MAY 2015 M A C H 993

Technique for Reducing the Effects of Nonlinear Terms on Electric Field Measurements of Electric Field Sensor Arrays on Aircraft Platforms

D. M. MACH Universities Space Research Association, Huntsville, Alabama

(Manuscript received 7 February 2014, in final form 25 November 2014)

ABSTRACT

A generalized technique has been developed that reduces the contributions of nonlinear effects that occur during measurements of natural electric fields around thunderstorms by an array of field mills on an aircraft. The nonlinear effects can be due to nearby charge emitted by the aircraft as it acquires and sheds charge, but the nonlinear effects are not limited to such sources. The generalized technique uses the multiple independent measurements of the external electric field obtained during flight to determine and remove nonlinear contaminations in the external vector electric field. To demonstrate the technique, a simulated case with nonlinear contaminations was created and then corrected for the nonlinear com- ponents. In addition, data from two different field programs utilizing two different aircraft and field mill configurations, each containing observable and different nonlinear effects, were also corrected for the significant nonlinear effects found in the field mill outputs. The expanded independent measurements in this new technique allow for the determination and correction of components in the field mill outputs from almost any measurable source. Alternate utilization of the technique can include removing effects in the aircraft charging such as aircraft altitude, cloud properties, engine power settings, or aircraft flap de- ployment. This technique provides a way to make more precise measurements of the true external electric field for scientific studies of cloud electrification.

1. Introduction often dominated by the aircraft charge (see example in Fig. 1), thus making it difficult to extract the ambient Measurements of electric fields within clouds have electric field. Note that all of the components of the been made using aircraft for many years (e.g., Winn electric field used in this study are in the reference frame 1993; Merceret et al. 2008; Mach et al. 2009). Retrieving of the aircraft. This study introduces a method that can electric field components from the raw aircraft field detect, and greatly reduce the effects of the nonlinear mill data while in cloud is perhaps the most difficult terms when they become large enough to mask the ex- aspect of the measurement process because of the usu- ternal electric fields. ally high charging of the aircraft (e.g., Jones 1990; Koshak et al. 1994; Koshak 2006; Koshak et al. 2006; Mach and Koshak 2007). The electric field as measured 2. Linear calibration method by an instrument on the aircraft includes linear com- Typically, aircraft are equipped with multiple elec- ponents from the external electric field eX , eY , eZ, tric field mills (Bateman et al. 2007) to indirectly sense charge on the aircraft (eQ), mill offset, and various other phenomenon, including measurement errors and the atmospheric electric field. Note that the field nonlinear terms (l). While in cloud, these terms are mills do not directly measure the external electric fields, but instead measure the fields induced on the aircraft by the external fields and charge on the air- i Denotes Open Access content. craft. The equation relating the output of the th mill (ai) to the external electric field components can be represented by Corresponding author address: Douglas Mach, Universities Space Research Association, 320 Sparkman Dr., Room 4022, Huntsville, 5 1 1 1 1 1 l 1« ai MiX eX MiY eY MiZeZ MiQeQ MiD i i , AL 35805. E-mail: dmach@nasa.gov (1)

