One-dimensional (1-D) continuous wavelet transform (CWT) calculations constitute the basic elements of the proposed CIPS wave tracking algorithm. The term “wave tracking” in the context of this paper refers to the operation of tracking all existing quasi-periodic wave displays in the CIPS PMCs over the scale range of ∼20–60 km, rather than tracking a specific wave event. We first demonstrate the effectiveness of the approach using an analytically composed series. When the algorithm is applied to CIPS (see Sect. 3.2 and later), each individual CWT calculation will be carried out along a presumably ∼400 km (or 80 grids) long CIPS PMC albedo segment and will deliver a total of 22 components spanning the scales 2.0–76.0 in grids (5 km per grid). The relevant scale range for this study is ∼4.0–12.0 grids (∼20–60 km), and the total power of the relevant CWT components is termed “CWT power” or “albedo power” in the following CIPS-related discussion. It is worth pointing out that a ∼20–60 km scale range is focused on in this study because the total albedo power spatial distribution corresponds well with the readily observed wave signatures in the albedo maps (see Figs. 2, 7–9, and 11). The 60 km threshold appears particular, but it is chosen simply because it is among the sequence of individual scales for the CWT calculations, which are 4.0, 4.7, 5.6, 6.7, 9.5, and 11.3 grid units. A radius of roughly ∼400 km is chosen because it is able to include many repeats (∼10) of the wave ridge and trough to provide the full extent of wave display. Yet the spatial span of the wave display should not be overly extended because we do not wish to go across several wave events or different types of variability along the path of CWT calculation. These calculations will be carried out in all 360∘ radial directions (3∘ increment) when being applied to CIPS PMCs.

A 6th-order Morlet wavelet is adopted in this study. A Morlet wavelet (Gabor, 1946) is a complex exponential (plane sinusoidal waves) windowed by Gaussian function so that both periodicity and localization can be realized, defined as eikx/se-x2/(2s2), where k is the (non-dimensional) order and s is the scale. Scale s determines both the width of the Gaussian function and the period of the sinusoidal signal. In a 6th-order Morlet wavelet (k=6) the scale s is almost precisely the period of the sinusoidal signal. We emphasize here that in the following main analysis applied to the CIPS PMC images the scales of the wave structures refer to the Morlet wavelet scales. However, we will first use an artificially created series to demonstrate that CWT and fast Fourier transform (FFT) deliver qualitatively consistent results.

The artificially created series, which is shown as the thick red curve in Fig. 1a, is composed of seven plane sinusoidal waves that have wavelengths of 4.0 to 12.0 grids and zero initial phases at the start point of the series. These wavelengths are the 5th–11th scales delivered in a CWT calculation mentioned above. The shortest and longest of these plane waves are shown by the gray and green dashed lines in Fig. 1a. The total CWT power over the scales of 4.0–12.0 grids is shown as the black curve in Fig. 1b, whereas the thick red curve in Fig. 1b is the reconstructed series using the Morlet wavelets and the corresponding CWT coefficients. It is notable that the black curve is smooth and exactly follows the magnitude change of the localized shorter scale signals. The basis vectors of CWT are not orthogonal, and therefore a reverse CWT does not exist in a strict sense, but we do find that the reconstructed series greatly resemble the original series although their magnitudes slightly differ. This suggests that the CWT and reverse CWT work efficiently on a quasi-periodic signal series. Especially, the fact that CWT almost precisely captures the local variability of the wave amplitude suggests that the CWT algorithm will be an effective approach to detecting the gravity waves in the CIPS PMCs.

Figure 1c shows the CWT spectrum of the created series. We just mentioned that the created series is the sum of only seven FFT components, but since FFT and CWT have different basis vectors, the CWT will project onto all the scales from 2.0 to 76.0 grids. Over the seven relevant scales from 4.0 grids to 12 grids the slope on the double-logarithm diagram is as weak as -0.54, reflecting the fact that we have adopted identical amplitudes for all the plane sinusoidal waves used to create the series. For scales <4.0 grids there is an extremely rapid decrease of the CWT power density with a slope of -12, indicating that these scales are almost non-existent. It is noteworthy that the created signal exhibits a quasi-periodic nature even though there are no spectral peaks. However, due to the involvement of multiple scales, the fluctuation washes out in a certain portion of the series (i.e., in the range of 3.0–23.0). In this latter case, quasi-periodicity is impaired by the involvement of multiple components.

