Infrared limb sounding from satellites is a well-established method for exploring the middle atmosphere . The basic geometry of limb sounding is depicted in Fig. . The instrument looks from its orbit towards the Earth's horizon, through the atmosphere and into cold space. Three exemplary lines of sight (LOS) are depicted in Fig.  by green dashed lines. The radiance measured by the instrument results from emission and reabsorption along the LOS. For optically thin emissions, reabsorption is weak and most of the radiance stems from the region around the tangent point (purple dots), where the LOS is closest to the Earth's surface. For this case, radiative transfer can be described by a Gaussian weighting function centered around the tangent point and, accordingly, measurements are associated with the tangent altitude (blue arrow) and the location of the tangent point. The precise viewing geometry varies for the individual instruments.

The SABER instrument uses broadband radiometers to detect limb radiance in the thermal infrared. Temperature is retrieved from the main CO2 ν2 emission at 15 µm . SABER was launched on 7 December 2001 onboard the TIMED (Thermosphere Ionosphere Mesosphere Energetics Dynamics) satellite into an orbit at an altitude of 625 km and inclination of 74.1∘ and is still in operation.

The angle between flight direction and LOS, called “view angle” below, is schematically shown in Fig. . It alternates between 90∘ for northward-looking mode and 270∘ for southward-looking mode in yaw maneuvers roughly every 60 days. In Fig. , the black arrow shows the flight direction, the green line (SABER-N) indicates the LOS of SABER in the northward-looking mode, while the red line (SABER-S) is the LOS in the southward-looking mode. The corresponding latitude coverage of northward- and southward-looking modes changes between 52∘ S to 83∘ N and 83∘ S to 52∘ N. More detailed information about the SABER instrument can be found, for instance, in and .

The orbital track and flight direction as well as satellite positions and corresponding tangent points for a typical southward-looking orbit of SABER are shown in Fig. a. Note that SABER views across the pole for the southern turning point. In Fig. a, green dots are the satellite positions, red triangles are the corresponding tangent points. Blue arrows along the satellite track show the flight direction, while the purple solid line indicates an example of a LOS. In addition, the latitude coverage during the year 2008 is shown in Fig. . Orange bands are coverages of the northward-looking mode, while blue bands indicate coverages of the southward-looking mode.

The HIRDLS (High Resolution Dynamics Limb Sounder) instrument is an infrared radiometer onboard the Aura satellite, which also measures thermal emissions from the atmospheric limb. The orbit altitude and orbit inclination of Aura are 710 km and 98.2∘, respectively. The HIRDLS instrument has a fixed view angle of 180 + 47 = 227∘, which leads to a latitude coverage from about 63∘ S to about 80∘ N. More detailed information about the HIRDLS instrument can be found, for instance, in .

The view angle of HIRDLS is schematically depicted in Fig.  where the purple line illustrates the LOS of the HIRDLS instrument. In addition, satellite positions (green dots) and corresponding tangent points (red triangles) for an exemplary orbit are shown in Fig. b. HIRDLS's flight direction is indicated by blue arrows and the purple solid line shows an exemplary LOS.

Our aim is to apply an observational filter to a simulated GW at a specific location. So far, we have seen in Fig.  the viewing geometry of SABER and HIRDLS with respect to the satellite. Now, in order to apply the observational filter, we need to determine the observation geometry with respect to the same local geophysical coordinate system in which the wave vector of the simulated GW is given. In Fig. , such an observation geometry is displayed for a short orbit segment. The instrument views in the direction of the LOS (blue solid arrows). The tangent points (blue crosses) are interpreted as the actual locations of the observations. The track of the tangent points, i.e., the track of the observations, is indicated by the green arrow. At one of the tangent points, a local coordinate system is shown (red axes). The angle between the LOS and the x direction of the local coordinate system is called β and the angle measured from the x direction to the tangent-point track is called γ. Dependences of the angles β and γ on latitude for the observation geometry of SABER and HIRDLS are shown in Appendix .

First, we consider the effects due to radiative transfer and retrieval, which also limit the waves that are visible to the instrument. We use an analytical approximation of the 2-D visibility filter for infrared limb sounding, which was derived by . This filter is based on two-dimensional cross sections through quasi-monochromatic waves. assumed that all LOSs of a given profile form a two-dimensional plane, consisting of the vertical and one horizontal axis in the viewing direction of the instrument. The similar approach was also applied for analyzing the visibility of gravity waves measured by radio occultation in the paper of .

