1. Introduction

Doppler Michelson Interferometry (DMI) is a well-established observational technique through which a field-widened Michelson interferometer is utilized to remotely detect atmospheric motions. To date, its primary implementation has been the observation of winds in the Mesosphere and Lower Thermosphere (MLT) using spectrally isolated airglow emissions [1]. The technique was pioneered during the development of the Wind Imaging Interferometer (WINDII) which flew on the UARS satellite [2]. Since the development of WINDII several other satellite instruments have been proposed [3–6] and several ground based instruments have been implemented in the field [7–11]. Surprisingly, although specific fringe analysis procedures have been described in these papers, to date, no general framework for the fringe analysis approach and consideration of the effects of instrument parameter uncertainties has been published. This deficiency is rectified with this paper.

DMI instruments operate in a somewhat similar manner to a Fourier Transform Spectrometer (FTS). The interferometer is scanned and the intensity is recorded at several phase steps spanning a fringe of path difference. The DMI approach takes advantage of the fact that when an isolated Doppler broadened emission is observed, the motion and temperature of the source result in detectable phase and fringe visibility variations in the interferograms. Instead of recovering the spectrum, fringe sampling permits the phase and visibility of the interferogram to be extracted and related to the source motion and temperature [8]. In earlier implementations of this technique for ground based observations, fringe analysis algorithms and calibration techniques for single point measurements and fixed $\pi /2$ phase steps [8] were developed. In WINDII [2] limb images were taken, but the fringe analysis was undertaken on a bin-by-bin basis with fixed $\pi /2$ phase steps, and wind and temperature profiles were determined through column averaging. In both cases, the full advantage of fringe imaging and the flexibility of using arbitrary phase steps was not implemented. In recent implementations of this technique for ground based imaging of gravity waves in airglow [11,12], a specific application of fringe imaging was used. In this paper, the general theory justifying this application is described for the first time.

To measure winds using DMI, variations in fringe phase relative to the background fringes of equal inclination (Haidinger fringes) for a motionless emission source must be determined. This is especially true when wind imaging is undertaken. Previous DMI instruments measured the phase associated with the fringes (hereinafter referred to as the background phase) using a calibration source emitting a wavelength close to that of the target emission [8]. Although this approach was sufficient for single-point wind measurements, as shown in this paper, this does not provide an accurate enough measurement of the background phase for high resolution wind imaging purposes. This is especially true for imaging winds to diagnose gravity wave signatures which require accuracies of the order of 1 m/s ($\sim 2$ fm). The general algorithms and fringe characterization approach presented in this paper result in improvements to the background phase determination, which increase the precision and accuracy of Doppler wind images.

Advanced ground based DMI instruments such as the E-Region Wind Interferometer-II (ERWIN-II) [10] and the Michelson Interferometer for Airglow Dynamics Imaging (MIADI) [11,13] implement an imaging system and specific bin-by-bin analysis approaches based on the algorithms described in this paper and achieve significant improvements in the sensitivity, temporal, and spatial resolution of the LOS Doppler wind measurements [7,10]. In this paper, results from these instruments were used to demonstrate the effectiveness of these new algorithms and background phase calibration techniques. There have been other Doppler wind instruments which image winds (e.g., Fabry-Perot interferometer) [12]. However, in this case, the geophysical source was aurora which is at least a factor of 10 brighter than airglow and the useful precision for thermospheric winds was $\sim 10$ m/s, a factor of 10 greater than the precision necessary for useful geophysical measurements of gravity waves. Therefore, this would require an integration time of approximately 15 minutes to match the wind precision of MIADI.

