Compensating for any systematic receiver channel phase offsets plays an important role in ensuring the accuracy of the position and height estimates of the detected meteors. To calibrate the phases of the receive channels for both of the meteor receiver interferometers used in this study, we have followed the approach suggested in .

The approach determines the offsets to apply to the phase differences between the centre and each of the other receive antenna channels that maximize the number of meteors within a range of heights that the meteors are expected to occur in (see , for a generalized approach to this). For the BP system, we have used minimum and maximum permissible heights of 70 and 110 km, respectively, and 70 and 120 km for the Mylor system. A slightly larger height interval has been used for the Mylor system to allow for the effect of the distribution of Bragg wavelengths (see Fig. c) on the meteor height distribution width (see, e.g. , and , Sect. 3 for a description of this effect).

The phase offsets applied to the Mylor system and the variability of the offsets for the BP system (for which a fixed calibration was used) are shown in Fig. . We note a stable calibration for BP but a few sudden shifts in the Mylor case; this has subsequently been determined to be due to a slight rotation of the antenna elements by local wildlife. We do not expect isolated shifts like this to have an adverse impact on the analysis performed in this paper, although to somewhat compensate for it we have performed a daily recalibration of the receiver channels (using the calibration results for each day) before subsequent processing of the data.

The aim in developing this simulation has been to quantify the uncertainties in the 〈u′w′〉 and 〈v′w′〉 covariance components derived from meteor echoes received in an arbitrary network of meteor radar transmitters and receivers, as well as to be able to characterize the dependence of those uncertainties on the network shape and the spectrum of the GWs constituting the input wind field. The basic workflow of the simulation (all components of which are elaborated upon in subsequent subsections) may be summarized as follows:

Produce a sample of meteors in space and time for each site under consideration, by sampling from realistic spatio-temporal meteor detections corresponding to each site.

To incorporate the dependence of the 〈u′w′〉 and 〈v′w′〉 uncertainties on the temporal and spatial characteristics of the meteor distribution, we have based the distributions used in the model on real measurements. For both the BP and Mylor sites, we constructed a composite day of 2-D histograms of the meteor position distributions at 5 km spatial and hourly time resolution, using measurements from April to July 2018. These 2-D histograms were taken to represent probability distributions for the meteor positions.

The sampling from these probability distributions at the beginning of each realization was done according the following process:

Prescribe a number of meteor detections for the day of measurements and altitude in question (e.g. 1340 d-1 at 90 km for the BP radar case).

The horizontal position coordinates assigned to each meteor in the probability distribution (and subsequently the model) are based on the distances from the receiver site in Transverse Mercator coordinates, calculated using the method of . The altitudes assigned to the meteors are derived from a uniform probability distribution, with a centre value of 90 km and a full width of 2 km (such that the simulation emulates the idea of analysing meteors from a single height bin).

To clarify the effect of a variable number of meteor radial-velocity–AOA pairs on the covariance error distribution, a variety of meteor detection rates have been simulated. We have endeavoured to make the detection rates used resemble the number of meteors detected across a range of heights by the combined BP–Mylor radar link (we note again though that the simulation itself is performed around a single altitude). The detection rates we have used for different heights, listed in Table , correspond to those averaged over April 2018 for the two receive sites, in 2 km wide bins.

The wind field in the simulation is comprised of tidal components and a superposition of monochromatic GWs whose amplitudes have a vertical wavenumber and frequency dependence. Diurnal and semidiurnal tidal components are assumed, with amplitudes of 25 and 10 m s-1 respectively. Random phases from a uniform distribution spanning the interval [0,2π) are added to the phase of the zonal component of the tides at the beginning of each realization, and the meridional component is set to be in quadrature with the zonal component. The 3-D wind velocity associated with the GWs at a given time t and Cartesian position vector r can be written as 1v=∑i=1nm∑j=1nωAmi,ωjv′ijsin⁡κi⋅r-ωjt+ϕij, where m is the vertical wavenumber, ω is the wave's angular frequency, nm and nω are the number of vertical wavenumbers and angular frequencies respectively in the spectral grid, A is the joint vertical-wavenumber–angular-frequency spectral amplitude, v′=[u′,v′,w′] is the vector of wind component fluctuation sizes, κ=[k,l,m] is the 3-D wave vector, and ϕ represents a (random for each unique [mi,ωj] pair) phase offset.