Following the analytical approach of , the instrument sensitivity of infrared limb sounders for temperature amplitude is S=λz22πΔz1-cos⁡2πΔzλzexp⁡-cb24c2+a2, where a = m/2RE = π/(λzRE), b = kh = 2π/λh, c = 1/(2HRE), RE is the Earth's radius, H scale height, kh horizontal wave number, and m vertical wave number. The values of RE and H are 6350 and 6.5 km, respectively. The vertical resolution Δz is 2 km for SABER and 1 km for HIRDLS.

As shown by , GWMF can be deduced from the temperature amplitude of the wave as follows: F=12ρkhmgN2T^T2, where ρ is the background atmosphere density, g the gravity acceleration, N the buoyancy frequency, T the background temperature and T^ is the temperature amplitude of the wave. The sensitivity function σ for GWMF, according to Eq. (), is therefore obtained by squaring the temperature amplitude ratio: σ=S2. Figure  illustrates the sensitivity function σ for GWMF from (a) SABER and (b) HIRDLS. Comparing these two sensitivities, it is evident that HIRDLS has higher sensitivity owing to its higher vertical resolution, especially at short vertical wavelengths. For HIRDLS, a reasonable sensitivity (0.3) can be found down to a vertical wavelength of about 2 km, whereas for SABER, this limit is approximately 3.5 km. Sensitivities of the two instruments in the horizontal direction are comparable.

The visibility function is a function of two variables: the vertical wavelength and the projection of the horizontal wavelength onto the LOS (see below). Figure  combines the viewing geometry of the satellite with the geometry of the observed GW in the horizontal plane. In this figure, part of an exemplary wave is shown by the dashed grey curve. The red arrow indicates the direction of the wave vector and purple lines indicate wave fronts. ψ is the angle between the wave vector and the x direction of the local coordinate system (ψ = arctan⁡(l/k) where k and l are wave numbers in the x and y directions, respectively). The horizontal wavelength λh is shown by the two-headed arrow, which is perpendicular to the wave fronts and parallel to the wave vector. The horizontal wavelength along LOS (λh, LOS), on the other hand, is parallel to the LOS (blue line) and is, in general, longer than λh.

Knowing λh and the angle β, the along-LOS horizontal wavelength λh, LOS can be calculated as follows: λh, LOS=λh|cos⁡(ψ-β)|. Figure b shows the spectrum of F with respect to λh, LOS and λz. It is referred to as “along-LOS spectrum” hereafter. This spectrum, as we would expect, spreads in the direction of longer horizontal wavelengths.

The application of the visibility filter as described above assumes infinite plane wave fronts. However, three dimensional simulations of CGWs from single convective towers exhibit concentric wave fronts . The assumption therefore is clearly non-realistic, in particular for short period, short horizontal wavelength CGWs. This is problematic in cases where the horizontal wave vector is almost perpendicular to the horizontal LOS, the along-LOS wavelength approaches infinity, and the wave would therefore be regarded as visible. However, in a three-dimensional consideration, the LOS would still intersect many wave fronts resulting in a vanishing net signal. Thus, these waves should not be regarded as visible. In order to mask all waves which have short horizontal wavelengths but are only seemingly visible, we firstly introduce a “stretching” factor: θstr=λh, LOSλh, and secondly, we simultaneously consider whether the horizontal wavelength is short compared to the shortest visible horizontal wavelength. Here, the shortest visible horizontal wavelength is determined as the value of λh from Eq. () corresponding to a temperature sensitivity of 0.3: λvis=λvisλz,S=0.3. We also introduce the visibility ratio as θvis=λhλvis, and threshold values of θstr and θvis are denoted as θstr, thresh and θvis, thresh, respectively. All waves, which have overly large stretching factors (θstr > θstr, thresh) and simultaneously have overly short horizontal wavelengths (θvis < θvis, thresh), are set to zero temperature amplitude as well as to a zero GWMF value. For this study, we chose θstr, thresh = 5 and θvis, thresh = 1. As shown later, results are not very sensitive on the choice of these threshold values (cf. Fig. ). This restriction was applied before the application of the instrument sensitivity function. GWMF values after considering this restriction are given by Frestr. A spectral distribution of Frestr with respect to λh, LOS and λz, which is called “λh-restriction spectrum” hereafter, is shown in Fig. c, again using λh, LOS for the x axis. Comparing with Fig. b, it can be seen that part of the spectral distribution at long λz and long λh, LOS is removed.