DOI: 10.1175/JTECH-D-14-00029.1

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FIG. 2. Electric field data components showing contamination of the (from top) fX (t), fY (t), and fZ(t) fields by eQ(t). Note how FIG. 1. Comparison of the magnitude of the electric field due to similar the black waveforms are compared to the f (t) fields. The aircraft charge to the actual external field components. (top) Raw Q similarities indicate that the other components [ f (t), f (t), and output of the starboard up (SU) field mill a from (1). The remaining X Y 4 f (t)] are contaminated by e (t). The gray plots are the same data panels are the values of the e (t), e (t), e (t), and e (t) fields, re- Z Q X Y Z Q as the black plots, except the e (t) contaminations have been re- spectively. Note that the enhancement factor for a cylinder is 2, so the Q 2 2 moved via the linear method of Mach and Koshak (2007). (bottom) 150 kV m 1 fields at the field mill are dominated by the 80 kV m 1 The f (t) term is the same between the two datasets. fields due to the aircraft charge. The external fields represent only Q about 1%–2% of the fields present at the field mill. 2 3 a1(t) 6 7 6 a (t) 7 6 2 7 6 a (3) 7 where the various M coefficients are the responses of the a(t) 5 6 3 7; (4) 6 . 7 field mills to the external field and charge on the aircraft, 4 . 5 MiD is the offset of the ith field mill, li are the nonlinear am(t) components, and «i are the measurement errors. The li term can include anything from aircraft corona effects e(t) is the vector external electric field (including the (i.e., aircraft in high electric fields may emit corona ions charge on the aircraft) as a function of time, from parts of the aircraft that are detected by the field 2 3 mills) to engine thrust or aircraft angle of attack effects; e (t) 6 X 7 these depend on engine and/or pilot input settings, not 6 e (t) 7 6 Y 7 the external electric field. e(t) 5 6 e (t) 7, (5) For most techniques (e.g., Koshak et al. 1994; Koshak 6 Z 7 4 e (t) 5 2006; Mach and Koshak 2007), the nonlinear terms are Q 1 neglected, so that the system can be solved with a linear algebraic approach, where the ‘‘1’’ is used to eliminate the effects of offsets in the field mill outputs (Mach and Koshak 2007); M is 5 1 1 1 1 1 « ai MiX eX MiY eY MiZeZ MiQeQ MiD i . (2) the m 3 5 calibration matrix,

This can be sufficient to solve for the external electric field (e.g., Mach and Koshak 2007), but at times, non- linear effects may create contaminations that mask the true external electric field. The resultant set of equations from (2) for all components of the electric field and field mill out- puts can be represented by the following matrix equation:

a(t) 5 Me(t) 1 «, (3)

FIG. 3. Field mill locations on the MU-300 aircraft. The desig- where a(t) is the vector of field mill outputs as a function nators are forward up (FU), forward down (FD), port up (PU), of time, starboard up (SU), port down (PD), and starboard down (SD).

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TABLE 1. Linear calibration matrix M for the MU-300. The two letters in the first column designate the location of the field mill: forward up (FU), forward down (FD), port up (PU), starboard up (SU), port down (PD), and starboard down (SD). The columns are for the nose–tail field (eX ), the wing–wing field (eY ), the up–down field (eZ), and the field due to charge on the aircraft (eQ).

eX eY eZ eQ FU 3.6419 20.6712 2.0712 1.6144 FD 1.2252 0.2366 21.8865 1.2584 PU 22.5803 0.6468 0.7650 0.9078 SU 22.4448 20.5988 1.0318 0.6819 PD 20.9059 0.4736 21.2189 1.4480 SD 20.7889 20.9259 21.0386 1.5910

2 3 FIG. 4. Electric field data from the same aircraft/field mill com- M M M M M 6 1X 1Y 1Z 1Q 1D 7 bination as in Fig. 2 but at a different time. The black plots use the 6 M M M M M 7 same calibrations as the black plots in Fig. 2, with the cross con- M 5 6 2X 2Y 2Z 2Q 2D 7; (6) 4 ⋯ 5 taminations of fX (t), fY (t), and fZ(t) fields by eQ(t). The gray plots have the same linear correction as the gray plots in Fig. 2. Notice MmX MmY MmZ MmQ MmD that the correction that removed the cross contaminations in the black plots in Fig. 2 did not remove the contaminations for this and again « are the measurement errors. As given in (3), period of time. For fX (t) and fZ(t), the linear correction (gray plots) the matrix M relates the field mill outputs to the external made the contaminations worse. The inability of the Mach and Koshak (2007) method to correct both time periods with different electric field and charge on the aircraft. Detailed de- amplitudes of eQ(t) contamination indicates the presence of non- scriptions of all of the components of (5) and (6) are in linear contaminations. Mach and Koshak (2007). The matrix M is constant and unique for each aircraft and field mill distribution. Given where the various Cs are the first row of the C matrix. M and assuming measurement errors («) are small, one The full version of (7), including all components, is can directly determine the outputs of the field mills on e(t) 5 Ca(t), (8) the aircraft based on the external electric field vector. Inverting this matrix, by applying the Moore–Penrose where a(t) and e(t) are defined in (4) and (5), re- ‘‘pseudoinverse’’ (Penrose 1955) produces the C matrix. spectively. The C matrix is defined analogous to the M The elements of C are similar to the elements of M in matrix as that they relate the field mill outputs (including offsets) 2 3 to the external electric field and aircraft charge. For C C C C ⋯ C 6 X1 X2 X3 X4 Xm 7 example, eX is related to the field mill outputs by 6 C C C C ⋯ C 7 6 Y1 Y2 Y3 Y4 Ym 7 C 5 6 C C C C ⋯ C 7 6 Z1 Z2 Z3 Z4 Zm 7. (9) 5 1 1 1 ⋯ 1 4 ⋯ 5 eX(t) CX1a1(t) CX2a2(t) CX3a3(t) CXmam(t), CQ1 CQ2 CQ3 CQ4 CQm ⋯ (7) CD1 CD2 CD3 CD4 CDm

