Bed topography and
stickiness inferred from surface measurements on ice streams
Ůr÷stur Ůorsteinsson*, R. Bindschadler||,
G. H. Gudmundsson#, C. F. Raymond*
* Geophysics Program, University of Washington, Seattle, WA 98195, USA
|| NASA Goddard Space Flight Center, Greenbelt, Maryland 20771, USA
# VAW-ETH Zentrum, Gloriastrasse 37/39, CH-8092, Zurich, Switzerland
Introduction. We attempt to relate observations of surface
elevation, S, and velocities, U, V and W, to bed topography,
B, and stickiness, C, at the bed, using transfer functions
(e.g. Gumundsson et al. ).
Using synthetic data we have shown that our techniques can separate the
effects of B and C, even with added white noise, when S and both components of
horizontal velocity (U,V) are given. This is easily satisfied
using synthetic data, but becomes tricky when using real data. We have
looked at small scale (~1 km) features in topography on the surface of ice
streams D and E, and used that information to invert for the bed topography, B, or basal
stickiness, C. One would like to determine both the bed topography and basal stickiness
using information measured at the surface. Since we only have good surface elevation data,
we have assumed two end member cases, that there is only B or C at the bed. In
order to partition correctly between the effects of B and C we need S and (U,V).
So far the (U,V) data has been irregularly spaced, which makes it impractical for inversion.
This study has shown that with surface and velocity data at about 0.25 km spacing we
can distinguish between the effects of bed topography and stickiness and resolve structures as
small as 0.5 km.
What is the source of roughness at the surface of fast flowing ice
Panel 1: Topography and Geographic setting
Small scale undulations on the surface of ice streams, and flow
variations associated with them are pervasive and characteristic of streaming flow.
A typical surface is shown in Figure 1. Here we use the theory
of glacier flow perturbations caused by small bedrock and/or basal stickiness
variations, to investigate the source of these surface features.
The theory has been effectively used to show that flow stripes on ice
streams can form from flow over bedrock bump (or sticky spot) [Gumundsson et al., 1998].
We infer conditions at the bed of ice streams, that is topography and
spatial pattern of basal stickiness, using this theory, which is most
effectively formulated using transfer-functions.
The theory of transfer-functions in the form
derived by G. Hilmar Gumundsson have made this method a practical tool in analysis of bed
perturbations. The transfer-functions relate bed topography and
basal resistance to surface topography and velocity (x,y and z components).
Most of the formulation here is done in
wave-number space, and we will use capital letters (S, B, C, U, V, W) for the
Fourier transform, F, of surface elevation, s(x,y), bed elevation, b(x,y), stickiness
pattern, c(x,y), longitudinal, u(x,y), transverse, v(x,y), and vertical, w(x,y),
velocity, respectively. That is F [s(x,y)] = S(k,l).
Figure 1. Surface topography, of area 4 (Fig. 2).
Figure 2. Ice Streams D and E, and areas 1-4.
Panel 2: The Transfer Functions
The method of transfer functions provides us with a simple way of
predicting surface topography, s(x,y), and velocity, (u,v,w) from
basal boundary conditions, topography b(x,y) and stickiness c(x,y).
The basic assumptions are that we are only looking
at small perturbations to the mean topography and basal resistance. The
ice is assumed linearly viscous, with no vertical variation of viscosity. The
surface topography due to basal topography is S = TSB B or from the
distribution of basal resistance, S = TSC
Both the velocity and surface topography may
be the result of some combination of topography and basal resistance, then the surface and
velocities are given by
S = TSB B + TSC C
U = TUB B + TUC C
V = TVB B + TVC C
We have to specify the ratio of sliding to internal deformation C, the surface slope a, and time (infinit in all our calculations here).
Figure 3 shows the transfer-functions as a function of longitudinal
(k) and transverse (l) wave number.
Gumundsson Transfer-functions, for C = 104 and a = 0.1 deg. (a) TSB, (b) TSC,
(c) TUB, (d) TUC, (e) TVB and (f) TVC.
|Assumptions and Geometry
- Internal deformation much smaller than sliding
- Linear Viscous Fluid Behavior of the Ice
- Bed topography and stickiness variations treated as perturbations
BOX 1: Filtering
The inverse transfer functions will amplify noise, particularly at short wavelengths, and along k = 0 and/or l=0, where the direct transfer functions tend to zero. It is therefore necessary to filter the surface data, so that those wavelengths aren't amplified in the inversion.
We construct our filters by using Wiener
optimal filtering. Given a signal S(f) and a noise N(f) as a function of frequency, the Wiener
F(f) = |S(f)|2/(|S(f)|2+|N(f)|2)
where F(f) is the optimal filter. Note that the measured signal
is C(f) = S(f)+N(f). We therefore have to find a model for the signal
We use the transfer-functions as a model of
the signal, since they determine how much of the true signal can be transferred. The
noise is assumed to be white. Our filters thus usually have the following
FSB = FSB(k,l) =
where FSB is used for the surface to bed inversion, as an example, and N is a constant.