After this consideration of the horizontal wavelength restriction, the sensitivity function was applied to GWMF. An example of this application of the SABER sensitivity function on MF1 is shown in Fig. d. This spectrum is referred to as “instrument-sensitivity spectrum” hereafter. In comparison with the previous spectrum (Fig. c), it is clear that a significant part of the spectrum associated with short vertical and horizontal wavelengths has been filtered out. The area of high-value GWMF has now shifted to the direction of longer horizontal as well as vertical wavelengths. GWMF values after applying the sensitivity function are denoted as Fvis.

Today's limb scanning satellite instruments provide information only along track. Therefore, from current limb sounders only the projection of the horizontal wavelength on the tangent-point track can be estimated . The horizontal sampling of current-day satellite observations is too sparse to directly infer the horizontal wavelength. This problem is circumvented by first analyzing vertical profiles and determining vertical wavelengths, amplitudes and phases dependent on altitude. The horizontal wavelength is then estimated from the phase difference of adjacent profiles at the same altitude and the distance between observations along the tangent-point track. The method was first introduced by . Although there are different applications with respect to the profile analysis they all rely on phase differences along the orbital track. In particular, if the phase difference is ΔΦ and the sampling distance between two altitude profiles is Δx, the horizontal wave number and horizontal wavelength along the tangent-point track (kh, track and λh, track) can be estimated as follows: kh, track=ΔΦΔx=2πλh, track. In our simulation, λh, track was calculated from the horizontal wavelength λh based on the geometric relation between them. This geometric relation is illustrated also in Fig. . In this figure, black dots are tangent points, and the green dashed line shows the tangent-point track. The horizontal wavelength along the tangent-point track (λh, track) is indicated by the two-headed arrow, which is parallel to the tangent-point track. It is clear that the angle between the wave vector and the tangent-point track is ψ - γ. From here, λh, track=λh|cos⁡(ψ-γ)|. Due to the projection, the horizontal wave number is changed in Eq. (), and as GWMF and horizontal wave number are proportional, the momentum flux calculated from λh, track is FtrackFvis=kh, trackkh=λhλh, track, or Ftrack=Fvisλhλh, track. A spectral distribution of Ftrack in terms of λh, track and λz is shown in Fig. e. This spectrum is called “projection-on-track spectrum” hereafter and contains both the effects of visibility filtering and along-track projection.

Altitude profiles sampled by most limb sounders are non-vertical, which is an effect that also has to be considered. For example, for SABER and HIRDLS this applies and the effect is investigated and taken into account in our simulations. In particular, we calculate the vertical wavelength, which would be observed by the satellite instrument. This wavelength is referred to as observed vertical wavelength hereafter.

From observations, the vertical wavelength is derived by analyzing altitude profiles as provided by the instrument teams. It is generally assumed that these altitude profiles are vertical and that therefore only the vertical wave structure contributes to the wave structure in the profile. However, for SABER and HIRDLS, scans are not strictly vertical: The change in altitude is performed by upward and downward scanning by the instrument. However, during upward and downward scanning, the satellite moves along its track. This leads to a slant of the profile in the direction along the tangent-point track. Also, when the LOS moves up (down), the tangent-point becomes closer to (further from) to the satellite (cf. Fig. ). This leads to another slant of the profile in the direction across the tangent-point track. Because of the slant of the altitude profiles, it can happen that during an altitude scan not only the vertical structure of an observed wave is sampled, but also to some extent the horizontal structure.