For ERWIN-II, the imaging technique and bin-by-bin analysis approach provides an increased sensitivity ($\sim 1$ m/s) of single point LOS Doppler wind observations using a high temporal cadence (5 minutes, for the green line (557.7 nm), O$_2$ (866 nm), and OH (843 nm) emissions; and the four cardinal directions and the vertical). The high temporal cadence is achieved through use of a quad mirror which allows for the simultaneous observation of the four cardinal directions and the zenith [10]. This is a factor of 2 increase in the sensitivity and a factor of 4 decrease in the temporal cadence as compared to the first implementation of the ERWIN instrument [8]. On the other hand, the MIADI measurement approach exploits the imaging aspect in order to extract images of the LOS Doppler wind field in the mesopause region with precisions of < 3 m/s across an 80 km x 80 km region of the night sky in less than 12 minutes. This provides information on the smaller scale spatial and temporal variability in the LOS Doppler wind field associated with the presence of gravity waves in the region. In addition, airglow intensity and wind images are obtained simultaneously, a feature that to our knowledge was first implemented for ground based observations with MIADI.

The main purpose of this paper is to present the DMI imaging approach and to demonstrate the increases to the sensitivity and spatial resolution that are provided using these new algorithms. In addition, approaches to determining the background zero wind using a diffuse source, essential for accurate wind measurements, are discussed in detail. The approaches discussed in this paper do not require the use of an external calibration source, which is very important for wind imaging since slight differences in the optical system used for calibration and that used to view the scene introduce significant differences in the determination of the background phase. A rigorous theoretical examination of the sampling of the Michelson fringes is explored, and expressions which provide estimates for the sensitivity of the LOS Doppler wind observations are derived.

The paper begins by reviewing the general fringe analysis approach and presents the bin-by-bin analysis algorithms and calibration techniques that are used to extract LOS Doppler winds from images of the interference fringes. The measurement approach is examined theoretically for different sampling cases and analytical expressions are derived that can be used to provide estimates of the sensitivity of the LOS Doppler wind observation based on instrument parameters, the source brightness and the integration time. Trade-offs between the temporal and spatial sampling capabilities are discussed and the overall approach is demonstrated using laboratory and field observations obtained with the ERWIN-II and MIADI. The use of a cloudy day to determine the background for the ERWIN-II [10] is discussed in significantly more detail than previous work, and a new technique for background phase determinations (use of a ground-glass diffuser implemented with the MIADI) is introduced.

2. Imaging Doppler Michelson interferometry

The imaging Doppler Michelson interferometry approach is a generalized extension of the single point measurement approach that was utilized in many earlier DMI instruments [8,14]. These instruments have been used for several decades to provide ground based and satellite based measurements of upper atmospheric dynamics using isolated airglow emissions. The advantage of the DMI approach for observing low light signals (such as airglow emissions) is the high spectral resolution and large throughput that can be achieved through field widening [15]. The field-widened Michelson interferometer operation is similar to that of a traditional FTS with the air gaps replaced with appropriately selected glasses to achieve field widening, achromaticity and thermal compensation. For each particular application, the design must be optimized for observations of specific target emissions [16–18].

An unfolded and simplified schematic of the DMI imaging configuration is shown in Fig. 1.Collimated light from the scene of interest is incident on the entrance pupil and the front optical system is configured to pass the image of the scene as collimated light through the interferometer. The aperture stop is imaged at the mirror location to maximize the throughput and to ensure uniform averaging over relative spatial path variations that are the result of deviations in the flatness of the mirrors. The exit optics are designed to image the fringes of equal inclination (Haidinger fringes) conjugate to the scene of interest onto an array detector. In this configuration, there is a one-to-one mapping of bins in the image to positions in the scene and the phase variation across the image depends on incident angle ($\Phi _0(\vec {u})=2\pi \sigma _0\Delta (\vec {u}(\theta ))$), where $\vec {u}$ is the position vector on the detector array, $\sigma _0$ is the rest (i.e., not Doppler shifted) wave number of the emission, $\Delta$ is the optical path difference, and $\theta$ is the incident angle.

The exact angular dependence is determined by the specific interferometer construction and the associated optical path traversed by the off-axis rays. For a field widened instrument, the variation in optical path with off axis angle is reduced to an acceptable level during the optimization process. In a manner analogous to the traditional FTS, the number of fringes crossing a particular bin is generally kept (by design) below 0.5 in order to minimize the reduction in contrast of the interfering beams due to averaging the signal over the associated phase gradient.