As per Sect. , the coordinate system used to specify horizontal position with respect to a reference location (i.e. that embodied by the r vector) is based on the Transverse Mercator distances evaluated using the method (which follow the Earth's surface and take into account its ellipsoidal shape). This is used in preference to line-of-sight distances, the use of which would result in stretching of the horizontal scales of the waves at large distances from the coordinate system origin. Furthermore, the calculated wind velocities are assumed to be in the local east–north–up (ENU) coordinates at the associated meteor positions.

To ensure that the correlations between the horizontal and vertical winds take on physically reasonable values, we have allowed the component fluctuation amplitudes to be related by the linear GW polarization relation w′=vhkhm, where vh=u′2+v′2 and kh=k2+l2. The horizontal components are determined by the wave propagation azimuth φ, through the relations [k,l]=kh[sin⁡φ,cos⁡φ] and [u′,v′]=vh[sin⁡φ,cos⁡φ].

In order to give the wind field a level of spatially correlated randomness akin to what is seen in mesospheric wind fields when no predominant wave scales are present, we have opted to let A(m,ω) take on values from a gravity wave spectral model. The vertical wavenumber spectrum we have used (, Eq. 7, and following their nomenclature) is given by 2Fu(m)=2παN2m*-3mm*sm≤m*m-3m*≤m≤mbmb-3mbm5/3mb≤m, where m is the vertical wavenumber of the wave, and following , Fig. 1, we let α=0.62, N=2π3×102 s-1, m*=2π1.5×104 m-1, mb=2π5×102 m-1, and s=2. The frequency spectrum we have used (, Eq. 24) is given by 3B(ω)=p-1ffωp, where ω is the angular frequency of the wave, and following , Fig. 2, we let f=2π7.2×104 s-1 and p=2. We then simply assume that the joint vertical-wavenumber–angular-frequency spectrum is given by the product of these two spectra, i.e. 4Am,ω=Fu(m)B(ω).

The 2-D spectrum we used for results presented in this paper consisted of 80 different vertical wavelengths and wave periods, spanning the ranges 0.5–20 km and 5–240 min (uniformly sampled in vertical wavenumber and frequency), respectively. These limits largely encompass the waves responsible for the majority of the momentum deposition in the mesosphere–lower-thermosphere (MLT) region (see e.g. ), whose momentum fluxes are of principal interest in this study.

The wave propagation azimuths were sampled from a uniform random distribution spanning [0,180∘] in bearing, with the intention being to emulate a wave field whose westward-propagating waves have been removed from the spectrum through selective filtering. This led to true values for the estimates of 〈u′w′〉 that were on average positive and values of 〈v′w′〉 that were on average zero. Testing a wider variety of wave field configurations was considered beyond the scope of the paper.

The absolute values taken by A(m,ω) were normalized in a way that resulted in mean values of 〈u′w′〉 in the vicinity of 20 m2 s-2, which is a typical value for this parameter in the MLT region (see e.g. the discussion in ). An example distribution of true covariances evaluated in the simulation is shown in Fig. b.

A diagram summarizing the bistatic reception geometry is shown in Fig. . Following the development of , the so-called “radial velocity” measured by a bistatic receiver corresponds to the projection of the 3-D wind velocity onto e (which is in the same direction as the Bragg vector in e.g. ), in turn projected onto b. Mathematically, this velocity is expressed as 5vrm=cos⁡(β/2)⋅vecef⋅e, which is the velocity that is used to produce a phase progression in the simulated receiver time series, discussed in Sect. . It should be noted that t, b, and e are expressed in Earth-centred, Earth-fixed (ECEF) coordinates and that the wind velocities v computed in the simulation are in the local ENU coordinates of each meteor. The “ecef” subscript on v is to denote v's rotation to the ECEF coordinate system; we have followed a slightly modified version of the approach discussed in detail by to do this (see Appendix ). We note that we apply the same procedure for both bistatic and monostatic receivers (though of course in the monostatic case t=b).

To ensure that realistic radial velocity and position estimation errors are propagated to the covariance estimation, we have opted to generate synthetic receiver time series based on the observables discussed in the previous sections and to then attempt to re-estimate the observables from the time series. The complex time series for the jth receiver is written as 6Vj(t)=ei2πA⋅d-4πvrmt/λ+Φje-t/τ+nj(t), where A=sin⁡θsin⁡ϕ,sin⁡θcos⁡ϕ,cos⁡θ (where θ and ϕ are the zenith and azimuth angles of the meteor, respectively, as measured from the receiver), d is a three-element vector of Cartesian displacements to the receiver antenna in question, λ is the radar wavelength, Φj is a phase calibration offset for the jth receiver, τ is the e-1 decay time of the meteor, and nj(t) is a background noise function.