In Fig. , two exemplary tangent points O1, O2 along an altitude profile are illustrated (purple dots). This could correspond to any pair of adjacent altitudes in a scan such as illustrated in Fig. . A local coordinate system at tangent point O1 is shown where the z axis indicates the vertical direction. The altitude difference dh between two tangent points O1 and O2 is small (we chose dh = 3 km), so that the vector O1O2 was considered to be the local profile vector. If p (blue vector) is the normalized vector of O1O2 and k (red vector) is the wave vector, then the wave number along the profile can be defined as the scalar product of k and p: mp=k⋅p. The wavelength along the profile is λz,p=2πmp. From λz,p and from the angle ζ between the normalized profile vector p and the z axis of the local coordinate system (cf. Fig. ), the observed vertical wavelength λz,obs is calculated: λz,obs=λz,pcos⁡ζ. Momentum flux corresponding to this vertical observed wavelength is symbolized as Fz,obs. Following Eq. (), GWMF is inversely proportional to the vertical wave number and thus proportional to the vertical wavelength: Fz,obs=Faliasλz,obsλz. In a statistical average, we will have as many upward-scanning observations as downward-scanning observations. Therefore, we calculate both solutions for each wave and show the average. The spectrum with observed vertical wavelength hereafter is referred to as “λz,obs spectrum” and an example for MF1 is shown in Fig. g. The spectrum was slightly redistributed towards longer vertical wavelengths. In particular, for vertical wavelengths longer than 6 km, GWMF was slightly enhanced.

For every wave, we also examined the difference between the observed vertical wavelengths for the upward and downward scans. If this difference is greater than 40 % of the average vertical wavelength, this wave will be rejected. We here follow the GWMF determination from real observations as described in , where such pairs of profiles are not used by the momentum flux (MF) calculation method. It should be noted that for other methods of MF calculation, these pairs may be used.

We symbolize GWMF after this restriction as Fz,obs, restr. The spectrum with this restriction, called “λz-restriction spectrum” later, is shown in Fig. h. In comparison with the previous spectrum (Fig. g), only minor changes were found. In particular, the magnitude of GWMF surrounding the spectral peak at vertical wavelength of about 5 km was reduced slightly.

In the last step of the observational filter, we applied an additional correction, which was used in . First, this correction removes dominant vertical oscillation of quasi-stationary planetary waves (which have a vertical wavelength ≥ 40 km) in the altitude profiles. Second, it helps to keep only those vertical wavelengths for which amplitudes can reliably be determined in the 10 km vertical window of the Maximum Entropy Method/Harmonic Analysis (MEM/HA) spectral analysis . The GWMF at this last step is denoted as Fobs.

This is the final step of our comprehensive observational filter. The resulting spectrum is therefore considered to represent the observed spectrum and is presented in Fig. i. In comparison with Fig. h, it can be seen that contributions of long vertical wavelength waves were somewhat reduced. However, the overall spectrum is changed only slightly.

A comparison of this observed spectrum and the true spectrum (Fig. a) shows that the spectral distribution of MF1 is significantly influenced by the observational filter in both shape and magnitude. In particular, the observed spectrum consists of horizontal wavelength for which MF1 did not generate any wave events and vice versa. This is due to the fact that MF1 has a small spatial scale and produces a large amount of short horizontal wavelength GWs, which can hardly be observed by limb sounders. However, as mentioned before, for demonstrating the different effects of the observational filter, MF1 was chosen because the different effects contributing to the observational filter can be demonstrated clearly. Later in the manuscript we will address other setups of the CGWS that produce wave spectra that can be better observed.

All steps of the observational filter are summarized by a flowchart in Fig. . The steps with significant changes are marked by bold characters. Additional examples of applying the observational filter to all three spectra MF1, MF2, MF3 using the observation geometries of SABER as well as HIRDLS will be presented in Sect.  below.

In Sect. , we illustrated the observational filter by applying it to the spectrum of MF1 and using SABER geometry. In this section, we provide further examples by applying the observational filter to all spectra MF1, MF2 and MF3 and using observation geometry of both instruments (SABER and HIRDLS).

For SABER geometry, the results of applying the observational filter are presented in Fig. . As shown by “true” simulated spectra (Fig. a, g, and m), MF3 provides GWs with the longest horizontal wavelength. The main spectral peak of MF3 is at a horizontal wavelength of about 220 km. It has some sub-structure and extends to λh as high as few hundred km. For MF2 and MF1, this peak is located at horizontal wavelengths of about 50 and 10 km, respectively. It should be noted that the spatial scale of the cloud tower specified in the source model is imagined to act as a single body force without substructure. Therefore, no waves with wavelength of the order or shorter than this body force are excited. Instead, the model produces a sharp onset at the wavelength of twice this size. This sharp onset is pronounced at the source altitude. When GWs propagate upward, the wavelength may be modified by horizontal refraction e.g., which slightly weakens this sharp onset. In the cases of MF1 and MF2, at the considered altitude of 25 km, this sharp onset still can be seen quite clearly.