FIG. 5. Plots of the components of f(t) as a function of fQ(t). There should be no patterns in the plots because the various components [ fX (t), fY (t), fZ(t), and fQ(t)] should be independent components of the field (Mach and Koshak 2007). Furthermore, if there were only linear contaminations in the data, each scatterplot would cluster around a line with zero intercept. However, that is not the case, indicating the presence of nonlinear contaminations.

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FIG. 6. Ideal example where there are no cross contaminations between the charge on the aircraft [eQ(t)] and the estimations of the external electric field [f(t)] component. FIG. 7. An example of a piecewise-linear curve (solid line) to remove the nonlinear cross contamination of aircraft charge [ fQ(t)] in fX (t).

The electric field solution in (8) is termed the least where eX , eY , eZ, and eQ are the true field components squares solution because it minimizes the residual (and aircraft charge equivalent); HXX is a factor that ja(t) 2 Me(t)j (Mach and Koshak 2007). The C matrix should approach 1.0 during the iterations; and the other allows the external electric field to be determined based on terms, HXY , HXZ, HXQ, and HXD, are the contamination the field mill outputs given a properly calibrated M matrix. factors that should approach 0.0 during the iterations Given the assumption that the field mill outputs are (such that fX approaches eX plus measurement errors). only a linear combination of the external electric fields The purpose of the iterations in the method of Mach and and aircraft charge, the intermediate estimates of the Koshak (2007) is to reduce the contamination terms external electric field components and aircraft charge (HXY, HXZ, HXQ,andHXD) to zero while setting the can be considered a summation of the true field com- HXX term to one so that ponent plus contaminations from other components. 0 f (t) 5 e (t) 1 « . (11) For example, the equation for the X component of the X X X estimated field is Note that in (10) and (11), if the random and systematic «0 5 1 1 errors in a are small ( X ), they can be neglected. In fX (t) HXX eX (t) HXY eY (t) HXZeZ(t) practice, the terms for the contamination by eY , eZ, and 1 1 1 «0 HXQeQ(t) HXD X , (10) HXD are easiest to remove using the results of the roll

FIG. 8. Same data in Fig. 5, but with the nonlinear cross contaminations of fQ(t) removed. Note that there still may be contaminations in fX (t) and fZ(t) from other unknown sources.

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FIG. 9. Comparison of the uncorrected data (blue plot), the lin- FIG. 10. Comparison of the uncorrected data (blue plot), the ear method corrected data (red plot), and the nonlinear method linear method corrected data (red plot), and the nonlinear method corrected data (green plot). Notice that for these data, both the corrected data (green plot). For this time period, only the nonlinear linear and the nonlinear techniques produce similar results. technique was able to remove the eQ(t) contamination.

