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[1] In this paper we present three methods for evaluating range rates of meteoroids passing through the ionosphere, using linear frequency modulated (LFM) chirped pulse data from the ALTAIR radar. The first method is based on the simple calculation of range differences divided by interpulse intervals. The second method utilizes the dual-frequency capability of ALTAIR to solve for range rates based on the difference in the measured ranges due to range-Doppler coupling. The third method utilizes a simplified form of integer programming in order to unwrap the phase differences of the matched filter time response, with reliance on the rough approximation available from the first method to disambiguate the solution set. The results of the three methods, with error bounds, are given for a large set of meteoroid head echoes taken from a data collection conducted with ALTAIR in 2007.

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[2] Data taken with the ALTAIR radar can be used to determine a variety of types of information about meteoroids as they pass through the ionosphere [Close et al., 2002; Hunt et al., 2004]. The range rates (i.e., radial velocities from the radar) of the meteoroids are of particular interest, both in themselves and as a basis for determining densities, etc. in other calculations. In this paper we present a brief discussion of related techniques, followed by a description of the ALTAIR radar and data collection, and the details of three methods for calculating range rates: (1) simple differencing and division, (2) dual-frequency matching utilizing range-Doppler coupling, and (3) phase unwrapping of the matched filter time response. This is followed by a comparison of the results of applying these methods to a large set of meteoroid head echoes.

2. Data

2.1. ALTAIR Radar

[3] ALTAIR is a 46 m diameter, high-power, two-frequency radar operating at 158 MHz (VHF) and 422 MHz (UHF). It resides in the central Pacific at 9°N and 167°E (geographic) on the island of Roi-Namur in the Kwajalein Atoll, Republic of the Marshall Islands. ALTAIR transmits a peak power of 6 MW simultaneously at the two frequencies with right-circularly (RC) polarized signal energy in a half power beam width of 2.8° and 1.1° at VHF and UHF, respectively. ALTAIR receives both right-circular and left-circular energy and has four additional receiving horns for the purpose of angle measurement, which gives the position of an object in three dimensions [Close et al., 2000].

2.2. Data Collection Parameters

[4] The two ALTAIR waveforms used to collect the data were a VHF LFM chirped pulse with a 100 μs pulse width (15 m range spacing), and a UHF LFM chirped pulse with a 400 μs pulse width (also 15 m range spacing). A 8.7 ms pulse repetition interval (PRI) was utilized in this collection. Using these waveforms, ALTAIR can detect a target as small as −74 decibels relative to a square meter (dBsm) at VHF and −80 dBsm at UHF at a range of 100 km. Few radars worldwide compare to the sensitivity achievable by ALTAIR [Janches et al., 2008]. Table 1 summarizes some of the UHF and VHF data collect parameters.

Table 1. Summary of Some of the UHF and VHF Data Collection Parameters^{a}

Parameter

VHF

UHF

a

T is the period of the LFM pulse, f_{0} is the central frequency, B is the bandwidth, and PRI is the pulse repetition interval.

T (μs)

100

400

f_{0} (MHz)

158

422

B (MHz)

7.06

5.00

PRI (ms)

8.6955

8.6955

3. Range Rate Calculation Methods

[5] Three different methods were used to calculate the range rates, varying widely in both complexity and generality.

3.1. Range Differencing Approach

[6] Range differencing is both the simplest and most general approach, with correspondingly lowest performance. In this case the range rate at time k, v_{k}, is estimated using (note that the effect of range-Doppler coupling is ignored because its effects are small here)

where the r_{k} are the measured range values interpolated from the peak of the match filter response at time t_{k}, with Δt_{k} being the pulse repetition interval (PRI). Assuming uncorrelated errors between the range and PRI, the error is then approximated by [Bevington, 1969]

For typical meteors σ/Δr_{k} ≈ 0.05, while σ_{Δt}/Δt ≈ 1.15 × 10^{−4}, making the second term in equation (2) negligible. (The first approximation is based on the assumption that σ ≈ 1 range bin, with Δr_{k} in the neighborhood of 20 range bins. The second approximation is probably quite conservative; the number given for the PRI has a value of 8.6955 ms, suggesting validity out to at least the fourth decimal place, so we take σ_{Δt} ≈ 0.0001 ms.) The error is then

The results of this technique on an example meteoroid streak are shown in Figure 1. It should be noted that while the percentage error is relatively low, the extreme quantization makes the determination of higher derivatives difficult.

3.2. Dual-Frequency Approach

[7] It was previously mentioned that ALTAIR pulses simultaneously in both UHF and VHF. Returns are generally much stronger for VHF, but some meteors also have a strong return for UHF pulses. For these pulses with significant returns in both bands, it is possible to infer range rates from the differing shifts in range values caused by range-Doppler coupling (RDC) [Cook and Bernfeld, 1967].