In terms of applying the algorithm to the PMC images from CIPS, a direct 2-D CWT routine would be preferred but does not exist in the standard numerical recipe. In addition, there is ambiguity in determining the phases in the CWT algorithm because there is always a tradeoff between the localization of the signal and a clear phase determination.

In this study we used an algorithm based on consideration of expediency as well as efficiency. The 1-D CWT calculations are carried out in all 360∘ radial directions (3∘ increment) centered at a given location within the CIPS albedo orbital strip. The resampling of the CIPS data is performed in the radial and angular directions centered at such a location. In the radial direction the increment step is 5 km, which is the same as the CIPS Level 2 resolution, while in the angular direction a step of 3∘ is chosen based on the consideration of yielding a sufficiently detailed yet smooth spatial map of the albedo wavelet power. This approach was inspired by the intent of detecting ideal concentric waves because such a design will result in the maximum CWT albedo power within a given radius. In addition, performing the CWT in all radial directions will efficiently capture the waves of all orientations. Since the basis vectors of 2-D FFT are straight (linear) waves of all orientations, the current algorithm is more or less a short version of the 2-D FFT in a localized area but has the merit of being straightforward in reflecting the local albedo wave power.

The specifics of the algorithm are described as follows. An elliptical region of 80 grids along-track and 80×0.65 grids cross-track is used to carry out the CWT calculations. The factor 0.65 is empirically chosen based on a few fitted examples of the concentric waves found in CIPS. The rationale of such a factor will be revisited in a following paragraph. Concentric waves found in CIPS appeared to be mostly elongated in the along-track direction. This factor has no qualitative effect on the results except that, when the elliptical region fits an actually existing concentric wave pattern, it achieves the largest albedo CWT power, which is a desired condition for the wave tracking operation. The elliptical region is moved around to fully cover the orbital strips. The steps of the movement are the half axial lengths in both along-orbit and cross-track directions. Except for the intent to achieve full coverage of the orbital strip, the “move-around” scheme will also ensure the capture of the albedo CWT power from varying orientations in the same region. The albedo CWT power map enclosed in the given elliptical region measures the total albedo fluctuation intensity in the scale range of ∼20–60 km for each spatial location.

Five examples of concentric waves and the corresponding albedo power maps are presented in Fig. 2 to demonstrate how the wave tracking algorithm works for CIPS. Concentric waves are chosen because they are well documented as possessing a unique morphology and meanwhile serving as a proof of the connection between the lower and higher atmosphere. These waves are extremely rare and were detected only a few times for a given PMC season. The percentage of detection is less than 5 % in terms of days per season, and by spatial coverage the fraction is even smaller. All examples have shown only partial rings. The model results by Vadas et al. (2009) simulating the concentric rings to compare with the observations near Fort Collins, Colorado, have shown that including realistic zonal winds can substantially disrupt the completeness of rings in the mesosphere, with about 50 % of the wave structure being disrupted. In the CIPS PMCs the rings are more severely disrupted. In some cases only a 20∘ section of the full 360∘ circle has survived (not shown). The model results by Vadas et al. (2009) also indicate that if a July zonal wind is adopted in the simulation the rings will be elongated to an axial ratio of about 0.6–0.7, which roughly agrees with the findings in CIPS. It appears that in summer the concentric waves will likely be substantially elongated compared to those in winter or spring.

