CMG Research: Boundary Inverse Problems in Glaciology Final Report

Transcription

1 CMG Research: Boundary Inverse Problems in Glaciology Final Report M. Truffer, D. Maxwell, S. Avdonin October 19, Introduction The primary purpose of this proposal was to develop inverse methods to deduce basal boundary conditions from surface velocity observations of glaciers. In previous work we had developed methods for a simple one-dimensional model that could be formulated as a linear inverse problem (Truffer, 24). In this work we proposed to extend these ideas to non-linear problems that arise when more dimensions are considered. The stated goals of the proposal were to derive robust inverse methods for finding basal velocity fields demonstrate the limits of such inverse methods (e.g. resolution) apply these methods to existing glaciological data sets It is the nature and intend of CMG proposals to not only make substantial contributions in the field of geophysics, but to also advance mathematical research. In the sections below, we separate our completed and some ongoing work along the lines of existing and planned publications. The material is organized into sections that range from almost purely geophysical to almost purely mathematical. A separate section addresses educational contributions. 2 Seasonal evolution of the basal boundary We used surface measurements from Taku Glacier, Alaska, to show that the observed seasonal evolution in surface velocity gradients can be explained by changes in the basal hydraulic gradient. This work was not based on any inverse methods per-se, but rather on exploring a greatly reduced parameter space (till friction angle and hydraulic gradient). However, it helped us develop tools for modeling full force balance ice flow (in 2D) with a plastic boundary condition, and showed that this could explain observed flow features (Fig. 1). This work was published in the Journal of Glaciology (Truffer and others, 29). 1

2 Figure 1: Modelling of a longitudinal section at Taku Glacier. The upper panel shows winter conditions with no water drainage and the lower panel shows summer conditions with elevated basal water pressure. The color bar shows velocities in ma 1. 3 Iterative methods for glacier basal boundary conditions Large non-linear problems are often most efficiently addressed via iterative methods. In Maxwell and others (28b) we introduced an ad-hoc accelerated version of an inverse method known as Kozlov-Maz ya iteration (Kozlov and Maz ya, 199). It operates by making an assumption about the basal boundary condition and then iterating between Neumann and Dirichlet conditions at the surface and the base. Both conditions are known at the surface, and the iteration has been shown to converge in the linear case. We have found it to be too slow in its originally proposed form, but introduced an accelerated version that worked well on synthetic examples, as well as applications to glacier. Figure 2 shows the application to Athabasca Glacier, where measurements of both surface velocities and internal deformation could be used to validate the method. We also applied the model to Glaciar Perito Moreno, where suitable ice thickness and surface velocities are known. The iterative method of Maxwell and others (28b) was applied and extended by others in Arthern and Gudmundsson (21). The method of Maxwell and others (28b) is fast, but suffers from spurious oscillations in grounded or near grounded regions of the glacier and hence has difficulty resolving sharp velocity transitions. These deficiencies make it difficult to infer so-called slipperiness coefficients that relate basal velocities and stresses. We have investigated two methods based on sequential quadratic optimization (SQO) that have the potential to do a better job of resolving these features. The first method infers the basal velocities directly (a constrained Dirichlet problem), while the second infers slipperiness coefficients (a constrained Robin problem). We presented a comparison of these methods with our original Kozlov-Maz ya scheme in Maxwell and others (28a) (Fig. 3). The SQO methods (and especially the constrained Robin problem) do a superior job of resolving basal boundary conditions, particularly when the basal velocities are governed by Coulomb friction. These methods are also significantly more expensive than Kozlov-Maz ya iteration. 2