Due to this difference in the spatial scale, the observational filter affects MF1, MF2 and MF3 differently. For example, the effect of the λh restriction (Sect. ) on MF1 is recognizable by comparing Fig. b and c, while for MF2 and MF3, this effect is minor and indicated by only an insignificant decrease in GWMF at long horizontal and vertical wavelengths (not shown).

However, differences can be seen much more clearly after the instrument sensitivity has been applied by comparing the second and third rows of Fig. . For MF1, a very large amount of GWMF corresponding to short horizontal as well as vertical wavelengths has been filtered out (cf. Fig. b and c). The spectral peak is shifted from a λh, LOS value of about 40 km (Fig. b) to a value of about 160 km (Fig. c). It should be noted that in this step, spectra are plotted with respect to the horizontal wavelength along LOS (λh, LOS). The shift of the spectral peak with respect to the true horizontal wavelength (λh) in general is shorter. For MF2, the reduction in GWMF is considerably smaller than for MF1 (cf. Fig. h and i). Nevertheless, the GWMF magnitude is reduced quite strongly. The spectral shape changes and the area of strong GWMF moves to the direction of longer horizontal and vertical wavelengths. For MF3, part of the spectrum related to short wavelengths has also been filtered out (cf. Fig. n and o). This part, however, is smaller than for MF2, and although GWMF magnitude has decreased, the main spectral peak of MF3 remains at the same position (at λh, LOS of about 500 km).

Figure d, j, and p show spectra of Ftrack with respect to λh, track and λz. For MF1 and MF2, GWs with a horizontal wavelength shorter than 100 km contribute quite strongly to the spectrum (cf. Fig. d and j). High values of GWMF are even found at horizontal wavelengths down to about 20–30 km. In contrast, the main part of the spectrum of MF3 arises from by GWs with a horizontal wavelength greater than 100 km (cf. Fig. p). The influence of the aliasing effect on MF3 is therefore weaker than on MF1 and MF2. This is shown in Fig. e, k, and q. Since MF1 and MF2 contain many more short horizontal-wavelength GWs, an essential part of their spectra is projected to the left. For MF1, the features of the spectrum are changed significantly, as described before in Sect. . For MF2, a strong alteration is also found, although no strong artificial spectral peaks appear, as they do in the case of MF1. In contrast, the part of MF3 projected to the left is minor in comparison with the originally long horizontal-wavelength part. Therefore, the strongest contribution to the spectrum in general, and the main peak in particular, still remains at the same position.

Figure f, l, and r shows observed spectra after the calculation of observed vertical wavelength, vertical wavelength restriction and additional correction. In comparison with aliasing-effect spectra, very minor changes were found for all spectra. In particular, spectra were redistributed slightly in the direction of longer vertical wavelengths, making them somewhat more homogeneous in this direction. The spectral peak at a vertical wavelength of about 30 km of MF1 was reduced in magnitude.

Briefly, the spectrum for MF3 was least influenced by the observational filter. For horizontal wavelengths longer than the Nyquist wavelength, major features were still conserved. The spectrum of MF1 was most influenced and significant changes were found in both shape and magnitude.

The observation geometry of HIRDLS has a shorter horizontal sampling distance. HIRDLS also has a higher vertical resolution. The results of applying the observational filter to the observation geometry of HIRDLS are presented in Fig. .

In the case of HIRDLS, “true” spectra (Fig. a, g, and m) are very similar to “true” spectra for SABER. Minor differences result from the different latitude coverage.

However, in contrast to SABER, along-LOS spectra of HIRDLS spread more strongly towards longer horizontal wavelengths (Fig. b, h, and n). This is an effect of the average orientation of the simulated GWs with respect to different view angles of the two instruments. This effect depends not only on the differences in viewing geometry, but also on the simulated distribution of GWs.

The effects of the horizontal wavelength restriction were similar to those observed for SABER observation geometry with minor reductions at long horizontal and vertical wavelengths for all three spectra (not shown).

However, HIRDLS possesses better sensitivity to short-wavelength GWs, particularly in the vertical direction. This weakens the influence of the instrument's sensitivity to all spectra. A comparison of Figs. c, i, o and c, i, o shows that in comparison with SABER, for HIRDLS, the amount of GWMF was not reduced as much by the instrument sensitivity. For HIRDLS, GWMF was still conserved quite well in the vertical direction down to λz of about 2 km, while for SABER this limit was about 4 km. In the horizontal direction, since spectra of HIRDLS geometry spread more strongly with respect to λh, LOS, GWs appeared to be more sensitive to the instrument. Hence, the reduction of GWMF in the horizontal direction was also lower than for spectra based on SABER geometry.