The background noise function consists of values derived from a Gaussian distribution, with a root-mean-square (rms) value derived from a probability distribution of meteor echo SNRs from the monostatic 55 MHz meteor radar at BP. The values used for τ are also derived from a probability distribution from this radar's data. In both cases, the data used to generate the probability distributions spanned 1–30 April 2018 and altitudes 70–110 km. Plots of these distributions are shown in Fig. .

The phase calibration offsets Φj, which are set for each receiver at the beginning of each simulation realization, are intended to embody the consequences of incorrectly estimating the true phase calibration offsets between the receiver channels. Based on the phase calibration offset time series shown in Fig. , we have chosen to apply to each receiver Gaussian-distributed phase offsets with an rms value of 2∘.

Radial velocities and meteor positions are estimated from the noise and phase-offset time series following the procedures outlined in (Sect. 3.11 and 3.12, respectively), with the exception that the radial velocity is corrected for the forward scatter angle in the case of bistatic reception. Using the definitions in Fig.  and following the approaches outlined in to compute the t and b (and e=t+b|t+b|) vectors, the forward scatter angle may be estimated using 7β=cos⁡-1-t⋅b|t||b|, and then Eq. () may be rearranged for vecef⋅e to get the radial velocity.

It should be noted that in rare (∼1 detection in every 3000) cases, we found it became impossible to estimate the AOA of the meteor unambiguously when the phase biases and noise were incorporated into the receiver time series (i.e. the error code 3 discussed in , was encountered). In these cases, the echo in question was simply discarded from the subsequent calculation of mean winds and covariances.

The way we have estimated mean horizontal winds in this simulation is similar to that typically applied to meteor radars in the literature (e.g. ). Our approach has been to use singular value decomposition (SVD) to solve the following inverse equation in the least-squares sense for v: 8vr=Av, where vr is a nmet×1 vector of radial velocities (nmet being the number of meteors in the time bin under consideration), v is a 2×1 vector of wind velocities, and A is a nmet×2 matrix whose rows take the same form as that described in Eq. () (without the vertical component). However, it is important to note in this case that the θ and ϕ defined in A represent the orientation of the e (or Bragg) vector (see Fig. ) in the ENU coordinate system at the location of the meteor. The velocities in vr of course also represent the projection of the wind vector on the e vector; i.e. vrm/cos⁡(β/2), where vrm is the radial velocity measured at the receiver.

In order to remove outliers from the input radial velocity distribution, we follow the iterative scheme proposed by (and subsequently used by e.g. ). This involves performing an initial fit for the wind velocities, removing the radial velocities whose value differs from the horizontally projected radial wind by more than 25 m s-1 and repeating the procedure until no outliers are found or until less than six meteors remain.

To remove the previously estimated mean winds and tides from the time series, we have calculated a low-pass-filtered version of the hourly averaged horizontal wind time series using an inverse wavelet transform with a Morlet wavelet basis, linearly interpolated a wind estimate at the time of each meteor, and subtracted the radial projection of the wind from the radial velocity time series. This is in principle similar to the approach of , who applied an S transform (in preference to a least-squares sinusoidal fit) in order to more completely remove transient spectral features around the tidal periods from the time series. The application of the inverse wavelet transform is described in Appendix .

To ensure that the filtered time series pertain to tidal-like (or longer) wind oscillations (and not short-period GWs), we select a minimum scale size in the reconstruction of 6 h and a total number of scales of 250. The reconstructed time series is then interpolated to the times of each of the meteors in question, and the radial component of this wind at each of the meteor positions is subtracted from the measured radial velocity.

Following the removal of the mean and tidal components of the horizontal wind from the radial velocities, covariances that pertain predominantly to gravity-wave-driven wind perturbations are estimated. The approach we apply is based on those presented by and ; much like in the wind estimation, it involves using SVD to least-squares solve the following inverse equation: 9v′r2=A′v′, where v′r2 is a nmet×1 vector containing the squares of the perturbation component of the radial velocities, v′=〈u′2〉,〈v′2〉,〈w′2〉,〈u′v′〉,〈u′w′〉,〈v′w′〉T is the vector of covariance components, and A′ is a nmet×6 matrix whose rows read [sin⁡2θsin⁡2ϕ,sin⁡2θcos⁡2ϕ,cos⁡2θ,sin⁡2θsin⁡2ϕ,sin⁡2θsin⁡ϕ,sin⁡2θcos⁡ϕ].