For a quasi-monochromatic source, such as a spectrally isolated airglow emission, the intensity recorded at a particular position on the array detector is given by

The fringe parameters $I_0(\vec {u})$, $V(\vec {u})$, $\Phi _0(\vec {u})$ and $\alpha (\vec {u})$ contain important information regarding dynamical properties of the source. In the case of airglow observations, the intensities are related to the vertical motion of the source and the constituent mixing ratios, and the visibility is related to the temperature. If the source is in motion with velocity $w(\vec {u})$, then this results in a small phase shift in the interferogram given by Eq. (2) where $D$ is the effective path difference.

It is clear that there are two basic configurations that can be realized to measure Doppler shift in well isolated emission lines. The first configuration is to take single point LOS wind measurements from each image by averaging over some region of the image or by utilizing a single element detector. This allows for a high signal to noise ratios to be achieved resulting in a high wind precision (by averaging), without sacrificing the temporal resolutions through long integration times [8,10]. The second method is to take full advantage of the imaging capability of the device. This allows for a much higher spatial resolution at the cost of a reduction in signal to noise ratio. This leads to a reduced wind precision that can be regained by increasing integration times or averaging adjacent measurements [13]. These trade-offs must be balanced against the scientific requirements of the particular instrument.

 

Fig. 1. Schematic of an unfolded imaging Michelson interferometer and the associated Haidinger fringes shown imaged onto a two-dimensional array.

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3. Bin-by-Bin analysis

Fully exploiting the imaging approach requires a bin-by-bin analysis of the phase, as well as, an ability to account for the variation of phase across the image due to the dependence of the phase on incident angle. Most of the previous single point DMI instruments used $\pi /2$ phase steps in order to use a simple four-point algorithm. The limitation of this algorithm is that it requires step sizes to be exactly $\pi /2$. Since different wavelengths would require different mirror steps to produce a $\pi /2$ phase shift, to acquire simultaneous scans with different wavelengths would necessitate at least one of the wavelengths having steps which are not multiples of $\pi /2$. The fringe analysis approach outlined below relaxes these restrictions by allowing non-$\pi /2$ phase steps and any combination of phase steps in a scan. It also provides closed analytical expressions for wind determinations so that the effects of the stepping profile and instrument uncertainties on the derived winds can be evaluated (see section 4).

A more robust algorithm is obtained using a non-linear least mean squares approach, and minimizing the merit function [7,10,19]. Arbitrary step sizes to be used in the scan, which is particularly useful when multiple wavelengths are to be observed simultaneously during a scan as is the case with MIADI.

The merit function can be written in terms of the Michelson equation (see Eq. (1)), and the observed intensity images, $I_s$, such that

The values for the $I_0(\vec {u})$, $V(\vec {u})$, $\Phi (\vec {u})$, and $\alpha (\vec {u})$, as contained in the vector $\vec {a}$, can be determined iteratively by

In the case of an imaging instrument, this algorithm is applied to each bin in the image of the interference fringes and an image of each of the fringe parameters is obtained. Therefore, an image of the LOS Doppler wind field is extracted from the images of $\Phi (\vec {u})$ by making use of Eqs. (1) and (2).

4. Uncertainty determination

In the ideal case, the primary factors limiting the sensitivity of the measurements are the signal to noise ratio (SNR) and the coherence of the interfering beams. In this case it is possible to determine the uncertainty in the phase, and hence wind measurements based on the intensity and visibility. The SNR is determined from photon counting statistics and the amplitude of the complex coherence of the interfering beams is determined from the visibility of the measurements $\left (V(\vec {u})\right )$.

The phase error can be derived by first assuming that each individual measurement (image) is Poisson noise limited such that the variance is equal to the intensity: $\sigma ^2\approx I_0(\vec {u})$, and that each measurement is independent [7]. For the further analysis of the uncertainty, it should be noted that each bin is considered to be a separate scan with the same stepping profile. Therefore, the dependence of the intensity, visibility and phase on the position at the imaging plane, $\vec {u}$, will be omitted in the following derivations.