3 Figure 2: Athabasca Glacier: (a) modeled velocity contour lines ( ma 1 ); (b) contour lines derived from measurements (Raymond, 1971); (c) measured (squares) and modeled (dashed curve) surface velocities and measurement-derived (asterisks) and modeled basal velocities (solid curve); and (d) modeled basal shear stress. 4 Stopping criteria for inverting surface velocities An integral part of any robust inverse method is the recognition that all data (and models) are subject to error. The fact that we are solving ill-posed problems means that it is not only unnecessary to fit data exactly, it is actually undesirable. The reason is that fitting data too well results in unrealistic features in the solutions. The general approach is, therefore, to optimize a certain property of the solution to the inverse problem subject to the condition of obtaining a desired fit to input data (tolerance). It can be shown that our iterative methods produce minimally featured (smooth) solutions. We applied these ideas to the problem of inverting velocity fields for basal stickiness, using the Shallow Shelf Approximation, which is a low-order theory for ice flow over a medium that offers little or no resistance (e.g. soft till or water). These equations have been inverted in previous work before, using a steepest descent method, but not much attention has been paid to the level of desired data fit. We used a series of synthetic ice stream models to demonstrate the effect of over- or underfitting data and showed how to find optimal inversion results, by fitting data to within a given tolerance. In this work we discovered that the steepest descent method converges very slowly and will often not reach optimal resolution in reasonable time. To alleviate this problem, we introduced a new, rapidly converging method, which we labeled the Incomplete Gauss-Newton method. It is based on using a series of regular Gauss-Newton minimizations of the linear problem, and has substantially better performance than either steepest descent or nonlinear conjugate gradient methods (Fig. 4). This work was written up by our graduate student, Marijke Habermann, and is now under review at the Journal of Glaciology. 3

4 Velocity.4.2 Velocity.4.2 Velocity a..2 b..2 c Stress.2 Stress.2 Stress.2 d. e. f. Figure 3: Comparison of inverse method performances for a test case involving a plastic boundary condition. The top row shows results for basal velocity and the bottom for basal stress. The control is shown with dashed lines. Kozlov-Maz ya iteration (a,d) introduces spurious oscillations, which are avoided by SQO methods. b and e shows constrained Dirichlet minimization and c and f constrained Robin minimization Figure 4: Comparison of steepest descent, nonlinear conjugate gradient and incomplete Gauss- Newton methods. The incomplete Gauss-Newton method performs much faster than the others, and the steepest descent method does not reach the desired level of misfit T, even after a large number of iterations. 4

5 Figure 5: Finding a derivative from noisy data using the siple library. 5 A public inverse methods library The work described above led to a variety of tools that are applicable to any (linear or nonlinear) inverse problem, and are thus of interest to a broader community. The tools were all developed in freely available Python software. The toolbox Small Inverse Problems (siple) together with an extensive tutorial and quick refresher on inverse problems is available at The tutorial works through the example of finding a derivative of a (inherently) noisy data series, which is also an ill-posed problems. It is used to demonstrate that these tools are not unique to the geophysical problem being solved, but can be applied quite generally (Fig. 5). Incidentally, the problem of finding derivatives from data has a direct applicability when finding velocities from position measurements, such as the high resolution GPS time series. We plan to explore this in future work. 6 Linking to ice sheet models We are currently working on incorporating the inverse tools developed in this grant to UAF s Parallel Ice Sheet Model (PISM, see This work was started under this NSF grant and is now being continued under a follow-up NASA modeling grant to E. Bueler. It is our goal to include parameter estimation as an integral part of the ice sheet model. The model is open source, and has consequently become one of the most used ice sheet models in the world. 7 Contributions to mathematics Several aspects of our work was published or is prepared for publication in the mathematics literature. It is in keeping with the spirit of the CMG program that contributions span from advances in geophysics to mathematics. While developing iterative inverse tools we have discovered a previously overlooked fact that Kozlov-Maz ya iteration can be posed as a more classical inverse method (Landweber iteration) when a suitable perspective is taken. This observation has lead to two insights: 5

