1. Introduction

Making in situ electrical observations of thunderclouds is a beneficial, albeit sometimes dangerous, means for investigating the detailed electrical state of a storm system. Using an aircraft equipped with multiple electric field mill sensors can provide measurements that offer scientists first-hand knowledge of the location and intensity of electrified regions of a storm and the occurrence of lightning. To obtain the maximum information from such flights, it is desired to retrieve the ambient vector storm electric field from the field mill data. To accomplish this, one needs to determine how the presence of the aircraft distorts the ambient field. There are many ways that an aircraft can distort the ambient field, but the main focus has been to determine how the geometrical shape of the aircraft locally distorts mill output. Hence, to first order, one determines a set of enhancement coefficients (or more generally, distortion coefficients), that define a “calibration matrix.” The aircraft is said to be calibrated when the calibration matrix is determined. Mathematical inversion of the calibration matrix allows one to retrieve storm electric fields from the mill data. Issues pertaining to calibration and inference of storm electric fields has been examined previously (e.g., see Bailey and Anderson 1987; Mazur et al. 1987; Binkley 1992; Winn 1993; Koshak et al. 1994; Mo et al. 1998).

More recently, the Lagrange multiplier method (LMM) for calibrating an aircraft has been developed in the first part of this study (Koshak 2006, hereafter Part I). The LMM offers a generalized approach for retrieving storm electric fields from the aircraft field mill data. It provides a practical and clear means for choosing additional physical constraints (i.e., beyond field mill data) to improve the calibration process. The study in Part I also introduced the pitch down method (PDM), which is a unique means of completing the absolute calibration (a subtask in the overall aircraft calibration effort).

In this article, we apply the LMM, the PDM, and conventional absolute calibration analyses to complete a full calibration of a Citation aircraft that is instrumented with six electric field mill sensors. Before this calibration is performed, computer simulations are run to test the LMM and assess expected retrieval errors. These simulations are carried out on two different applications of the LMM that involve different side constraint strategies. Next, actual fair-weather calibration maneuvers performed by the Citation are analyzed with the LMM, and we use the side constraint strategy that provides the best relative calibration results. Both the PDM and conventional analyses are then applied to complete an absolute calibration. Finally, the calibration results are used to retrieve storm electric fields from a 2 June 2001 Citation flight. The storm field results are compared with results derived from earlier calibration studies that are based on iterative methods.

2. Computer simulated tests of the LMM

The basic result from Part I is the Lagrange multiplier equation set given by

Here, λ_k are Lagrange multipliers, and the G_k are side constraint functions. Practical equivalent matrix forms for ∂r ²/∂b have been derived in Part I so that the system of equations in (1) can be solved either by analytic, quasi-analytic, or purely numerical means depending on the form of the side constraints chosen. For an aircraft equipped with m field mills, the relative calibration problem is reduced to finding the optimum value for the unknown 2m vector b. We shall test two solution processes (solution 1 and solution 2) derived in Part I that employ different side constraint strategies. Solution 1 involves the constraints

and solution 2 involves the constraints

Here, s_j = a_j/(cosα_j cosβ_j), where a_j is the m vector of mill outputs, α_j is aircraft roll angle, and β_j is aircraft pitch angle. The j subscript refers to the jth aircraft orientation at time t_j during the aircraft fair-weather calibration maneuvers. The variables F and E_q refer to the fair-weather field value and aircraft charge field, respectively. Bars over variables indicate averages over all j = 1, . . . , n aircraft orientations. The quantity ϕ² is an error tolerance. The unknown b is split into two unknown m vectors; that is, b → (b_z, b_q). Owing to lack of information content, retrieval of b_q is not feasible, but is also not strictly required for retrieving the three components of the storm electric field. So with both solution 1 and solution 2, the main interest is to determine b_z. Finally, the variable ξ_j represents the well-known Gish (1944) model for the fair-weather field [see Part I for additional details on all variables introduced here].