It is noted that, as per the wind estimation case, the θ and ϕ terms represent the orientation of the e vector in ENU coordinates at the location of each meteor and that the velocities in v′r2 are based on the wind velocities' projection onto e.

A two-step radial velocity outlier rejection procedure is utilized to remove meteors with dubious square radial-velocity–AOA pairs from the input distribution in an attempt to reduce the bias in the resulting covariance estimates. The first step is to discard all radial-velocity–AOA pairs that have a projected horizontal velocity of ≥200 m s-1 (by virtue of which we argue that measured horizontal velocities above this threshold are nonphysical). The second step iteratively discards the pairs that satisfy the following criterion: 10|v′ri2-v′rpi2|≥[median|v′r2-v′rp2|+5×1.4826×MAD|v′r2-v′rp2|]2, where v′rpi2=Ai*′v′ is the ith projected square radial velocity, MAD indicates the median absolute deviation operator, and 1.4826 is the factor to convert a MAD to a standard deviation, assuming the input has a Gaussian distribution. In practice, we have found that the 5-standard-deviations criterion removes outliers that are large enough to substantially bias the resulting covariance estimates, without iteratively removing an excessive number of samples that are good. The intention of using the median and MAD statistics (as opposed to mean and standard deviation) has been to reduce the bias outlying points inflict on the measured standard deviation of the distribution of |v′ri2-v′rpi2|.

The performance of the second outlier rejection criterion on simulated data is briefly summarized in Sect. .

To evaluate the truth value of the simulated covariances – i.e. that used to estimate the accuracy and precision of the covariances derived through inversion of Eq. () – we have opted to compute the covariances at the origin of the coordinate system (in the meteor region directly above the receiver at BP) at 2 min time resolution. We found this estimate to agree extremely closely with that computed at the positions and times of the meteors incorporated in the simulation, which in turn represents the most accurate and precise estimate one could hope to obtain when inverting Eq. ().

In the case of using wave fields generated from the previously discussed gravity wave spectral model, we found that the covariances estimated by inverting Eq. () are more correlated with those calculated using the above two methods than those computed by summing the covariances associated with each wave in the spectrum. Therefore, while the latter method gives the covariances that would be measured over an infinitely large sampling area/time (in a sense the expectation value of the covariances), we have refrained from using it as a truth value with a view to not overestimating the size of the simulated technique's measurement errors.

Plots of the mean horizontal winds and the 〈u′w′〉 and 〈v′w′〉 covariance terms from 17 March through to 9 September 2018 are shown in Fig. . Both quantities have been sampled using 2 km, non-oversampled altitude bins. We chose to evaluate the covariance terms using 10 d long windows, with a time shift of 2 d between the centres of adjacent windows, in an attempt to resolve the planetary-wave-induced modulation of the covariances. A low-pass wavelet filter with a cut-off of 2 d and a 10 d moving average has been applied to the hourly horizontal winds to evaluate the winds shown; the filtering was performed to avoid the aliasing of GW activity and tides into the wind's variability, as well as the moving average in order to more closely match the temporal sampling of the two parameters. Therefore, the winds shown should provide a good measure of the background mean winds responsible for selective filtering of the gravity wave spectrum.

As is expected for this time of year at a mid-latitude SH site (see e.g. ), the eastward winds around 80 km generally increase with time from the autumnal equinox to the winter solstice (∼ days 80 and 170 respectively) and decrease toward the vernal equinox (∼ day 265). A wavelet analysis (not shown here) reveals that much of the shorter term zonal wind variability evident in the figure is transient and encompasses a spectrum of periods between about 10 and 60 d. The meridional wind, conversely, has a mean much closer to zero. Much of its variability is confined to periods around 10, 20, 25, and 40–50 d below 90 km, with variability in the 50–100 d period becoming increasingly dominant above 90 km.