[8] RDC is an artifact of the LFM pulse compression scheme, which results in a shift in the measured range value of [Mahafza, 1998]

where the subscript x represents either UHF or VHF values. These are combined into the constant c_{x} for notational convenience. The measured range values are then

where the are the true range values. Knowing that while the r_{x,k} and c_{x,k} vary between UHF and VHF, the ranges and range rates v_{k} are identical, we use equation (5) to estimate

Note that while the previous range differencing technique used a Δr_{k} based on measurements from times k and k-1, the Δr_{d,k} here is based on UHF and VHF measurements taken simultaneously at time k.

[9] Using a process similar to that conducted previously to find the error, assuming that the values underlying the constants c_{x} are precisely known compared to the range uncertainties, we obtain

We can relate this error estimate to that of the previous technique by relating the error σ and range Δ.

[10] Since the same noise affects the correct estimate of the return pulse peak locations in both cases, σ = σ. Fortunately, however, Δr_{d,k} is larger than Δr_{k} by about a factor of 4. To see this, consider that for a given range rate, with no noise,

Rearranging,

The relation between σ_{vd} and σ_{vk} is then found using equations (3) and (7) to be

resulting in an error reduction of about a factor of 4 from the simple differencing technique. This is clearly an improvement, but it should be noted that the applicability of this technique is limited to those cases where the signal-to-noise ratio (SNR) is high enough in both UHF and VHF to allow the detection of the meteoroid. In practice this limitation is severe enough to make this approach of limited value.

[11] The results of this technique on an example meteoroid streak are shown in Figure 1. It is clear that this is an improvement on the results of the first method, but that the quantization is still a problem for determining higher-order derivatives.

3.3. Phase Unwrapping Approach

3.3.1. Background

3.3.1.1. Intrapulse Doppler

[12] The two approaches discussed previously are based on using range values inferred from the location of the peak of the magnitudes of the matched filter time response, y(t), given by Skolnik [2008] as

Note that the location of the peak utilizes only the magnitude of equation (11). The phase,

also contains information about the velocity through f_{d}, the Doppler frequency shift, which is related to the range rate by f_{d} = 2 · v/λ, where λ is the central wavelength of the LFM pulse.

[13] Unfortunately, intrapulse phase variation that can be meaningfully determined from samples with sufficiently high SNR is too low to be useful. To see this consider the following scenario with typical values. For a typical meteoroid with range rate of ∼40 km/s, with a VHF center frequency of 158 MHz, we obtain a Doppler frequency of ∼42 kHz, with a corresponding Doppler period of ∼23 μs. We find that in practice in the data the portion of the matched filter return that can be discerned above the noise floor is ∼250 ns long; this corresponds to the length of time during which a meaningful phase value can be measured. Then, given that the ratio of this length of time to the Doppler period is about 0.01 for the given scenario, we expect to see a roughly constant phase from the matched filter return for a given pulse. This matches what is seen in the data.

3.3.1.2. Interpulse Doppler

[14] Between pulses, however, the phase change is quite significant for the same scenario. From Skolnik [2008], the phase change due to the Doppler shift is

Rearranging, inserting frequency, and using the sampling interval T (i.e., the PRI), the formula relating Δϕ, the phase difference between two adjacent pulses, to the range rate, v, is given by

where c is the speed of light. The mod ( ) operation is included because we are dealing with phase; it must be reversed by “unwrapping” in order to find v.

[15] Using the example scenario values, we expect a phase change of ∼2300 rad between pulses. This level of wrap is unsurprising and is in good agreement with the well known ambiguities arising in range and/or Doppler as a result of various choices of PRI. Methods for addressing these ambiguities using multiple PRI waveforms are discussed by, for example, Trunk and Brockett [1993] and Trunk and Kim [1994]. These data were acquired well in advance of our analysis, however, so that no waveform modification could be performed.

[16] The interpulse phase change is clearly large enough to be problematic to unwrap, but knowing that the change in velocity is small, we can expect nearly all of this to remain constant over the streak. Due to the mod(), this constant will only show as a remainder with value <2π. If we concentrate instead on Δv,

Based on a meteoroid with approximately 4 km/s Δv over 35 samples, a representative Δv over a single PRI is ≈117 m/s. Using equation (16), neglecting the mod(), we obtain Δ^{2}ϕ ≈ 6.7 rad. Thus we see that the actual change in Δϕ values between samples is low enough to allow for the possibility of unwrapping. The Δϕ of a particularly slowly decelerating meteoroid is used to illustrate this in Figure 2.

3.3.2. Algorithm

[17]Section 3.3.1 illustrates that it may be possible to find a solution for the overall range rate curve by determining individual Δv, and subsequently combining these together. In order to do this, the associated Δ^{2}ϕ must be unwrapped. To accomplish this, we can utilize two constraints resulting from the underlying physics: (1) the range rate curve must be smooth, and (2) the curve should roughly match the results of the range differencing approach. This in turn suggests a two-step approach of (1) unwrapping the phase to arrive at a family of possible answers which satisfy the smoothness criterion and (2) using the results of the range differencing approach to select the single correct answer.

