Adjusting Two-Dimensional Velocity Data to Obey Continuity

Abstract An algorithm is developed for adjusting glacier surface-velocity vectors, given on the nodes of a square grid, so that they obey a central-difference approximation of the continuity equation. Also required on the grid nodes are the glacier thickness, the ratio of the surface-velocity to the average velocity in the column, and the difference between the mass balance and the thickness change. All these other variables are assumed to be known exactly, and only the surface-velocity field is adjusted. The result is optimum in the sense that the magnitude of the adjustment is minimized. Either the relative or the absolute adjustment can be minimized, depending on how weights are specified. No restriction is placed on the shape of the solution region, and no boundary condition is required. The algorithm is not iterative. The algorithm first forms a parallel flow field that satisfies the continuity equation, and then uses a stream function to add a divergenceless field to it. The stream function that leads to the minimum velocity adjustment is obtained as four independent, interlacing solutions covering the solution region. For each of the four, a well-conditioned, sparse-matrix system of simultaneous linear equations is solved. A compact, sub-optimum, well-behaved iterative procedure is also developed for transforming part of the velocity adjustment into an adjustment of the thickness field.

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Adjusting Two-Dimensional Velocity Data to Obey Continuity

Journal of Glaciology ◽

10.3189/s0022143000006341 ◽

1985 ◽

Vol 31 (108) ◽

pp. 115-119

Author(s):

L. A. Rasmussen

Keyword(s):

Stream Function ◽

Continuity Equation ◽

Sparse Matrix ◽

Linear Equations ◽

Surface Velocity ◽

Difference Approximation ◽

Central Difference ◽

Glacier Surface ◽

Thickness Change ◽

Solution Region

AbstractAn algorithm is developed for adjusting glacier surface-velocity vectors, given on the nodes of a square grid, so that they obey a central-difference approximation of the continuity equation. Also required on the grid nodes are the glacier thickness, the ratio of the surface-velocity to the average velocity in the column, and the difference between the mass balance and the thickness change. All these other variables are assumed to be known exactly, and only the surface-velocity field is adjusted. The result is optimum in the sense that the magnitude of the adjustment is minimized. Either the relative or the absolute adjustment can be minimized, depending on how weights are specified. No restriction is placed on the shape of the solution region, and no boundary condition is required. The algorithm is not iterative. The algorithm first forms a parallel flow field that satisfies the continuity equation, and then uses a stream function to add a divergenceless field to it. The stream function that leads to the minimum velocity adjustment is obtained as four independent, interlacing solutions covering the solution region. For each of the four, a well-conditioned, sparse-matrix system of simultaneous linear equations is solved. A compact, sub-optimum, well-behaved iterative procedure is also developed for transforming part of the velocity adjustment into an adjustment of the thickness field.

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Influence of recent warming and ice dynamics on glacier surface elevations in the Canadian High Arctic, 1995–2014

Journal of Glaciology ◽

10.1017/jog.2018.37 ◽

2018 ◽

Vol 64 (245) ◽

pp. 450-464 ◽

Cited By ~ 2

Author(s):

COLLEEN A. MORTIMER ◽

MARTIN SHARP ◽

WESLEY VAN WYCHEN

Keyword(s):

Land Surface ◽

High Arctic ◽

Surface Elevation ◽

Surface Velocity ◽

Glacier Surface ◽

Ice Dynamics ◽

Thickness Change ◽

Near Surface ◽

Elevation Changes ◽

Moderate Resolution Imaging Spectroradiometer

ABSTRACTRepeat airborne laser altimetry measurements show widespread thinning (surface lowering) of glaciers in Canada's Queen Elizabeth Islands since 1995. Thinning rates averaged for 50 m elevation bins, were more than three times higher during the period 2005/06 to 2012/14 pentad than during the previous two pentads. Strongly negative thickness change (dh/dt) anomalies from 2005/06 to 2012/14, relative to the 1995–2012/14 mean, suggest that most of the measured thinning occurred during the most recent 5–6 year period when mean summer land surface temperatures (LSTs) were anomalously high and the mean summer black-sky shortwave broadband albedos (BSA) were anomalously low, relative to the 2000/01–15/16 period, and upper-air (700 hPa) and near surface (2 m) air temperatures were between 0.8°C and 1.5°C higher than 1995–2012 mean. Comparisons of dh/dt with mean summer LST and BSA measurements from the Moderate Resolution Imaging Spectroradiometer and with surface longitudinal strain rates computed from surface velocity fields derived from RADARSAT 1/2 and Landat-7 ETM + data suggest that surface elevation changes were driven mainly by changes in climate. An exception to this occurs along many fast-flowing outlet glaciers where ice dynamics appear also to have played an important role in surface elevation changes.