The level of (anti)correlation between the covariance terms and the winds is highly variable. The 〈u′w′〉 term appears to be anticorrelated with the zonal wind between 80 and 84 km around the winter solstice, as does 〈v′w′〉 with the meridional wind above 88 km across a similar time interval. While pronounced levels of anticorrelation between these quantities in the mesospheric region arising from the selective filtering mechanism are typical (see e.g. the recent summary provided by ) – particularly in the zonal component – departures from these predictions are also not uncommon. As explains, it is difficult to conceive a mechanism for departures from this theory in the zonal component (given the dominance of eastward winds in the lower mesosphere during winter), aside from considering that the GWs may have propagated through a region with weak eastward mesospheric winds.

The feature we focus the remainder of this discussion on concerns the coincident enhancement in the 〈u′w′〉 and 〈v′w′〉 terms in the interval spanning days 100 to 120, around 90–94 km. Peak values of ∼50 and 100 m2 s-2 for 〈u′w′〉 and 〈v′w′〉 respectively are obtained during this interval. Interestingly, they coincide with a brief enhancement in the zonal winds at the same height and the peak of the northward phase of an oscillation in the meridional winds with periods spanning 50–100 d.

Figure  shows an inset of Fig. , spanning April 2018 (which the aforementioned covariance enhancement is centred on). In an attempt to increase the temporal resolution, the covariances in this figure have been evaluated with 1 d windows, with a time shift of 6 h between adjacent windows. Tidal components have also been removed from the winds as per Fig.  (i.e. in order to not alias tidal/GW activity into the winds), and for a closer match to the time sampling of the covariances, no moving average has been applied.

This figure shows evidence of a pronounced periodicity around 10 d in the zonal wind, which attains its highest amplitude at approximately day 110 around 85 km. At this time and in the same altitude region, the mean meridional winds abruptly (over a period of a few days) switch from northward to southward. All of this variability is likely attributable to a superposition of planetary waves. Albeit noisy (owing to the relatively short integration time), the 〈u′w′〉 covariance term shows an enhancement between days 105 and 110, and attains especially high positive values (exceeding 100 m2 s-2) at around 90 km altitude. Interestingly, the 〈v′w′〉 enhancement lags that of 〈u′w′〉 by several days, with a peak again in excess of 100 m2 s-2 around day 110.

We have also noted that this interval is associated with an abrupt enhancement of the amplitudes of the diurnal and semidiurnal tides. Figure  shows the amplitude of the horizontal wind time series reconstructed from a inverse wavelet transform (see Eq. ), for scales between 0.4 and 0.6 d for the semidiurnal tide and between 0.8 and 1.2 d for the diurnal tide. The diurnal tide in the zonal wind is seen to reach an amplitude of ∼50 m s-1 during day 107 at a height of around 92 km and of 35–40 m s-1 in the meridional component around 88 km during day 109. It should be noted that the hourly averaged zonal wind velocity (not shown here) reached a maximum of about 140 m s-1 at 92 km during this period. The semidiurnal tide, whose amplitude is known to rarely exceed 10 m s-1 at Adelaide's location (e.g. ), also reached an amplitude of 35–40 m s-1 during day 104 in both the zonal and meridional components, at a height of around 94 km. The figure additionally shows that the phase of the diurnal tide is modulated, with the timescale of those modulations appearing to follow the phases of the planetary wave activity in Fig.  – although there are no noteworthy phase changes at the times of the sudden amplitude enhancements. The semidiurnal tidal phase is persistent, and also has a well-defined vertical progression, during the few days in which its amplitude is large but clearly has little meaningful structure at other times.

The large tidal amplitudes during this period lead us to expect the propagation directions of the GWs removed from the wave spectrum by the winds to exhibit a diurnal variation. A complicating factor is that these waves may also amplify, dampen, or shift the phase of the tide, depending on the waves retained in the spectrum at the wave breaking height; the large variability in the tidal amplitudes during this period indicates that this may have indeed occurred. To provide some clarity on the extent to which the GWs have been modulated by the tide and vice versa, in the next section we examine a composite day of the tidal winds, covariances, and the implied flow accelerations over a 20 d interval spanning the interval in which the diurnal tide has a reasonably consistent phase and an enhanced amplitude.

Figure  shows a composite day of the horizontal winds, covariances, and flow accelerations implied by the covariances, over 5–25 April 2018 (i.e. days 95–115). The composite day consists of time windows of width 3 h, with a time shift of 30 min between the centres of adjacent windows. The height binning again consists of 2 km width bins with centres separated by 1 km.

