1. INTRODUCTION

Fabry–Perot interferometers (FPIs) are commonly used to measure line-of-sight atmospheric neutral wind and temperature and have become common in the fields of upper-atmospheric and ionospheric science. Winds and temperatures largely influence the dynamics of the upper atmosphere, playing a significant role in the generation of ionospheric electric fields and currents, instabilities [1], and joule heating [2].

Over the years, many different techniques have been used to extract winds and temperatures from FPI data. Early methods often used Fourier fits on individual fringes measured using photomultiplier tubes and pressure scanning of the etalon [3]. As detector technology improved, many systems moved toward instrumentation based on charge-coupled device (CCD) image sensors because of the ability to simultaneously image multiple fringes. However, it was discovered that using single-fringe Fourier fits leads to biases in temperature and an increase in measurement uncertainties caused by cross contamination between neighboring fringes [4]. Additionally, different temperatures can be inverted depending on which individual fringe is used [5]. Assumptions of an instrumental function having a Gaussian shape have also been shown to result in reasonable wind measurements [6]; however, temperature reconstructions are often inaccurate [7]. Combinations of (and/or adjustments to) various techniques, e.g., using a Gaussian fit for wind reconstruction and a Fourier fit for temperatures [8] or using an Airy function that has been modified to account for etalon distortions and optical aberrations [9] have been used to address the various issues related to each fitting methodology.

More recently, [10] developed a CCD-based nonlinear regression method that fits a modified Airy function (accounting for various nonidealities such as blurring and intensity falloff) to a laser exposure to characterize the system function. The sky exposures are then modeled as the convolution of the system function and a Gaussian sky spectrum. In doing so, all fringes are considered in the inversion simultaneously. This method has been further explored in [11], where certain terms in the model are given slightly different functional forms.

A commonality among all of these methods is an implicit assumption that the system characterization, derived from exposures of a frequency-stabilized laser, and the inversion of wind and temperature from a sky exposure are performed independently. Because sky exposures often require different integration times and are taken after each laser exposure, the cadence at which the optical path length is characterized by the laser is influenced by the conditions that determine the sky integration time (e.g., sky brightness). As this integration time increases, variation of the optical path length between laser exposures (caused by temperature fluctuation in the instrument housing, for example) may not be fully captured by the information available from the before and after path-length characterizations and will lead to a larger total velocity uncertainty.

Here, we propose an extension to the regression technique outlined in [10] used for recovery of neutral wind and temperature using data from an FPI system. As we will demonstrate, this technique permits simultaneous retrieval of wind, temperature, and optical path length between laser exposures. By accounting for changes in the optical path length between each laser exposure, the rate at which a laser calibration is required will be significantly reduced.

In the formulation from [10] (henceforth referred to as the standard method or standard deconvolution), the forward model of the CCD fringe pattern (where $A$ and $Y$ represent the instrument function and source spectrum, respectively)

Depending on the emission brightness, the exposure time needed to obtain a suitable inversion of $Y(\lambda ,t)$ from a sky fringe can reach as high as 10 min. The large time gap between characterizations of a fluctuating optical path length during such long exposures amplifies the error introduced by temporal interpolation into the neutral wind measurement. Additionally, in the case of a bright emission, the time it takes to perform the calibration exposure sets a lower bound on the cadence of the wind and temperature measurement that would otherwise not exist. Practical hardware implementations also often require that the mirror system be mechanically turned to face the laser during calibration, further limiting the minimum time between measurements.

Here, we consider the determination of $Y(\lambda ,t)$ and the optical path length simultaneously. To facilitate this, we modify the original signal $S(r,t)\to \tilde{S}(r,t)$ to include a pilot signal near the operating point. The pilot signal can be added, for example, by allowing scattered light from the reference laser to enter the instrument field-of-view during a sky exposure. In Sections 2 and 3, we mathematically formulate a (forward and inverse) model for piloted sky exposures containing a single emission and test the inversion performance relative to standard deconvolution using Monte Carlo simulations. In Section 4, we characterize the additional velocity uncertainty present when using isolated calibration exposures in the presence of a time-varying optical path length and show that deconvolution using a pilot signal (which we will henceforth call piloted deconvolution) often leads to a lower variance velocity estimate. Finally, in Section 5, we present several pragmatic physical configurations that would allow for construction of a piloted signal in a real system. The configurations are tested and compared using the FPI installation at the Urbana Atmospheric Observatory (UAO; 40.17°N, 88.16°W).

