Centre for Computational Geography, School of Geography, University of Leeds, Leeds LS2 9JT, United Kingdom

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Abstract
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Water evacuation and storage beneath a glacier control glacial dynamics and stability. Rapid glacier sliding is facilitated by high pressure within the basal system which decouple the ice from its bed. However, measurements of water pressure beneath Trapridge Glacier, Yukon Territory, Canada show that the basal water system is highly heterogeneous. Regions of the bed can be divided into two systems, *connected* and *unconnected*, dependent on whether they form part of the evacuation route for basal water. Large spatial pressure gradients can occur within the unconnected system, and between the unconnected and connected systems. During the study period three types of behaviour are seen by instrumenting the bed with water pressure sensors: records which display strong correlation; records which are strongly anticorrelated; records which alternate between strong correlation and strong anticorrelation. We are led to the conclusion that water pressure within the connected system can be viewed as a forcing while other pressures are a response to this forcing. Previous work (Murray & Clarke, 1995) has shown that this input-output relationship can be modelled by low-order nonlinear differential equations, and developed optimised models using repeated application of a simplex algorithm. However, despite optimising the *model parameters* we cannot be sure that the final *model forms* are themselves optimal. In this paper we describe alternative techniques for fitting models which are robust to missing or noisy data, applicable to non-smooth models, and further attempt to derive optimal model forms as well as optimal model parameters.
Four techniques have been used and the results compared with more conventional mathematical models:

*Genetic programming*is a technique that uses model breeding to "evolve" a set of initially random models selecting those models with good fits to become parents of new generation models. If the component model parts are chosen to represent the physical processes occurring then the final result is a single best model that can be examined to give insights into the physics of the subglacial environment. This model was run on the Cray T3D parallel supercomputer at EPCC.*Artificial neural networks*are general problem-solvers capable of providing solutions to nonlinear transformations between vector spaces. Their wide applicability derives from the variety of problems that can be expressed as such transformations. The multilayer perceptron (MLP), the most-widely applied network, consists of layers of processing nodes with weighted interconnections. The MLP may be trained to solve a nonlinear modelling problem by repeated presentation of experimental data, using the errors between calculated and expected results to modify the weighted connections. The trained network may then be used to model previously unseen data. This model was run on a fast workstation with training times lasting several days.*Fuzzy logic*is a multivalued logic system based on fuzzy set theory that provides a means of modelling the behaviour of complex and non-linear systems. A fuzzy model consists of a set of rules expressed linguistically rather than mathematically and an inference engine that executes the rules in response to model inputs. The structure of the model can be derived through expert knowledge or directly from the data through the use of genetic algorithms or artificial neural networks. The rule base and the membership functions of the fuzzy logic models described in the paper have been optimised using a genetic algorithm. This model was run on a PC.*Self organising maps.*This is an unsupervised artificial neural network that will learn how to map a set of inputs onto a set of outputs without being trained. It acts as an associative fuzzy memory. It is similar to a multi-dimensional lookup table driven by fuzzy inputs. It seems to work well in controlling robotic arms, and the question is how well does it perform in modelling the glacial data. This model was run on an old workstation with run times of a few minutes.

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References
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Murray, T. and Clarke, G.K.C. 1995. "Black-box modeling of the subglacial water system", Journal of Geophysical Research, 100(B7), 10231-10245.