(for fX , that would be HXQ, or the lowest plot in Fig. 2). and pitch maneuvers (Mach and Koshak 2007). The con- It is now a simple matter at this point to determine the tamination from e is more difficult to remove because the Q aircraft charge cross terms (HXQ, HYQ, and HZQ), to field produced by a given aircraft charge is difficult to remove them from the calibration matrix (M), and to predict. Consequently, the equation usually reduces to complete the iteration process (gray curves in Fig. 2). f (t) 5 H e (t) 1 H e (t), (12) At a different time and relative eQ magnitude (Fig. 4, X XX X XQ Q black plots), however, the correction to remove the aircraft charge cross terms described in the previous where H is small and H is close to 1.0. Much of the XQ XX paragraph has actually increased the contamination of work in refining the f term in Mach and Koshak (2007) X the fields by aircraft charge in the components (f , f , is determining and removing the e component. This X Y Q and f , gray plots). Note that the data at 0600 UTC 11 work is further compounded when there are significant Z February 2013 have the same calibration factor that nonlinear components in the system. eliminated the aircraft charge cross terms at 0730 UTC 5 February 2013 in Fig. 2 (gray plots). It is the same air- 3. Nonlinear effect removal technique craft with the same configuration of mills. If the aircraft Figure 2 (0730 UTC 5 February 2013) shows an in- charge cross terms varied linearly with aircraft charge termediate step (black plots) in the calibration method and the external electric field, then the M matrix that described in Mach and Koshak (2007) for the field mills works at 0730 UTC 05 February 2013 should work at mounted on a Japanese MU-300 (Whitaker 1981; Y. Saito 0600 UTC 11 February 2013. If one plots fX , fY , and fZ et al. 2013; T. Saito et al. 2013).Thedataarefromthe as functions of fQ (Fig. 5), then the problem is apparent. Rocket Launch Atmospheric Electricity Investigation by JAXA in Cooperation with Academia (RAIJIN) program based at Tanegashima, Japan (Y. Saito et al. 2013; T. Saito et al. 2013). The aircraft was flown in and around cloud features of interest to the RAIJIN project, such as developing cumulus, layered clouds, mature thunder- storms, anvils, and decaying clouds (debris clouds) in the Tanegashima area. The recorded data were then calibrated using the method of Mach and Koshak (2007). The location of the field mills on the aircraft are shown in Fig. 3,andthe linear calibration matrix (M)isshowninTable 1. By this point in the iteration process, the cross-term components (for fX , that would be HXY ,HXZ) have been reduced to a negligible value, with the exception of the FIG. 11. Simulated fields and aircraft charge [e(t)] profile for the cross-term components resulting from aircraft charge example analysis. The time and field amplitudes are arbitrary.

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0 0 FIG. 12. Simulated fields plus typical noise values [e (t)] to create FIG. 13. Estimated fields f(t) with nonlinear eQ(t) components. a more realistic example. This is the starting point of the ideal example analysis using the technique described in this study.

The plots in Fig. 5 should show no pattern, as fX , fY , and 4. Application and results fZ should not have dependency on fQ. At most, there should be a linear dependence if some of the cross terms a. First real-world example (MU-300) (HXQ, HYQ, and HZQ) have not yet been eliminated. The general pattern of li seen in Fig. 5 for this aircraft From (12), it is clear that in a linear system, any re- is for a linear relationship between the field component lationship between the estimated field components and 21 and eQ in the range of about 610 kV m . Outside of 2 fQ are the results of residual HiQ components. Such 610 kV m 1, there is a different relationship with a relationship should be linear in fQ. Clearly, the as- a nonzero intercept for each external field component. sumption that the nonlinear components (l) are negli- Most of the dependence of f(t)onli can be removed by gible is false. There are measureable nonlinear effects fitting the data to a set of piecewise-linear curves (an l ( ) present in the data. example of which is shown in Fig. 7). Figure 5 actually illustrates the core concept of the The equations relating the field components and eQ technique that can be used to remove the nonlinear ef- that produced the li piecewise-linear curves are sub- fects from the f(t) function. The scatterplots in Fig. 5 tracted from the components in f(t) to produce scatter- should show no pattern. Each of the components (fX , fY , plots that have very little dependency on eQ (Fig. 8). For fZ, and fQ) should be independent of each other (like eX , example, the equations for this specific fX example are eY , eZ, and eQ). If all cross-term HiQ components are reduced to 0.0, then the scatterplots in Fig. 5 should 0 5 1 : .2 : 21 fX fX 0 176eQ, eQ 11 71 kV m (13) ideally look more like Fig. 6. The technique uses the patterns found in Fig. 5 to determine the nonlinear re- 0 5 1 : 1 : 21 fX fX 0 2196(eQ 10 77 kV m ), lationship between the two components. The technique ,2 : 21 then removes that nonlinear relationship to create a plot eQ 11 71 kV m . (14) more like Fig. 6. The next section details the steps to remove the nonlinear components.