This better conservation of GWMF for HIRDLS was also found in spectra of Ftrack with respect to λh, track and λz (cf. Fig. d, j, and p). The contribution of short vertical-wavelength GWs to these spectra is more pronounced than in the case of SABER (Fig. d, j, and p).

Moreover, due to the shorter sampling distance, spectra in the case of HIRDLS were less influenced by aliasing than for SABER. Comparing aliasing-effect spectra of HIRDLS (Fig. e, k, and q) and SABER (Fig. e, k, and q), it is evident that for HIRDLS, a smaller part of the respective spectrum for MF1 was cut and for MF3 projected towards longer horizontal wavelengths (before the aliasing effect could take effect). The remaining part of each spectrum is therefore larger and more features are conserved.

In particular, the spectrum for MF3 and HIRDLS including the observational filter shown in Fig. q is the only one which has a well-resolved maximum that also decreases at short horizontal wavelength, similar to the observations of . For this case (MF3), the spectral peak of the “true” spectrum is indeed captured by the observations.

Concerning the effect of “λz restriction”, only an insignificant variation was found in the number of wave events for SABER (not shown). In the case of HIRDLS, this variation was more pronounced, and the variation of the number-of-wave-event spectrum for HIRDLS is shown in Fig. . In this figure, the spectrum of the ratio r = n2/n1 is plotted with respect to the true horizontal and vertical wave numbers. Here, n1 is the number of wave events in one bin before considering λz restriction, n2 is the number of wave events in the same bin after considering this restriction. Reduced ratios were found in the lower right corner of the spectrum for all MF1, MF2 and MF3. This indicates that most of the filtered-out waves have short horizontal wavelength and long vertical wavelength. This can be explained as follows: when the horizontal wavelength is much longer than the vertical wavelength, the wave fronts are almost parallel to the horizon. In this case, the angle between the wave vector k and the normalized profile vector p is almost the same for upward scanning and downward scanning. Following Eqs. ()–(), the difference between observed vertical wavelengths in those two cases (upward and downward scan, respectively) is therefore insignificant. However, when horizontal wavelength and vertical wavelength are of the same order of magnitude, angles between k and p for upward and downward scanning are strongly different. This leads to a considerable difference in the observed vertical wavelengths. Figure  shows that even in the bins which were most affected, maximum profile loss was only about 10 %. On the other hand, profile loss in deriving data from HIRDLS observations using the method of was about 50 % . This indicates that the λz restriction step cannot be the major reason for the observed loss of about 50 % of altitude profiles in real observations.

Figure f, l, and r shows “observed” spectra in the case of HIRDLS observation geometry. Again, in comparison with the aliasing-effect spectra, only minor changes were found and these changes were analogous to the case of SABER.

Overall, similar to the case of SABER, MF3 was least affected, while MF1 was most affected by aliasing. In particular for MF3, with observation geometry of HIRDLS, it was shown that almost all spectral features are preserved.

To conclude, for both cases of observation geometry (SABER and HIRDLS), all spectra (MF1, MF2 and MF3) shifted to the direction of longer horizontal as well as vertical wavelengths. A rather large part of each spectrum associated with short horizontal wavelengths was projected to longer horizontal wavelengths. The spectrum for MF3 has the longest spatial scale and was least influenced by the observational filter. In contrast, the spectrum for MF1 has the smallest spatial scale and was most influenced by the observational filter. The better sensitivity of HIRDLS helps to decrease the reduction of GWMF due to instrument sensitivity. In addition, HIRDLS's shorter sampling distance allows us to see a larger part of spectra after aliasing.

As shown above, the magnitude of GWMF is decreased after applying filters mimicking λh restriction, instrument sensitivity, and aliasing. Moreover, the magnitude of GWMF also changes by calculating the observed vertical wavelength, the observed-vertical-wavelength restriction and additional correction. The changes during these last three steps, however, were minor, as we have seen from the spectra. In order to quantify the change in GWMF during the process of filtering, GWMF were integrated over all horizontal wave numbers and afterwards plotted against the vertical wave number in a base 10 logarithmic scale. The effects of the last three steps are discussed as one common step.