We begin by describing the computer simulator program used to test solution 1. For this solution there are two primary sources of errors that affect solution retrieval error: 1) the mill measurement errors e_j associated with the j = 1, . . . , n aircraft maneuvers in fair weather, and 2) the uncertainty (call it ϑ) in the mean fair-weather field F. The typical magnitude of each element of e_j is one or more volts per meter, but can vary depending on the specific mill instrumentation employed by a particular aircraft experiment.

Given the natural spatial and diurnal variability of the fair-weather field, we expect ϑ to range anywhere from about 1 to 20 V m⁻¹ or larger. The low end corresponds to direct and independent measurements of the fair-weather field, and the high end corresponds to maximum errors associated with unsophisticated guesses. Presumably, guesses based on empirical models of the fair-weather field profile (such as the Gish field) lie somewhere within the 1–20 V m⁻¹ interval.

Rather than choosing a specific value for the typical mill measurement error, and a specific value for ϑ, we chose a broad range of error values to support a broader range of aircraft mill types and F estimation techniques. To ensure that our simulation was as robust as possible, we also ran the simulation with no errors to confirm the mathematical validity of the LMM solution, and to confirm that there is very little retrieval error under such conditions (there will always be some retrieval error due to computer truncation). To accommodate a broad range of conditions, simulated errors were assumed to be normally distributed with a mean μ = 0, and a standard deviation σ that varied. Specifically, we simulated five different types of mill measurement errors having standard deviations of (0., 0.5, 1.0, 1.5, and 2.0 V m⁻¹), and 21 values of ϑ having standard deviations of (0, 1, 2, . . . , 20 V m⁻¹).

The simulation consisted of defining 100 distinct computer-simulated aircrafts, each aircraft performing unique maneuvers in a unique fair-weather field. The maneuvers included roll, pitch, and altitude changes. In addition, each aircraft had unique charging characteristics.

Each aircraft was defined by simply defining its calibration matrix 𝗠 (see Part I). We assumed each aircraft had six mills. To begin with, a base matrix given by

was used; the base matrix was obtained by an iterative calibration analyses of a Citation aircraft (D. Mach 2003, unpublished manuscript). The accuracy of this base matrix is immaterial in this simulation; it is simply a reasonable way to define a hypothetical base aircraft that looks, in some respects, like a Citation aircraft. Next, random errors selected from a normal distribution (mean zero, standard deviation 0.5) were added to the base matrix to generate the 100 distinct 𝗠 matrices. To build each of the 100 unique fair-weather fields, the Gish field was used as a base, upon which random errors selected from a normal distribution (mean zero, standard deviation 1 V m⁻¹) were added. Similarly, realistic base functions for roll, pitch, altitude, and charge were defined (see Table 1). The standard deviations applied to these base functions were 1° (for roll and pitch), 2 m (for altitude), and 1 V m⁻¹ for charge. As an extra test of the effects of aircraft charge, an additional charge base function was employed that simulates turning on and off a high voltage stinger (see second set of numbers in column 4 of Table 1).

For each of the 100 individual fair-weather field calibration experiments described above (one experiment for each of the 100 simulated aircraft), a distinct matrix 𝗕 was retrieved. (Of course, from the discussion in Part I, the fourth or “q row” of this matrix is meaningless due to lack of information content.)

Next, each simulated aircraft is flown through a unique computer-simulated foul weather environment wherein the aircraft collects data at N individual locations. Each aircraft therefore makes the following mill measurements: a_s = 𝗠E_s + e_s, s = 1, . . . , N. Here, the ambient storm electric field vector is E_s = (E_magû_s, 1), where E_mag is a constant and û_s is a random unit vector over the s = 1, . . . , N locations. For all our simulations, we chose N = 1000. From the simulated storm measurements, the following root-mean-square (rms) storm field retrieval errors are computed as

where the squared retrieval error δ²_s is given by

Hence, D is the magnitude of the total typical storm field retrieval error expected from a specific aircraft, while P is the associated typical percent retrieval error. The median, mean, and standard deviation of D and P are calculated for the 100 aircraft. In addition, each value of (D, P) for a particular aircraft depends on mill measurement error and ϑ. Figures 1 –3 summarize the storm retrieval error results.