The flow accelerations (e.g. in the case of the zonal direction) have been evaluated using the expression (e.g. ): 11〈Fx〉=-1ρ(z)∂∂zρ(z)〈u′w′〉, where ρ(z) represents the neutral density as a function of height z. The density climatology we have used has been derived from the Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) satellite instrument (see Appendix  for details). Similarly to , we also apply a low-pass filter with a cut-off wavelength of 10 km to the vertical profile of the covariance prior to evaluating its density-weighted derivative, in order to remove small-scale fluctuations from it that are clearly not associated with tidal modulation.

As expected from the amplitudes in Fig. , both horizontal wind components show a predominantly diurnal variation, with the meridional component lagging the zonal's by approximately 6 h across the observed height region. The time of the zonal wind maximum occurs around 00:00 UT at 92 km and 08:00–09:00 UT at 82 km.

In contrast, the 〈u′w′〉 covariance term shows a predominantly semidiurnal variation with little vertical phase progression, maximizing at around 00:00 and 12:00 UT and minimizing around 05:00 and 20:00 UT. The 〈v′w′〉 term is more variable with altitude, exhibiting a semidiurnal variation between 82 and 84 km and a largely diurnal variation above this. The semidiurnal variation between 82 and 84 km is associated with positive covariances for the entire day except between about 18:00 and 24:00 UT, and the diurnal variation above is associated with negative covariances between about 08:00 and 15:00 UT and positive otherwise.

Between about 88 and 92 km, the zonal flow acceleration shows a pronounced minimum between 04:00 and 06:00 UT, a maximum around 13:00 UT at about 88 km, and a weaker minimum around 19:00 UT. The maximum occurs at a similar time to the corresponding zonal wind minimum, whereas the first minimum lags the zonal wind maximum by about 5 h, and the second minimum precedes it by about 5 h. Conversely, there is little flow acceleration structure below 87 km, other than a broad maximum at about 85 km around 01:00 UT. These observations are difficult to reconcile for three reasons: (1) the wave forcing is consistent with a rapid deceleration of the zonal wind from 04:00 to 06:00 UT at around 90 km, but there appears to be no positive forcing around 20:00 UT to accelerate the wind; (2) the strong positive forcing which does occur around 13:00 UT appears to result in little wind variability; and (3) the positive forcing around 85 km between 23:00 and 04:00 UT is associated with an acceleration of the zonal wind, but this acceleration is much smaller than that around 90 km.

From 88 to 92 km, the meridional flow acceleration shows a small maximum around 04:00 UT, a minimum at about 10:00 UT, and a large maximum around 20:00 UT. As per the zonal case, this leads to a peculiar relationship with the meridional wind; the forcing's large maximum occurs at a similar time to the wind minimum, the minimum corresponds roughly with a rapid wind deceleration, and the smaller maximum corresponds with a rapid wind acceleration. As for the zonal component, there is little meridional flow acceleration structure below around 86 km.

In the simulations section of this paper, we have tried to conclusively define estimates for the absolute and relative uncertainties of the 〈u′w′〉 and 〈v′w′〉 covariance terms as measured by the multistatic BP–Mylor meteor radar, for typical time and height sampling cases. We subsequently replicated these sampling schemes on the case study data. Even with this replication, we have noticed that there are three main caveats in applying the uncertainties directly to the observations:

As shown by , the covariance estimation uncertainty is proportional to the geometric mean of the horizontal and vertical variances, in the case of sampling the wind field using a perfect anemometer. Assuming this holds for a meteor-radar-like detection distribution, this means that the absolute uncertainties of 〈u′w′〉 and 〈v′w′〉 reported in this paper should be similar for a given wave field, regardless of the value of 〈u′w′〉/〈v′w′〉. Therefore, the likelihood of correctly estimating the sign of one of the components in the presence of an anisotropic wave field may not be the same as for the other component.

Despite these caveats, we can broadly conclude that the 10-day integrated covariances (Fig. ), except where the absolute values are smaller than about 10–15 m2 s-2, are likely to be of the correct sign. The correlation length of the features in both the time and height domains also indicates that the noise component in the signal is considerably smaller than the sum of all the modes of geophysical variability. Additionally, at this time integration there is likely to be little difference in the uncertainty at the peak and edges of the height region analysed.