2. FORWARD MODEL

Piloted sky exposures are modeled by extending Eq. (1) to include a pilot signal, i.e.,

Table 1. Model Parameters

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3. INVERSE MODEL

To develop the inverse model, we consider $\tilde{S}(r,t)$ around the operating point ${\lambda }_0$ and assume that the angular dimension has been collapsed using the summation scheme from [10] in order to increase SNR. This makes the measurement a 1D signal and a function of radius only. We expand Eq. (2) in terms of the Riemann sum

Mathematically, we seek to minimize the covariance-weighted sample error in the 2-norm, i.e.,

Gradient-based nonlinear inversion methods (e.g., Levenberg-Marquardt) of this equation are quite sensitive to optical path length changes and require a reasonable initialization. To address this, the system is first run through a procedure, whereby a standard set of exposures is collected (i.e., unpiloted laser and sky exposures) and inverted using exactly the same methodology as in [10]. This initial inversion serves to characterize the operating point of the parameters in Table 1. Following this, piloted exposures are collected, and inversions of the sky parameters (and also the optical path length) are performed on each image until the next set of calibration exposures. Fluctuation of the optical path length, $nd$, is determined by assuming the etalon gap, $d$, to be a free parameter ($n$ is fixed to unity).

In general, straightforward inversion of Eq. (14) to determine $\widehat{\mathbf{q}}$ has a much greater computational complexity (orders of magnitude) than the standard deconvolution method. This is because the sky parameters are estimated at the same time as the system matrix $\mathbf{A}$, and iterative minimization routines must repopulate the entries of $\mathbf{A}$ for each evaluation of Eq. (14). This issue is not present with the standard methodology because the system function and sky parameters are determined independently using separate images. To overcome this issue, we employ a customized version of coordinate descent. This is similar to the strategy employed by [10], where only certain parameters are allowed to vary within each inversion step. The parameter inversions from a given step are used as the initialization for the next step. Each stage of coordinate descent uses Levenberg–Marquardt on the free parameters but fixes the system matrix $\mathbf{A}$ to reduce computational complexity. The system matrix is updated between inversion stages. Due to the large number of possible combinations, we did not comprehensively investigate an optimal order in which to invert the parameters. In general, however, it did appear that first inverting variables affecting fringe position (e.g., $d$ and $v$) prior to the rest of the parameters (which affect fringe shape) was important for a good solution. This is likely because gradient steps resulting from changes to fringe shape do not tend toward local minima that lead to accurate values of velocity and temperature if the fringe centers are not initialized reasonably close to their true positions.

A. Monte Carlo Simulations

In order to test inversion performance, we ran several sets of 500-trial Monte Carlo simulations. The first set of simulations tested the ability of piloted deconvolution to retrieve a 100 m/s wind having a temperature of 800 K. We took ${\mathbf{\Sigma }}_s=\mathbf{0}$, ${\mathbf{\Sigma }}_{es}=\mathbf{0}$, and ${\mathbf{\Sigma }}_e=\mathrm{diag}(\mathbf{e})$ where $e_i\sim \mathcal{N}(0,\sigma )$ is a Gaussian distributed random variable with mean 0 and standard deviation $\sigma$. We define SNR as $Y_{\mathrm{line}}/\sigma$. We also ran the simulations using the standard method from [10], i.e., first a laser image was generated and the system parameters were inverted. This was then followed by an inversion of the sky parameters. Results were generated using the piloted method by first performing the laser/sky calibration for initialization. A simulated piloted exposure was then generated with a different optical path length than what was used during calibration. This was done by changing the etalon gap, i.e., $d\to d+\mathrm{\Delta }d$, where $\mathrm{\Delta }d\sim \mathcal{U}(-a,a)$ is a random variable that is uniformly distributed between $-a$ and $a$ (we took $a=2.5\mathrm{nm}$). Piloted deconvolution was then carried out on the simulated exposure for each trial. The results are shown in Fig. 1 and Table 2 for SNRs of 5, 12.5, and 25. The top row in Fig. 1 shows the results using standard deconvolution, and the bottom row for piloted deconvolution. In general, standard deconvolution exhibits a better inversion performance; at a given SNR, velocity and temperature uncertainties (and also biases) are smaller for standard deconvolution. This is not surprising; the piloted signal model is more complex and the assumptions made (e.g., fixing the system matrix at each inversion stage) should be expected to lead to a suboptimal inversion procedure. In [10], a similar Monte-Carlo simulation at $\mathrm{SNR}=5$ resulted in uncertainties of 1.8 m/s and 6.5 K for velocity and temperature, respectively. We were not able to reproduce uncertainties this low at $\mathrm{SNR}=5$ (see Table 2), despite using the same methodology. The crude initial values in Table 1 were based on average values observed at UAO over a multiyear period, which might be different than the initialization used in [10]; our standard inversion could be converging to a nearby less-optimal local minimum.

