The influence of correlated crustal signals in modelling the main geomagnetic field

Abstract

Algorithms used in geomagnetic main-field modelling have for the most part treated the noise in the field measurements as if it were white. A major component of the noise consists of the field due to magnetization in the crust and it has been realized for some time that such signals are highly correlated at satellite altitude. Hence approximation by white noise, while of undoubted utility, is of unknown validity. Langel, Estes & Sabaka (1989) were the first to evaluate the influence of correlations in the crustal magnetic field on main-field models. In this paper we study two plausible statistical models for the crustal magnetization described by Jackson (1994), in which the magnetization is a realization of a stationary, isotropic, random process. At a typical satellite altitude the associated fields exhibit significant correlation over ranges as great as 15" or more, which introduces off-diagonal elements into the covariance matrix, elements that have usually been neglected in modelling procedures. Dealing with a full covariance matrix for a large data set would present a formidable computational challenge, but fortunately most of the entries in the covariance matrix are so small that they can be replaced by zeros. The resultant matrix comprises only about 3 per cent non-zero entries and thus we can take advantage of efficient sparse matrix techniques to solve the numerical system.We construct several main-field models based on vertical-component data from a selected 5" by 5" data set derived from the Magsat mission. Models with and without off-diagonal terms are compared. For one of the two Jackson crustal models, k3, we find significant changes in the main-field coefficients, with maximum discrepancies near degree 11 of about 27 per cent. The second crustal spectrum gives rise to much smaller effects for the data set used here, because the correlation lengths are typically shorter than the data spacing. k, also significantly underpredicts the observed magnetic spectrum around degree 15. We conclude that there is no difficulty in computing mainfield models that include off-diagonal terms in the covariance matrix when sparse matrix techniques are employed; we find that there may be important effects in the computed models, particularly if we wish to make full use of dense data sets. Until a definitive crustal field spectrum has been determined, the precise size of the effect remains uncertain. Obtaining such a statistical model should be a high priority in preparation for the analysis of future low-noise satellite data.

Revised magnetic power spectrum of the oceanic crust

“…That means correlation of the magnetization is given only for separations smaller than approximately 2200 km. Correlation functions associated with the theoretical models of Jackson [1994] and Langel et al [1989] have been calculated by Rygaard‐Hjalsted et al [1997]. Comparing our result to those, it agrees well with that of Jackson [1994].…”

Maximum entropy regularization of time-dependent geomagnetic field models

“…Moreover, the statement is reliable only if both correlation in the errors and error estimates are perfectly represented; unfortunately this is not the case for the geomagnetic problem. For instance anisotropy in the error treatments as well as correlation between errors could be taken into account (Holme & Jackson 1997; Rygaard‐Hjalsted et al 1997). Furthermore, a model m will in practise not simultaneously satisfy the collection of conditions for all the subsets of data γ k , of size N k .…”

Post-processing scheme for modelling the lithospheric magnetic field

“…From a practical point of view, scientists have been relatively successful in estimating a priori the noise in gravity or magnetic data sets, however correlations between errors have been mostly ignored. This is partly because, when known, the full covariance matrix for the data errors is generally so large that it cannot be handled easily, even on modern computers (but see for example Langel et al (1989); Holme and Bloxham (1996); Rygaard-Hjalsted et al (1997); Holme (2000), where correlated errors are accounted for in geomagnetism).…”