Figure  shows GWMF for SABER (left column) and HIRDLS (right column). The cyan dashed-dot line indicates GWMF of the true spectrum, the black solid line is GWMF after considering λh restriction, the blue dashed line presents GWMF after the instrument sensitivity has been considered, the orange line is GWMF after projecting on tangent-point track, the red line shows GWMF after the aliasing effect and the green line with crosses shows GWMF of the observed spectrum.

For both SABER and HIRDLS, the reduction due to the whole filtering process is largest for MF1 and smallest for MF3. It is indicated by the notable difference between GWMF of the true spectrum (cyan dashed-dot line) and the observed spectrum (green line with crosses). It is about 2.5 orders of magnitude for MF1 viewed by SABER (Fig. a) and about 2 orders of magnitude for MF1 viewed by HIRDLS (Fig. d). This difference is smaller in the case of MF2 (Fig. b and e) and is smallest in the case of MF3 (Fig. c and f). For MF3, the difference is only about half an order of magnitude. This agrees well with the fact that the spectrum for MF1 is most influenced and the spectrum for MF3 is least influenced by the observational filter, as discussed above.

Moreover, for all spectra and for both observation geometries, it is clear that the instrument sensitivity is the factor that reduces GWMF the most. This reduction can be seen by comparing the black line and the dashed blue line. The difference between these two lines is the largest difference between two adjacent lines in all panels. Moreover, this reduction was strongest for MF1, decreasing from MF1 to MF3 due to the increase in spatial scales. Again, this finding is in agreement with the change in spectra described above in this section.

In addition, the reduction in the case of HIRDLS was weaker than in the case of SABER, which is explained by the better sensitivity of the HIRDLS instrument. For example, after considering the instrument sensitivity of HIRDLS, GWMF of MF1 and MF2 (Fig. d and e) was about 2.1–2.2 (in the unit of base 10 logarithmic scale), while for SABER, the value of GWMF dropped to about 1.8 (Fig. a and b). The contribution of short vertical-wavelength GWs from about 1 to about 3 km was also much larger in the case of HIRDLS than for SABER. For MF3 (Fig. c and f), the difference between these two observation geometries was lower than for MF1 and MF2; however, it is still recognizable even in the base 10 logarithmic scale.

The second strongest factor of GWMF reduction for SABER is aliasing, as can be seen by comparing the orange and the red lines, which are separated quite clearly from each other (except in the case of MF3). Again, the effect of aliasing decreases from MF1 to MF3 due to the increase in the spatial scales of the waves. Moreover, since the sampling distance of HIRDLS is shorter (70 km) than for SABER (185 km), less GWMF reduction by aliasing was found for HIRDLS.

The process of projecting the horizontal wavelength on the tangent-point track reduces GWMF less than instrument sensitivity and the aliasing effect in most cases; the exceptions are MF2 and MF3 for HIRDLS. Furthermore, the reduction by this factor was very similar for all spectra MF1, MF2 and MF3. This is due to the fact that the reduction is mainly induced by |cos⁡(ψ - γ)|, which does not depend on the spatial scale of the individual waves.

Minor redistribution of the spectra by the last three steps is shown by the difference between the red line and the green line with crosses. GWMF values at the spectral peak (at a vertical wavelength of about 30 km) were reduced by the additional correction. This can be seen clearly in Fig. a and d.

The step of λh restriction affected GWMF least. In the base 10 logarithmic scale, GWMF of true spectrum (the cyan dashed-dot line) and GWMF after considering λh restriction (the black solid line) were nearly the same in almost all panels.

In addition, cyan dashed-dot lines in Fig.  (true spectra) show that GWMF given by MF1 is the largest with a peak at about 3.5 (in the base 10 logarithmic scale). For MF2, this value is about 2.7 and for MF3 it is only about 2.4. The relative importance of these different spectra (MF1, MF2, MF3) in the whole GWMF spectrum is, however, still unknown and may be adjusted (e.g., by intermittency or efficiency factors), as the relative importance of various convective process in exciting GWs is still badly constrained.

More details about the reduction in GWMF during the observational filter are presented in Table . Here, the total GWMF of the true spectrum is 100 %. The percentages of the remaining GWMF in other steps of the observational filter (instrument sensitivity, projection on track, aliasing effect and observed spectrum) are shown for all spectra.