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 $Q=\sqrt {B^2+A^2}$ 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 T are 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  = 2 A or A /2 and 0.2 and 5.8% when B  = 10 A or A /10. If the directions, relative phase, and polarizations of the two fields are unknown, Q has maximum bias and random errors of ≈ 2.6 and ≈ 23%, respectively, when B  =  A . These errors decrease to ≈ 1.5 and ≈ 18% when B  =2  A or A /2 and ≈ 0.08 and ≈ 4.0% when B  = 10 A 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  = 2 A or A /2 and 0.2 and 7.0% when B  = 10 A 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. © 2002 Wiley‐Liss, Inc.