Table 2. Inversion Performance

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Fig. 1. Monte Carlo simulations testing the ability of both standard and piloted deconvolution to retrieve a 100 m/s wind having a temperature of 800 K. Retrieval using standard deconvolution (top row). Retrieval using piloted deconvolution (bottom row). Each column is for a different SNR. In general, standard deconvolution exhibits better inversion performance; at a given SNR, velocity and temperature uncertainties (and also biases) are smaller for standard deconvolution.

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The second set of simulations investigated wind and temperature biases. The piloted exposures were generated in the same manner as in the first simulation, including the perturbation to the optical path length. Additionally, true velocity and temperature were chosen such that $v\sim \mathcal{U}(-b,b)$ and $T\sim \mathcal{U}(f,g)$, where $b$ is 300 m/s, $f$ is 200 K, and $g$ is 1200 K. The results are shown in Fig. 2. Notice the presence of a velocity bias when the true velocity is negative. The magnitude of the bias increases at higher negative true velocities, reaching around 20 m/s at $-300\mathrm{m}/\mathrm{s}$ (panel a). This corresponds to the situation where the sky fringes Doppler shift toward the nearby pilot signal, leading to a one-sided overlap between the Airy peaks and the tail of the sky signal. Additionally, both the inverted velocity and temperature variance increase with true temperature (panels b and d). A velocity bias appears with increasing true temperature (panel b), reaching around $-15\mathrm{m}/\mathrm{s}$ at a true temperature of 1200 K. The temperature error did not appear to appreciably change with true velocity (panel c).

 

Fig. 2. Monte Carlo simulation results demonstrating biases over velocity and temperature. Notice the presence of a velocity bias when the true velocity is negative. The magnitude of the bias increases at higher negative true velocities, reaching around 20 m/s at $-300\mathrm{m}/\mathrm{s}$ (panel a). Both the inverted velocity and temperature variance increase with true temperature (panels b and d). A velocity bias appears with increasing true temperature (panel b), reaching around $-15\mathrm{m}/\mathrm{s}$ at a true temperature of 1200 K. The temperature error did not appear to appreciably change with true velocity (panel c).

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In the third set of Monte Carlo simulations, we investigated the dependence of the biases on SNR. The piloted exposures were generated in the same manner as in the second simulation, i.e., the perturbation to optical path length was applied, and true velocity and temperature were drawn from the aforementioned uniform distributions. In addition to this, the signal-to-noise ratio of each simulated exposure was chosen such that $\mathrm{SNR}\sim \mathcal{U}(0.5,25)$. The results are shown in Fig. 3 for 500 trials. Both velocity and temperature uncertainty tend to decrease with increasing SNR. There appears to be a velocity bias of around $-10\mathrm{m}/\mathrm{s}$ that appears at high SNR. No temperature biases over SNR were apparent. In Fig. 4, the sample and estimated uncertainties are compared. Velocity uncertainties tend to be underestimated by a factor of around 3 at higher SNRs. Temperature uncertainties are slightly underestimated but more closely reflect the sample uncertainties. The source of the bias is unclear at this point; however, our leading hypothesis is that a higher SNR leads to a higher-curvature residual manifold [i.e., Eq. (14)]. We use a gradient-based inversion methodology, and so it is likely that the initialization provided by the laser/sky calibration is not sufficiently close to the global minimum in order to converge on the true velocity. This may also be the reason why the velocity uncertainties are underestimated, i.e., the curvature information used to propagate the uncertainties would not be calculated around the global minimum.