The example shown in Fig. 2a has a wavelength of about 60–80 km, which is longer than most other concentric waves in the CIPS PMCs, but since the (bright) ridges are much narrower than the (dim) troughs the albedo power still reaches notable magnitudes. Remember that in this study only the CWT power within the wavelength range of ∼20–60 km is calculated, and what is shown in Fig. 2a again reminds us that wave patterns in PMCs only achieve quasi-periodicity. The example in Fig. 2b is a set of highly distinct concentric waves. The albedo power reaches notably larger values when the waves are the most distinct in the upper-right quadrant of the albedo map (see Fig. 2b, upper panel). In the upper-left quadrant of the albedo map the waves become blurry, but one can still tell that they are concentric. In the blurry part of the map the albedo power decreases sharply, as is clearly seen from the corresponding lower panel. Given the confined scale range of ∼20–60 km, being visually blurry can be interpreted as possessing a lower variance level, which may have been a result of diffusive processes. In the spectral space, if the leading scales stand out poorly from the neighboring smaller scales, then the structure will be less organized, which also makes it less distinct. This topic will be further addressed in Sect. 5. Within the same wave display, varying from being distinct to blurry must be controlled by some physical process yet to be unraveled. In the lower-right quadrant of the albedo map there are some straight waves that have cross-interfered with each other but are much less distinct than the main part of the concentric waves. This suggests that multiple wave packets of different morphology often coexist right next to each other in the PMCs. Figure 2c shows an example of extremely faint concentric waves which are characterized by much lower albedo power than those in the rest of the examples. Figure 2d shows a slightly weaker but also fairly distinct partial concentric wave pattern that also coexists with some blurry straight-wave patterns. Figure 2e is an example of concentric waves in a brighter background that occurred in 2009. In 2009 CIPS orbits appear twisted due to the camera's turned position.

Lastly, we must also point out an inherent drawback of the CWT wave tracking algorithm. It is apparent that in all the examples shown in Fig. 2 the clouds (e.g., >5.0×10-6 sr-1) do not fill up the entire elliptical region, and furthermore the wave signatures are mostly partial rings. If such an inhomogeneity is strong, the mean albedo power or background brightness can misrepresent the characteristics of the region. We therefore have chosen a relatively small radius (∼400 km) to carry out the calculations to minimize the effect of inhomogeneity. Due to the high complexity of the PMC signatures, it is unlikely to simply eliminate such an effect. Nevertheless, so far we have not run into any noteworthy problem due to such effect when analyzing the wave tracking results.

The statistical ensemble consists of the albedo power values averaged within all elliptical regions used to carry out the wave tracking. We need to emphasize here that the albedo power refers to the total power within the scale range of ∼20–60 km. The wave tracking procedure has been carried out throughout the two northern hemispheric summers in 2007 and 2010 from 1 June to 31 August. We take a particular interest in the wave display in the brighter PMCs because the previously identified waves mostly reside in the relatively dim cloud environment. We split the cloud population into two subsets, one containing only 0–2 % bright clouds (with a threshold of 25×10-6 sr-1) in the elliptical region, which is the overall dimmer cloud group, and the remaining set containing a systematically larger fraction of bright clouds. The 25×10-6 sr-1 threshold is chosen empirically. The bright-cloud presence frequency (denoted by freq25 hereinafter) is a better index than the mean cloud albedo in characterizing a systematically brighter cloud background.

Figure 3 shows the histograms of the albedo power values for the full set of wave detections (in red) and those residing in the brighter background (in black), and both show a peak number density being close to zero and then a rapid decrease as the albedo power increases. Peak locations approaching zero indicate that the majority of the waves are in the range of low albedo power. It is especially worth noting that the full set and the brighter set collapse as the albedo power is greater than ∼20×10-12 sr-2, whereas before this point they separate substantially. This indicates that brighter clouds mostly coincide with the higher albedo power and that the waves that have caused the difference in the two curves reside in the dimmer group. In addition, the collapsing part of the curves forms a straight line under a logarithmic vertical axis, suggesting an exponential decrease of the wave detection number density as the albedo power exceeds the value where the peak number density occurs. For the full set of the wave detections the straight line proceeds to a much smaller albedo power value, while for the brighter set it shows a peak at a higher albedo power. Removal of the dimmer cloud group by a given threshold caused this. In general, the exponential decrease of the wave detection number density toward increasingly higher albedo power is a robust result, but how rapidly the number density decreases may show interannual variability (not shown). It is worth mentioning that the analysis of the PMC ice water content measured by the Solar Backscatter Ultraviolet (SBUV) instruments (DeLand and Thomas, 2015) yielded the same type of distribution. These authors further investigated the apparent interannual variability of the distribution slope and concluded that population ratios between the hierarchy of particle sizes may have been different for individual years to cause this variability. Later in this paper we will find that PMC albedo and the corresponding wave power hold a statistically linear relationship, and therefore it is within expectation that both the PMC intensity and the wave power follow similar distributions. Figure 3 also shows that when the albedo power reaches its upper limit the curve flattens out, but such a behavior is simply caused by the low sample number, which is of no significance.

