding the geographic distribution in accumulation rate and (2) correctly allowing for enhanced shearing due to structural variations with depth.

scholarly journals Investigation of Bottom Mass-Balance Rates by Electrical Resistivity Soundings on the Ross Ice Shelf, Antarctica

1979 ◽  
Vol 24 (90) ◽  
pp. 331-343 ◽  
Author(s):  
Sion Shabtaie ◽  
Charles R. Bentley
Keyword(s):  
Activation Energy ◽  
Electrical Resistivity ◽  
Steady State ◽  
Mass Balance ◽  
Apparent Resistivity ◽  
Ice Thickness ◽  
Seasonal Temperature ◽  
Ice Shelf ◽  
Near Surface ◽  
Ross Ice Shelf

AbstractElectrical resistivity sounding, using the four-electrode Schlumberger array, has been carried out at 11 locations on the Ross Ice Shelf. The apparent resistivity curves generally show four characteristic zones. The first, at distances from 1 to 10 m, reflects the near-surface zone of seasonal temperature changes and inhomogeneities. The second zone, from 10 m to 100 m, reflects primarily the increasing density with depth in the upper 50 m of the ice shelf, modified, in some locations, by temperature variations. The third zone, from 100 m to a distance roughly equal to the ice thickness, is affected principally by the temperature gradient in the solid ice. In the fourth zone, at distances greater than approximately twice the ice thickness, the apparent resistivity usually decreases rapidly with distance, owing to the highly conductive sea-water beneath the ice shelf. At some stations associated with ice streams and outlet glaciers, however, an increase at large spacings indicates much more resistive basal ice.Using data from seven locations on the grid eastern half of the shelf that do not show obvious evidence of a basal resistive zone, including temperatures to 100 m at two of the sites, the mass-balance rate at the bottom of the ice is estimated to be within a few tenths of a meter per year of zero at distances between 90 and 530 km from the ice front, assuming steady-state condition over most of the ice shelf. However, the assumption of steady-state is questionable at locations close to outlet glaciers, and must be treated with great caution. The temperature measurements at the two sites, along with previously observed temperatures at the RISP drill site, make it possible to estimate the activation energy in the solid ice. The models fitted to the observed values suggest an “apparent” activation energy in the solid ice closer to 0.15 eV (14 kJ mol−1) than to 0.25 eV (24 kJ mol−1). This difference is believed to be due to a decrease in the ionic impurity content with increasing depth in the ice by a factor of two or more.

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