[18] The desired answer is a set of k_{i}, k_{i} ∈ , i ∈ {1.N}, with N equal to the number of samples in the streak, such that

[19] The smoothness constraint is addressed first by minimizing the higher-order derivatives. Specifically, we

[20] 1. Let k_{1},k_{2} = 0.

[21] 2. For i = 3.N, step 1 find the slopes Δ^{2}ϕ_{i−1} and Δ^{2}ϕ_{i}, step 2, over a range of j ∈ {−10…10}, find j* such that ∥(Δ^{2}ϕ_{i} + j2π) − Δ^{2}ϕ_{i−1})∥ is minimized and step 3, let {k_{i}…k_{N}} = {k_{i}…k_{N}} + j*.

[22] Note that step 3, which adds the correction to both the current and all successive points, is effectively equivalent to an integration. This results in a smooth curve of Δ_{i}, but there are other curves that are equally smooth that can be arrived at by adding in lines with slopes of integer multiples of 2π. Thus, an entire family of solutions exist,

Since wrapping is limited to multiples of 2π, m must be an integer, and the guaranteed low values of deceleration constrain it to be small.

[23] The proper value for m can be found by finding the corresponding solution for v that best matches the rough range rates arrived at using the differencing method. In order to do this, a v_{0} must be estimated, since the previous technique is based on integrating Δv and therefore does not provide this. For now this is estimated by simply averaging the first four points of the rough range rates, designated v_{R}, so

The solution for m is then found by minimizing the RMS error between phase-based and rough differencing range rates, over m,

Finally, this results in a range rate curve solution of

The results of this process are shown in Figure 3. The red line is the final selected range rate solution.

[24] It was experimentally observed that for a few meteoroids with particularly large changes in deceleration the smoothness assumption breaks down. In order to counter the resulting ambiguity in k_{i} selection, a check was inserted into the algorithm which stops the phase unwrapping method when the difference between the two closest answers for adjacent k_{i} values becomes <0.5 rad/(PRI)^{2}. The final answer is then constructed by fitting a fifth-order polynomial to the combined set of valid unwrapped points and difference based v_{R,i} for the remainder of the streak. An example streak illustrating this is shown in Figure 4.

3.3.3. Validation Case

[25] As a check on the validity of the algorithm we tried using it independently in both UHF and VHF on a special case with sufficient signal strength in UHF. The results are shown in Figure 5; agreement is good with both bands producing nearly identical results for deceleration. The slight vertical offset between the two results from a combination of slightly different v_{0} values and RDC.

[26] It should be noted that the phase-based method is more liable to fail in the first part of the algorithm (i.e., the initial unwrapping) in UHF. This is because the constant for UHF used in equation (14) is larger than that for VHF by a factor of 2.7, which in turn makes Δϕ greater for a given value of Δv and thus easier to confuse with a corresponding wrapped possibility.

3.3.4. Error Analysis

[27] Two kinds of errors occur with the phase-based technique. The first is catastrophic, in the sense that if the wrong value of k_{i} is selected early in the streak, or an ambiguity occurs which results in an early termination of the phase-based method, the resulting range rate estimate is totally wrong. These types of errors are mostly discernible by eye; a particularly obvious example is shown in Figure 6.

[28] The second type of error is much smaller, occurring as a result of misestimating v_{0}, combined with noise in the phase measurements. If we assume that v_{0} induced error is much larger than the phase measurement induced error, and that v_{0} was basically constant over the first 4 measurements, then the error corresponding to equation (19) becomes

Figure 7 shows the 1 σ error bars for the differencing method and the phase method for a given meteoroid trace based on these assumptions.

[29] Also of interest is the deceleration (i.e., Δv) error, which has no v_{0} component overshadowing the noise of the phase measurement. Making a good estimate of this is difficult because (1) there is no “ground truth” available through alternative data from direct measurements, (2) there is no sufficiently specific analytical form available to fit the range curve to, and (3) σ_{Δϕ} is a (an unknown) function of the SNR. A characterization of the relation between σ_{Δϕ} and the SNR might be addressed using the observed intrapulse phase variation, however, a full investigation of this cannot be supported as part of the current effort, and will therefore be postponed.

4. Conclusion

[30] The evaluation of meteoroid range rates is necessary in order to calculate a variety of meteoroid characteristics including mass, density, etc. In this work we have presented three methods and have shown that the novel interpulse Doppler approach in particular provides unprecedented measurement of velocities and accelerations.

[31] Future work will include an investigation of the interpulse Doppler method's error dependence on SNR so that proper error bounds can be found for both velocity and acceleration. Application of the method to a large number of meteoroids will also be conducted in order to examine statistical results for a large population. Algorithmic improvements will also be investigated in order to overcome the current breakdown of the method when accelerations become too large. Starting the Δϕ calculation from the middle of the streak will also be investigated in order to mitigate the sensitivity of the technique to the low SNR values present at the ends of the streak.

Acknowledgments

[32] The authors would like to acknowledge the provision of data by William Cooke of the NASA Marshall Space Flight Center and the Los Alamos National Laboratory LDRD office for funding this effort.