In Fig. 1, the median, mean, and standard deviation of D are provided as a function of mill measurement error and ϑ (which again is the uncertainty in F). The leftmost (rightmost) charge base function shown in Table 1 was used in the left (right) column of plots of Fig. 1. Because E_mag is chosen as 100 kV m⁻¹ in Fig. 1, the plots for D and P are identical as indicated by (5). Therefore, all the plots in Fig. 1 also correspond to the median, mean, and standard deviation, respectively, for P. For example, if a value for the median D is 12 kV m⁻¹, the associated median P is 12%. In addition, note that the two base charge functions in Table 1 are substantially different, but the difference had little effect on the storm retrieval results; that is, the left and right column of plots in Fig. 1 are fairly similar.

Figure 2 shows results for E_mag = 10 kV m⁻¹, and Fig. 3 shows results for E_mag = 1 kV m⁻¹. In each of these figures, the leftmost charge base function in Table 1 was used. Note from Figs. 1, 2 and 3 that the overall error patterns (as well as the magnitude of P statistics) do not change appreciably across the large range in E_mag.

We performed the same computer simulation tests on solution 2. The storm field retrieval errors for D and P are shown in Figs. 4 and 5; once again, since the results in Fig. 4 are for E_mag = 100 kV m⁻¹, the D results are identical to P results as was explained relative to Fig. 1. Since solution 2 does not involve the parameter ϑ, the retrieval errors in Figs. 4 and 5 are simply plotted as a function of mill measurement errors. What solution 2 does involve is selecting ξ_j (the estimate of the fair-weather field function). Columns 1 and 2 of Fig. 4 represent retrieval errors when the function ξ_j deviates by typically 0 and 1 V m⁻¹, respectively, from the true (simulated) fair-weather field. Columns 1 and 2 of Fig. 5 represent retrieval errors when the deviation is typically 3 and 5 V m⁻¹, respectively. As expected, retrieval error decreases when a better estimate in the fair-weather field function is used. The first column in Fig. 5 shows that, for a reasonable deviation of 3 V m⁻¹ in the fair-weather field estimate and a mill measurement error of 1 V m⁻¹, the storm field retrieval errors are reasonably small D ∼ 27 kV m⁻¹ (P ∼ 27%). The computer tests show that the method retrieves the 3D storm field to within an error of about 8% if the fair-weather field estimate is typically within about 1 V m⁻¹ of the true fair-weather field (see column 2 of Fig. 4). Finally, note from Fig. 4 that retrieval errors increase with increasing mill measurement errors. However, this dependence is not present in Fig. 5 because the large deviation errors (3 and 5 V m⁻¹, respectively) in the fair-weather field estimate fully control retrieval error, and mask the dependence on mill measurement error.

3. Relative calibration

The LMM is now applied to obtain a relative calibration of a Citation aircraft that was equipped with six electric field mill sensors. The aircraft field measuring system is shown in Fig. 6. The mill output resolution as measured in the laboratory was 1.9 V m⁻¹. If such a mill is mounted on an infinite perfect conducting cylinder (which has an enhancement coefficient of 2) the mill would detect field changes as small as ½ (1.9) = 0.95 V m⁻¹. The labeling of mills is as follows: mill 1 (port down), mill 2 (port up), mill 3 (starboard down), mill 4 (starboard up), mill 5 (aft down), mill 6 (aft up).

