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.

Abstract

“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)

(where I 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

“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

Cookbook Procedure to Determine M

1. Estimate MUse 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

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

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)

Figure 1Figure 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)

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

Figure 3

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

Cookbook Procedure to Determine M(continued)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

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

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)

Comparison to “K” Matrix Method 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

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

Figure 5

Mill Emphasis/De-emphasis with “M” matrix and MP

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(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)

Mill Emphasis/De-emphasis with “M” matrix and MP (continued) 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.)

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

Figure 6

Future Work 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

Figure 7

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

Future Work(Continued)