Simultaneous stochastic inversion for geomagnetic main field and secular variation: 2. 1820–1980

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Abstract
We apply the method of simultaneous stochastic inversion for the geomagnetic main field and secular variation to data from the period 1820–1980, producing two time‐dependent field models, one for the period 1900–1980, the other for 1820–1900. We introduce several adaptations of our original formalism of the inverse problem: we choose to represent the time dependency of the field by an expansion in Chebychev polynomials; we apply an alternative regularization condition in the time domain; and we adopt a different method from that used in our earlier study for solving the resulting system of equations. Whereas our earlier study used only twentieth century observatory data, we consider almost all the available data from the time interval 1820–1980. We examine the effects of the crustal field on our observations and find that existing methods of accounting for these fields as sources of random noise are inadequate in two circumstances. The first is sequences of measurements made at one particular site (for example, at permanent magnetic observatories), in which case the crustal field produces temporal correlation of the errors in the observations: we devise a method for accounting for these correlations. The other is measurements made at satellite altitude, the crustal field producing spatial correlation of the errors. We show how, using a very simple statistical model of the crustal field, these correlations can be calculated. Rather than attempting to account directly for these correlations (which would require, without further assumptions, massive computation) we instead limit our data selection to points sufficiently well separated in space that the effect of the crustal field is minimal. We are able to fit the data to within our estimates of the noise in the data, yet the resulting models have lower power in the secular variation than previously postulated by some authors. Many of the features in the field at the core‐mantle boundary described in recent work by Bloxham and Gubbins are seen most clearly; indeed this time‐varying representation of the field reinforces many of their earlier conclusions.

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