Although the dimmer subset takes up a major fraction of the cloud population, which exceeds 65 %, we take just as much interest in the wave display in the brighter cloud background in this study. Figure 4 shows the scatterplot of the wave detections in the brighter cloud background on the plane of albedo power versus freq25. In Fig. 4a, within equally spaced bins of freq25 and the albedo power, the rainbow-colored squares represent the number density of the wave detections. The dimmer group of PMCs are collapsed into the first bin of freq25=0–0.02 and is not shown in this scatterplot to avoid any discontinuity induced by the artificially chosen threshold (i.e., 25×10-6 sr-1).

Figure 4a shows that the wave detections (see rainbow colors) are grouped more densely in the low-albedo-power as well as the low-brightness region. This generally agrees with what Fig. 3 shows, but we should note that the wording “low” is in a relative sense because we have taken away the dimmer set in this analysis. For any given freq25 the wave detection number density distribution resembles a normal distribution, but the outliers are strongly asymmetric, showing a much-further-reaching albedo power value at the upper limits. In addition, we find that as freq25 increases the peak number density moves toward an increasingly larger albedo power, shown by the dashed black curve. Both suggest that in a statistical sense larger albedo power corresponds to brighter cloud background. At the dimmer end where freq25=0.02 the albedo power is within 50×10-12 sr-2 (amplitude ∼7.0×10-6 sr-1), while for freq25=1.0 it reaches values greater than 150×10-12 sr-2 (corresponding to an amplitude of ∼12.0×10-6 sr-1).

We next adopt an analytic form to parameterize the relationship between the albedo power and freq25 in order to resample the wave detections into a consistent normal distribution. This is a preliminary step taken for the future removal of the apparent dependence of the wave power on the background cloud brightness. The mechanism that controls such an apparent dependence is not pursued in this paper, and a future study will be required to understand this since we have learned that the previous modeling studies do not seem to directly interpret it. For example, Chandran et al. (2012) have shown that both the short-period and long-period gravity waves ultimately reduce the domain-averaged PMC brightness.

A set of square root sectioning curves (albedo power = factor × √freq25) are used to split the wave detections into subsets to achieve the normal distribution. Such an analytic form is chosen because it roughly coincides with the dashed black curve in Fig. 4a that shows how peak number density of wave detections varies with freq25. The interval of the sectioning curves is by a factor of 1/2i/4, with index i varying from 0 to 21, as shown in Fig. 4a. In total, 23 sections are used to produce a smooth probability distribution, with the first index i=-1 being the closest to the albedo power axis. The 2i/4 rather than a linear sequencing is chosen to account for the fact that the data points become increasingly denser from being close to the albedo power axis toward the freq25 axis. Figure 4b confirms that the resampling produces almost a precise normal distribution, and at the 11th interval it reaches the peak, which splits the wave detections in half.

In this subsection we reexamine the albedo power dependence on the background cloud brightness using the mean albedo within the elliptical region as the horizontal axis. This angle of investigation provides a smoother picture because it does not use any imposed threshold (25×10-6 sr-1). Two purposes are served by doing so. First, we tend to include the wave detections with dimmer background since they take a major fraction of the cloud population as is mentioned above. Second, we tend to examine whether the full set of wave detections and those residing in the brighter cloud background follow a consistent statistical relationship between the albedo power and the background cloud brightness.