On 29 June 2001 the Citation performed calibration maneuvers in fair weather. Figure 7 shows the roll, pitch, altitude, and mill outputs (mills 1–3, Fig. 7a; mills 4–6, Fig. 7b) for two selected time intervals. Here, time is discretized as t_j, for j = 1, . . . , n aircraft orientation in fair weather. The first interval (1559:46.7–1603:19.2 UTC) corresponds to the roll maneuvers. Following this interval, a period of almost 22 min (indicated by the dark vertical line “time break” in Fig. 7) is omitted because the calibration data during this period offered little if any additional information content, and also had some undesirable features associated with aircraft charging. In particular, the pitch up maneuver occurred at the tail end of this period and was contaminated with aircraft charging due to engine throttle up. (Since pitch ups were contaminated with excessive aircraft charging due to throttle up, we omitted all pitch up data from our analyses; this omission does not adversely affect the LMM analyses.) After the omitted period is our second period of analysis (1625:17.9–1626:25.0 UTC). During this interval, the aircraft performed a very good pitch down maneuver that was not only needed for the LMM analyses, but was also well suited for the PDM analyses.

The data in Fig. 7 has a time resolution of 0.1 s and there are a total of n = 2778 data points. Because the elements of the 𝗭 matrix (see Part I) contain quadruple sums, a fair amount of computer time is required to compute each element of 𝗭 if n is large. Therefore, it was beneficial to decimate the selected calibration data shown in Fig. 7. We resampled this data at evenly spaced points to generate 0.50-s resolution data as shown in Figs. 8a and 8b. However, we retain the 0.1-s time resolution during the pitch down, so as to optimize the PDM analyses (section 4a). The decimated calibration data in Fig. 8 contain a total of n = 571 data points, yet still contain the critical information content associated with full range roll and pitch data. Using the LMM (solution 2, with constraint ξ_j equal to the Gish field), one obtains for the first three rows of 𝗕, hereafter called the submatrix 𝗕*, the following result (rounded at the sixth decimal place):

This matrix is associated with a Lagrange multiplier λ₁ = 7841.6555. The rms error between the Gish field and the retrieved field is ɛ_rms = ϕ²/n = 1.179 V m⁻¹. The matrix 𝗕* is next altered by the absolute calibration.

4. Absolute calibration

a. Pitch down method

We begin with some preliminary comments about the pitch down data, and the nature of the coefficients M_ix of 𝗠. Figures 9a and 9b zoom in on the brief time interval associated with the pitch down and zero crossing. Computations of the time derivatives of several variables (including roll, pitch, altitude, and output from the four front mills) are provided in Table 2. To show how generally stable the derivative calculation is, we computed the numerical derivatives for several different values of the time interval Δt that is approximately centered on the zero crossing time t_o. We were most comfortable using a Δt = 1.9 s for the absolute calibration calculation (but many smaller values of Δt could have been used with not much difference in the final results).

Note that, as predicted in the PDM theory of Part I, the four front mills of the Citation aircraft each have positive increases in the mill outputs. The altitude of the zero crossing is z_o = 2683.94 m. Evaluating (45) of Part I using the Gish field gives the order-of-magnitude estimates as follows:

The first term dominates since it is about two orders of magnitude larger than the second term. Since M_ix > 0 for the four front mills, the derivatives for the four front mills should indeed be positive.

The relative magnitudes of the M_ix can also be assessed. For convenience, pertinent variables introduced in Part I are provided here:

The first expression in (9) is simply the estimate of the fair-weather field at the zero crossing. Given ℕ = 4 front mills on the Citation, there are four estimates (F_1o, . . . , F_4o) associated with the four values of the enhancement coefficients (M_1x, . . . , M_4x). The average and standard deviation of the F_io estimates are given in the second and third lines of (9). To understand what relative values of M_ix for the four front mills are appropriate, Table 3 indicates a few different estimates, and the associated values of F_io from the first equation in (9). The first estimate of the M_ix is derived from an iterative method that attempts to find 𝗠 directly as discussed in section 4 of Part I. This particular iterative method is currently under revision (D. Mach 2004, unpublished manuscript). Column 2 of Table 3 shows that the values of M_ix for the front mills produce highly different estimates of the zero-crossing fair-weather field; the value of σ = 13.1 V m⁻¹ is unacceptably high given the collective measurement errors in the mill and pitch data. In fact, a coarse grid search (Table 3, column 3) brings the value of σ down to acceptable levels. When this coarse grid search is followed by a Powell minimization (Press et al. 1988) the various estimates of zero-crossing fair-weather field can be made to converge to effectively identical values (Table 3, column 4).