An analytic expression for the wind error can be determined for a general set of steps, using the expression for $\Phi$ for a general stepping profile derived using the least mean squares (LMS) equations, namely,

By considering the summation over $i$, the index for each individual image, it is possible to re-write this using only a single constant multiplied by each image such that

Using trigonometric identities, each individual intensity becomes

This is then substituted into the equations for $\sigma _a^2$ and $\sigma _b^2$ resulting in

Given Eq. (9), which specifies the functional form of $\tan {\Phi }$, the numerator will only contain $\sin {\Phi }$ terms, and the denominator only $\cos {\Phi }$ terms. Hence, the numerator can be thought of as some constant, $\xi$, times $\sin \Phi$, and the denominator must be $\cos \Phi$ times the same constant, $\xi$.

Therefore, the variance of $\Phi$ is

If the phase step profile has a range of $2\pi$, and is evenly sampled, the resulting $\gamma$ and $\delta$ terms with subscripts of 2 and 3 are 0, and $\gamma _1=\delta _1$, removing any dependence on $\Phi$. In the case of uneven sampling, the derivation becomes more complicated and the constants must be calculated for each specific case. Care must be taken in the selection of phase steps to minimize any dependence on $\Phi$ that may arise. Here we examine three specific cases to demonstrate the capacity and scope of this generalized approach. The first two cases address situations previously published in the literature and the third situation for arbitrary step sizes.

4.1 Four-point algorithm

For the simple four-point algorithm, as shown in Gault et al. [8], the mirror steps are 0, $\pi /2$, $\pi$, and $3\pi /2$, and the resulting four intensities are

Substitution in Eq. (9) results in the standard 4-point algorithm demonstrating that it is both an analytical solution to the determination of $\Phi$ and the LMS solution (i.e., robust to the presence of noise)

If one defines $a$ and $b$ such that

It follows that the total uncertainty in the phase measurement, $\Phi$ is

4.2 Even sampling with N>4

Following a similar approach, but considering a phase step profile of (0, $\pi /2$, $\pi$, $3\pi /2$, $3\pi /2$, $\pi$, $\pi /2$, 0), such as that of the ERWIN-II, the phase error becomes

This result can be verified by comparing to the experimentally determined phase uncertainties determined for the ERWIN-II measurements. The observed wind error determined from the statistics of the measurements for a measurement window obtained on Jan 8, 2019 are shown in Fig. 2 along with the theoretical errors calculated using the intensity and visibility.These results are of the standard error, which is $\sigma _\Phi /\sqrt {N}$, where $N$ is the number of bins in the measurement. There is a good agreement between the two measurements; however, a small bias of $\sim 0.15 m/s$ is observed. This bias may be due to geophysical variability during the integration times, which would result in a slight increase in the observed uncertainties. The theoretical results (red dots) determined using the observed intensity and visibility, therefore, provide a lower bound for the experimental results, and is the minimum error that one would expect the measurements to have.

 

Fig. 2. The ERWIN-II wind errors in m/s, where plot a) is the wind error determined from statistics (blue) and wind error from Eq. (25) (red), b) is the intensity in ADU which is 1.6 times the number of electron counts (electron counts is the number used in determination of the SNR), and c) is the visibility [10].

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4.3 Uneven sampling

In the lab, measurements were performed with a scanning field widened interferometer using a filtered Krypton lamp line as an emission source. This provides additional validation for these theoretical error calculations, and allows different stepping scenarios to be formulated and tested. As an example, an experiment was performed whereby the scanning mirror of the Michelson was stepped by 1.1667 radians eight times, to generate the stepping profile. The resulting uncertainties are calculated using Eq. (16) with the following constants

This approximation removes the dependence on the calculated $\Phi$. The two theoretical uncertainties, and the experimentally determined uncertainties are shown in Fig. 3.