Figure 5a shows a similar scatterplot except using the mean albedo (within the elliptical region) as the horizontal axis. The isopleths of the wave detection number density suggest a linear relationship between the albedo (fluctuation) power and the background albedo, and we also note a strong asymmetry of the albedo power distribution between the lower and upper limits, suggesting an apparent dependence of the albedo power on the background mean albedo. This confirms that albedo power monotonically increases with the background cloud brightness, except that a different analytic form will be used to carry out the sectioning procedure.

Linear sectioning lines are applied to the plane of the albedo power versus the mean background albedo to yield a normal distribution. The sectioning lines are emitted from the (0, 0) point, and the angular interval gradually increases from the lower-right to upper-left corner to achieve symmetry of the distribution. A total of 27 intervals are used. Such a sectioning and resample procedure makes both the full set and the dimmer subset achieve the normal distribution, shown in Fig. 5b, which confirms a consistent behavior between the dimmer set and the full set. The peak of the normal distribution is reached at the 12th sectioning interval. Both Fig. 5 and Fig. 3 (i.e., the albedo power histograms) indicate that the dimmer subset and the full set follow a similar behavior.

The dots of three different colors in Fig. 4a are the sample selections of the wave detections for future-demonstration purposes. Wave tracking yields a large number of detections, and we aim at obtaining a universal law that governs the full set of wave display. The selections are made roughly at the two tails and the peak of the normal distribution (Fig. 4b), labeled the high-, medium-, and low-albedo-power categories. Note that these categories are chosen based on the combination of both albedo power and the background cloud brightness. Three brightness levels (freq25=0, 0.4, and 0.8) are used, and therefore a total of nine selections are made. The cases at freq25=0.4 and 0.8 almost precisely follow the sectioning curves. While at freq25=0, the three selections are made based on a linear relationship with the two selections at freq25=0.4 and 0.8. This is because the dimmer subset (freq25<0.02) is not included in Fig. 4a. We should point out that the sectioning curves serve only as the guidance to select the cases that reasonably cover the full set of detections, but our conclusions are not sensitive to what exact cases are chosen.

We place the same nine selected cases in Fig. 5a and find that the three albedo power categories also roughly follow the linear sectioning lines and are also located approximately at the lower and higher tails and at the peak of the normal distribution.

At last we check on where the concentric waves shown in Fig. 2 are distributed on the scatterplots. From looking at the turquoise-colored triangles we find that they do not seem to preferably occur at any specific combinations of albedo power and background brightness. Rather, the five cases are approximately evenly spread over the core region, and none has occurred in either high-albedo-power or high-background-brightness ranges. Also, it is worth mentioning that the case of the most distinct concentric waves shown in Fig. 2b has a combination of freq25=0.016 and albedo power = 25.4×10-12 sr-2, suggesting that the combination of relatively low background brightness and relatively high albedo power seem to correspond to the best clarity of wave display. Although rare in occurrence and possessing a known particular form of driving mechanism, concentric waves seem to have shown a regular behavior in terms of the correspondence between the albedo power and the background cloud brightness.

The albedo power spectra of the nine selected detections are shown in Fig. 6, with the three brightness levels being collapsed to each other for each albedo power category. These power spectra are calculated to extract a universal law that governs the wave display. The power spectra of all orientations (full 360∘ with 3∘ step) within a given elliptical region are averaged for each individual scale in the range of ∼20–60 km. Although inhomogeneity exists between different orientations, the spectra of the main wave signatures will dominate the mean spectrum. Unlike the FFT power spectra that are prone to exhibiting spikes (not shown), the CWT power spectra have blunter features. Especially, due to the spatial average it is even less likely that spectral peaks will stand out unless a given scale within the ∼20–60 km range is consistently dominant throughout the gravity wave population, which is not the case.

