Electric Field Measurement Using Rotating Vane Mills near Kennedy Space Center

1. Abstract 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, (EX, EY, EZ) and the field, EQ, due to charge on the aircraft. Our goal is to determine the "full field" E = (EX, EY, EZ, EQ) 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 = C*m, 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 = M*E) 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, EXest = 1*EXtrue + 2*EYtrue + 3*EZtrue + 4*EQtrue where 1>> 2, 3, 4). We use this information to correct the estimated electric field. We then use the Moore-Penrose pseudoinverse and the mill responses again to create a better M matrix. 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.

2. “M” Theory • Each field mill output can be considered as a linear sum of the external electric field, field due to charge on the aircraft, and “other” terms: mi = Mxi*Ex + Myi*Ey + Mzi*Ez + Mqi*Eq + (?) (a) •  can depend on anything other than Ex, Ey, Ez, or Eq • The set of equations (a) for all mills on an aircraft can be represented as a matrix equation: m = M*E (b) • where m (mill outputs) & E (vector electric field and field due to charge on the aircraft) are vectors, M is a 6x4 matrix, and we are “neglecting” the “other” () terms for now • However, we need the 4x6 matrix C which satisfies the equations: E = C*m (c) C*M = I (d) • (whereI is the 4x4 identity matrix) to calculate the electric fields from the mill outputs • So we need a way to get from M to C in a logical way • To calculate C from M, we use the Moore-Penrose pseudoinverse [Penrose, R., A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51,406-413, 1955] (MP) • One major advantage of using the MP is that when properly invoked, it is not sensitive to DC offsets in the data

3. “M” Theory(continued) • Although we need C to determine E from the mill outputs, m, the unique properties of the M matrix drive our method • There is only one M, that satisfies (b) for all possible values of E and m • In the process of determining C from M, we can manipulate the MP to emphasize or de-emphasize individual mills in the determination of E • To determine M, we follow a “cookbook” type procedure: • Estimate M • Determine C from M • Calculate the estimated E from C and m • “Fix” E based on knowledge of flight conditions • Use “fixed” E and m to determine new M • Repeat • Final M scaling • We will use examples from the latest ABFM program • UND Citation II aircraft • Flew in and around study clouds near KSC, FL during 2000 & 2001

4. Cookbook Procedure to Determine M • 1. Estimate M Use aircraft symmetry arguments to estimate M -Enhancement factor for EY, EZ, EQ around 2 (cylinder) for front mills -Angle of top forward mills 47° down from vertical centerline -Angle of bottom forward mills 114° down from vertical centerline -Enhancement factor for EX around 3 (guess) for front mills -Top aft mill +EZ & -EX only -Bottom aft mill -EZ & -EX only

5. Cookbook Procedure to Determine M(continued) • 1. Estimate M (continued) • First guess estimate of M: • 3.00 1.83 -0.81 2.00 • 3.00 1.36 1.46 2.00 • 3.00 -1.83 -0.81 2.00 • 3.00 -1.36 1.46 2.00 • -3.00 0.00 -2.00 2.00 • -3.00 0.00 2.00 2.00 • 2. Determine C from M • MP pseudoinverse of first guess M: • 0.0463 0.0370 0.0463 0.0370 -0.0751 -0.0916 • 0.1760 0.1308 -0.1760 -0.1308 0.0000 0.0000 • -0.0863 0.0863 -0.0863 0.0863 -0.1521 0.1521 • 0.0695 0.0555 0.0695 0.0555 0.1374 0.1126 • 3. Calculate E from C & m • Select set of m values where the electric field components are relatively simple to deduce • For example, we chose m values during a set of roll/pitch maneuvers in fair weather that also had some aircraft charging • External field constant • The EX & EY fields are 0 except during the roll/pitch maneuvers • There is significant aircraft charge to determine EQ components

6. Cookbook Procedure to Determine M(continued) 3. Calculate E from C & m (continued) • The example data (Figure 1): • For the fair weather case chosen, the external electric field (EFW) is vertical, constant over the calibration period, and decreases with altitude • When the aircraft is flying straight & level: EX = EY = 0 (e) EZ = EFW (f) EQ is low (g) • During roll maneuvers: EX = 0 (h) EY = EFW*sin(roll) (i) EZ = EFW*cos(roll) (j) EQ should be low (k) • During pitch maneuvers: EY = 0 (l) EX = EFW*sin(pitch) (m) EZ = EFW*cos(pitch) (n) EQ should be low (o)