The two theoretical uncertainty curves are very similar, demonstrating that this is a good approximation for this type of scan, and that it is possible to approximate the uncertainties with just the intensity and visibility measurements, removing the dependence on the phase measurements. Furthermore, the theoretical results provide a good approximation for the lower bound of the statistical uncertainties and provide a suitable means for linking science goals and instrument performance when designing new instruments. This is similar to the results in Fig. 2 from ERWIN-II results.

 

Fig. 3. The uncertainty in the phase measurement determined via the standard deviation (blue), the approximate theoretical error from Eq. (26) (red) and the exact theoretical error from Eq. (16) (black).

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5. Phase calibration

In practice, the Doppler phase shift image, $\delta \phi (\vec {u})$, associated with the motion of the source (see Eq. (2)) is determined as a difference between an observed phase field, $\Phi (\vec {u})$, and a zero wind background phase field, $\Phi _0(\vec {u})$, corresponding to the case of a motionless atmosphere. Instrument phase drifts, $\delta T(t)$, associated with the dependence of optical path difference on the environmental conditions (pressure, temperature etc.) will result in $\Phi _0(\vec {u})$ varying in time, $t$. Hence, the phase image associated with atmospheric motion is given by:

The phase calibration procedure can be broken down into several steps. First, the observations are corrected for instrument phase drift $\delta T(t)$ by performing periodic measurements using an isolated calibration lamp emission with a wavelength close to that of the target emission. Second, to obtain the angular dependence of $\Phi _0(\vec {u})$ relative to the phase at normal incidence a zero wind background phase image is used. Third, the remaining systematic offset between the calibration lamp phase background phase and the true zero wind phase is determined (by taking the mean of the vertical wind over a day) and corrected. The sensitivity of the phase shift determination using the bin-by-bin analysis approach hinges on accurate calibration of these parameters since errors in these parameters cannot be distinguished from the geophysical variations. To ensure measurements are of geophysical relevance for gravity wave observations, uncertainties in the phase calibration must be less than 1 m/s (i.e. $10^{-6}$ nm).

The calibration procedures developed for the bin-by-bin analysis approach build on previous ground based DMI instruments that performed single point wind measurements, such as ERWIN-I. For those observations, the measured intensity at each step in the scan consisted of an integration over the angular field imaged at the detector and the four point algorithm was applied (see previous section). In this case, the instrument phase drift, $\delta T(t)$, and zero wind background phase, $\bar {\Phi }(\vec {u})$ can be combined as a single measurement and accomplished by taking 4-point phase measurements using a filtered calibration line emitting close in wavelength to the target emission. For a thermally compensated instrument, the instrument phase is easily tracked by taking periodic measurements of the calibration emission phase and interpolating to the times of the observations to correct for these phase variations. The zero wind phase was determined by periodically viewing and observing the wind in the vertical assuming the vertical wind is significantly smaller than the horizontal winds [8].

Measurements of LOS wind are acquired in the four cardinal look directions (N, E, S, W) at a known angle relative to zenith. After correcting for instrument phase drift and the background zero wind phase, the average horizontal wind vector can be determined from the multi-directional set using the approach discussed by Gault et al. [8] and Kristoffersen et al. [10].

For a wind imaging instrument, the background phase varies across the detector and must be corrected on a bin-by-bin basis, and removed from the wind images on a bin-by-bin basis. This is essential for obtaining geophysical winds for each bin since different bins in the image have different LOS wind values so that each bin is effectively a single point measurement. These variations are in addition to the systematic differences between the shape of the scene background phase and the calibration background phase. Therefore, implementing an imaging capability requires removal of the angular dependence of the phase across the image and minimizing systematic variations between this phase image, $\Phi (\vec {u})$ and the true background phase. The generalized fringe analysis approach discussed earlier in this paper facilitates an accurate background phase determination and correction.

To obtain a perfect background phase image, the distribution of the radiation (centerline wavenumber/line shape and intensity distribution) should be identical to that of the source emission. In this case, the optical path variations associated with the interfering beams would be identical. If the spatial intensity distribution differs slightly from the source emission, then the presence of slight spatial phase variations introduced between the interfering wave fronts due to slight imperfections in the mirror surfaces and inhomogeneities in the glass will result in slightly different background phase images.