The 1 d integrated covariances (Fig. ), in contrast, are far more affected by measurement noise. There is still some degree of temporal-height correlation, especially in the region of consistently high values of 〈u′w′〉 between days 105 and 110 above about 86 km, but very little below 84 km. The excursions below 84 km are of the same order as the simulations predict for 1 d of integration in a spectral wave field, so it may be that the noise component at these heights is considerably larger than the signal.

The 20 d composite covariances (Fig. ), while clearly affected by measurement noise, do not show fluctuations from bin to bin of the same size as the uncertainties predicted in the corresponding simulation. This gives weight to the covariance structures observed and also suggests that the wave field being observed over the 20 d period was not as complex as the simulation's or particularly variable.

Unfortunately, it is impossible to know (using the meteor observations alone) if the discrepancies between the 1 and 10 d integration (for example, the absolute values of the covariances during the enhancement between days 105 and 110) are a result of statistical noise in the 1 d estimate or a precise estimate of a strong, transient monochromatic wave event using the 1 d integration. The observation of waves in the MLT airglow may aid in the interpretation of how monochromatic the background wave field is; in the future, we intend to complement these meteor radar case studies with images of the sodium and hydroxyl airglow taken nearby the BP site. This, in conjunction with the random resampling method employed by , may lead to more refined uncertainty estimates.

In the 10 d integrated results, the small difference in measurement error at the peak and lower edge of the height distribution (around 20 %, for an order of magnitude increase in detections) places an important question on the usefulness of further increasing the integration times/detection rates. On this point, argued that the covariance measurement error should decrease with the square root of the number of detections and, by extrapolating from the 250 % error for a 1 h integration presented in , concluded that their relative error for a 1-month composite should have been as low as 10 %. Our simulations suggest that an increase in precision of this magnitude cannot occur. Moreover, using a similar detection rate and a 3 h bin in our 20 d composite of a spectral model-derived wave field shown in Fig. , we obtain a minimum relative error of about 70 %. In saying this, we note of course that a relative error of 10 % is possible for a considerably less complex wave field.

All of our simulations have shown that a systematic underestimation of non-zero covariances arises when an attempt is made to remove tidal effects. This clearly becomes more of a problem in the presence of large-amplitude GWs with ground-based periods close to those of the tides. A number of questions about the process of tidal removal could be raised:

What is the importance of incorporating the momentum fluxes of gravity waves with ground-based periods close to the tides in climate models?

With regard to 3, it may be that a wavelet/S transform has insufficient frequency resolution to define solely tidal features; a long-windowed harmonic fitting (as used by e.g. ) may be more appropriate if there is a specific interest in GW features close to or between the tidal periods. Of course, this method assumes no variability in the tidal amplitudes, tidal periods, or in the GW spectrum. The best way forward may be to simply apply both of the methods independently and contrast their effects.

In Sect.  we showed that the radial velocity outlier rejection scheme of Eq. () substantially increases the covariance measurement precision in the presence of outliers. However, we note that the criterion used (especially the 5-standard-deviations aspect) has not been rigorously tested; we merely selected it on the basis of it removing points in the distribution of |v′ri2-v′rpi2| (real and simulated) that we had noticed were spuriously affecting the covariance estimates. A more rigorous scheme would adaptively modify the thresholding based on observed characteristics of the wind field rather than simply the residual of the fit.

A complication arises from the fact that the criterion results in a more precise (albeit less accurate) covariance estimate in the absence of outliers. This also illustrates an important point about the sensitivity of the Eq. () inversion to the input: it is as though the data that contribute to the accuracy of the measurement actually increase the measurement's uncertainty, if they are associated with large radial velocity perturbations.

A subject we have not addressed in this paper is the application of weights to the meteors in the inversion of Eqs. ()/() to minimize the errors in the resulting winds/covariance estimates. In particular (as discussed by ), at the midpoint between the transmitter and receiver sites the e vector (see Fig. ) is vertical, meaning that the measured radial velocity corresponds to the true wind velocity projected onto the vertical. Large errors in the inverted horizontal winds/covariances may result in the presence of radial velocity errors here and at nearby locations where e is close to vertical. We decided to ignore the issue on the basis of there being a small number of meteors with sufficiently oblique entrance angles to be detected in this region; at Mylor, we found about 0.3 % of all detected meteors to have effective zenith angles (that is, the zenithal orientation of the e vector) of less than 20∘. Nevertheless, there is still a need to quantify the usefulness a weighting scheme may have in minimizing errors arising from these meteors.

