Numerical Evaluation of magnetic absolute Measurements with
arbitrary distributed DI-Fluxgate Theodolite Positions
Abstract. At geomagnetic observatories so called absolute measurements are used to determine the calibration parameters of the main three-axis-magnetometer and their long term drift. This allows to get the vector of the geomagnetic field in an absolute geographic reference frame over long periods of time in order to study the secular variation of the Earth's magnetic field. Absolute measurements of the magnetic declination D and inclination I are done by means of a nonmagnetic theodolite with a fluxgate sensor mounted on its telescope parallel to the optical axis. A fluxgate measures the magnetic field component along its sensor axis. The reading S of this magnetometer vanishes if it points towards a direction perpendicular to the field. For absolute measurements standard measuring scheme using six to eight such theodolite positions are established routines in magnetic observatories. These standard DI schemes allow for a simple numeric evaluation and cancel out the influence of instrument parameters like sensor offset and misalignment angles between fluxgate sensor and telescope. We present a numerical method that allows to evaluate measurements of an arbitrary number (minimum 5 as there are 5 independent parameters) and of arbitrary theodolite positions and exploit it to this end. We implement an instrument model to calculate the fluxgate reading S in dependence of field, instrument parameters and telescope direction (herein after referred to as theodolite position). Inserting actual measured values gives one nonlinear equation for each theodolite position. Eventually this is resulting in an overdetermined system of nonlinear equations. This system is solved in the sense of a least square solution using the Gauss-Newton-method generalized to an overdetermined system. The accuracy of the resulting D, I and base values is given in terms of estimated variances. The accuracy of the resulting D and I values depends on both, the choice of used theodolite directions and on the accuracy of the measurements. The quality of each individual measurement can be assessed by means of calculated residuals. A general approach has significant advantages. The method allows to seamlessly incorporate additional measurements for higher accuracy. Individual erroneous readings are identified and discarded without invalidating the entire data set. We show how a-priory information can be incorporated and how that allows to even evaluate a very reduced data set. We expect the general method to ease requirements for both manual and automated DI-flux measurements. It can reveal certain properties of the DI-theodolite which are not captured by the conventional method. Based on the new method, a new measuring schema is presented. It avoids the need to calculate the magnetic meridian prior to the inclination measurements. Adjustment is always done with the same fine adjustment wheel, the one for the horizontal circle. Leveling of the telescope is not necessary and thus leveling errors are avoided. All these makes the measurements faster and less prone to errors. The option of using measurements off the normal DI positions makes measurements in the vicinity of the magnetic equator possible for theodolites without zenith ocular.