In practice, a pragmatic approach to determining the background phase is needed. The best option is the use of a high quality integrating sphere and a calibration source; however, DMI instruments are typically operated remotely and the calibrations change slightly over time, which makes this approach less appealing. For the instruments discussed in this paper, ERWIN-II and MIADI, we developed and tested novel and relatively simple calibration approaches in combination with the generalized fringe analysis framework that can be utilized to get an accurate phase correction that can be applied to an imaging DMI instrument.

To determine the background phase, one simply needs a source that is not Doppler shifted due to wind and which fills the aperture and field of view in the same way as during measurements. This is accomplished by taking measurements with a diffuse source. This can be done in two ways. First, for an operational instrument in the field measurements can be taken during a time of low level cloud cover. This cloud cover acts as a diffuser so that light from all directions contributes equally to the light detected on each bin and a net Doppler shift of zero results. The observed Doppler shift image of the airglow emissions corresponds to a zero wind image, and these phase images can be used to provide a suitable background zero wind image. In principle a calibration lamp could be used to obtain background phases, but in practice (as demonstrated below) this was not sufficiently precise. Second, a ground glass diffuser can be appropriately placed in the optical train to perform the same task as the cloud cover measurements. In this case, one needs to place the diffuser at a location in the optical system where the light from the scene is well collimated. Both of these methods are discussed in more detail below.

5.1 Cloud cover approach: ERWIN-II

As was mentioned in Section 1, the ERWIN-II has a quad mirror which allows for the simultaneous observation of the four cardinal directions and the zenith. This is achieved by imaging the quad mirror onto the detector, with each section of the quad mirror image viewing small regions of nightglow (roughly an 8 km by 8 km cross section) in different cardinal directions in the sky. By taking the mean of each section (typically $\sim$180 bins are imaged per section), the average line of sight winds for that direction is determined. The standard deviation provides a means to determine the uncertainty in the wind measurement. It should be noted that bins are rejected based on visibility and intensity conditions: if the visibility is too low (<0.05) or if the visibility is unphysical (>1) or if the intensity is too large (>10 standard deviations from the mean).

For the ERWIN-II, with the addition of the quad mirror, it was no longer sufficient to simply take a single average background phase value using the calibration lamp, because the quad mirror was being imaged. Using this method would result in a different error for the background phase and zero wind phase for each section of the imaged quad mirror, and hence each viewing direction. Therefore, it is necessary to determine the background phase image so that it can be removed from the airglow wind images, and provide a more accurate measure of the wind in the different viewing directions. This also allows for a statistical determination of the wind errors. It was discovered that the use of a calibration lamp, and a calibration optical train resulted in differences from the airglow background phase and the calibration background phase.

The initial design and implementation of ERWIN-II used several calibration lamps (noble gas discharge lamps) to determine the background phase, which would be removed from the total phase leaving only the phase due to the wind and the zero wind phase. However, it was discovered by Kristoffersen et al. [10], that these calibration lamps did not provide a sufficiently accurate background phase (Fig. 4). Instead the results from airglow observations from a cloudy/foggy day (Jan 28/29, 2009) were used to determine the background phase. These cloudy/foggy days are fairly infrequent, occurring typically only a few times over the observing season, and result in a diminished airglow signal, with no gradient in the observed Doppler shift.

As shown in Fig. 5(b), the average wind for the entire day using the background phase determined from the cloudy Jan 28 observations is much more consistent compared to the daily averaged wind using the calibration lamp (Fig. 5(a)). Since ERWIN-II is not focused on the sky, but on a quad mirror located in the front optics, there are five observed wind sectors (four in each of the cardinal directions and one in zenith). For a daily average a constant wind value should be measured across each sector. It is clear in Fig. 5(a) that this is not the case; the calibration lamp background phase does not provide uniform winds accross each viewing direction. The large variations in the wind measurements in Fig. 5(a) are due to differences between the emission and calibration background phases. As illustrated in Fig. 5(b), these variations do not appear when the background phase measured during a cloudy day (Jan 28, 2009) is used to determine the daily average winds.