Note that the true coefficients are given by the unique values: M_ix = ȧ_io/F_oβ̇_o. Since F_o is only known to within some estimation error ɛ_o, one can write F′_o = F_o + ɛ_o ≡ cF_o so that the best we can do is find a set of coefficients M′_ix = ȧ_io/(F′_oβ̇_o) = ȧ_io/(cF_oβ̇_o) = M_ix/c, where c varies and is given by c = F′_o/F_o. However, for any fixed nonzero estimate F′_o, the ratio of any two coefficients is correct since M′_ix/M′_jx = M_ix/M_jx. For example, with F′_o = −20.238 722 V m⁻¹ assumed, the results in column 4 of Table 3 give the correct relative values of the M_ix. Any other assumed fixed value for F′_o would give the same relative values of the M_ix. To obtain the best estimate of the true values of the M_ix, the value of F′_o must be fixed reasonably close to F_o; that is, c ≅ 1 must hold.

Thus far we have verified that the PDM theory correctly predicts the sign of the rate of change of certain mill outputs, and we have also shown in detail how the PDM theory determines the relative magnitudes of the M_ix. This builds confidence in, and highlights the advantages of, the PDM.

In the LMM formalism, however, it is not necessary to consider the values of the M_ix, or any other part of 𝗠, to complete the absolute calibration. As introduced in Part I, the absolute calibration is simply accomplished by computing F*_o = −|𝗕*a(r_o, t_o)| and by making an estimate (preferably a measurement of) the zero-crossing fair-weather field F_o. In the absence of any fair-weather field balloon sounding data, we use the Gish field to estimate F_o, which gives F′_o = −21.531 064 V m⁻¹. For the Citation aircraft, we find that F*_o = −21.104 813 V m⁻¹. Hence, one estimate of the final (absolutely calibrated) matrix is

As can be seen, there is not much correction here; that is, 1.020 197 ≅ 1.0. This is to be expected since in the LMM we had already used the Gish field as a side constraint to obtain 𝗕*. Hence, the elements of this matrix were already biased toward a Gish field.

b. Ground-based field mill overpass (GBFMO) method

The conventional approach for completing an absolute calibration is to fly low-level overpasses of a ground-based electric field mill sensor. On 28 June 2001, the Citation aircraft performed low-level overpasses of ground-based field mill number 10 located at the National Aeronautics and Space Administration (NASA) Kennedy Space Center (KSC). Six overpasses of this mill have been analyzed to complete the absolute calibration. As discussed in Part I, the absolute calibration is performed by multiplying 𝗕* by the ratio F′/F*. Here, F′ is the ground-based field mill estimate of the fair-weather field at the aircraft as it overpasses the ground-based mill, and F* = −|𝗕*a(r, t)| is the associated aircraft-measured fair-weather field using the relative calibration matrix 𝗕*. During an overpass, the aircraft charge distorts the ground-based mill measurement; this error was identified by the variable ɛ_a in (43) of Part I. We found that five of the six overpasses were associated with a negative excursion in the electric field of about 10 V m⁻¹; one of the six overpasses showed little or no excursion. A negative excursion in field implies that the aircraft is carrying a net positive charge, and this sign was consistent with the sign of the mill outputs during each overpass. To improve our absolute calibration, we avoid the error ɛ_a by picking the ground-based mill value near, but outside, the excursion so that the ground-based mill value we use is more representative of the true surface field, rather than an aircraft-distorted surface value. So our improved absolute calibration is performed by multiplying 𝗕* by the ratio ρ ≡ (F′ − ɛ_a)/F*. The value of ρ still contains errors that are somewhat difficult to remove by estimation methods. For example, we do not attempt to account for discrepancies in the field between the surface and the aircraft overpass altitude h (where h ∼ 4 m) and, of course, there is ground-based mill instrument measurement error. The dc offsets in aircraft mill output could also drift between the time of the absolute calibration (28 June 2001) to the time of the relative calibration (29 June 2001). Despite these possible errors, we found for the six overpasses the following values of ρ: 1.406 251, 1.094 580, 1.069 726, 1.135 467, 1.428 291, 1.128 382. The average value of ρ is 1.210 449, with a standard deviation of 0.162 098. Hence, this approach gives an absolute calibration to within about 13.4% (=0.162 098/1.210 449), and an estimate of the final calibration matrix is

