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Constable, C, Parker R.  1991.  Deconvolution of Long-Core Paleomagnetic Measurements - Spline Therapy for the Linear Problem. Geophysical Journal International. 104:453-468.   10.1111/j.1365-246X.1991.tb05693.x   AbstractWebsite

The magnetization of long cores of sedimentary material is often measured in a pass-through magnetometer, whose output is the convolution of the desired function with the broad impulse response of the system. Because of inevitable measurement noise and the inherent poor conditioning of the inverse problem, any attempt to estimate the true magnetization function from the observations must avoid unnecessary amplification of small-scale features which would otherwise dominate the model with deceptively large undulations. We propose the construction of the smoothest possible magnetization model satisfying the measured data to within the observational error. By means of a cubic spline basis in the representations of both the unknown magnetization and the empirically measured response, we facilitate the imposition of maximum smoothness on the unknown magnetization. For our purposes, the smoothest model is the one with the smallest 2-norm of the second derivative, the same criterion used in the construction of cubic spline interpolators. The approach is tested on a marine core that was subsequently sectioned and measured in centimetre-sized individual specimens, with highly satisfactory results. An empirical estimate of the resolution of the method indicates a three-fold improvement in the processed record over the original signal. We illuminate the behaviour of the numerical scheme by showing the relation between our smoothness-maximizing procedure and a more conventional filtering approach. Our solution can indeed be approximated by convolution with a special set of weights, although the approximation may be poor near the ends of the core. In an idealized system we study the question of convergence of the deconvolution process, by whether the model magnetization approaches the true one when the experimental error and other system parameters are held constant, while the spacing between observations is allowed to become arbitrarily small. We find our procedure does in fact converge (under certain conditions) but only at a logarithmic rate. This suggests that further significant improvement in resolution cannot be achieved by increased measurement density or enhanced observational accuracy.

Constable, C.  2007.  Dipole moment variation. Encyclopedia of geomagnetism and paleomagnetism. ( Gubbins D, Herrero-Bervera E, Eds.).:159-161., Dordrecht: Springer Abstract