Based on the nine representative cases, the general form of power spectra can be expressed as A × (1/wavelength)α, where coefficient A=1.42×10-4, 1.45×10-4, and 1.49×10-4, and the spectral exponent α=-2.5, -2.9, and -3.2 for the low-, medium-, and high-power categories, respectively. Note that the three categories are defined in Sect. 4.5. For a confined range of wavelengths, i.e., ∼20–60 km, the higher wave power or larger variance level of the display will correspond to higher clarity of wave display or to sharper features. More rapid decay toward the smaller scales will also contribute to higher display clarity because the leading scale will be more dominant over the smaller scales, and therefore the wave signature will be better organized. The actual wave displays shown in Sects. 5.2 and 5.3 will verify these arguments.

The black lines in Fig. 6 are for the high-power category (see the black dots in Fig. 4a). The three brightness levels possess a consistent exponent α=-3.2, and therefore via simple adjustment of A (in the analytic form) a normalization or collapse between the three spectra is achieved. The factors of normalization toward the brightest level are defined as follows: ratio0.4/0.8=albedo poweratfreq25=0.4/albedo poweratfreq25=0.8,ratio0.0/0.8=albedo poweratfreq25=0.0/albedo poweratfreq25=0.8, where albedo power refers to the total wave power over the scales of ∼20–60 km. As is argued above, the normalization is carried out to examine the wave morphology beyond the wave power dependence on the background brightness. After applying these factors, the three brightness levels will achieve a nearly fully collapsed power spectrum and exhibit the same level of display clarity (see Sect. 5.2). The black line under the linear axes shown in Fig. 6b indicates that the high-power category possesses both the highest overall power level and the most rapid decay rate toward the smaller scales.

The error bars in Fig. 6 are the uncertainty ranges over the three brightness levels for the different scales, and the exact match occurs at the middle data point (i.e., ∼33 km of wavelength or 0.03 of spatial frequency). It is noteworthy that both the high- and medium-albedo-power categories have very small error bars, suggesting a strong collapse of spectra at the three brightness levels. The low-albedo-power category, however, has notably larger error bars. This means that when the albedo power reaches very low values the exponent α maintains poorer consistency between the three brightness levels. Measurement noise may play a role in causing this. For waves of small amplitudes and especially with very dim background, the noise will be strong enough to affect the determination of the wave amplitude.

Before examining the wave displays, we first set the rules of presentation. First, a white–blue color scheme with a linear red–green–blue (RGB) code system is used to generate the color bars. Second, the mean albedo within the elliptical region, or the background cloud brightness, is subtracted. Third, the maximum and minimum albedo deviation is set to be ±20.0 × 10-6 sr-1 for freq25=0.8 and is then reduced by factors √ratio0.4/0.8 and √ratio0.0/0.8 for freq25=0.4 and freq25=0.0, respectively. As is argued above, these factors are expected to unify the display clarity between different brightness levels.

The three lines with different colors in Fig. 6a have a hierarchy of albedo power levels as well as different slopes characterized by varying A and α. They, however, appear parallel to each other because the standard deviation of the exponent α (-3.2 to -2.5) only reaches 0.3. We next systematically raise the medium power level to match the high power level by increasing the coefficient A but maintain the exponent α to examine how the display clarity improves. We have argued that parameters A and α both have a control over the display clarity. Carrying out this experiment is a way to test the role of the coefficient A in determining the display clarity. The previously determined A for the medium-power category is 1.45×10-4, and in this experiment a new parameter Anew=2.778×A is used to draw closer the spectra of the medium toward the high power levels. The factor 2.778 is chosen to maximally match the high- and medium-power spectra over the scales of ∼20–60 km.

The result of the amplified case is shown in Fig. 10, and we used the case presented in Fig. 8a to conduct this experiment. Figure 10a shows the albedo map and the power map for the amplified case, and Fig. 10b and c are repeats of Figs. 8a and 7a, to make comparisons. It is shown that Fig. 10a exhibits notably improved clarity compared to its previous version, and as a result Figs. 10a and c show clarity levels drawn closer, but the amplified case is still blurrier. This indicates that the difference between the two exponents (-3.2 versus -2.9) has a fundamental effect on the wave display clarity. Lastly, we need to point out that it is not straightforward why exponent α varies between different albedo power levels and why it remains roughly consistent along the sectioning curves used to resample the wave detections. The physical mechanism that governs such variability is worth a further investigation. It is probable, as we have argued above, that noise contamination is one cause of the smaller slopes for lower albedo power.

