A proposal for regional modelling at the Earth's surface, R-SCHA2D

Abstract

S U M M A R YModelling the magnetic field at a regional scale has always been challenging. Several techniques were proposed in the past but few solve Laplace equation and meet the requirements of potential theory. Among the regional techniques, the Spherical Cap Harmonic Analysis (SCHA) that solve Laplace equation has become widespread. However, recognizing the inherent difficulties of SCHA to represent the main field over a small spherical cap, a revision of this technique the R-SCHA was proposed to deal with multilevel magnetic measurements. For the reasons explained in this paper, R-SCHA is not suitable either when data are available at one surface only. The present paper proposes an alternative to SCHA and R-SCHA, named R-SCHA2D, that is suitable for modelling geomagnetic data available at a quasi-constant altitude. The efficiency of R-SCHA2D is successfully tested on a set of repeat station data. The magnetic field over France and its secular variation are reliably represented between 1965 and 2007.5. The regional parameters are obtained from the joint inversion in space and time of 29 metropolitan repeat stations and 1 observatory vector data using time power series as a temporal basis function. This approach does not require additional constraints like adding synthetic data, regularizing the inverse problem, or removing a trend prior to the inversion to be stable. It is shown that the geomagnetic jerks over France are detected in the regional parameters themselves and that it is possible to estimate the crustal bias. Considering the real data distribution in space and time, and the parameters used for this modelling, the R-SCHA2D model compared with the CM4 comprehensive model showed no indisputable evidence proving the existence of a secular variation anomaly over France. The main field and the secular variation are sketched for various epochs, in particular for epoch 2007.5. The present approach does neither address the problem of internal/external fields separation nor the upward continuation of the model to satellite altitudes.

A statistical spatial power spectrum of the Earth's lithospheric magnetic field

“…Beggan et al 2013), on combination of band-limited SHs (e.g. Lesur 2006), or on revised spherical cap harmonic analysis (Thébault 2008), for instance.…”

Toward a better representation of the secular variation. Case study: The European network of geomagnetic observatories

“…However, in case of the data set available from European observatories, the piecewise fit of second degree polynomials to data, be they the field or its first time derivative, can be applied to map the internal field secular change, as was done in Section 3, only for a limited time interval , in which the observatory network providing data is dense enough. An alternative solution, using main field models at global (see recent reviews by Olsen et al (2007) and Jackson and Finley (2007)) or at regional scales (Korte and Holme, 2003;Thébault, 2008;Verbanac et al, 2009), spanning a long enough time interval to get meaningful information on the secular variation, has potential in mapping the SV for areas and time intervals with a less dense distribution of observatory data. Among drawbacks of available models based only or mainly on observatory data that could be used in SV mapping are: assigning distorted information to areas or time spans uncovered with data and leakage of external signal into main field models.…”

“…Because at the observing point these effects comprise both the direct and the induced fields, and having in view the frequency difference of the SC variation in comparison to the secular variation, a suitable filter applied to data would eliminate both the external fields and their induced effects. Recent works (Korte and Holme, 2003;Thébault, 2008;Verbanac et al, 2009) use a quite different way to deal with the external signal, modeling the internal field from data by means of a spherical cap harmonic analysis (SCHA). We think that the problem of external effects leaking into the main field model, which we discuss in the present paper in relation with global field models, might be a problem for the SCHA too.…”

Building the second version of the World Digital Magnetic Anomaly Map (WDMAM)

“…The global WDMAM model is first used to compute synthetic vector measurements to SH 900 on a dense equal area grid located at the Earth's mean radius. A modelling method relying on 2D spherical local functions (Thébault 2008) is then applied to fit the synthetic measurements to an equivalent SH degree 900 within 2000 caps homogeneously distributed on the reference sphere. The power spectrum is computed for each cap, and the cutoff harmonic degree is defined as the first degree from which the power spectrum value is lower than the misfit value between the regional model and the synthetic measurements.…”