6 The body of existing work on Landweber iteration can be applied to Kozlov-Maz ya iteration. In particular, our acceleration method for Kozlov-Maz ya iteration can be understood in terms of the conjugate gradient method for accelerating Landweber iteration. There are three other inverse schemes (and accelerated variants) that are closely related to Kozlov-Maz ya iteration. A literature search shows that two of the related schemes have been discovered, the third is not known, and the four methods are not already known to be related to each other. We have a manuscript in progress (Maxwell et al., in prep.) for the mathematics literature that develops these ideas. This is an interesting mathematical result with direct implications for geophysics. In particular showing the relation between these different methods clears up some of the confusion for the applied geophysicist to decide which inverse method is the correct one. We also made some advances on the purely mathematical side. The main geophysical goal of our project is solving boundary inverse problems for real glaciers. Even a simple mathematical model of such a problem involves a nonlinear partial differential equation (PDE). More sophisticated models are described by systems of several nonlinear PDEs. Therefore, our project gives rise to many difficult mathematical problems, and its mathematical part is closely related to several areas of mathematics such as the theory of inverse problems for partial differential equations, control theory, approximation theory, and signal processing (sampling and interpolation theory). In Avdonin and others (29b); Maxwell and others (28b) we were able to adjust a very efficient Kozlov Maz ya iteration method (which was originally proposed for systems of linear PDEs) to our nonlinear equations. There are other efficient methods for solving inverse problems for linear PDEs which could potentially serve for our purposes. First of all, we consider the Boundary Control (BC) method in inverse theory. This method is based on controllability of the corresponding partial differential equations. Avdonin and others (29a); Avdonin and Mikhaylov (28); Avdonin (28) contain new results on controllability of partial differential equations. In particular, these are results on graphs; graphs are reasonably considered as a good approximation of 2-dimensional and 3-dimensional problems. On the other hand, for numerical realization of solutions to our problems, it important to understand how to represent a solution to a PDE in a form of a series of its samples. Avdonin and Ivanov (28); Avdonin and others (28) present some new results in this direction. 8 Educational activities As a direct result of this grant, we taught a one-semester graduate short course on Inverse Methods in Glaciology at UAF. The course material was developed during a sabbatical stay and was taught at ETH Zürich as well. The material together with Matlab code can be found at truffer/inverse. This same material was also used to teach a module and supervise a project on inverse theory at the McCarthy Summer School of Glaciology in 21, which will be repeated in 212 ( While the siple library (see above) is well-suited to teaching inverse methods (from theory to application), it has not (yet) been used in that fashion. It is publicly available though and perfectly suited for self-study. This grant has also supported a PhD student in Geophysics (Marijke Habermann) and aspects of PhD mathematics students Victor Mikhaylov and Anna Bulanova. 6

7 9 Dissemination of results In addition to the peer-reviewed literature mentioned in the previous sections we presented results at a variety of meetings and seminars. Habermann has presented aspects of her work at several meetings (Northwest Glaciologists 21, 211; AGU 21; SIRG New Zealand, 21; Karthaus, 29) and at seminars at the New Zealand Antarctic Center and the University of Washington. Maxwell has presented a poster (29) and an invited talk (21) at AGU. Truffer presented at a WAIS meeting, as well as at the 28 IGS meeting in Limerick. We also helped organize a first-of-its-kind special session at the 28 AGU meeting, which attracted 16 abstracts. References Arthern, Robert J and G. H. Gudmundsson, 21. Initialization of ice-sheet forecasts viewed as an inverse Robin problem, J. Glaciol., 56(197), Avdonin, S., 28. Control problems on quantum graphs, Analysis on Graphs and Its Applications, Proceedings of Symposia in Pure Mathematics, vol. 77, Avdonin, Sergei, B.P. Belinskiy and S.A. Ivanov, 29a. Exact controllability of an elastic ring, Applied Math. Optim, 6(1), Avdonin, S., A. Bulanova and D. Ovsyannikov, 28. Optimal cubature formulae related to solutions of initial boundary value problems, Vestnik St. Petersburg Univ., (2), Avdonin, S. and S. Ivanov, 28. Sampling and interpolation problems for vector valued signals in the Paley Wiener spaces, IEEE Trans. Signal Proc., 56(11), Avdonin, S., V. Kozlov, D. Maxwell and M. Truffer, 29b. Iterative methods for solving a nonlinear boundary inverse problem in glaciology, J. Inverse and Ill-Posed Problems. Avdonin, S. and V. Mikhaylov, 28. Controllability of partial differential equations on graphs, Appl. Math., 35, Kozlov, V A and V G Maz ya, 199. On iterative procedures for solving ill-posed boundary value problems that preserve differential equations, Lenningr. Math. J., 1, Maxwell, David, Martin Truffer and Sergei Avdonin, 28a. Inverse Methods for Reconstructing Basal Boundary Data, American Geophysical Union 28 Fall Meeting, (poster). Maxwell, David., Martin Truffer, Sergei Avdonin and Martin Stuefer, 28b. Determining glacier velocities and stresses with inverse methods: an iterative scheme, J. Glaciol., 54, Raymond, C. F., Flow in a transverse section of Athabasca Glacier, Alberta, Canada, J. Glaciol., 1, Truffer, M., 24. The basal speed of valley glaciers: an inverse approach, J. Glaciol., 5(169), Truffer, M., R. J. Motyka, Michael Hekkers, I. M. Howat and M. A. King, 29. Terminus dynamics at an advancing glacier: Taku Glacier, Alaska, J. Glaciol., 55(194),