0 FIG. 14. Plots of f(t)vsfQ(t) used to determine the cross-contamination nonlinear fits. The solid 0 curves are the fits used to remove the nonlinear fQ(t) contaminations.

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0 0 0 FIG. 15. Plots of f (t) verses fQ(t) after removing the nonlinear fQ(t) contaminations using the technique detailed in this report.

Figure 9 shows the same data from Fig. 2 but with the Fig. 12) and the ‘‘corrected’’ fields (Fig. 16). Note that nonlinear components (li) from (13) and (14) removed the technique was able to remove the nonlinear con- (green plots). Notice that the large eQ terms that were taminations to the various simulated field components in present in all three components have been eliminated. the presence of small amounts of measurement errors Figure 10 shows the same data as in Fig. 4, only with the (random noise). The differences in Fig. 17 are very small nonlinear li correction from (13) and (14) removed and on the order of the noise introduced into the data. (green plots). This section of data had a larger, errone- The differences are due to errors in determining the ous eQ component after the linear correction was used to slope and intercept point for the nonlinear terms. eliminate the eQ components from the data in Figs. 2 and 5. The nonlinear technique was able to remove the eQ c. Second real-world example components from both time periods simultaneously. A second real-world example from a different aircraft b. Forward problem simulated example and a different field program are shown in Figs. 18 and 19. The data are from the 2000–01 Airborne Field Mill II To more clearly illustrate how this technique works, experiment (ABFM II) that was conducted near Ken- a simulated dataset was constructed that consists of an nedy Space Center, Florida, to measure the electric field, overpass of several model charge structures. The simu- reflectivity, and microphysics in thunderstorm anvils lated ideal fields, e(t)inFig. 11 characterize an aircraft (and other clouds) produced by deep convection with flying to one side, then through, and then to the other a Citation aircraft (Dye et al. 2007). The locations of the side of an arbitrary vertical charge distribution. The field mills are shown in Fig. 20, while Table 2 shows the fields at the mills from aircraft charge eQ are arbitrarily 2 linear calibration matrix. Figure 18 shows the un- varied from 10.0 to 280 kV m 1 as is typically found in corrected and corrected electric field vector compo- the MU-300 dataset. Figure 12 shows the same data as in nents’ time series for a cloud penetration on 22 May Fig. 11, except with a small amount of random noise 2001 with the North Dakota Citation (Dye et al. 2007). added to each component to simulate measurement errors. Figure 13 shows the data from Fig. 12 with added nonlinear contaminations that depend on eQ. The data in Fig. 13 are now the f(t) data that will be analyzed using the method in this report. A major assumption at this point is that the method of Mach and Koshak (2007) has been used to eliminate all cross terms in (8) except those produced by the charge on the aircraft (HiQ). Figure 14 corresponds to the same plot as Fig. 5. For each component, the estimated piecewise-linear fits are shown in the panels. These curves are subtracted from the components to produce Fig. 15, which has the esti- mated nonlinear components removed. Figure 16 shows the results of the removal of the nonlinear dependency. 0 FIG. 16. Corrected fields [f (t)] as a function of time. The non- Figure 17 shows the differences between the simulated 0 linear fQ(t) components were removed using the technique de- fields (with noise but without the nonlinear components, scribed in this study.