Longer-wavelength (>100 km) waves are not as visually detectable as the shorter waves because the CIPS orbital strip is not able to embrace many repeats of the ridge and trough of such waves. Chandran et al. (2010) used a wave detection algorithm to yield a peak population of waves at scales of ∼250 km from the cross-track traces. Zhao et al. (2015) yield a peak wavelength at ∼400 km from the along-track traces. Different traces may be the cause of the different peak wavelengths. Based on these studies, gravity waves of all scales are likely widespread in the PMCs. The waves at <100 km wavelengths are more visually detectable because their full displays are well captured. But these small-scale waves possess a lower variance level than the larger-scale waves. Using a spectral analysis the larger scales often stand out as the dominant wave events. In addition, wave event counting is not a deterministic procedure. For example, in the current study, the wave detections are forcibly confined within the elliptical regions. Generally speaking, we have to cope with a lot of challenge and uncertainty in the wave tracking study.

Figure 11a presents an example of longer and shorter waves observed together. They are a set of bright and dim cloud bands that suggest wavelengths of ∼150–200 km and shorter waves with wavelengths of ∼20–60 km, indicated by the pairs of magenta arrows with solid heads and thin heads, respectively. In this case the CIPS orbital strip achieves pretty satisfying capture of the waves because ∼150–200 km is still a short wavelength relative to the cross-track span (∼900 km) of the CIPS orbital strip. The albedo map reveals that the longer and shorter waves are nested together and the longer waves appear to have larger amplitudes. The albedo power map in the lower panel shows that the wave power is primarily distributed in the upper-right and lower-left quadrants, which reflects the orientation of the wave ridges and troughs that is perpendicular to this.

The albedo power spectra for this particular case are shown in Fig. 11b, revealing a -3.0 slope over the scale range of ∼20–150 km. The correspondence of freq25=0.34 and albedo power = 25.0 × 10-12 sr-2 makes this case the closest to the medium-albedo-power category shown above, and the -3.0 spectral slope is also close to -2.9. The spectra are not reliable in either the longwave limit (>150 km) or the shortwave limit (<20 km). In the longwave limit the elliptical region does not capture enough repeats of ridge and trough, and as a result the power spectra in this range often readily change when the CWT is applied to a much expanded region (>400 km) (not shown). In the shortwave limit the measurement noise will contaminate the PMC signals. It is worth mentioning that the ∼20–60 km scale range focused on in this study is for the shortest waves CIPS can resolve due to the ∼5 km spatial resolution and the signal noise levels (also see Randall et al., 2017). If 1.0×10-6 sr-1 is taken as the threshold of the noise level in the CIPS measurements, the wave signatures will be systematically contaminated for the low-albedo-power category based on approximate amplitudes shown in Fig. 6b.

The albedo map and the power spectra in the presumed valid scale range (∼20–150 km) together indicate that longer and shorter waves are nested together with decreasing albedo power. This reminds us of the term “self-similarity”, which is used to describe the possible fractal nature of the PMC albedo field with a fractal perimeter dimension of 1.3 (Brinkhoff et al., 2015; von Savigny et al., 2011). Self-similarity generally refers to the condition of the small- and large-scale structures resembling each other in morphology. We observed a hint of self-similarity in the gravity wave manifestation described by a -2.9 ± 0.3 law of albedo power, but such an analogy is still preliminary and it requires a further investigation to confirm. It is also worth mentioning that the cascading power spectra from large to small scales (3000–100 km) of periodic PMC structures were also reported by Carbary et al. (2000) via analyzing the middle ultraviolet (210–252 nm) images from the Mid-course Space Experiment (MSX) (Carbary et al., 1994). However in their study no structures of <100 km wavelengths were detected, and the authors attributed this to the small amplitudes of the shorter-wavelength waves that are not able to rise above the noise level.