Estimating the magnitude of the sum of two magnetic fields with uncertain spatial orientations, polarizations, and/or relative phase

A problem frequently encountered when modeling the power frequency magnetic fields, B and A, produced by two sources is the necessity of estimating the root mean square (rms) magnitude of their sum, i.e., T = \B + A\, when the rms magnitudes, B and A, of the fields are specified by the model, but not necessarily their spatial directions, polarizations, and/or relative phase. The estimator Q = rootB(2) +A(2) was proposed many years ago for this purpose. The accuracy of this estimator is characterized in this paper. If it is known that B and A are approximately linearly polarized and in phase, the maximum bias (i.e., systematic) and random errors for Q used to estimate Tare 6.1 and 35%, respectively, when B = A. These errors decrease as the difference between B and A increases. The bias and random errors are, respectively, 3.2 and 26% when B = 2A or A/2 and 0.2 and 5.8% when B = 10A or A/10. If the directions, relative phase, and polarizations of the two fields are unknown, Q has maximum bias and random errors of approximate to2.6 and approximate to 23%, respectively, when B = A. These errors decrease to approximate to 1.5 and 18% when B = 2A or A/2 and approximate to 0.08 and approximate to 4.0% when B = 10A or A/10. If B and A are known to be linearly polarized and collinear, but with unknown phase between them, the maximum bias and random errors are 11 and 48%, respectively, when B = A. The errors are 5.1 and 32% when B = 2A or A/2 and 0.2 and 7.0% when B = 10A or A/10. Estimators for T with zero bias can be derived, but they are more complicated and increase overall accuracy very little. Bioelectromagnetics 23:59-67, 2002. (C) 2002 Wiley-Liss, Inc.