7. Figure 1 • Figure 1 shows the aircraft & mill data from the set of roll/pitch/charging maneuvers during June of 2001 • Note that the mill field changes during the roll/pitch maneuvers are much smaller than during the aircraft charging • Note there is aircraft charging during the pitch maneuver due to turbulence/engine RPM changes • The top two plots are the roll/pitch data while the bottom 6 plots are the mill outputs during the roll/pitch maneuvers (Port Down, Port Up, Starboard Down, Starboard Up, Aft Down, and Aft Up)

8. Figure 3 Figure 2 • Example of “fixing” E: • Figure 2 is a plot of the estimated E from the initial M & C matrices • Note that all three components are dominated by the EQ • Figure 3 is a plot of the same estimated E from the initial M & C matrices (Figure 2) with the estimated EQ subtracted • Note that the charging due to aircraft turbulence/engine RPM is still present to some degree • Note that the EX has some contamination of EY

9. Cookbook Procedure to Determine M(continued) 4. “Fix” E • Each estimated E component will contain the “true” component plus contaminations from the other 3 “true” components • For example: EXest = 1*EXtrue + 2*EYtrue + 3*EZtrue + 4*EQtrue (p) • If our first guess of M was “reasonable”: 1 >> 2, 3, & 4 (q) • Using our knowledge of the true external fields and the estimated external fields, we can determine approximate values for 1, 2,3, &4 • We use these estimated values of the various ’s to “correct” the estimated E value determined from the initial M & C matrices

10. 5. Use “Fixed” E and m to Determine new M & C We take the mill outputs and “divide” them by the “fixed” E values: M = m/E (r) Note that since we are using multiple values of both m & E to determine M, we are actually performing a matrix multiplication of the m “matrix” with the MP inverse of the E “matrix” Example first iteration M: 3.7671 0.7418 -2.3653 2.4892 2.4985 2.7971 3.1282 2.4905 5.3042 -0.8989 -1.7117 2.4650 0.4301 -2.6400 2.2488 2.6914 -4.7430 0.0892 -0.6057 1.7765 -1.2570 -0.0892 0.6057 1.1015 6. Repeat Steps 1-5 until M converges After 3 iterations, the M matrix is: 2.2868 2.4493 -0.0099 2.4116 8.2428 1.5658 2.1827 2.7197 2.4925 -2.1202 -0.5461 2.3643 7.0092 -2.5815 2.3365 2.7062 -7.8235 0.0832 -3.2351 1.7508 -0.9226 0.1958 2.0617 1.1227 Cookbook Procedure to Determine M(continued)

11. Figure 4 • Example data: • The electric fields for the M matrix is shown in blue in Figure 4 • The red plot in Figure 4 is the “ideal” fields based on roll/pitch information and a uniform vertical fair weather field (scaled to EFW) • Note that there is a lot of fine structure in the “true” electric field

12. Cookbook Procedure to Determine M(continued) 7. Final scaling of M matrix • After the iteration process has stabilized, the M matrix is internally consistent • The relative magnitudes of the EX, EY, EZ, & EQ components are correct relative to each other • For the E values obtained with the M matrix to represent the actual external E, we must compare them to a known field value (i.e., find c): Mtrue= c*Mrelative (s) • For our dataset, we do a low fly-by of the KSC ground based field mill system • We compare the ground field with the value obtained with the M matrix at the time when the aircraft passed by the ground based mill • The ratio between the field at the ground and aircraft is the constant we multiply with the intermediate M matrix to get the final M matrix Eg = c-1*Ea (t)

13. Comparison to “K” Matrix Method Figure 5 • The blue plot in Figure 5 is the final M based E data while the red plot is the K based E data • Note: • The DC offsets are left in both datasets to help differentiate the two sets of results • K method does not calculate EQ • The two methods handle the charge spike around 21:16 differently • The K and M methods derive similar EY & EZ fields but different EX field • On this aircraft & mill configuration, the EX values have been difficult to derive • To verify our results, we compare our M matrix E with those of the “K” matrix method [Koshak, W., J. Bailey, H. Christian, and D. Mach, Aircraft electric field measurements: Calibration and ambient field retrieval, J. Geophys. Res., 99, 22781-22792, 1994] on the same dataset