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The Causes of Debris-Covered Glacier Thinning: Evidence for the Importance of Ice Dynamics From Kennicott Glacier, Alaska

Frontiers in Earth Science ◽

10.3389/feart.2021.680995 ◽

2021 ◽

Vol 9 ◽

Author(s):

Leif S. Anderson ◽

William H. Armstrong ◽

Robert S. Anderson ◽

Dirk Scherler ◽

Eric Petersen

Keyword(s):

Mass Balance ◽

Flow Direction ◽

Surface Velocity ◽

Glacier Surface ◽

Ice Dynamics ◽

Ice Flow ◽

Thickness Change ◽

Air Temperatures ◽

Compressive Flow ◽

Wide Range

The cause of debris-covered glacier thinning remains controversial. One hypothesis asserts that melt hotspots (ice cliffs, ponds, or thin debris) increase thinning, while the other posits that declining ice flow leads to dynamic thinning under thick debris. Alaska’s Kennicott Glacier is ideal for testing these hypotheses, as ice cliffs within the debris-covered tongue are abundant and surface velocities decline rapidly downglacier. To explore the cause of patterns in melt hotspots, ice flow, and thinning, we consider their evolution over several decades. We compile a wide range of ice dynamical and mass balance datasets which we cross-correlate and analyze in a step-by-step fashion. We show that an undulating bed that deepens upglacier controls ice flow in the lower 8.5 km of Kennicott Glacier. The imposed velocity pattern strongly affects debris thickness, which in turn leads to annual melt rates that decline towards the terminus. Ice cliff abundance correlates highly with the rate of surface compression, while pond occurrence is strongly negatively correlated with driving stress. A new positive feedback is identified between ice cliffs, streams and surface topography that leads to chaotic topography. As the glacier thinned between 1991 and 2015, surface melt in the study area decreased, despite generally rising air temperatures. Four additional feedbacks relating glacier thinning to melt changes are evident: the debris feedback (negative), the ice cliff feedback (negative), the pond feedback (positive), and the relief feedback (positive). The debris and ice cliff feedbacks, which are tied to the change in surface velocity in time, likely reduced melt rates in time. We show this using a new method to invert for debris thickness change and englacial debris content (∼0.017% by volume) while also revealing that declining speeds and compressive flow led to debris thickening. The expansion of debris on the glacier surface follows changes in flow direction. Ultimately, glacier thinning upvalley from the continuously debris-covered portion of Kennicott Glacier, caused by mass balance changes, led to the reduction of flow into the study area. This caused ice emergence rates to decline rapidly leading to the occurrence of maximum, glacier-wide thinning under thick, insulating debris.

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Antarctic ice shelf thickness change from multimission lidar mapping

The Cryosphere ◽

10.5194/tc-13-1801-2019 ◽

2019 ◽

Vol 13 (7) ◽

pp. 1801-1817 ◽

Cited By ~ 2

Author(s):

Tyler C. Sutterley ◽

Thorsten Markus ◽

Thomas A. Neumann ◽

Michiel van den Broeke ◽

J. Melchior van Wessem ◽

...

Keyword(s):

