Accurate determination of magnetic field gradients from four point vector measurements .1. Use of natural constraints on vector data obtained from a single spinning spacecraft

Cluster introduces a new generation of spacecraft that will measure the spatial gradients of the magnetic field in the Earth's magnetosphere. As gradients require knowledge of differences, small errors resulting from an inadequate knowledge of the orientations, zero levels and the scale factors of the magnetometer sensors affect the calculation of field gradients disproportionately and must be removed with high accuracy, We show that twelve calibration parameters are required for each of the spacecraft (for a total of 48 for the four spacecraft) to correctly infer the measured magnetic fields at each of the spacecraft. By application of a Fourier transform technique, some of the parameters can be recovered, We will show that errors in eight of the twelve calibration parameters generate coherent monochromatic signals at the first and the second harmonics of the spin frequency in the despun data. These narrow-band signals can be readily characterized because of the natural constraint that low frequency geophysical signals in the Earth's magnetosphere have a broad-band character, We relate the real and the imaginary parts of the monochromatic signals to the eight calibration parameters. We then present a least squares scheme that improves the eight calibration parameters by iteration until the power of the coherent signal superimposed above the broad-band background is minimized. In an accompanying paper, we report on another technique that determines the rest of the calibration parameters by utilizing the natural constraints that del . B is zero everywhere and del x B is vanishingly small in certain regions of the magnetosphere.