What is Predicted by the Model ?
Panel 3: The Forward Problem
Here we specify a bed topography, b(x,y) and basal stickiness pattern,
c(x,y), and use the theory explained in previous panel to calculate
the surface elevation and horizontal velocities. Figure 4 shows the surface topography and velocities generated by a bedrock bump at the bed. Figure 5 shows the topography and velocities generated on the surface by an assumed stickiness pattern.
In Panel 4 we will use the surface topography, S, and velocity, U, V, obtained here to create a surface and velocity with added noise for
the inverse problem, simply by adding them together.
Figure 4. Given a bed topography, we calculate the surface
elevation and the longitudinal, U, and transverse, V, velocities. Here C=104 and a=0.1 deg.
Figure 5. Given a stickiness pattern, we calculate the surface
elevation and the longitudinal, U, and transverse, V, velocities. Here C=104 and a=0.1 deg.
Panel 4: The Inverse Problem
The inverse problem involves inferring the bed condition from
measurements at the surface. If the surface topography is generated from bed
topography alone, then the inverse is simply B = S/TSB. Equations for U and V can also be solved for B, which gives over-determinacy which can
be used in the inverse procedure. If there is both topography and stickiness, the inverse isn't quite as trivial. We have tried several methods for inversion, and find that an iterative procedure (currently) works the best (see below). The transfer functions are very
sensitive to noise, particularly at short wavelengths, and along k = 0 and/or l = 0, where they tend to zero. It is therefore necessary to filter the data, so that those wavelengths aren't amplified (Box 1).
B-inverse: In the first step we assume that the surface is
entirely due to bed topography, B. We then use the difference in measured
velocities and those predicted to calculate the basal stickiness pattern, C.
The surface due to C is then subtracted off the real surface, and the loop
C-inverse: Assume that the velocities, U and V, are entirely due
to basal stickiness, C, then calculate the surface due to C and use the
difference between predicted and real surface to calculate B. Then we find
the velocity difference and the loop repeats itself.
Figure 6 shows the result of the iterative
method for surface and velocities generated by adding together the respective variables from
the forward problem (Figs. 4 and 5) and adding a 50% random noise.
Since there is over-determinacy in the problem, and due to the wild behavior
of the transfer-functions (Fig. 3), we cannot expect to recover the input
variable completely. Figure 7 shows surface elevation profiles
down the center of the perturbed, unperturbed (without noise) and predicted
||Figure 6. Inverting for the bed topography and stickiness
pattern given in Figs. 4 and 5. We added random noise, 50% of maximum signal
for elevation and velocities, before inverting (see Fig. 7).
||Figure 7. The input bed (blue line) and the bed topography predicted by the inversion (red line).
Panel 5: Application to Ice Stream E
With surface elevation and limited surface velocity data extracted from sequential satellite imagery of ice streams D and E [Bindschadler et al.,1996] we invert for the basal conditions. Since the velocity measurements are sparse and not evenly spaced, we can not use them in the inversion. Therefore we only consider two end member cases, surface due to bed topography (Fig. 8) or stickiness (Fig. 9) only.
The bed relief (Fig. 8) is about 1.5 times the surface relief, or
about ~100m. The velocity perturbation is ~6% of the total surface velocity along flow, and ~3% in the transverse direction (V).
||Figure 8. Assuming that there is only bed topography at the base of area 4 (Fig. 1), we get the bed shown. The velocities at the surface predicted by this bed are shown in the lower two panels. We use C=6000 and a = 0.01 deg.
||Figure 9. Assuming that there is only stickiness at the base of area 4 (Fig. 1), we get the bed shown. The velocities at the surface predicted by this bed are shown in the lower two panels. We use C=6000 and a = 0.01 deg.
Panel 6: Discussion
We have shown that we can retrieve the basal conditions for complex
geometry's even if we add random noise to the data (Fig. 6).
From the discussion in Panel 5 we note that we have to know the surface to within ~10 m and the velocities within better than 3% of mean surface
velocity to be able to resolve both the bed topography and stickiness pattern.
Since we work in wave-number space, it is crucial that the data are evenly spaced, both the elevation and velocity data. Spacing closer than 0.25 km is necessary if we want resolution of about 0.5 km.
- Structures down to ~0.5 km wavelength can be detected
- Good data on surface topography and velocities allows one to
distinguish between bed topography and stickiness
- Evenly spaced surface data, at ~0.25 km spacing is the key for successful
Bindschadler, R., P. Vornberger, D. Blankenship, T. Scambos, and R.
Jacobel, 1996. Surface velocity and mass balance of Ice Streams
D and E, West Antarctica. Journal of Glaciology, 42(142), p. 461-475.
Gumundsson, G. H., C. F. Raymond, and R. Bindschadler, 1998. The origin
and longevity of flow-stripes on Antarctic Ice Streams. Annals of
Glaciology, 27, p. 145-152.