Motion-induced noise can be an overwhelming source of inteference in vector magnetic field sensors designed for submarine communications in the 30-130-Hz portion of the extremely low frequency (ELF) band. In the case of an ELF antenna constructed using three orthogonal superconducting quantum interference devices (SQUID's) as receiving elements, the removal of motion noise is complicated by the inability of SQUID sensors to determine the absolute value of the earth's field component along the axis of the sensor. An equation is derived that expresses the projection of the ELF signal vector on the earth's field in terms of the SQUID outputs, and an adaptive vector that is approximately equal to the earth's field vector. Two adaptive schemes using the methods of process parameter estimation are described which determine the adaptive vector. The Least Squares method requires no initial estimate of the adaptive vector but is an open-loop method which tends to be slow. The second adaptive scheme, the steepest descent method, is a close-loop technique that has the potential of being rapid, but requires an initial value of the adaptive vector that is close to the true value. Both techniques remove, to first order, sources of noise arising from sensor nonorthogonality, sensitivity differences and dc offset errors. From a computer simulation of both techniques, it is found that the optimum system is a combination of the two methods, using the Least Squares method for estimation of the coarse value of the adaptive vector, and using the Steepest Descent method for fine adjustment.