### Fundamental laws of electromagnetism impose constraints that can be exploited.

A method of reducing errors in noisy magnetic-field measurements involves exploitation of redundancy in the readings of multiple magnetometers in a cluster. By "redundancy" his meant that the readings are not entirely independent of each other because the relationships among the magnetic-field components that one seeks to measure are governed by the fundamental laws of electromagnetism as expressed by Maxwell ofs equations.

Assuming that the magnetometers are located outside a magnetic material,that the magnetic field is steady or quasi-steady, and that there are no electric currents flowing in or near the magnetometers,the applicable Maxwell ofs equations are *∇×B = 0* and *∇-B = 0*, where *B* is the magnetic-flux-density vector. By suitable algebraic manipulation, these equations can be shown to impose three independent constraints on the values of the components of *B* at the various magnetometer positions.

In general, the problem of reducing the errors in noisy measurements is one of finding a set of corrected values that minimize an error function. In the present method, the error function is formulated as (1) the sum of squares of the differences between the corrected and noisy measurement values plus (2)a sum of three terms, each comprising the product of a Lagrange multiplier and one of the three constraints. The partial derivatives of the error function with respect to the corrected magnetic-field component values and the Lagrange multipliers are set equal to zero, leading to a set of equations that can be put into matrix vector form. The matrix can be inverted to solve for a vector that comprises the corrected magnetic-field component values and the Lagrange multipliers.

The method was tested in computational simulations of random noise superimposed on readings of a dipole magnetic field by four magnetometers in a cluster like the one shown in the figure. The numerical results of the simulations showed that errors in the magnetometer readings were reduced by values ranging from about 20 to about 40 percent.

*This work was done by Igor Kulikov and Michail Zak of Caltech for NASA's Jet Propulsion Laboratory. For further information, access the Technical Support Package (TSP) free on-line at www.techbriefs.com/tsp under the Physical Sciences category. NPO-40695*

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