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0 FIG. 17. Difference between the ideal fields [e (t)] and the fields 0 determined with the technique described in this study [f (t)]. The differences are on the order of the noise introduced into the data and are from the errors in determining the slope and intersection point for the nonlinear terms. FIG. 18. Plot of aircraft fields for the Citation aircraft and field mill Note that the uncorrected fX , a term that should be configuration (Dye et al. 2007). The black curves are the uncorrected more or less symmetric about zero, is unrealistically values, while the gray curves are corrected using the nonlinear unipolar (black curve in the top panel). Figure 19 shows method described here [ fQ(t) is the same for both versions]. Al- though the nonlinear effects are smaller in this case, the technique the relationship between fX and fQ,indicatingthe described in this study fixes the unrealistic unipolar fX (t) values. presence of nonlinear charging of the aircraft. The corrected fX data in Fig. 18 (gray curve in the top panel) e. Alternate uses of technique are now much closer to a realistic pattern with sym- metry about zero. The technique can also be used to determine linear correction factors. Figure 22 is a variation of Fig. 8 from d. Independent measurements Mach and Koshak (2007). It shows the fX and fY esti- mates of the e and e fields, with contamination of e In Mach and Koshak (2007), the linear method re- X Y X quires at least as many field mills as components. For example, with four components (eX , eY , eZ, and eQ), at least four field mills are required. Determining higher- order components, such as nonlinear ones, was impos- sible because there simply were too few field mills. The method described in this work uses the same number of field mills, yet it is able to determine many more com- ponents (the various nonlinear terms) of the electric field than there are mills. This is possible because there are multiple in- dependent measurements of the electric field during a flight. Figure 21 shows the autocorrelation functions for the ABFM Citation dataset (Dye et al. 2007) and the MU-300 (Whitaker 1981). The autocorrelation co- efficient (r) drops below 1/e for both sets of data by about 150 s, indicating that measurements made at lon- ger intervals than that are uncorrelated (Leith 1973; Rudlosky and Fuelberg 2013). Electric field measure- ments made at intervals longer than ;2.5 min are es- sentially independent; subsequently, in a 90-min flight, there are over 200 independent samples of the electric FIG. 19. Uncorrected plot of fQ(t) verses fX (t) for the data from field (one sample every 2.5 min and six field mills). More Fig. 18. Although the nonlinear components are smaller for this independent data can be obtained across multiple flights aircraft and field mill configuration, the nonlinear terms are still as long as the aircraft configuration is constant. These present and can be corrected using the technique from this study. For the corrected data in Fig. 18, the piecewise-linear correction numbers of independent measurements are sufficient to (solid curve) was used. It is possible that a quadratic fit (dashed be able to detect and remove nonlinear components of curve) might provide a better fit. Either nonlinear correction can be the electric field by this technique. detected with the technique from this study.

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TABLE 2. Linear calibration matrix M for the Citation. The two letters in the first column designate the location of the field mill: port down (PD), port up (PU), starboard down (SD), starboard up (SU), aft down (AD), and aft up (AU). The columns are for the nose–tail field (eX ), the wing–wing field (eY ), the up–down field (eZ), and the field due to charge on the aircraft (eQ).

eX eY eZ eQ PD 1.4034 0.4055 20.8679 2.0769 PU 3.7311 0.3511 1.7768 2.2777 2 2 FIG. 20. Field mill locations on the Citation aircraft. Shown are SD 4.0447 2.5493 1.4986 2.0353 2 the locations of the port down (PD), port up (PU), aft down (AD), SU 1.6499 2.4773 2.0238 2.3006 2 2 2 and aft up (AU). The starboard down (SD) and starboard up (SU) AD 2.3170 0.0145 1.0017 1.6112 2 field mills are in the same locations as the PD and PU mills but on AU 0.7488 0.0979 1.1178 0.9607 the other side of the aircraft. Note that the AD mill is near the centerline of the aircraft on the tail section. inadequate to remove all cross terms and nonlinear ef- in fY . In that paper, the cross contamination was esti- fects. It should be noted that this technique can also be mated from the amplitudes of the ‘‘wrong’’ features on used to determine more complex nonlinear components the waveforms. An alternative way to determine the than were used here. An example is the parabolic cross contamination is to plot fX against fY during this function as seen in the dashed curve in Fig. 19. Given the period (Fig. 23), to determine the amount of cross con- number of independent measurements found in a typical tamination. The slope of the line indicates the cross- flight, there is ample data to fit a quadratic nonlinear contamination term that can then be subtracted from the term to correct the contaminations in the external fX and fY estimates to produce more accurate estimates electric field measurements. It is also possible to de- of eX , eY , and eZ. In this case, the relationship is linear termine (and separate) linear combinations of the eX , and has a zero intercept. eY , and eZ components in the initial stages of the linear method of Mach and Koshak (2007) with the technique described here. 5. Summary In this report, only contaminations in the external The technique described in this report utilizes the electric fields resulting from aircraft charging were large number of independent measures of the electric detected and removed. Other sources of contamination, field across all field mills that are available on an aircraft. such as engine power settings or flap positions, that can The independent measurements are used to detect and contribute to the aircraft self-charging variations can be then remove contaminations in the estimates of the true external electric field in the presence of small random measurement errors. The method is applied after the best linear calibration matrix (M) is found using the technique of Mach and Koshak (2007). The M matrix results are then used as a basis for determining and then removing the remaining linear and nonlinear terms that might still be present. Note that the nonlinear terms were not from the field caused by simple charge on the aircraft. The basic charge on the aircraft is linear and is corrected using the method of Mach and Koshak (2007). The technique was demonstrated on two very different (see Figs. 7 and 19) aircraft field datasets and one sim- ulated dataset. The datasets collected from the two air- craft were under nearly all cloud conditions, ranging from pure water clouds to mixed phase clouds to pure ice clouds. The technique worked for all cloud types and conditions. Any effect of charge on local ice particles FIG. 21. Autocorrelation of the aircraft electric field component was simply summed in with the other effects. [ fX (t)], indicating a lack of correlation for both datasets after only This technique can be used on any aircraft/field mill ;150 s (r , 1/e). The bottom curve is from the Citation data, while combination where the linear method proves to be the top curve is from the MU-300 data.