 

Fig. 3. Monte Carlo simulation results demonstrating biases over SNR. There appears to be a velocity bias of around $-10\mathrm{m}/\mathrm{s}$ that appears at high SNR. No temperature biases over SNR were apparent.

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Fig. 4. Sample and estimated uncertainties as a function of SNR. Velocity uncertainties tend to be underestimated by a factor of around 3 at higher SNRs. Temperature uncertainties are slightly underestimated but more closely reflect the sample uncertainties.

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In the final set of Monte Carlo simulations, we investigated the ability of piloted deconvolution to recover changes to wind and temperature in the presence of changes to optical path length, laser intensity, and sky brightness (these parameters are likely to change from exposure to exposure in a realistic scenario). After the standard laser/sky calibration, random perturbations to $v$, $T$, $d$, $I_0$, and $Y_{\mathrm{line}}$ were drawn from the distributions

 

Fig. 5. Velocity and temperature error as a function of single-step optical path length fluctuation in the presence of changes to wind, temperature, laser intensity, and sky brightness. Velocity error appears to slightly depend on $\mathrm{\Delta }d$, most visible for extreme single-step fluctuations (notice the structure near $\mathrm{\Delta }d=\pm 2\mathrm{nm}$ at higher SNR); however, the source of the dependence is currently unclear. The velocity uncertainties do not tend to decrease between $\mathrm{SNR}=5$ and $\mathrm{SNR}=25$. Temperature errors are well-behaved, exhibiting no biases as a function of $\mathrm{\Delta }d$ and exhibiting a decrease in uncertainty at higher SNRs.

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4. CHARACTERIZING REFERENCE UNCERTAINTY

Typically, systems employing the standard deconvolution method will use laser exposures before and after a sky exposure (with the corresponding inversions $A(r,\lambda ,t-\mathrm{\Delta }{t}_1)$ and $A(r,\lambda ,t+\mathrm{\Delta }{t}_2)$, respectively) and interpolation to estimate the optical path length at the sky exposure timestamp $t$ for inversion of wind and temperature [10]. Under this scheme, long sky exposure times decrease the effective rate at which the optical path length is characterized. This leads to an additional source of uncertainty arising from time variation of the optical path length between laser exposures, typically caused by fluctuation of the ambient temperature within the instrument housing. Variation of the optical path length directly corresponds to a shift in the position of the Airy peaks and, subsequently, the velocity measurement.

Because piloted deconvolution simultaneously measures the optical path length and sky state, it does not suffer from this type of uncertainty; no interpolation needs to be performed. A fair comparison between the standard and piloted velocity estimator variances should therefore (in the least) account for the error introduced into the velocity measurement from a sparse temporal characterization of optical path length when using standard deconvolution. The amount of velocity error caused by a small change to the etalon gap $\mathrm{\Delta }d$ (representing an optical path length change) is given by $\mathrm{\Delta }v=\frac{\mathrm{\Delta }d}{d}c$, where $d$ is the baseline value of the etalon gap and $c$ is the speed of light. This formula can easily be derived by taking the derivative of the constructive interference condition for an Airy function with respect to $d$ and $\lambda$ and using the Doppler shift relationship $\mathrm{\Delta }\lambda =\mathrm{\Delta }v\frac{\lambda }{c}$.

In order to characterize the aforementioned etalon gap temporal interpolation error (which we will henceforth refer to as reference uncertainty) arising from the standard deconvolution method, we perform the following procedure. For a constant laser exposure time $\mathrm{\Delta }{t}_L$, the effective cadence of etalon gap retrievals will be $\mathrm{\Delta }{t}_L+\mathrm{\Delta }{t}_S$, where $\mathrm{\Delta }{t}_S$ is the sky exposure time. Denoting the true etalon gap variation as $\mathrm{\Delta }d(t)$, a set of knots is created at times $(\mathrm{\Delta }{t}_L+\mathrm{\Delta }{t}_S)k$ for integer $k$ and used to generate the series $\mathrm{\Delta }d([\mathrm{\Delta }{t}_L+\mathrm{\Delta }{t}_S]k)$. Denoting the linear interpolation associated with these knots as $\mathrm{\Delta }{d}_S(t)$, the interpolation error is mathematically (in terms of velocity)

 

Fig. 6. Standard deconvolution interpolation error assuming a 5 nm amplitude, 20 min period sinusoidal gap variation. Panel a shows $\mathrm{\Delta }d(t)$, and panels b, c, and d show the interpolation error $e_{vd}$ for sky exposure times of 3, 6, and 9 min, respectively.