High Resolution ◽

Climate Models ◽

Regional Climate ◽

Regional Climate Models ◽

Surface Velocity ◽

Ice Thickness ◽

Flux Divergence ◽

Ice Shelf ◽

Thickness Change ◽

Ice Shelves

Abstract. We calculate rates of ice thickness change and bottom melt for ice shelves in West Antarctica and the Antarctic Peninsula from a combination of elevation measurements from NASA–CECS Antarctic ice mapping campaigns and NASA Operation IceBridge corrected for oceanic processes from measurements and models, surface velocity measurements from synthetic aperture radar, and high-resolution outputs from regional climate models. The ice thickness change rates are calculated in a Lagrangian reference frame to reduce the effects from advection of sharp vertical features, such as cracks and crevasses, that can saturate Eulerian-derived estimates. We use our method over different ice shelves in Antarctica, which vary in terms of size, repeat coverage from airborne altimetry, and dominant processes governing their recent changes. We find that the Larsen-C Ice Shelf is close to steady state over our observation period with spatial variations in ice thickness largely due to the flux divergence of the shelf. Firn and surface processes are responsible for some short-term variability in ice thickness of the Larsen-C Ice Shelf over the time period. The Wilkins Ice Shelf is sensitive to short-timescale coastal and upper-ocean processes, and basal melt is the dominant contributor to the ice thickness change over the period. At the Pine Island Ice Shelf in the critical region near the grounding zone, we find that ice shelf thickness change rates exceed 40 m yr−1, with the change dominated by strong submarine melting. Regions near the grounding zones of the Dotson and Crosson ice shelves are decreasing in thickness at rates greater than 40 m yr−1, also due to intense basal melt. NASA–CECS Antarctic ice mapping and NASA Operation IceBridge campaigns provide validation datasets for floating ice shelves at moderately high resolution when coregistered using Lagrangian methods.

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Variations in the surface velocity of an alpine cirque glacier

Journal of Glaciology ◽

10.1017/jog.2018.85 ◽

2018 ◽

Vol 64 (248) ◽

pp. 969-976 ◽

Cited By ~ 1

Author(s):

J. W. SANDERS ◽

K. M. CUFFEY ◽

K. R. MACGREGOR ◽

J. L. KAVANAUGH ◽

C. F. DOW

Keyword(s):

Water System ◽

Heavy Rain ◽

Temporal Variations ◽

Surface Velocity ◽

Channel Network ◽

Glacier Surface ◽

Ice Dynamics ◽

Speed Up ◽

Water Volumes ◽

One Year

ABSTRACTFollowing pioneering work in Norway, cirque glaciers have widely been viewed as rigidly rotating bodies. This model is incorrect for basin-filling cirque glaciers, as we have demonstrated at West Washmawapta Glacier, a small glacier in the Canadian Rocky Mountains. Here we report observations at the same glacier that assess whether complex temporal variations of flow also occur. For parts of three summers, we measured daily displacements of the glacier surface. In one year, four short-duration speed-up events were recorded. Three of the events occurred during the intervals of warmest weather, when melt was most rapid; the fourth event occurred immediately following heavy rain. We interpret the speed-up events as manifestations of enhanced water inputs to the glacier bed and associated slip lubrication by increased water volumes and pressures. No further speed-ups occurred in the final month of the melt season, despite warm temperatures and several rainstorms; the dominant subglacial water system likely transformed from one of poorly connected cavities to one with an efficient channel network. The seasonal evolution of hydrology and flow resembles behaviors documented at other, larger temperate glaciers and indicates that analyses of cirque erosion cannot rely on simple assumptions about ice dynamics.

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Correlation dispersion as a measure to better estimate uncertainty of remotely sensed glacier displacements

10.5194/tc-2021-202 ◽

2021 ◽

Author(s):

Bas Altena ◽

Andreas Kääb ◽

Bert Wouters

Keyword(s):

Large Data ◽

Surface Velocity ◽

Dominant Factor ◽

Fast Method ◽

Imaging Data ◽

Velocity Data ◽

Correlation Peak ◽

Glacier Surface ◽

Fast Processing ◽

Single Number

Abstract. In recent years a vast amount of glacier surface velocity data from satellite imagery has emerged based on correlation between repeat images. Thereby, much emphasis has been put on fast processing of large data volumes. The metadata of such measurements are often highly simplified when the measurement precision is lumped into a single number for the whole dataset, although the error budget of image matching is in reality not isotropic and constant over the whole velocity field. The spread of the correlation peak of individual image offset measurements is dependent on the image content and the non-uniform flow of the ice. Precise dispersion estimates for each individual velocity measurement can be important for inversion of, for instance, rheology, ice thickness and/or bedrock friction. Errors in the velocity data can propagate into derived results in a complex and exaggerating way, making the outcomes very sensitive to velocity noise and errors. Here, we present a computationally fast method to estimate the matching precision of individual displacement measurements from repeat imaging data, focussing on satellite data. The approach is based upon Gaussian fitting directly on the correlation peak and is formulated as a linear least squares estimation, making its implementation into current pipelines straightforward. The methodology is demonstrated for Sermeq Kujalleq, Greenland, a glacier with regions of strong shear flow and with clearly oriented crevasses, and Malaspina Glacier, Alaska. Directionality within an image seems to be dominant factor influencing the correlation dispersion. In our cases these are crevasses and moraine bands, while a relation to differential flow, such as shear, is less pronounced.