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FIG. 22. Plot of fX (t) and fY (t) with fY (t) contaminated by eX (t) [the excursions in fY (t) around 1614 and 1617 UTC] marked with the dashed circles. The plot is based on data from Fig. 8 of Mach and Koshak, (2007). detected and removed using this technique. The only requirement is that the source of the contamination be FIG. 23. Plot of fX (t) against fY (t) with a line indicating the measurable as a function of time against the external contamination of eX (t)infY (t). The slope of the curve indicates the electric field components. A possible example is illus- magnitude of the cross contamination. In this case, the contami- nation is linear and has a zero intercept, unlike the contaminations trated in Fig. 8. The multiple vertical lines in the fX plot in Figs. 5, 7, 14, and 19. However, the technique can still help de- indicate the presence of a term that depends on aircraft termine the correction factor. parameters other than eQ. If a plot of the external source and the electric field component indicates a pattern, then the effect of the contamination can be detected and REFERENCES removed. Bateman, M. G., M. F. Stewart, R. J. Blakeslee, S. J. Podgorny, This technique can also be used to remove self-charging H. J. Christian, D. M. Mach, J. C. Bailey, and D. Daskar, 2007: effects from electric field measurements from rockets, A low-noise, microprocessor-controlled, internally digitizing dropsondes, or balloon-borne instruments (e.g., Marshall rotating-vane electric field mill for airborne platforms. J. Atmos. Oceanic Technol., 24, 1245–1255, doi:10.1175/JTECH2039.1. et al. 1995; Stolzenburg et al. 1998; Zawislak and Zipser Dye, J. E., and Coauthors, 2007: Electric fields, cloud microphysics, 2014). Once the basic calibration of the instrument is and reflectivity in anvils of Florida thunderstorms. J. Geophys. completed, the techniques can be applied to remove any Res., 112, D11215, doi:10.1029/2006JD007550. nonlinear components due to instrument charging or other Jones, J. J., 1990: Electric charge acquired by airplanes penetrating sources. As long as there are enough independent mea- thunderstorms. J. Geophys. Res., 95, 16 589–16 600, doi:10.1029/ JD095iD10p16589. surements to plot self-charging effects against the mea- Koshak, W. J., 2006: Retrieving storm electric fields from aircraft sured electric field, the method will work. field mill data. Part I: Theory. J. Atmos. Oceanic Technol., 23, 1289–1302, doi:10.1175/JTECH1917.1. Acknowledgments. The author gratefully acknowl- ——, J. C. Bailey, H. J. Christian, and D. M. Mach, 1994: Aircraft edges the support received from the Japan Aerospace electric field measurements: Calibration and ambient field re- trieval. J. Geophys. Res., 99, 22 781–22 792, doi:10.1029/94JD01682. Exploration Agency (JAXA) Space Transportation ——, D. M. Mach, H. J. Christian, M. F. Stewart, and M. Bateman, Mission Directorate, the sponsor of one of the airborne 2006: Retrieving storm electric fields from aircraft field mill field measurement campaigns. In addition, the author data. Part II: Applications. J. Atmos. Oceanic Technol., 23, would like to thank the individuals from the collabo- 1303–1322, doi:10.1175/JTECH1918.1. rating organizations that contributed to the data used in Leith, C. E., 1973: The standard error of time-average estimates of climate means. J. Appl. Meteor., 12, 1066–1069, doi:10.1175/ this analysis. Collaborators include the University of 1520-0450(1973)012,1066:TSEOTA.2.0.CO;2. Alabama in Huntsville, the Universities Space Research Mach, D. M., and W. J. Koshak, 2007: General matrix inversion Association, the NASA Marshall Space Flight Center technique for the calibration of electric field sensor arrays on (MSFC), Mitsubishi Heavy Industries, and Diamond aircraft platforms. J. Atmos. Oceanic Technol., 24, 1576–1587, Air Service. I would like to also thank Dr. William doi:10.1175/JTECH2080.1. ——, R. J. Blakeslee, M. G. Bateman, and J. C. Bailey, 2009: Koshak of MSFC for his help in checking the mathe- Electric fields, conductivity, and estimated currents from air- matical formulas employed in the study. This work was craft overflights of electrified clouds. J. Geophys. Res., 114, funded under NASA Contract NNM13AA43C. D10204, doi:10.1029/2008JD011495.