14. Once you have determined the M matrix, it is possible to de-emphasize selected mills when calculating the C matrix By properly incorporating a “weight factor” matrix (W) in the MP, one or more mills will be minimized or not used at all in the C matrix calculation of E The W matrix is very similar to the 6x6 identity matrix with the diagonal term corresponding to the de-emphasized mill set to a value much less than 1 (or even 0 to completely “turn off” that mill in the C matrix) Mill Emphasis/De-emphasis with “M” matrix and MP

15. Mill Emphasis/De-emphasis with “M” matrix and MP(continued) • For example: • Given the simplified MS matrix (r), the nominal CN matrix derived with the MP is (s): • 1.4000 0.4000 -0.9000 2.1000 3.7000 0.4000 1.8000 2.3000 MS = 4.0000 -2.5000 -1.5000 2.0000 (r) 1.6000 -2.5000 2.0000 2.3000 -2.3000 0.0000 -1.0000 1.6000 -0.7000 -0.1000 1.1000 1.0000 0.0181 0.1010 0.0792 -0.0574 -0.1429 -0.0680 CN = 0.1487 0.1885 -0.1415 -0.2045 0.0062 -0.0020 (s) -0.1073 0.1046 -0.1543 0.1504 -0.0872 0.0869 0.1539 0.0785 -0.0112 0.0449 0.2052 0.0870 • The W5 matrix (t) will de-emphasize the data from mill #5 by a factor of 10 (W5(5,5) = 0.1) by using the equation (u): 1.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 W5 = 0.0 0.0 1.0 0.0 0.0 0.0 (t) 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 1.0 • CW = pinv(W5*MS)*W5 (u) • -0.1495 0.2145 0.1037 -0.0861 -0.0054 -0.1801 • CW = 0.1559 0.1836 -0.1425 -0.2032 0.0002 0.0025 (v) • -0.2096 0.1739 -0.1393 0.1329 -0.0033 0.0184 • 0.3946 -0.0845 -0.0465 0.0862 0.0078 0.2480 • The weighted CW matrix (v) shows much lower values in the 5th column (the contributions to the field values from mill #5) than the the C matrix (s) yet will still produce nearly the identical E as the C in equation (s)

16. Mill Emphasis/De-emphasis with “M” matrix and MP (continued) Figure 6 • The weight factor allows us to remove mills from the determination of E if there is a problem with a mill output (bad mill, excessive corona, etc.) • Figure 6 shows examples of mill de-emphasis data • The blue plot in Figure 6 is the example data with all mills equally weighted (using nominal C matrix) • The red plot is the same data using the C matrix with mill # 3 de-emphasized by a factor of 10 (weight factor 0.1) • The green plot is the same data using the C matrix with mill #5 “turned off” (weight factor 0.0) • Note that we did not remove the DC offsets from the two weighted plots so that the three example plots could be distinguished • The weight factor allows us to remove mills from the determination of E if there is a problem with a mill output (bad mill, excessive corona, etc.)

17. Future Work Figure 7 • Nonlinear effects • In our original formulation of the M matrix problem [Equation (a)], we had a term [(?)] that did not depend linearly on the external electric field or aircraft charge • For the rest of the derivation, we neglected that term assuming it was small • We see evidence in our dataset that the (?) term is not always small • The EQ features shown in Figure 7 do not effect each mill equally • The first is caused by aircraft turbulence/engine RPM while the others are caused by artificial charging of the aircraft (stinger) • Given the linear EQ derived from the mill data, it is not possible to null all “EQ” features in the EX, EY, and EZ plots of Figure 7 (see blue plot vs. red plot) • If our assumption of (?) being always small were true, we should be able to null all EQ features shown in Figure 7 with the same EQ term • We need to determine how to model this (?) term • For a further example of (?) in our Citation ABFM data, EX is often unipolar while traversing anvil clouds

18. Future Work (Continued) • Detecting failing mills • Using the M and C matrices along with the other mill outputs, we can determine what each mill’s output “should” be and detect when it is not working correctly • Using self-consistency to check validity of M matrix • Using a “round robin” approach, we can verify each mill’s calibration with the outputs of the other mills • Each mill’s output can be determined from the output of the other mills • We can use this information to check the original M matrix data