Given the likely errors in the Gish field, we consider the result in (11) more realistic than the PDM result in (10). Nonetheless, two overpasses had values of ρ equal to 1.069 726 and 1.094 580 that do not greatly differ from the PDM result of 1.020 197 given in (10).

5. Storm electric field retrieval

Figure 10 shows storm electric field retrievals derived from the LMM analyses using 𝗕*_final from both (10) and (11). The LMM results are compared to two earlier iterative techniques: 1) the 𝗠-theory iterative method (D. Mach 2004, unpublished manuscript) and 2) a slightly modified version of the 𝗞-theory iterative method described in Koshak et al. (1994).

The rigorous approach of the LMM serves as a quantitative validation of the earlier iterative techniques. Overall, the three methods agree reasonably well for the E_y and E_z components given all errors and differences in retrieval methods. This is encouraging support for the validity of the iterative methods. However, retrieval of E_x is evidently more difficult for the iterative approaches. Here, the LMM and 𝗞-theory results agree in polarity, but not particularly well in magnitude. The magnitude and polarity of E_x derived from the 𝗠 theory does not agree with the other two methods.

6. Summary

The Lagrange multiplier theory developed in Part I of this study for calibrating an aircraft equipped with several electric field mill sensors has been rigorously tested and applied. We developed a computer model that simulates the flight of uniquely shaped aircraft through distinct fair-weather field environments. Each model aircraft performs unique roll and pitch maneuvers and has distinct charging characteristics; even charging due to high-voltage stinger probes were simulated. These simulations were run for two types of Lagrange side constraints proposed in the Part I investigation, and the statistics of retrieval errors were obtained. We determined from our simulations (and subsequent real-life calibration analyses) that a Gish field side constraint was optimum. Application of the method was applied to fair-weather field maneuvers performed on 29 June 2001 by a Citation aircraft that was equipped with six field mill sensors. The analysis allowed us to complete a (relative) calibration of the Citation.

To obtain an absolute calibration of the Citation, we applied the technique developed in Part I that involves a simple pitch down maneuver at high (>1 km) altitude. In addition to providing an estimate of the absolute calibration, the method also provided us direct insight into what should be the appropriate values of some enhancement coefficients. This is valuable for independently checking iterative calibration method results that directly retrieve elements of 𝗠. Indeed, column 2 of Table 3 revealed that previous iterative methods can produce enhancement coefficients that result in contradictory predictions for the value of the fair-weather field at the aircraft “zero crossing,” an instant that occurs as the aircraft passes through zero pitch during a pitch down maneuver.

Since the pitch down method of absolute calibration was limited by using coarse Gish field estimates (rather than preferred direct measurements) of the zero crossing fair-weather field, we completed an independent absolute calibration using conventional low-level overpasses of a ground-based field mill sensor. Six such overpasses were examined to obtain our best estimate of the absolute calibration.

With the completion of both the relative and absolute calibrations, we were able to retrieve storm electric field values along the flight path taken by the Citation aircraft on 2 June 2001. The retrieved storm fields were found to be physically reasonable in most regards, and in reasonable agreement with earlier iterative methods of solution (for retrieved values of the storm E_y and E_z components).