Parameter trade-off in one-dimensional magnetotelluric modelling
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The problem of inverting or modelling one-dimensional magnetotelluric data can, today, be considered as largely resolved. Attention now focuses on the class of acceptable models. Viewed in the space of model parameters this class occupies a singly connected volume, bound with a surface where the standard deviation ε between measured and calculated response exceeds the minimum εo of the best-fitting model by a constant factor (typically ε ≈ 1.10 εo). This volume of acceptable models is described by its intersections with the parameter axes, and also by the extreme excursions possible for any of the model parameters when all the other parameters are adjusted accordingly. These extreme excursions therefore represent "trade-off" conditions among the model parameters and are summarized in the "trade-off matrix". In a sense this is a generalization of the parameter correlation matrix, which gives only local information in the vicinity of a proposed model. The trade-off matrix, however, is independent of any initial model. Another important question considered deals with the correct choice of the number of layers with which to model a data set. Whereas a single minimum of ε is found with the correct number no, when this number is too small the information contained in the data is spread among several isolated minima. When n > no the problem becomes "illposed ". There are too many degrees of freedom and it becomes possible, then, to move in model space in directions at right angles to the meaningful dimensions without finding a clear minimum. The problem is analogous to a vanishing determinant in linear algebra. To find a regular problem again it is necessary to specify auxiliary constraints provided, for example, by other soundings or prior geological knowledge, to compensate for the increased number of variables.
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