Using the ERWIN-II measurements it was found that the use of a calibration lamp to determine the background phase led to systematic errors in the wind determinations [10]. The phase calibrations (determined using a cloudy day), developed for the wind imaging application, eliminate these systematic errors and provide simple improvements to the background phase determination.

 

Fig. 4. Background phase determined using (a) the Noble gas discharge lamps, and the calibration optical train; (b) the cloudy sky, and the sky viewing optical train.

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Fig. 5. Mean green line wind image for January 26, 2009 using (a) the Krypton discharge lamp as the background calibration source, and (b) the cloudy night of January 28, 2009 for the background phase.

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5.2 Ground-glass diffuser approach: MIADI

The ground-glass diffuser approach was examined using the setup shown in Fig. 6. Light from a He-Ne laser was focused onto a fiber and the fiber directed through a beam splitter to illuminate a retro-reflecting disk. The disk was connected to a motor that could be used to control the rotation rate of the disk. An optical system configured to collect the retro-reflected light and form an image of the illuminated cross section in the field of view of the MIADI instrument.

When the disk is rotating, a predictable LOS Doppler wind gradient is produced across the field of view. First, several observations are taken with the wheel not rotating in order to obtain a zero wind calibration. Second, several observations are taken with the wheel rotating. Third, the diffuser is then placed at the entrance aperture of the MIADI optical system and several observations are taken with the wheel rotating.

An image of the stationary wheel is shown in Fig. 7(a). An image of the observed LOS Doppler wind field after correcting for the zero wind background phase obtained with a stationary wheel and the zero wind background phase obtained with the diffuser is shown in Fig. 7(b) and Fig. 7(c) respectively. The precision of the wind measurements shown in Fig. 7(b) are < 0.5 m/s and the expected linear gradient in the wind field associated with the rotating wheel is observed.

With the diffuser present, the observed LOS Doppler shifts are effectively randomized and the LOS gradient is not observed (see Fig. 7(c)). Since the diffuser is placed conjugate to the aperture stop, which corresponds to the angular field which is averaged at the detector, the resulting Doppler winds are averaged, removing any observable wind gradient. The diffuser technique provides a highly practical approach that can be easily implemented for use in a field instrument. The primary drawback is the reduction in the signal level and the associated increase in the derived wind uncertainties. In practice, the issue is dealt with by taking a sufficient number of images so that the associated standard error for each bin across the image to be suitably small for the background phase uncertainty to be negligible in the wind determinations.

 

Fig. 6. Apparatus used to produce a predictable gradient in the LOS wind field.

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Fig. 7. Images of (a) the brightness of the stationary wind wheel in the MIADI field of view, (b) the observed LOS wind gradient across the image of the wheel and (c) the same as (b) but with a diffuser placed at the field stop (and the mean velocity removed to centre this image around 0 m/s). Note the lack of a gradient across the wind wheel when the diffuser is in place.

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6. Discussion

The bin-by-bin analysis approach discussed in this paper results in several significant improvements to the sensitivity and sampling capabilities of DMI instruments. More geophysical information can be extracted from these measurements than was previously possible using earlier DMI techniques, or other instrument techniques. Incorporating the imaging capability allows the wind and irradiance fields to be simultaneously sampled. As demonstrated with the MIADI, this allows gravity waves to be unambiguously diagnosed in wind and brightness observations. The technique also provides the means to take measurements at a higher measurement cadence - as demonstrated by the rapid sampling of the wind field in the case of ERWIN-II. However, increases to the spatial sampling capability come at a cost to the temporal resolution of the wind measurements given a geophysically defined wind resolution for useful science.

In practice, the spatial and temporal sampling is designed to enable the observation of the specific geophysical variabilility of interest. The ERWIN-II utilizes the bin-by-bin analysis approach to increase the sensitivity of single point wind measurements, providing better than 1 m/s precision for each meridional and zonal wind sample in less than 5 minutes. This observation cadence allows for waves with periods down to approximately 10 minutes to be detected. However, the spatial sampling is such that only waves with horizontal wavelengths on the order 100 km or greater can be unambiguously detected.

