International Journal of Applied Electromagnetics and Mechanics 15 (2001/2002) 207–212 207 IOS Press The least square deviations criterion and inverse-source magnetostatic problems Z. Haznadar, ˇ Z. ˇ Stih ∗ and S. Berberovi´ c Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, 10000 Zagreb, Croatia Abstract. A procedure for the determination of the distribution of magnetization in an object of interest using measured disturbances of external magnetic field is described in the paper. The object is divided into a finite number of brick-shaped elements and we prescribe uniform distribution of magnetization within each element. The contribution of each element to the total magnetic field may be calculated analytically in terms of unknown coefficients of magnetization distribution. These coefficients are determined from a set of measured magnetic field values by application of the least square deviations criterion. Such an approach enables a priori analysis of stability of computation. The application of the procedure is illustrated by an example. 1. Introduction Placing of an object with ferromagnetic properties into external magnetic field (i.e.,the Earth’s magnetic field) results in the perturbation of the external field caused by permanent and/or induced magnetization. This deviation in the external field may be measured and used for the detection of the presence of the object. Therefore, the determination of magnetic sources that produce specified perturbation of external magnetic field in a specified region of space plays an important role in detection and demagnetization of ferromagnetic object of interest (such as ships, submarines, vehicles etc.). Magnetic sources in these object are most frequently unknown and we have to determine them using measured disturbances of external magnetic field, so this is a typical inverse-source magnetostatic problem. In order to solve such a problem we use the method of moments [1]. The magnetized object is divided into a finite number of elements that are magnetized by a prescribed distribution of magnetization. Contribution of each element to the total field at a given point may be calculated numerically or analytically as a function of unknown coefficients of magnetization distribution. These coefficients are determined from a set of total magnetic field values that are measured at a set of prescribed measuring points. In an ideal noiseless situation the unknown coefficients may be calculated using simple inversion techniques. In real situations, even negligible noise may cause an ill-posed problem. Accordingly, it is desirable to measure the field at more points than necessary in order to smooth the effect of noise. In this paper we describe the determination of the unknown coefficients from the measured fields using the least square deviations criterion. Such an approach enables a priori analysis of measuring configuration and we may choose a configuration that ensures the computation stability. * Corresponding author: ˇ Z. ˇ Stih, Tel.: +385 1 6129 656; Fax: +385 1 6129 616; E-mail: zeljko.stih@fer.hr. 1383-5416/01/02/$8.00  2001/2002 – IOS Press. All rights reserved