Resolution and informational aspects of surface inversion on ice streams

G. Hilmar Gudmundsson

British Antarctic Survey, Cambridge, UK

Melanie Raymond,

Section of Glaciology, ETH, Switzerland

The Bayesian approach

Forward model: (s,u,w)=f(b,c)

The Bayesian approach to inverse problem is to determine the

conditional probability P(b,c|s,v,w)

We look for a system state that maximizes this probability, and determine the pdf of that system

state. Requires covariance matrixes of measurements errors and a priorie.

Forward model: A full-system non-linear FE modelModel derivatives: Linear transfer functions

Can slipperiness/bedrock perturbations be

misinterpreted as bedrock/slipperiness perturbations (mixing effects)?

Any fundamental limits to retrieval?

How is retrieval affected by the accuracy of measurements?

Questions:

Bed-to-surface transfer function for a slip-ratio of 10 and surface slope 3 deg.

Raymond & Gudmundsson, JGR, 2005

Forward problem

Bed-to-surface transfer amplitudes on ice streams are large. This is good news for inversion

surface data model bed properties errors

prior cov. model derivatives error cov.

Retrieval method

Model derivatives approximated using linear transfer functions

Case study #1: Measurements of high quality • Data errors 1% of mean values

• Data points every 0.25 ice thickness

Kernels give information about smoothing/resolution and mixing aspects of the retrieval methods. Ideally the bb and cc kernels would be diagonal

(perfect resolution), and bc and cb kernels zero (no mixing) Conclusion: b-perturbations resolved spatially about 10 times

better than c-perturbations. Mixing not a problem

Kernels for `high quality’ data in Fourier space

Kernels for `high quality’ data in real space

Ideally the bb and cc kernels would be delta functions (perfect resolution), and bc and cb kernels zero (no mixing). The area (number in boxes) gives

the sensitivity of the retrieval to the data, as opposed to the a priori.

Case study #2: Measurements of `average’ quality • Thickness errors 10%

• Velocity errors 1%• Data points every 0.25 ice thickness

Kernels for `average quality’ data in Fourier space

b-perturbation retrieval similar to the `high quality’ case, but resolution of c-perturbation considerably worse than before and about 100 times poorer than for the b-perturbation. Mixing still

not a problem

An example of a non-linear (n=3) finite-amplitude inversion

It is not obvious that using the linear-transfer functions as approximation of model derivatives of a non-linear forward model can work.

However, recent results obtained by Melanie Raymond (shown to the right) suggest that this is nevertheless the case. This makes inversion using a full-system non-linear forward model a feasible option.

Conclusion

•Mixing effects not a problem when solving for both

bedrock and slipperiness simultaneously

•Usefulness of basal slipperiness retrieval seems limited unless ice thickness known to better than

about 10%

•Spatial resolution of slipperiness retrieval about an

order of magnitude less than for a bed-line retrieval

Work that remains to be done: Amplitude effects, and effects due to rheological non-linearities