Conversely, the MIADI utilizes this analysis approach to increase the spatial information that can be extracted from measurements. MIADI takes advantage of the imaging capability to obtain a field of view covering an 80 km by 80 km region of the night sky (see Fig. 8). Each cycle of images is obtained in roughly 40 minutes. In practice, each image is binned into 5 km by 5 km bins in order to achieve a precision of <2 m/s for each bin. MIADI is therefore able to detect wave activity with wavelengths greater than 10 km, and periods greater than 40 minutes.

 

Fig. 8. An example MIADI line-of-sight wind image obtained on the evening of July 14, 2014 [11].

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An example of the LOS wind images obtained with MIADI during a campaign in July, 2014 is provided in Fig. 8. The image is geo-referenced to a Cartesian coordinate system with (0,0) corresponding to zenith. In order to isolate wave activity, the background horizontal wind field is first extracted by performing a LMS fit to the LOS Doppler wind image. Perturbations to this background wind associated with wave activity are then isolated and examined. More information on this approach can be found in Langille et al. [11].

A critical aspect of the bin-by-bin analysis approach is the associated calibration measurement procedure described earlier in this paper. It was demonstrated in laboratory tests that the use of a ground glass diffuser is sufficient to provide the background phase measurements. This method was implemented in the field, for airglow measurements with the MIADI [11], and resulted in a more accurate determination of the background phase, and as a result, a more accurate measure of the background Doppler winds.

Comparisons between the ERWIN-II winds as determined using the calibration lamps, and using a cloudy day (similar in concept to the use of a diffuser) to measure the background phase, provide further evidence for the improvements in wind determinations associated with using the cloudy day to determine the background phase. There is a slight decrease in the observed variances for each direction when the cloudy day as opposed to the calibration lamp is used to determine the background phase (see Fig. 9). However, the comparison of the meridional and zonal winds, shown in Fig. 10, highlights the need to use the cloudy day for background phase determination. There is a significant systematic offset ($\sim 10$ m/s) between the meridional winds determined using the calibration lamp versus the cloudy day background phase, and a smaller offset between the zonal winds. Until the exact instrument field and aperture can be duplicated with the calibration lamp optical train, using background phase determinations using cloudy days is the more accurate approach.

 

Fig. 9. The observed statistical uncertainties on Jan 26, 2009 for the (a) meridional and (b) zonal winds. The blue dots are the uncertainties determined using the calibration lamp to determine the background phase, and the red dots are using the cloudy night.

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Fig. 10. The ERWIN-II observations of (a) Meridional and (b) zonal winds, on Jan 26, 2009. The blue dots denote the winds determined using the calibration lamp, and the red dots denote the winds determined using the cloudy night to determine the background phase. Note the systematic offsets between the two calculations of the winds.

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7. Conclusion

In this paper, we present for the first time a bin-by-bin analysis and calibration approach that is used to extract images of the LOS wind field using well isolated emission lines. Novel general LMS expressions for fringe phase determinations are stated for the first time. These allow the dependence of the LOS wind uncertainties on the instrument parameters and source signal to be examined theoretically and directly determined from the recorded interference fringes.

It has been demonstrated that the use of diffuse light of the exact same wavelength as the desired atmospheric emission and the same optical train provides a much more reliable measure of the background phase compared to the use of a calibration wavelength that is both slightly different than that of the desired emission, and with a slightly different optical path. This new method of background phase determination was necessary to provide winds of suitable temporal cadence, spatial scale and accuracy for useful geophysical information on gravity wave fields to be determined.

The results presented in this paper demonstrate that this bin-by-bin analysis and calibration approach significantly improves the sensitivity and sampling capabilities of the DMI approach. It also improves the accuracy of the wind measurements. These improvements are demonstrated using observations from two field instruments, MIADI and ERWIN-II.