Calibration of an Airborne Field Mill Array

We measured the vector electric field in and around targets of interest near the Kennedy Space Center, Florida using a set of six rotating vane electric field mills mounted on a University of North Dakota Citation II aircraft. Each mill on the aircraft responds to the ambient vector field, (E_X, E_Y, E_Z), and the field, E_Q, due to charge on the aircraft. Our goal is to determine the "full field" E = (E_X, E_Y, E_Z, E_Q) from the column vector of mill outputs m = (m₁, m₂, m₃, m₄, m₅, m₆). In matrix form, the linear retrieval for the full field can be written E = Cm, where C is a (4x6) calibration matrix. To determine C, we first estimate its inverse M, where M is a (6x4) matrix (and where m = ME) based on aircraft shape and symmetry arguments. We then use the Moore-Penrose pseudoinverse to estimate C. If the initial estimates of the mill responses M are close, the resultant electric field component estimates will be dominated by the proper component with "contaminations" from the other three (for example, E_Xest = E_Xtrue + \epsilon₁*E_Ytrue + \epsilon₂*E_Ztrue + \epsilon₃*E_Qtrue where \epsilon₁, \epsilon₂, and \epsilon₃ are small). We use this information to correct the calibration matrix. We then use the Moore-Penrose pseudoinverse again to create a better mill response matrix, M. We correct M for any known symmetries in the electric field responses of the mills and continue the iterative process. Once the iterative process has been completed, the final C matrix can then be used to determine E for all cases. We also compare our results to an independent method developed by one of the authors (Koshak). Although the two methods involve very different approaches, they achieve similar results. One advantage of our method using the Moore-Penrose pseudoinverse is that it is simple to emphasize or de-emphasize mills when we calculate the calibration matrix.