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Combined measurements of Subglacial Water Pressure and Surface Velocity of Findelengletscher, Switzerland: Conclusions about Drainage System and Sliding Mechanism

Journal of Glaciology ◽

10.3189/s0022143000006936 ◽

1986 ◽

Vol 32 (110) ◽

pp. 101-119 ◽

Cited By ~ 35

Author(s):

Almut Iken ◽

Robert A. Bindschadler

Keyword(s):

Functional Relationship ◽

Water Pressure ◽

Drainage System ◽

High Water ◽

Water Levels ◽

Surface Velocity ◽

Glacier Surface ◽

Measured Velocity ◽

Sliding Mechanism ◽

Velocity Variations

AbstractDuring the snow-melt season of 1982, basal water pressure was recorded in 11 bore holes communicating with the subglacial drainage system. In most of these holes the water levels were at approximately the same depth (around 70 m below surface). The large variations of water pressure, such as diurnal variations, were usually similar at different locations and in phase. In two instances of exceptionally high water pressure, however, systematic phase shifts were observed; a wave of high pressure travelled down-glacier with a velocity of approximately 100 m/h.The glacier-surface velocity was measured at four lines of stakes several times daily. The velocity variations correlated with variations in subglacial water pressure. The functional relationship of water pressure and velocity suggests that fluctuating bed separation was responsible for the velocity variations. The empirical functional relationship is compared to that of sliding over a perfectly lubricated sinusoidal bed. On the basis of the measured velocity-pressure relationship, this model predicts a reasonable value of bed roughness but too high a sliding velocity and unstable sliding at too low a water pressure. The main reason for this disagreement is probably the neglect of friction from debris in the sliding model.The measured water pressure was considerably higher than that predicted by the theory of steady flow through straight cylindrical channels near the glacier bed. Possible reasons are considered. The very large disagreement between measured and predicted pressure suggests that no straight cylindrical channels may have existed.

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Investigating the potential to determine the upstream accumulation rate, using mass-flux calculations along a cross-section on a small tributary glacier in Heimefrontfjella, Dronning Maud Land, Antarctica

Annals of Glaciology ◽

10.3189/172756404781814555 ◽

2004 ◽

Vol 39 ◽

pp. 175-180 ◽

Cited By ~ 3

Author(s):

Veijo Allan Pohjola ◽

Jim Hedfors ◽

Per Holmlund

Keyword(s):

Cross Section ◽

Catchment Area ◽

Accumulation Rate ◽

Surface Velocity ◽

Velocity Data ◽

High Accumulation ◽

Glacier Surface ◽

Mass Fluxes ◽

Orographic Enhancement ◽

Dronning Maud Land

AbstractHow well can we estimate the incoming ice flux by calculating the ice flux through a well-defined cross-section? We test this by comparing calculated ice flux out from the small glacier Bonnevie-Svendsenbreen with the measured accumulation rate integrated over the well-defined catchment area in the Sivorgfjella plateau, Dronning Maud Land, Antarctica (74˚45’ S, 11˚10’ W). The ice flux is calculated using ice-dynamical properties from an ice temperature model and the distribution of forces calculated using a force-budget model. The input we use includes velocity data of the glacier surface, combined with ice-thickness measurements. The result is an accumulation rate on the Sivorgfjella plateau of 0.50 ± 0.05 mw.e. a–1. We find that this is similar to the accumulation rate recorded by ground-penetrating radar work in the area. We therefore find the balance-flow method, in combination with the force-budget technique and ice temperature modeling, to be a useful tool for studies of mass fluxes in a catchment area. The most important source of uncertainty in these calculations is the quality and the spatial distribution of the ice surface velocity data. The high accumulation rate shows the effect of orographic enhancement on accumulation in montane areas in Antarctica.

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