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We ran the aforementioned reference uncertainty characterization procedure across a realistic range of variation amplitudes, periods, and sky exposure times. The results are shown in Fig. 7 for three characteristic gap variation periods. Longer exposure times, larger amplitude variations, and shorter characteristic periods lead to an increased inability to describe the true variation with linear interpolation and are associated with increased reference uncertainties.

 

Fig. 7. Reference uncertainties calculated across a realistic range of variation amplitudes and sky exposure times for three characteristic gap variation periods. In general, longer exposure times, larger amplitude variations, and shorter characteristic periods lead to an increased inability to describe the true variation with linear interpolation and are associated with increased reference uncertainties.

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Having a method for characterization of reference uncertainty, we can now provide a comparison of the total uncertainty in velocity between standard and piloted deconvolution. The comparison assumes that the output of both methods is Gaussian distributed, and that the reference and inversion uncertainties add in quadrature. The true variation of the etalon gap is assumed to be a 5 nm sinusoid with a 20 min period. We show the results for SNRs of 5, 12.5, and 25 in Table 3 under the same conditions as the first Monte Carlo simulation (i.e., a 100 m/s wind with temperature 800 K). The bolded values of ${\sigma }_v$ are cases in which the piloted method has a lower total variance in the velocity estimate. Indeed, for longer exposure times, our results suggest that piloted deconvolution is the lower variance estimator for velocity. This remains true even when the relatively larger velocity biases from Table 2 are added into the uncertainties (in general, these biases may be removable with a more refined initial guess; this would be a good topic for future study).

Table 3. Total Velocity Uncertainty

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5. HARDWARE IMPLEMENTATION

In this section, we discuss important aspects of the full physical system needed to take measurements of wind and temperature using a Fabry–Perot interferometer. We provide a brief overview of a typical system setup used in conjunction with standard deconvolution, and then we discuss modifications of the setup that will permit use of piloted deconvolution.

In Fig. 8, we show the hardware configuration used at UAO (which measures the 630.0 nm redline emission). This system, often referred to as the MiniME FPI [13], alternates between taking calibration and sky exposures and uses the standard deconvolution method to extract wind and temperature from the recorded data. In this system, light enters the sky scanner, a scanning mirror system, (see Fig. 8) and is directed through a 630.0 nm filter and etalon. The resulting fringe pattern is then focused using a lens onto a CCD. During calibration exposures, the sky scanner is directed toward diffuse light from the laser (created by first scattering the laser light into a scattering chamber), and a 30 s exposure is taken. During sky exposures, the sky scanner is turned toward a specified direction facing the sky. In order to avoid laser contamination during sky exposures, the laser is equipped with a USB-controlled shutter that opens and closes at appropriate stages throughout the process of collecting data.

 

Fig. 8. Typical FPI hardware configuration.

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A. Modifications for Piloted Deconvolution

In order to simultaneously obtain laser and sky fringes, modifications are required to the aforementioned setup. The system must allow both laser light and light from the sky emission to enter the system aperture. Several possible methods for doing so are shown in Fig. 9.

 

Fig. 9. Proposed modified configurations for piloted deconvolution.

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The first proposed modification (referred to as the dome scatter method) is shown in Fig. 9(a). The hardware setup is identical to what is used for standard deconvolution, with the only difference being that the laser shutter is kept open for the entirety of the sky exposure. The laser light exiting the scattering chamber enters into the sky scanner by scattering off of the plastic dome that separates the instrument from the open sky. Assuming that the dome scattering occurs uniformly, the aperture should remain uniformly lit and keep the new method as similar to the current method as possible. No hardware modifications are required, and only slight modifications to the operating code would be needed.