Our aim in analysing the GW-induced flow accelerations in Sect.  has been to verify that the estimated momentum fluxes were physically reasonable and devoid of tide-induced biases, as well as to contribute to the well-known gap in knowledge of GW effects on tides. Our analysis, which was centred on a 20 d interval containing an abrupt enhancement in tidal amplitudes, has yielded inconclusive results on whether the GW momentum deposition has on the whole enhanced, dampened, or changed the phase of the tidal motions. Nevertheless, the expected uncertainties in the flow accelerations based on the bias mean and standard deviations in the Fig.  covariances, shown in Fig. , indicate that the signal components between 84 and 90 km shown in Fig.  will have well exceeded the noise levels.

The results are complex, illustrating tidal enhancement at some times of day, dampening at others, and that there are also times in which a forcing is present but no apparent effect on the tide is clear. A broad observation is that the forcing components have a more pronounced diurnal variability between about 86 and 92 km, with the result that the forcing dampens the tide at the tide's minimum (i.e. westward and southward phase) and shifts its phase at its maximum. Of course, our interpretation is complicated by the fact that we have no knowledge of what the tidal features may have looked like without any GW forcing.

It is widely accepted in modelling studies that GW forcing plays a role in the observed seasonal variation of the migrating diurnal tide (DW1) amplitudes (i.e. equinoctial maxima and solstitial minima) and that whether amplification or dampening of the amplitude occurs depends on the GW source spectrum (e.g. ). However, there is still ongoing debate about whether or not the forcing is responsible for all of DW1's observed amplitude and phase variability. For example, both and have concluded that the forcing is in phase with DW1 during the equinoxes and out of phase during the solstices, leading to DW1's amplification at the equinoxes and dampening at the solstices. reached the same conclusion for the September equinox but stated that may have significantly underestimated the magnitude of the forcing. In contrast, for the March equinox has argued that the tidal variability is caused by a superposition of GW forcing and advection terms that varies with altitude and latitude and that GW forcing exclusively dampens tidal amplitudes in the MLT/I. Moreover, has reported considerably larger GW forcing magnitudes than in a related modelling study by .

The small number of recent observational studies that have sought to quantify the effect of GW forcing on the DW1 amplitude and phase have also yielded contradictory results. For example, using TIMED satellite data showed that while the zonal and meridional GW forcing maximizes at the equinoxes and minimizes at the solstices, the zonal forcing is in quadrature with the zonal tidal wind, and the meridional forcing is out of phase with the meridional tidal wind, leading to a zonal tide with advanced phase and a dampened meridional tide. They noted that the zonal advection due to variability in the meridional DW1 amplitude also, like the GW forcing, maximized at the equinoxes and minimized at the solstices but were not able to reconcile if this variability was a cause or an effect of the seasonal DW1 variation. Also using TIMED data, concluded that the GW-induced dampening of tidal amplitudes is largest during equinoxes and therefore that dampening cannot cause the observed seasonal variation in tidal amplitudes. In contrast, using measurements from a ground-based meteor radar in Hawaii (20.7∘ N, 156.3∘ W), noted that GW forcing tends to slightly dampen the DW1 amplitude below 90 km but enhance it above 90 km. Using a similar approach on lidar data from Starfire Optical Range (35.0∘ N, 106.5∘ W), also noted that GW forcing can amplify or dampen the DW1 amplitudes, depending on the altitude.

Tides may also interact with GWs through the diurnal variations in atmospheric stability they induce (i.e. making conditions more favourable for GW breaking and hence GW forcing at particular times of day). For example, showed from observations at Scott Base, Antarctica, that the highest levels of turbulence due to convective instability occurred at the times that the vertical component of the tidal wind induced the most negative value of dT/dz (the vertical temperature gradient). Using temperature perturbations from the GSWM-98 model for the BP site, also showed that maximum negative values of dT/dz were in phase with the maximum values of the turbulent velocity measured by the BP MF around the autumnal equinox. Using GSWM-00 output, we have noted that the maximum negative dT/dz (of ∼-1 K km-1) should occur between 01:00 to 03:00 UT across the 85–92 km region at the BP site during the period of our composite day analysis; curiously, we observe large positive values of 〈Fx〉 at this time just below this region and an abrupt shift in the sign of 〈Fx〉 above it. As notes, while a dT/dz of this size is too small to result in static instability, it still corresponds with a large level of GW forcing and the maximum eastward phase of the diurnal tide, which we have observed to be particularly large during this interval.