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Marshall, T. C., W. Rison, W. D. Rust, M. Stolzenburg, J. C. Willett, Saito, Y., T. Saito, and K. Okita, 2013: Proposal of new trig- and W. P. Winn, 1995: Rocket and balloon observations of gered lightning launch commit criteria for Japan’s safety electric field in two thunderstorms. J. Geophys. Res., 100, rocket launch. Sixth IAASS Int. Space Safety Conf., 20 815–20 828, doi:10.1029/95JD01877. Montreal, QC, Canada, International Association for the Merceret, F. J., J. G. Ward, D. M. Mach, M. G. Bateman, and J. E. Advancement of Space Safety, 12.3.1. [Available online at Dye, 2008: On the magnitude of the electric field near http://iaassconference2013.space-safety.org/wp-content/uploads/ thunderstorm-associated clouds. J. Appl. Meteor. Climatol., sites/26/2013/06/0900_Saito.pdf.] 47, 240–248, doi:10.1175/2007JAMC1713.1. Stolzenburg, M., W. D. Rust, B. F. Smull, and T. C. Marshall, 1998: Penrose, R., 1955: A generalized inverse for matrices. Proc. Cambridge Electrical structure in thunderstorm convective regions: 1. Philos. Soc., 51, 406–413, doi:10.1017/S0305004100030401. Mesoscale convective systems. J. Geophys. Res., 103, 14 059– Rudlosky, S. D., and H. E. Fuelberg, 2013: Documenting storm 14 078, doi:10.1029/97JD03546. severity in the mid-Atlantic region using lightning and radar Whitaker, R., 1981: Diamond 1: Mitsubishi’s first business jet. information. Mon. Wea. Rev., 141, 3186–3202, doi:10.1175/ Flight Int., 120, 163–170. MWR-D-12-00287.1. Winn, W. P., 1993: Aircraft measurement of electric field: Saito, T., Y. Saito, and K. Okita, 2013: Results of RAIJIN2012, the Self-calibration. J. Geophys. Res., 98, 7351–7365, doi:10.1029/ intensive observation program to relax the triggered lightning 93JD00165. launch commit criteria. 16th Conf. on Aviation, Range, and Zawislak, J., and E. J. Zipser, 2014: Analysis of the thermodynamic Aerospace Meteorology, Austin, TX, Amer. Meteor. Soc., properties of developing and nondeveloping tropical distur- J3.5. [Available online at https://ams.confex.com/ams/ bances using a comprehensive dropsonde dataset. Mon. Wea. 93Annual/webprogram/Paper222202.html.] Rev., 142, 1250–1264, doi:10.1175/MWR-D-13-00253.1.

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