The second proposed modification (henceforth referred to as the direct method) is shown in Fig. 9(b). Here, the laser light directly enters the sky scanner during the sky exposure. The immediate disadvantage of this method is the loss of diffuseness in the laser light and the introduction of nonuniformity in the fringe pattern on the CCD. However, this method requires the least amount of laser power, and many lasers will deliver a sufficient amount of light after only a few seconds of exposure (in our tests, a subsecond laser “shutter blink” was more than sufficient). Implementation of this method would require hardware modification because the fiber optic cable would need to be mounted onto the sky scanner. Additionally, the fiber would need precise positioning in order to obtain a uniform fringe pattern. The mounting system would also need to move with the sky scanner and not block the instrument’s view of the sky. It is also quite easy to overexpose the laser with this method; thus, it would be important to ensure that the laser fringe intensity will not overwhelm the sky fringe through proper control of the laser shutter.

The final proposed modification (which we have named the specular method) is shown in Fig. 9(c), and represents a trade-off between the two previous cases. The fiber is pointed toward the sky such that the light scatters off of a small part of the dome and enters into the sky scanner. Less laser light enters the system than with the direct method but more consistently and uniformly spreads across the aperture. As with the direct method, the specular method would require a mounting system. If the dome is spherically symmetric and different sky exposures require only azimuthal changes in pointing direction, a correctly positioned fiber would not need to be reoriented for different exposures. An additional complication arises, however, due to the construction of the sky scanner. The pointing head consists of two mirrors whose angles are independently controlled. The top mirror (the first mirror that light reflects from when entering the system aperture) is able to rotate freely over 360°. Thus, the final mounting solution would need to account for the sky scanner range of motion.

B. Field Experiments

The FPI located at UAO was utilized to test each of the proposed modifications. The system uses a $1024\times 1024$ Andor CCD with 2-by-2 binning and a class-3R 632.8 nm frequency-stabilized He–Ne laser. The tests were carried out over the span of three nights.

The dome scatter method was tested by simply leaving the laser shutter open and attempting to take an exposure. This was performed for exposure times ranging from 30 to 240 s. Shorter exposure times yielded images with no visible signature of the pilot signal. Tests with higher exposure times (greater than or equal to 120 s) yielded a visible pilot signal but with an amplitude likely too weak for tractable application of piloted deconvolution. An example of a piloted fringe pattern collected using the dome scatter method is shown in Fig. 10(a). Notice the low intensity of the pilot signal. We conclude that a relatively high-powered laser would be needed for the dome scatter method to be feasible.

 

Fig. 10. Images obtained for each of the three modified data collection methods. Overlaid in white is a plot of the $\theta$-integrated fringes.

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The direct method was tested by disconnecting the fiber optic cable from the scattering chamber and pointing it directly into the sky scanner by hand. The sky exposure was collected normally, and the laser shutter was opened for a small period of time during sky collection. The resulting pilot signal was strong for essentially any shutter-open duration. In fact, exposure times over just a few seconds resulted in a significant amount of blooming on the CCD, rendering the exposure useless. A 1 s “shutter-blink” was more than sufficient to get the pilot signal into the image alongside the sky fringes. However, achieving uniform intensity across the CCD was difficult to do consistently. Slight misalignment in the positioning resulted in severe image nonuniformity. This effect would likely be reduced in practice because the fiber would be on a mount. The mount would require good stability and likely need precise adjustments for nonuniformity to be avoided. An example of a piloted fringe pattern collected using the direct method is shown in Fig. 10(b).

Finally, the specular method was tested by manually pointing the fiber optic cable at the dome as depicted in Fig. 9(c) for part of the sky exposure. As expected, this method resulted in a compromise between the dome scatter and direct methods; further, sufficiently high intensities were achievable by keeping the shutter open throughout the duration of the exposure, and the nonuniformity was less severe than with the direct method (and also less susceptible to small changes in the pointing direction). An example of a piloted fringe pattern collected using the specular method is shown in Fig. 10(c). The result of the piloted inversion procedure applied to the fringe shown in Fig. 10(c) is shown in Fig. 11. The fit agrees with the data quite well. Overall, our tests suggest that the positioning of the laser does not need to be as precise as the direct method to achieve the same level of uniformity on the CCD.

 

Fig. 11. Example inversion on integrated data collected at the Urbana Atmospheric Observatory using the specular method. The associated fringe is shown in Fig. 10(c). The data are shown in black. The reconstructed signal is shown in green.

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An additional result of our experiments worth noting is that the laser needs to be sufficiently bright in order for these methods to work. We repeated the experiments with a significantly dimmer laser (by a factor or 150 or more in terms of integrated counts), and only the direct method was able to generate a visible pilot signal, albeit with an extremely nonuniform intensity. Therefore, the laser needs to be of sufficient quality and brightness in order to obtain a desired fringe amplitude.

For all methods, further experimentation would be beneficial. Most systems use dynamic integration times in order to obtain a desired sky brightness (emission intensity can change throughout the night). It is unclear at this time how the parameter estimates behave as a function of the laser-to-sky peak ratio, a quantity fundamentally tied to both sky exposure time and the duration that the system is allowed to collect light from the laser in order to build the pilot signal. Based on our current experimentation, we conclude that the specular method is the most pragmatic overall of the three methods for achieving a piloted sky fringe at a quality sufficient for recovery of wind and temperature.

6. CONCLUSIONS

In this study, we have provided a proof-of-concept implementation of piloted deconvolution and demonstrated its feasibility for inversion of atmospheric neutral wind and temperature. This was done by extending the forward model developed in [10] to include a pilot signal. We discussed the increased computational complexity required to optimally invert system and sky parameters simultaneously; we also presented an alternative inversion method that, while suboptimal, is computationally tractable.

Several sets of Monte Carlo simulations were carried out to test inversion performance. Standard deconvolution had smaller inversion variances and smaller biases for both wind and temperature. With piloted deconvolution, the magnitude of the velocity bias tended to increase in situations where the true velocity caused the sky fringes to shift toward and slightly overlap with the pilot signal. Velocity uncertainties were also underestimated at high SNRs by a factor of around 3. The performance of piloted deconvolution in the presence of realistic changes to the sky parameters remained competitive.

When comparing total velocity uncertainty, however, we found that piloted deconvolution is the lower variance estimator for exposure times exceeding around 3 min (the exact threshold will be SNR and system dependent). This is because piloted deconvolution does not suffer from reference uncertainty arising from sparse temporal characterization of the etalon optical path length. This is a limitation of the standard deconvolution method that is avoided with inclusion of a pilot signal. The standard method remained the lower variance estimator for temperature but only by a small amount (uncertainty differences were $⪅3\mathrm{K}$).

We also discussed several possible modifications that could be made to an existing FPI installation in order to construct a piloted sky exposure. Each method had specific advantages and disadvantages. Based on field experiments carried out at the Urbana Atmospheric Observatory, we concluded that a method based on targeted dome scattering (which we named the specular method) was the most pragmatic overall of the methods for achieving a piloted sky fringe at a quality sufficient for recovery of wind and temperature. The piloted inversion algorithm developed in Section 3 was applied to the real data collected using the specular method; qualitatively, the results were shown to fit the data quite well.

There are some facets of this study that would benefit from future work. First, in practice, frequency-stabilized lasers often “lock on” and stabilize around wavelengths that are slightly different than nominal (e.g., 632.8 nm in the case of a He–Ne laser used for calibration of a redline FPI). This effect is uncontrollable, i.e., although the temporal stability of the center frequency can be generally guaranteed once a lock is achieved on power-up, the center frequency itself does not always lock onto the same value after the laser power has been cycled. This effect manifests itself as a constant shift in the relative position of the sky and laser fringes, and may affect the fidelity of the inversion if the sky and laser fringes have appreciable overlap. Second, we make no claims about the optimality of the inversion procedure we developed to perform piloted deconvolution. It may be possible to reduce or eliminate the velocity biases by using global minimization techniques (e.g., basin hopping or differential evolution) and/or using a better initialization procedure. Last, further testing of the proposed physical system modifications presented in Section 5 and implementation of a fully engineered solution for construction of a piloted sky fringe will ultimately be necessary in order to test the method over a long-term campaign for validation against currently accepted methods for recovery of wind and temperature.